Download Mathematical foundation of quantum annealing

JOURNAL OF MATHEMATICAL PHYSICS 49, 125210 共2008兲
Mathematical foundation of quantum annealing
Satoshi Morita1,a兲 and Hidetoshi Nishimori2
1
International School for Advanced Studies (SISSA), Via Beirut 2-4, I-34014 Trieste, Italy
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku,
Tokyo 152-8551, Japan
2
共Received 4 June 2008; accepted 10 September 2008; published online 15 December 2008兲
Quantum annealing is a generic name of quantum algorithms that use quantummechanical fluctuations to search for the solution of an optimization problem. It
shares the basic idea with quantum adiabatic evolution studied actively in quantum
computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence
conditions of quantum annealing to the target optimal state after an infinite-time
evolution following the Schrödinger or stochastic 共Monte Carlo兲 dynamics. It is
proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrödinger dynamics and the stochastic dynamics in spite of
the essential difference of these two types of dynamics. Also described are the
prescriptions to reduce errors in the final approximate solution obtained after a long
but finite dynamical evolution of quantum annealing. It is shown there that we can
reduce errors significantly by an ingenious choice of annealing schedule 共time
dependence of the control parameter兲 without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity
using a classical-quantum mapping. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2995837兴
I. INTRODUCTION
An optimization problem is a problem to minimize or maximize a real single-valued function
of multivariables called the cost function.1,2 If the problem is to maximize the cost function f, it
suffices to minimize f. It thus does not lose generality to consider minimization only. In the
present paper we consider combinatorial optimization, in which variables take discrete values.
Well-known examples are satisfiability problems, exact cover, maximum cut, Hamilton graph, and
traveling salesman problem. In physics, the search of the ground state of spin systems is a typical
example, in particular, systems with quenched randomness such as spin glasses.
Optimization problems are classified roughly into two types, easy and hard ones. Loosely
speaking, easy problems are those for which we have algorithms to solve in steps 共=time兲 polynomial in the system size 共polynomial complexity兲. In contrast, for hard problems, all known
algorithms take exponentially many steps to reach the exact solution 共exponential complexity兲. For
these latter problems it is virtually impossible to find the exact solution if the problem size exceeds
a moderate value. Most of the interesting cases as exemplified above belong to the latter hard
class.
It is therefore important practically to devise algorithms which give approximate but accurate
solutions efficiently, i.e., with polynomial complexity. Many instances of combinatorial optimization problems have such approximate algorithm. For example, the Lin–Kernighan algorithm is
often used to solve the traveling salesman problem within a reasonable time.3
a兲
Electronic mail: [email protected].
0022-2488/2008/49共12兲/125210/47/$23.00
49, 125210-1
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125210-2
S. Morita and H. Nishimori
J. Math. Phys. 49, 125210 共2008兲
In the present paper we will instead discuss generic algorithms, simulated annealing 共SA兲 and
quantum annealing 共QA兲. The former was developed from the analogy between optimization
problems and statistical physics.4,5 In SA, the cost function to be minimized is identified with the
energy of a statistical-mechanical system. The system is then given a temperature, an artificially
introduced control parameter, which, by reducing slowly from a high value to zero, we hope to
drive the system to the state with the lowest value of the energy 共cost function兲, reaching the
solution of the optimization problem. The idea is that the system is expected to stay close to
thermal equilibrium during time evolution if the rate of decrease in temperature is sufficiently
slow, and is thus led in the end to the zero-temperature equilibrium state, the lowest-energy state.
In practical applications SA is immensely popular due to its general applicability, reasonable
performance, and relatively easy implementation in most cases. SA is usually used as a method to
obtain an approximate solution within a finite computation time since it needs an infinitely long
time to reach the exact solution by keeping the system close to thermal equilibrium.
Let us now turn our attention to QA 共see Refs. 6–11兲.1 In SA, we make use of thermal
共classical兲 fluctuations to let the system hop from state to state over intermediate energy barriers to
search for the desired lowest-energy state. Why then not try quantum-mechanical fluctuations
共quantum tunneling兲 for state transitions if such may lead to better performance? In QA we
introduce artificial degrees of freedom of quantum nature, noncommutative operators, which induce quantum fluctuations. We then ingeniously control the strength of these quantum fluctuations
so that the system finally reaches the ground state, just like SA in which we slowly reduce the
temperature. More precisely, the strength of quantum fluctuations is first set to a very large value
for the system to search for the global structure of the phase space, corresponding to the hightemperature situation in SA. Then the strength is gradually decreased to finally vanish to recover
the original system hopefully in the lowest-energy state. Quantum tunneling between different
classical states replaces thermal hopping in SA. The physical idea behind such a procedure is to
keep the system close to the instantaneous ground state of the quantum system, analogously to the
quasiequilibrium state to be kept during the time evolution of SA. Similarly to SA, QA is a generic
algorithm applicable, in principle, to any combinatorial optimization problem and is used as a
method to reach an approximate solution within a given finite amount of time.
The reader may wonder why one should invent yet another generic algorithm when we
already have the powerful SA. A short answer is that QA outperforms SA in most cases, at least
theoretically. Analytical and numerical results indicate that the computation time needed to
achieve a given precision of the answer is shorter in QA than in SA. Also, the magnitude of error
is smaller for QA than for SA if we run the algorithm for a fixed finite amount of time. We shall
show some theoretical bases for these conclusions in this paper. Numerical evidence is found in
Refs. 9–11 and 14–22.
A drawback of QA is that a full practical implementation should rely on the quantum computer because we need to solve the time-dependent Schrödinger equation with a very large scale.
Existing numerical studies have been carried out either for small prototype examples or for large
problems by Monte Carlo simulations using the quantum-classical mapping by adding an extra
关Trotter or imaginary-time 共IT兲兴 dimension.23–25 The latter mapping involves approximations,
which inevitably introduces additional errors as well as the overhead caused by the extra dimension. Nevertheless, it is worthwhile to clarify the usefulness and limitations of QA as a theoretical
step toward a new paradigm of computation. This aspect is shared by quantum computation in
general whose practical significance will be fully exploited on the quantum computer.
The idea of QA is essentially the same as quantum adiabatic evolution 共QAE兲, which is now
actively investigated as an alternative paradigm of quantum computation.26 It has been proved that
QAE is equivalent to the conventional circuit model of quantum computation,27 but QAE is
sometimes considered more useful than the circuit model for several reasons including robustness
1
The term quantum annealing first appeared in Refs. 12 and 13 in which the authors used quantum transitions for state
search and the dynamical evolution of control parameters were set by hand as an algorithm. QA in the present sense using
natural Schrödinger dynamics was proposed later independently in Refs. 6 and 7.
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125210-3
Mathematical foundation of quantum annealing
J. Math. Phys. 49, 125210 共2008兲
against external disturbance. In the literature of quantum computation, one is often interested in
the computational complexity of the QAE-based algorithm for a given specific problem under a
fixed value of acceptable error. QAE can also be used to find the final quantum state when the
problem is not a classical optimization.
In some contrast to these situations on QAE, studies of QA are often focused not on computational complexity but on the theoretical convergence conditions for infinite-time evolution and
on the amount of errors in the final state within a fixed evolution time. Such a difference may have
led some researchers to think that QA and QAE are to be distinguished from each other. We would
emphasize that they are essentially the same and worth investigations by various communities of
researchers.
The structure of the present paper is as follows. Section II discusses the convergence condition
of QA, in particular, the rate of decrease in the control parameter representing quantum fluctuations. It will be shown there that a qualitatively faster decrease in the control parameter is allowed
in QA than in SA to reach the solution. This is one of the explicit statements of the claim more
vaguely stated above that QA outperforms SA. In Sec. III we review the performance analysis of
SA using quantum-mechanical tools. The well-known convergence condition for SA will be rederived from the perspective of quantum adiabaticity. The methods and results in this section help
us strengthen the interrelation between QA, SA, and QAE. The error rate of QA after a finite-time
dynamical evolution is analyzed in Sec. IV. There we explain how to reduce the final residual error
after evolution of a given amount of time. This point of view is unique in the sense that most
references of QAE study the time needed to reach a given amount of tolerable error, i.e., computational complexity. The results given in this section can be used to qualitatively reduce residual
errors for a given algorithm without compromising computational complexity. Convergence conditions for stochastic implementation of QA are discussed in Sec. V. The results are surprising in
that the rate of decrease in the control parameter for the system to reach the solution coincides
with that found in Sec. II for the pure quantum-mechanical Schrödinger dynamics. The stochastic
共and therefore classical兲 dynamics shares the same convergence conditions as fully quantum
dynamics. Summary and outlook are described in Sec. VI.
The main parts of this paper 共Secs. II, IV, and V兲 are based on the Ph.D. thesis of Morita28 as
well as several original papers of the present and other authors as will be referred to appropriately.
The present paper is not a comprehensive review of QA since an emphasis is given almost
exclusively to the theoretical and mathematical aspects. There exists an extensive body of numerical studies and the reader is referred to Refs. 9–11 for reviews.
II. CONVERGENCE CONDITION OF QA: REAL-TIME SCHRÖDINGER EVOLUTION
The convergence condition of QA with the real-time 共RT兲 Schrödinger dynamics is investigated in this section, following Ref. 29. We first review the proof of the adiabatic theorem30 to be
used to derive the convergence condition. Then introduced is the Ising model with a transverse
field as a simple but versatile implementation of QA. The convergence condition is derived by
solving the condition for adiabatic transition with respect to the strength of the transverse field.
A. Adiabatic theorem
Let us consider the general Hamiltonian which depends on time t only through the dimensionless time s = t / ␶,
H共t兲 = H̃
冉冊
t
⬅ H̃共s兲.
␶
共1兲
The parameter ␶ is introduced to control the rate of change in the Hamiltonian. In natural quantum
systems, the state vector 兩␺共t兲典 follows the RT Schrödinger equation,
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125210-4
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
d
兩␺共t兲典 = H共t兲兩␺共t兲典,
dt
i
共2兲
or, in terms of the dimensionless time,
i
d ˜
兩␺共s兲典 = ␶H̃共s兲兩˜␺共s兲典,
ds
共3兲
where we set ប = 1. We assume that the initial state is chosen to be the ground state of the initial
Hamiltonian H共0兲 and that the ground state of H̃共s兲 is not degenerate for s ⱖ 0. We show in Sec.
III that the transverse-field Ising model 共TFIM兲, to be used as H共t兲 in most parts of this paper, has
no degeneracy in the ground state 共except possibly in the limit of t → ⬁兲. If ␶ is large, the
Hamiltonian changes slowly and it is expected that the state vector keeps track of the instantaneous ground state. The adiabatic theorem provides the condition for adiabatic evolution. To see
this, we derive the asymptotic form of the state vector with respect to the parameter ␶.
Since we wish to estimate how close the state vector is to the ground state, it is natural to
expand the state vector by the instantaneous eigenstates of H̃共s兲. Before doing so, we derive useful
formulas for the eigenstates. The kth instantaneous eigenstate of H̃共s兲 with the eigenvalue ␧k共s兲 is
denoted as 兩k共s兲典,
H̃共s兲兩k共s兲典 = ␧k共s兲兩k共s兲典.
共4兲
We assume that 兩0共s兲典 is the ground state of H̃共s兲 and that the eigenstates are orthonormal,
具j共s兲 兩 k共s兲典 = ␦ jk. From differentiation of 共4兲 with respect to s, we obtain
冓 冏冏 冔
j共s兲
−1
d
k共s兲 =
ds
␧ j共s兲 − ␧k共s兲
冓 冏 冏 冔
j共s兲
dH̃共s兲
k共s兲 ,
ds
共5兲
where j ⫽ k. In the case of j = k, the same calculation does not provide any meaningful result. We
can, however, impose the following condition:
冓 冏冏 冔
k共s兲
d
k共s兲 = 0.
ds
共6兲
This condition is achievable by the time-dependent phase shift: If 兩k̃共s兲典 = ei␪共s兲兩k共s兲典, we find
冓 冏冏 冔
k̃共s兲
冓 冏冏 冔
d␪
d
d
k̃共s兲 = i + k共s兲
k共s兲 .
ds
ds
ds
共7兲
The second term on the right-hand side is purely imaginary because
冋冓 冏 冏 冔册 冓 冏 冏 冔
k共s兲
d
k共s兲
ds
ⴱ
+ k共s兲
d
d
k共s兲 = 具k共s兲兩k共s兲典 = 0.
ds
ds
共8兲
Thus, condition 共6兲 can be satisfied by tuning the phase factor ␪共s兲 even if the original eigenstate
does not satisfy it.
Theorem 2.1: If the instantaneous ground state of the Hamiltonian H̃共s兲 is not degenerate for
s ⱖ 0 and the initial state is the ground state at s = 0 , i.e., 兩˜␺共0兲典 = 兩0共0兲典 , the state vector 兩˜␺共s兲典
has the asymptotic form in the limit of large ␶ as
兩˜␺共s兲典 = 兺 c j共s兲e−i␶␾ j共s兲兩j共s兲典,
共9兲
c0共s兲 ⬇ 1 + O共␶−2兲,
共10兲
j
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125210-5
J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
i
c j⫽0共s兲 ⬇ 关A j共0兲 − ei␶关␾ j共s兲−␾0共s兲兴A j共s兲兴 + O共␶−2兲,
␶
共11兲
where ␾ j共s兲 ⬅ 兰s0ds⬘␧ j共s⬘兲 , ⌬ j共s兲 ⬅ ␧ j共s兲 − ␧0共s兲 , and
A j共s兲 ⬅
1
⌬ j共s兲2
冓 冏 冏 冔
j共s兲
dH̃共s兲
0共s兲 .
ds
共12兲
Proof: Substitution of 共9兲 into Schrödinger equation 共3兲 yields the equation for the coefficient
c j共s兲 as
dc j
dH̃共s兲
ei␶关␾ j共s兲−␾k共s兲兴
= 兺 ck共s兲
具j共s兲兩
兩k共s兲典,
ds k⫽j
␧ j共s兲 − ␧k共s兲
ds
共13兲
where we used 共5兲 and 共6兲. Integration of this equation yields
c j共s兲 = c j共0兲 + 兺
k⫽j
冕
s
ds̃ck共s̃兲
0
ei␶关␾ j共s̃兲−␾k共s̃兲兴
␧ j共s̃兲 − ␧k共s̃兲
具j共s̃兲兩
dH̃共s̃兲
ds̃
兩k共s̃兲典.
共14兲
Since the initial state is chosen to be the ground state of H共0兲, c0共0兲 = 1 and c j⫽0共0兲 = 0. The second
term on the right-hand side is of the order of ␶−1 because its integrand rapidly oscillates for large
␶. In fact, the integration by parts yields the ␶−1 factor. Thus, c j⫽0共0兲 is of order ␶−1 at most. Hence
only the k = 0 term in the summation remains up to the order of ␶−1,
c j⫽0共s兲 ⬇
冕
s
ds̃
ei␶关␾ j共s̃兲−␾0共s̃兲兴
⌬ j共s̃兲
0
具j共s̃兲兩
dH̃共s̃兲
ds̃
兩0共s̃兲典 + O共␶−2兲,
共15兲
and the integration by parts yields 共11兲.
䊐
Remark: The condition for the adiabatic evolution is given by the smallness of the excitation
probability. That is, the right-hand side of 共11兲 should be much smaller than unity. This condition
is consistent with the criterion of the validity of the above asymptotic expansion. It is represented
by
␶ Ⰷ 兩A j共s兲兩.
共16兲
Using the original time variable t, this adiabaticity condition is written as
冏
冏
1
dH共t兲
兩0共t兲典 = ␦ Ⰶ 1.
2 具j共t兲兩
⌬ j共t兲
dt
共17兲
This is the usual expression of the adiabaticity condition.
B. Convergence conditions of quantum annealing
In this section, we derive the condition which guarantees the convergence of QA. The problem
is what annealing schedule 共time dependence of the control parameter兲 would satisfy adiabaticity
condition 共17兲. We solve this problem on the basis of the idea of Somma et al.31 developed for the
analysis of SA in terms of quantum adiabaticity as reviewed in Sec. III.
1. Transverse-field Ising model
Let us suppose that the optimization problem we wish to solve can be represented as the
ground-state search of an Ising model of general form
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125210-6
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
N
HIsing ⬅ − 兺 Ji␴zi − 兺 Jij␴zi ␴zj − 兺 Jijk␴zi ␴zj ␴zk − ¯ ,
i=1
ij
共18兲
ijk
where the ␴i␣ 共␣ = x , y , z兲 are the Pauli matrices, components of the spin 21 operator at site i. The
eigenvalue of ␴zi is +1 or −1, which corresponds the classical Ising spin. Most combinatorial
optimization problems can be written in this form by, for example, mapping binary variables 共0
and 1兲 to spin variables 共⫾1兲. Another important assumption is that Hamiltonian 共18兲 is extensive,
i.e., proportional to the number of spins N for large N.
To realize QA, a fictitious kinetic energy is introduced typically by the time-dependent transverse field
N
HTF共t兲 ⬅ − ⌫共t兲 兺 ␴xi ,
共19兲
i=1
which induces spin flips, quantum fluctuations, or quantum tunneling between the two states ␴zi
= 1 and ␴zi = −1, thus allowing a quantum search of the phase space. Initially the strength of the
transverse field ⌫共t兲 is chosen to be very large, and the total Hamiltonian
H共t兲 = HIsing + HTF共t兲
共20兲
is dominated by the second kinetic term. This corresponds to the high-temperature limit of SA.
The coefficient ⌫共t兲 is then gradually and monotonically decreased toward 0, leaving eventually
only the potential term HIsing. Accordingly the state vector 兩␺共t兲典, which follows the RT
Schrödinger equation, is expected to evolve from the trivial initial ground state of transverse-field
term 共19兲 to the nontrivial ground state of 共18兲, which is the solution of the optimization problem.
An important issue is how slowly we should decrease ⌫共t兲 to keep the state vector arbitrarily close
to the instantaneous ground state of total Hamiltonian 共20兲. The following theorem provides a
solution to this problem as a sufficient condition.
Theorem 2.2: Adiabaticity (17) for TFIM (20) yields the time dependence of ⌫共t兲 as
⌫共t兲 = a共␦t + c兲−1/共2N−1兲
共21兲
for t ⬎ t0 (for a given positive t0 ) as a sufficient condition of convergence of QA. Here a and c are
constants of O共N0兲 and ␦ is a small parameter to control adiabaticity appearing in (17).
The following theorem proved by Hopf32 will be useful in proving this theorem. See Appendix
A for the proof.
Theorem 2.3: If all the elements of a square matrix M are strictly positive, M ij ⬎ 0 , its
maximum eigenvalue ␭0 and any other eigenvalues ␭ satisfy
兩␭兩 ⱕ
␬−1
␭ ,
␬+1 0
共22兲
M ik
.
M jk
共23兲
where ␬ is defined by
␬ ⬅ max
i,j,k
Proof of Theorem 2.2: We show that power decay 共21兲 satisfies adiabaticity condition 共17兲
which guarantees convergence to the ground state of HIsing as t → ⬁. For this purpose we estimate
the energy gap and the time derivative of the Hamiltonian. As for the latter, it is straightforward to
see
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125210-7
Mathematical foundation of quantum annealing
冏
具j共t兲兩
J. Math. Phys. 49, 125210 共2008兲
冏
d⌫共t兲
dH共t兲
兩0共t兲典 ⱕ − N
dt
dt
共24兲
since the time dependence of H共t兲 lies only in the kinetic term HTF共t兲, which has N terms. Note
that d⌫ / dt is negative.
To estimate a lower bound for the energy gap, we apply Theorem 2.3 to the operator M
⬅ 共E+ − H共t兲兲N. We assume that the constant E+ satisfies E+ ⬎ Emax + ⌫0, where ⌫0 ⬅ ⌫共t0兲 and Emax
is the maximum eigenvalue of the potential term HIsing. All the elements of the matrix M are
strictly positive in the representation that diagonalizes 兵␴zi 其 because E+ − H共t兲 is non-negative and
irreducible, that is, any state can be reached from any other state within at most N steps.
For t ⬎ t0, where ⌫共t兲 ⬍ ⌫0, all the diagonal elements of E+ − H共t兲 are larger than any nonzero
off-diagonal element ⌫共t兲. Thus, the minimum element of M, which is between two states having
all the spins in mutually opposite directions, is equal to N ! ⌫共t兲N, where N! comes from the ways
of permutation to flip spins. Replacement of HTF共t兲 by −N⌫0 shows that the maximum matrix
element of M has the upper bound 共E+ − Emin + N⌫0兲N, where Emin is the lowest eigenvalue of
HIsing. Thus, we have
␬ⱕ
共E+ − Emin + N⌫0兲N
.
N ! ⌫共t兲N
共25兲
If we denote the eigenvalue of H共t兲 by ␧ j共t兲, 共22兲 is rewritten as
关E+ − ␧ j共t兲兴N ⱕ
␬−1
关E − ␧ 共t兲兴N .
␬+1 + 0
共26兲
Substitution of 共25兲 into the above inequality yields
⌬ j共t兲 ⱖ
2关E+ − ␧0共t兲兴N!
⌫共t兲N ⬅ A⌫共t兲N ,
N共E+ − Emin + N⌫0兲N
共27兲
where we used 1 − 共共␬ − 1兲 / 共␬ + 1兲兲1/N ⱖ 2 / N共␬ + 1兲 for ␬ ⱖ 1 and N ⱖ 1. The coefficient A is estimated using the Stirling formula as
A⬇
冉
2冑2␲N关E+ − ␧max
N
0 兴
N
Ne
E+ − Emin + N⌫0
冊
N
,
共28兲
is maxt⬎t0兵␧0共t兲其. This expression implies that A is exponentially small for large N.
where ␧max
0
Now, by combination of estimates 共24兲 and 共27兲, we find that the sufficient condition for
convergence for t ⬎ t0 is
−
d⌫共t兲
N
= ␦ Ⰶ 1,
A2⌫共t兲2N dt
共29兲
where ␦ is an arbitrarily small constant. By integrating this differential equation, we obtain 共21兲.
䊐
Remark: The asymptotic power decay of the transverse field guarantees that the excitation
probability is bounded by the arbitrarily small constant ␦2 at each instant. This annealing schedule
is not valid when ⌫共t兲 is not sufficiently small because we evaluated the energy gap for ⌫共t兲
⬍ ⌫0共t ⬎ t0兲. If we take the limit t0 → 0, ⌫0 increases indefinitely and the coefficient a in 共21兲
diverges. Then result 共21兲 does not make sense. This is the reason why a finite positive time t0
should be introduced in the statement of Theorem 2.2.
2. Transverse ferromagnetic interactions
The same discussions as above apply to QA using the transverse ferromagnetic interactions in
addition to a transverse field,
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125210-8
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
冉兺
N
HTI共t兲 ⬅ − ⌫TI共t兲
i=1
冊
␴xi + 兺 ␴xi ␴xj .
ij
共30兲
The second summation runs over appropriate pairs of sites that satisfy extensiveness of the Hamiltonian. A recent numerical study shows the effectiveness of this type of quantum kinetic energy.18
The additional transverse interaction widens the instantaneous energy gap between the ground
state and the first excited state. Thus, it is expected that an annealing schedule faster than 共21兲
satisfies the adiabaticity condition. The following theorem supports this expectation.
Theorem 2.4: The adiabaticity for the quantum system HIsing + HTI共t兲 yields the time dependence of ⌫共t兲 for t ⬎ t0 as
⌫TI共t兲 ⬀ t−1/共N−1兲 .
共31兲
Proof: The transverse interaction introduces nonzero off-diagonal elements to the Hamiltonian
in the representation that diagonalizes ␴zi . Consequently, any state can be reached from any other
state within N / 2 steps at most. Thus, the strictly positive operator is modified to 共E+ − HIsing
− HTI共t兲兲N/2, which leads to the lower bound for the energy gap as a quantity proportional to
䊐
⌫TI共t兲N/2. The rest of the proof is the same as that of Theorem 2.2.
The above result implies that additional nonzero off-diagonal elements of the Hamiltonian
accelerate the convergence of QA. It is thus interesting to consider the many-body transverse
interaction of the form
N
HMTI共t兲 = − ⌫MTI共t兲 兿 共1 + ␴xi 兲.
共32兲
i=1
All the elements of HMTI are equal to −⌫MTI共t兲 in the representation that diagonalizes ␴zi . In this
system, the following theorem holds.
Theorem 2.5: The adiabaticity for the quantum system HIsing + HTMI共t兲 yields the time dependence of ⌫共t兲 for t ⬎ t0 as
⌫MTI共t兲 ⬀
2N−2
.
␦t
共33兲
Proof: We define the strictly positive operator as M = E+ − HIsing − HMTI共t兲. The maximum and
minimum matrix elements of M are E+ − Emin + ⌫MTI共t兲 and ⌫MTI共t兲, respectively. Thus we have
␬=
E+ − Emin + ⌫MTI共t兲
,
⌫MTI共t兲
E+ − Emin
2⌫MTI共t兲
␬−1
,
=
ⱖ1−
␬ + 1 E+ − Emin + 2⌫MTI共t兲
E+ − Emin
共34兲
共35兲
The inequality for strictly positive operator 共22兲 yields
⌬ j共t兲 ⱖ
2⌫MTI共t兲共E+ − ␧max
0 兲
⬅ Ã⌫MTI共t兲,
E+ − Emin
共36兲
where à is O共N0兲. Since the matrix element of the derivative of the Hamiltonian is bounded as
冏
具j共t兲兩
冏
d⌫MTI
dH共t兲
,
兩0共t兲典 ⱕ − 2N
dt
dt
共37兲
we find that the sufficient condition for convergence with the many-body transverse interaction is
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J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
−
2N
Ã2⌫MTI
d⌫MTI
= ␦ Ⰶ 1.
共t兲2 dt
Integrating this differential equation yields annealing schedule 共33兲.
共38兲
䊐
3. Computational complexity
The asymptotic power-law annealing schedules guarantee the adiabatic evolution during the
annealing process. The power-law dependence on t is much faster than the log-inverse law for the
control parameter in SA, T共t兲 = pN / log共␣t + 1兲, to be discussed in Sec. III, first proved by Geman
and Geman.33 However, it does not mean that QA provides an algorithm to solve NP 共nondeterministic polynomial兲 problems in polynomial time. In the case with the transverse field only,
the time for ⌫共t兲 to reach a sufficiently small value ⑀ 共which implies that the system is sufficiently
close to the final ground state of HIsing whence HTF is a small perturbation兲 is estimated from 共21兲
as
tTF ⬇
冉冊
1 1
␦ ⑀
2N−1
.
共39兲
This relation clearly shows that the QA needs a time exponential in N to converge.
For QA with many-body transverse interactions, the exponent of t in annealing schedule 共33兲
does not depend on the system size N. Nevertheless, it also does not mean that QA provides a
polynomial-time algorithm because of the factor 2N. The characteristic time for ⌫MTI to reach a
sufficiently small value ⑀ is estimated as
tMTI ⬇
2N−2
,
␦⑀
共40兲
which again shows exponential dependence on N.
These exponential computational complexities do not come as a surprise because Theorems
2.2, 2.4, and 2.5 all apply to any optimization problem written in generic form 共18兲, which
includes the worst cases of most difficult problems. Similar arguments apply to SA.34
Another remark is on the comparison of ⌫共t兲共⬀t−1/共2N−1兲兲 in QA with T共t兲共⬀N / log共␣t + 1兲兲 in
SA to conclude that the former schedule is faster than the latter. The transverse-field coefficient ⌫
in a quantum system plays the same role qualitatively and quantitatively as the temperature T does
in a corresponding classical system at least in the Hopfield model in a transverse field.35 When the
phase diagram is written in terms of ⌫ and ␣ 共the number of embedded patterns divided by the
number of neurons兲 for the ground state of the model, the result has precisely the same structure
as the T-␣ phase diagram of the finite-temperature version of the Hopfield model without a
transverse field. This example serves as a justification of the direct comparison of ⌫ and T at least
as long as the theoretical analyses of QA and SA are concerned.
III. CONVERGENCE CONDITION OF SA AND QUANTUM ADIABATICITY
We next study the convergence condition of SA to be compared with QA. This problem was
originally solved by Geman and Geman33 using the theory of the inhomogeneous Markov chain as
described in the quantum Monte Carlo context in Sec. V. It is quite surprising that their result is
reproduced using the quantum adiabaticity condition applied after a classical-quantum mapping.31
This approach is reviewed in this section, following Ref. 31, to clarify the correspondence between the quasiequilibrium condition for SA in a classical system and the adiabaticity condition in
the corresponding quantum system. The analysis will also reveal an aspect related to the equivalence of QA and QAE.
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J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
A. Classical-quantum mapping
The starting point is an expression of a classical thermal expectation value in terms of a
quantum ground-state expectation value. A well-known mapping between quantum and classical
systems is to rewrite the former in terms of the latter with an extra IT 共or Trotter兲 dimension.24 The
mapping discussed in the present section is a different one, which allows us to express the thermal
expectation value of a classical system in terms of the ground-state expectation value of a corresponding quantum system without an extra dimension.
Suppose that the classical Hamiltonian, whose value we want to minimize, is written as an
Ising spin system as in 共18兲:
N
H = − 兺 Ji␴zi − 兺 Jij␴zi ␴zj − 兺 Jijk␴zi ␴zj ␴zk − ¯ .
i=1
ij
共41兲
ijk
The thermal expectation value of a classical physical quantity Q共兵␴zi 其兲 is
具Q典T =
1
兺 e−␤HQ共兵␴i其兲,
Z共T兲 兵␴其
共42兲
where the sum runs over all configurations of Ising spins, i.e., over the values taken by the z
components of the Pauli matrices, ␴zi = ␴i共⫾1兲 共∀i兲. The symbol 兵␴i其 stands for the set
兵␴1 , ␴2 , . . . , ␴N其.
An important element is the following theorem.
Theorem 3.1: Thermal expectation value (42) is equal to the expectation value of Q by the
quantum wave function
兩␺共T兲典 = e−␤H/2 兺 兩兵␴i其典,
兵␴其
共43兲
where 兩兵␴i其典 is the basis state diagonalizing each ␴zi as ␴i . The sum runs over all such possible
assignments.
Assume T ⬎ 0 . Wave function (43) is the ground state of the quantum Hamiltonian
Hq共T兲 = − ␹ 兺 Hqj共T兲 ⬅ − ␹ 兺 共␴xj − e␤H j兲,
j
共44兲
j
where H j is the sum of the terms of Hamiltonian (41) involving site j ,
H j = − J j␴zj − 兺 J jk␴zj ␴zk − 兺 J jkl␴zj ␴zk␴zl − ¯ .
k
共45兲
kl
The coefficient ␹ is defined by ␹ = e−␤p with p = max j兩H j兩 .
Proof: The first half is trivial:
具␺共T兲兩Q兩␺共T兲典
1
=
兺 e−␤H具兵␴i其兩Q兩兵␴i其典 = 具Q典T .
具␺共T兲兩␺共T兲典
Z共T兲 兵␴其
共46兲
To show the second half, we first note that
␴xj 兺 兩兵␴i其典 = 兺 兩兵␴i其典
兵␴其
兵␴其
共47兲
since the operator ␴xj just changes the order of the above summation. It is also easy to see that
␴xje−␤H/2 = e␤H je−␤H/2␴xj
共48兲
because
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Mathematical foundation of quantum annealing
␴xje−␤H/2␴xj = e−␤共H−H j兲/2␴xje−␤H j/2␴xj = e−␤共H−H j兲/2e␤H j/2 = e␤H je−␤H/2
共49兲
as both H and H j are diagonal in the present representation and H − H j does not include ␴zj , so
关H − H j , ␴xj 兴 = 0. We therefore have
Hqj共T兲兩␺共T兲典 = 共␴xj − e␤H j兲e−␤H/2 兺 兩兵␴i其典 = 0.
兵␴其
共50兲
Thus 兩␺共T兲典 is an eigenstate of Hq共T兲 with eigenvalue 0. In the present representation, the nonvanishing off-diagonal elements of −Hq共T兲 are all positive and the coefficients of 兩␺共T兲典 are also all
positive as one sees in 共43兲. Then 兩␺共T兲典 is the unique ground state of Hq共T兲 according to the
䊐
Perron–Frobenius theorem.36
A few remarks are in order. In the high-temperature limit, the quantum Hamiltonian is composed just of the transverse-field term,
Hq共T → ⬁兲 = − 兺 共␴xj − 1兲.
共51兲
j
Correspondingly the ground-state wave function 兩␺共T → ⬁兲典 is the simple summation over all
possible states with equal weight. In this way the thermal fluctuations in the original classical
system are mapped to the quantum fluctuations. The low-temperature limit has, in contrast, the
purely classical Hamiltonian
Hq共T ⬇ 0兲 → ␹ 兺 e␤H j
共52兲
j
and the ground state of Hq共T ⬇ 0兲 is also the ground state of H as is apparent from definition 共43兲.
Hence the decrease in thermal fluctuations in SA is mapped to the decrease in quantum fluctuations. As explained below, this correspondence allows us to analyze the condition for quasiequilibrium in the classical SA using the adiabaticity condition for the quantum system.
B. Adiabaticity and convergence condition of SA
The adiabaticity condition applied to the quantum system introduced above leads to the
condition of convergence of SA. Suppose that we monotonically decrease the temperature as a
function of time, T共t兲, to realize SA.
Theorem 3.2: The adiabaticity condition for the quantum system of Hq共T兲 yields the time
dependence of T共t兲 as
T共t兲 =
pN
log共␣t + 1兲
共53兲
in the limit of large N . The coefficient ␣ is exponentially small in N .
A few lemmas will be useful to prove this theorem.
Lemma 3.3: The energy gap ⌬共T兲 of Hq共T兲 between the ground state and the first excited
state is bounded below as
⌬共T兲 ⱖ a冑Ne−共␤p+c兲N ,
共54兲
where a and c are N -independent positive constants in the asymptotic limit of large N .
Proof: The analysis of Sec. II B 1 applies with the replacement of ⌫共t兲 by ␹ = e−␤p and ␧0共t兲
= 0. This latter condition comes from Hq共T兲兩␺共T兲典 = 0. The condition ⌫共t兲 ⬍ ⌫0共t ⬎ t0兲 is unnecessary here because the off-diagonal element ␹ can always be chosen smaller than the diagonal
elements by adding a positive constant to the diagonal. Equation 共27兲 gives
⌬ j共t兲 ⱖ Ae−␤pN
共55兲
and A satisfies, according to 共28兲,
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J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
A ⬇ b冑2␲Ne−cN ,
共56兲
䊐
with b and c positive constants of O共N 兲.
Lemma 3.4: The matrix element of the derivative of Hq共T兲 , relevant to the adiabaticity
condition, satisfies
0
具␺1共T兲兩⳵THq共T兲兩␺共T兲典 = −
⌬共T兲具␺1共T兲兩H兩␺共T兲典
,
2kBT2
共57兲
where ␺1共T兲 is the normalized first excited state of Hq共T兲 .
Proof: By differentiating the identity
Hq共T兲兩␺共T兲典 = 0,
共58兲
we find
冉
冊
冉
冊
1
⳵
⳵
H 兩␺共T兲典.
Hq共T兲 兩␺共T兲典 = − Hq共T兲 兩␺共T兲典 = Hq共T兲 −
⳵T
⳵T
2kBT2
共59兲
This relation immediately proves the lemma if we notice that the ground-state energy of Hq共T兲 is
䊐
zero and therefore Hq共T兲兩␺1共T兲典 = ⌬共T兲兩␺1共T兲典.
Lemma 3.5: The matrix element of H satisfies
兩具␺1共T兲兩H兩␺共T兲典兩 ⱕ pN冑Z共T兲.
共60兲
Proof: There are N terms in H = 兺 jH j, each of which is of norm of at most p. The factor 冑Z共T兲
䊐
appears from normalization of 兩␺共T兲典.
Proof of Theorem 3.2: The condition of adiabaticity for the quantum system Hq共T兲 reads
1
⌬共T兲 冑Z共T兲
2
冏
具␺1共T兲兩⳵THq共T兲兩␺共T兲典
冏
dT
= ␦,
dt
共61兲
with sufficiently small ␦. If we rewrite the matrix element by Lemma 3.4, the left-hand side is
冏 冏
兩具␺1共T兲兩H兩␺共T兲典兩 dT
.
2kBT2⌬共T兲冑Z共T兲 dt
共62兲
By replacing the numerator by its bound in Lemma 3.5 we have
冏 冏
pN
dT
= ˜␦ Ⰶ 1
2kBT2⌬共T兲 dt
共63兲
as a sufficient condition for adiabaticity. Using the bound of Lemma 3.3 and integrating the above
differential equation for T共t兲 noticing dT / dt ⬍ 0, we reach the statement of Theorem 3.2.
䊐
C. Remarks
Equation 共53兲 reproduces the Geman–Geman condition for convergence of SA.33 Their
method of proof is to use the theory of the classical inhomogeneous 共i.e., time-dependent兲 Markov
chain representing nonequilibrium processes. It may thus be naively expected that the classical
system under consideration may not stay close to equilibrium during the process of SA since the
temperature always changes. It therefore comes as a surprise that the adiabaticity condition, which
is equivalent to the quasiequilibrium condition according to Theorem 3.1, leads to Theorem 3.2.
The rate of temperature change in this latter theorem is slow enough to guarantee the quasiequilibrium condition even when the temperature keeps changing.
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Mathematical foundation of quantum annealing
J. Math. Phys. 49, 125210 共2008兲
Also, Theorem 3.2 is quite general, covering the worst cases, as it applies to any system
written as the Ising model of 共41兲. This fact means that one may apply a faster rate of temperature
decrease to solve a given specific problem with small errors. The same comment applies to the QA
situation in Sec. II.
Another remark is on the relation of QA and QAE. Mathematical analyses of QA often focus
their attention to the generic convergence conditions in the infinite-time limit as seen in Secs. II
and V as well as in an earlier paper,7 although the residual energy after finite-time evolution has
also been extensively investigated mainly in numerical studies. This aspect may have led some
researchers to think that QA is different from QAE since the studies using the latter mostly
concern the computational complexity of finite-time evolution for a given specific optimization
problem using adiabaticity to construct an algorithm of QAE. As has been shown in the present
and the previous sections, the adiabaticity condition also leads to the convergence condition in the
infinite-time limit for QA and SA. In this sense QA, QAE, and even SA share essentially the same
mathematical background.
IV. REDUCTION IN ERRORS FOR FINITE-TIME EVOLUTION
In Sec. II, we discussed the convergence condition of QA implemented for the TFIM. The
power decrease in the transverse field guarantees the adiabatic evolution. This annealing schedule,
however, does not provide practically useful algorithms because an infinitely long time is necessary to reach the exact solution. An approximate algorithm for finite annealing time ␶ should be
used in practice. Since such a finite-time algorithm does not satisfy the generic convergence
condition, the answer includes a certain amount of errors. An important question is how the error
depends on the annealing time ␶.
Suzuki and Okada16 showed that the error after adiabatic evolution for time ␶ is generally
proportional to ␶−2 in the limit of large ␶ with the system size N kept finite. In this section, we
analyze their results in detail and propose new annealing schedules which show smaller errors
proportional to ␶−2m 共m ⬎ 1兲.37 This method allows us to reduce errors by orders of magnitude
without compromising the computational complexity apart from a possibly moderate numerical
factor.
A. Upper bound for excitation probability
Let us consider general time-dependent Hamiltonian 共1兲. The goal of this section is to evaluate
the excitation probability 共closely related with the error probability兲 at the final time s = 1 under
adiabaticity condition 共16兲.
This task is easy because we have already obtained the asymptotic form of excitation amplitude 共11兲. The upper bound for the excitation probability is derived as
兩具j共1兲兩˜␺共1兲典兩2 = 兩c j⫽0共1兲兩2 ⱗ
1
关兩A 共0兲兩 + 兩A j共1兲兩兴2 + O共␶−3兲.
␶2 j
共64兲
This formula indicates that the coefficient of the ␶−2 term is determined only by the state of the
system at s = 0 and 1 and vanishes if A j共s兲 is zero at s = 0 and 1.
When the ␶−2 term vanishes, a similar calculation yields the next order term of the excitation
probability. If H̃⬘共0兲 = H̃⬘共1兲 = 0, the excitation amplitude c j⫽0共1兲 is at most of order ␶−2 and then
c0共1兲 ⬇ 1 + O共␶−3兲. Therefore we have
c j⫽0共1兲 ⬇
冕
1
ds
0
1
ei␶关␾ j共s兲−␾0共s兲兴
dH̃共s兲
具j共s兲兩
兩0共s兲典 + O共␶−3兲 ⬇ 2 关A共2兲
共0兲 − ei␶关␾ j共s兲−␾0共s兲兴A共2兲
j 共1兲兴
⌬ j共s兲
ds
␶ j
+ O共␶−3兲,
共65兲
where we defined
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125210-14
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
FIG. 1. 共Color online兲 Examples of annealing schedules with reduced errors listed in 共70兲–共73兲.
A共m兲
j 共s兲 ⬅
1
dmH̃共s兲
兩0共s兲典.
m+1 具j共s兲兩
⌬ j共s兲
dsm
共66兲
To derive the second line of 共65兲, we used integration by parts twice and 共5兲 and 共6兲. The other ␶−2
terms vanish because of the assumption H̃⬘共0兲 = H̃⬘共1兲 = 0. Thus the upper bound of the next order
for the excitation probability under this assumption is obtained as
兩具j共1兲兩˜␺共1兲典兩2 ⱗ
1 共2兲
2
−5
关兩A 共0兲兩 + 兩A共2兲
j 共1兲兩兴 + O共␶ 兲.
␶4 j
共67兲
It is easy to see that the ␶−4 term also vanishes when H̃⬙共0兲 = H̃⬙共1兲 = 0. It is straightforward to
generalize these results to prove the following theorem.
Theorem 4.1: If the kth derivative of H̃共s兲 is equal to zero at s = 0 and 1 for all k
= 1 , 2 , . . . , m − 1 , the excitation probability has the upper bound
兩具j共1兲兩˜␺共1兲典兩2 ⱗ
1
2
−2m−1
关兩A共m兲共0兲兩 + 兩A共m兲
兲.
j 共1兲兩兴 + O共␶
␶2m j
共68兲
B. Annealing schedules with reduced errors
Although we have so far considered the general time-dependent Hamiltonian, the ordinary
Hamiltonian for QA with finite annealing time is composed of the potential term and the kinetic
energy term,
H̃共s兲 = f共s兲Hpot + 关1 − f共s兲兴Hkin ,
共69兲
where Hpot and Hkin generalize HIsing and HTF in Sec. II, respectively. The function f共s兲, representing the annealing schedule, satisfies f共0兲 = 0 and f共1兲 = 1. Thus H̃共0兲 = Hkin and H̃共1兲 = Hpot. The
ground state of Hpot corresponds to the solution of the optimization problem. The kinetic energy is
chosen so that its ground state is trivial. The above Hamiltonian connects the trivial initial state
and the nontrivial desired solution after evolution time ␶.
The condition for the ␶−2m term to exist in the error is obtained straightforwardly from the
results of Sec. V because Hamiltonian 共69兲 depends on time only through the annealing schedule
f共s兲. It is sufficient that the kth derivative of f共s兲 is zero at s = 0 and 1 for k = 1 , 2 , . . . , m − 1. We
note that f共s兲 should belong to Cm, that is, f共s兲 is an mth differentiable function whose mth
derivative is continuous.
Examples of the annealing schedules f m共s兲 with the ␶−2m error rate are the following polynomials 共Fig. 1兲:
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J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
½
¼
½
¼
½
´
µ
´
µ
´
µ
´
µ
¼
½
½
¼
½
¼
¼
½
¼
¼
¼
½
¼
¼
¼
¼
FIG. 2. 共Color online兲 The annealing-time dependence of the excitation probability for two-level system 共74兲 using
schedules 共70兲–共73兲. The curved and straight lines show 共75兲 and 共76兲 for each annealing schedule, respectively. The
parameters in 共74兲 are chosen to be h = 2 and ␣ = 0.2.
f 1共s兲 = s,
共70兲
f 2共s兲 = s2共3 − 2s兲,
共71兲
f 3共s兲 = s3共10 − 15s + 6s2兲,
共72兲
f 4共s兲 = s4共35 − 84s + 70s2 − 20s3兲.
共73兲
The linear annealing schedule f 1共s兲, which shows the ␶−2 error, has been used in the past studies.
Although we here list only polynomials symmetrical with respect to the point s = 1 / 2, this is not
essential. For example, f共s兲 = 共1 − cos共␲s2兲兲 / 2 also has the ␶−4 error rate because f ⬘共0兲 = f ⬘共1兲
= f ⬙共0兲 = 0 but f ⬙共1兲 = −2␲2.
C. Numerical results
1. Two-level system
To confirm the upper bound for the excitation probability discussed above, it is instructive to
study the two-level system, the Landau–Zener problem, with the Hamiltonian
HLZ共t兲 = −
冋 冉 冊册
1
t
−f
2
␶
h␴z − ␣␴x .
共74兲
The energy gap of HLZ共t兲 has the minimum 2␣ at f共s兲 = 1 / 2. If the annealing time ␶ is not large
enough to satisfy 共16兲, nonadiabatic transitions occur. The Landau–Zener theorem38,39 provides the
excitation probability Pex共␶兲 = 兩具1共1兲 兩 ˜␺共1兲典兩2 as
冋
Pex共␶兲 = exp −
册
␲ ␣ 2␶
,
f ⬘共sⴱ兲h
共75兲
where sⴱ denotes the solution of f共sⴱ兲 = 1 / 2. On the other hand, if ␶ is sufficiently large, the system
evolves adiabatically. Then the excitation probability has upper bound 共68兲, which is estimated as
Pex共␶兲 ⱗ
4h2␣2
␶2m共h2 + 4␣2兲m+2
冋冏 冏 冏 冏册
dm f
dm f
共0兲
+
共1兲
dsm
dsm
2
.
共76兲
We numerically solved Schrödinger equation 共2兲 for system 共74兲 with the Runge–Kutta
method.40 Figure 2 shows the result for the excitation probability with annealing schedules
共70兲–共73兲. The initial state is the ground state of HLZ共0兲. The parameters are chosen to be h = 2 and
␣ = 0.2. The curved and straight lines show 共75兲 and 共76兲, respectively. In the small and large ␶
regions, the excitation probability perfectly fits to those two expressions.
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J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
0.29
−0.15
−0.54
0.08
0.77
0.66
−0.45
−0.73
0.31
0.29
125210-16
−0.44
−0.6
FIG. 3. 共Color online兲 Configuration of random interactions 兵Jij其 on the 3 ⫻ 3 square lattice which we investigated and spin
configuration of the target state. The solid and dashed lines indicate ferromagnetic and antiferromagnetic interactions,
respectively.
2. Spin-glass model
We next carried out simulations of a rather large system, the Ising spin system with random
interactions. The quantum fluctuations are introduced by the uniform transverse field. Thus, the
potential and kinetic energy terms are defined by
N
Hpot = − 兺 Jij␴zi ␴zj − h 兺 ␴zi ,
具ij典
共77兲
i=1
N
Hkin = − ⌫ 兺 ␴xi .
共78兲
i=1
The initial state, namely, the ground state of Hkin, is the all-up state along the x axis.
The difference between the obtained approximate energy and the true ground-state energy
共exact solution兲 is the residual energy Eres. It is a useful measure of the error rate of QA. It has the
same behavior as the excitation probability because it is rewritten as
Eres ⬅ 具˜␺共1兲兩Hpot兩˜␺共1兲典 − ␧0共1兲
共79兲
= 兺 ⌬ j共1兲兩具j共1兲兩˜␺共1兲典兩2 .
共80兲
j⬎0
Therefore Eres is expected to be asymptotically proportional to ␶−2m using the improved annealing
schedules.
We investigated the two-dimensional square lattice of size 3 ⫻ 3. The quenched random coupling constants 兵Jij其 are chosen from the uniform distribution between −1 and +1, as shown in Fig.
3. The parameters are h = 0.1 and ⌫ = 1. Figure 4 shows the ␶ dependence of the residual energy
using annealing schedules 共70兲–共73兲. Straight lines representing ␶−2m 共m = 1 , 2 , 3 , 4兲 are also
shown for comparison. The data clearly indicate the ␶−2m law for large ␶. The irregular behavior
around Eres ⬇ 10−25 comes from numerical rounding errors.
3. Database search problem
As another example, we apply the improved annealing schedule to the database search problem of an item in an unsorted database. Consider N items, among which one is marked. The goal
of this problem is to find the marked item in a minimum time. The pioneering quantum algorithm
proposed by Grover41 solves this task in time of order 冑N, whereas the classical algorithm tests
N / 2 items on average. Farhi et al.26 proposed a QAE algorithm and Roland and Cerf42 found a
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125210-17
J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
FIG. 4. 共Color online兲 The annealing-time dependence of the residual energy for the two-dimensional spin-glass model
with improved annealing schedules. The solid lines denote functions proportional to ␶−2m 共m = 1 , 2 , 3 , 4兲. The parameter
values are h = 0.1 and ⌫ = 1.
QAE-based algorithm with the same computational complexity as Grover’s algorithm. Although
their schedule is optimal in the sense that the excitation probability by the adiabatic transition is
equal to a small constant at each time, it has the ␶−2 error rate. We show that annealing schedules
with the ␶−2m error rate can be constructed by a slight modification in their optimal schedule.
Let us consider the Hilbert space which has the basis states 兩i典 共i = 1 , 2 , . . . , N兲, and the marked
state is denoted by 兩m典. Suppose that we can construct Hamiltonian 共69兲 with two terms,
Hpot = 1 − 兩m典具m兩,
N
共81兲
N
1
Hkin = 1 − 兺 兺 兩i典具j兩.
N i=1 j=1
共82兲
The Hamiltonian Hpot can be applied without the explicit knowledge of 兩m典, the same assumption
as in Grover’s algorithm. The initial state is a superposition of all basis states,
兩␺共0兲典 =
1
N
兩i典,
冑N 兺
i=1
共83兲
which does not depend on the marked state. The energy gap between the ground state and the first
excited state,
⌬1共s兲 =
冑
1−4
N−1
f共s兲关1 − f共s兲兴,
N
共84兲
has a minimum at f共s兲 = 1 / 2. The highest eigenvalue ␧2共s兲 = 1 is 共N − 2兲-fold degenerate.
To derive the optimal annealing schedule, we briefly review the results reported by Roland
and Cerf.42 When the energy gap is small 关i.e., for f共s兲 ⬇ 1 / 2兴, nonadiabatic transitions are likely
to occur. Thus we need to change the Hamiltonian carefully. On the other hand, when the energy
gap is not very small, too slow a change wastes time. Thus the speed in parameter change should
be adjusted adaptively to the instantaneous energy gap. This is realized by tuning the annealing
schedule to satisfy adiabaticity condition 共16兲 in each infinitesimal time interval, that is,
兩A1共s兲兩
= ␦,
␶
共85兲
where ␦ is a small constant. In the database search problem, this condition is rewritten as
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125210-18
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
FIG. 5. 共Color online兲 The optimal annealing schedules for the database search problem 共N = 64兲. The solid line denotes
original optimal schedule 共87兲 and the dashed lines are for the modified schedules.
冑N − 1
df
= ␦.
␶N⌬1共s兲3 ds
共86兲
After integration under boundary conditions f共0兲 = 0 and f共1兲 = 1, we obtain
f opt共s兲 =
1
2s − 1
+
.
2 2冑N − 共N − 1兲共2s − 1兲2
共87兲
As plotted by a solid line in Fig. 5, this function changes most slowly when the energy gap takes
the minimum value. It is noted that the annealing time is determined by the small constant ␦ as
冑N − 1
␶=
␦
共88兲
,
which means that the computation time is of order 冑N, similarly to Grover’s algorithm.
Optimal annealing schedule 共87兲 shows the ␶−2 error rate because its derivative is nonvanishing at s = 0 and 1. It is easy to see from 共87兲 that the simple replacement of s with f m共s兲 fulfills the
condition for the ␶−2m error rate. We carried out numerical simulations for N = 64 with such
共m兲
annealing schedules, f opt
共s兲 ⬅ f opt共f m共s兲兲, as plotted by dashed lines in Fig. 5. As shown in Fig. 6,
共m兲
共s兲 is proportional to ␶−2m. The characteristic time ␶c for the ␶−2m error
the residual energy with f opt
FIG. 6. 共Color online兲 The annealing-time dependence of the residual energy for the database search problem 共N = 64兲 with
the optimal annealing schedules described in Fig. 5. The solid lines represent functions proportional to ␶−2m 共m
= 1 , 2 , 3 , 4兲.
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125210-19
J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
共m兲
rate to show up increases with m: Since the modified optimal schedule f opt
共s兲 has a steeper slope
at s = 1 / 2 than f opt共s兲, a longer annealing time is necessary to satisfy adiabaticity condition 共86兲.
共m兲
共s兲 is only a factor of O共1兲, and therefore ␶c is still
Nevertheless, the difference in slopes of f opt
scaled as 冑N. Significant qualitative reduction in errors has been achieved without compromising
computational complexity apart from a numerical factor.
D. Imaginary-time Schrödinger dynamics
So far, we have concentrated on QA following the RT Schrödinger dynamics. From the point
of view of physics, it is natural that the time evolution of a quantum system obeys the RT
Schrödinger equation. Since our goal is to find the solution of optimization problems, however, we
need not stick to physical reality. We therefore investigate QA following the IT Schrödinger
dynamics here to further reduce errors.
The IT evolution tends to filter out the excited states. Thus, it is expected that QA with the IT
dynamics can find the optimal solution more efficiently than RT-QA. Stella et al.20 investigated
numerically the performance of IT-QA and conjectured that 共i兲 the IT error rate is not larger than
in the RT and that 共ii兲 the asymptotic behavior of the error rate for ␶ → ⬁ is identical for IT-QA and
RT-QA. We prove their conjectures through the IT version of the adiabatic theorem.
1. Imaginary-time Schrödinger equation
The IT Schrödinger equation is obtained by the transformation t → −it in the time derivative of
the original RT Schrödinger equation:
−
d
兩⌿共t兲典 = H共t兲兩⌿共t兲典.
dt
共89兲
The time dependence of the Hamiltonian does not change. If the Hamiltonian is time independent,
we easily see that the excitation amplitude decreases exponentially relative to the ground state,
兩⌿共t兲典 = 兺 c je−it␧ j兩j典 → 兺 c je−t␧ j兩j典 = e−t␧0 兺 c je−t共␧ j−␧0兲兩j典.
j
j
共90兲
j
However, it is not obvious that this feature survives in the time-dependent situation.
An important aspect of the IT Schrödinger equation is nonunitarity. The norm of the wave
function is not conserved. Thus, we consider the normalized state vector
兩␺共t兲典 ⬅
1
冑具⌿共t兲兩⌿共t兲典 兩⌿共t兲典.
共91兲
The equation of motion for this normalized state vector is
−
d
兩␺共t兲典 = 关H共t兲 − 具H共t兲典兴兩␺共t兲典,
dt
共92兲
where we defined the expectation value of the Hamiltonian,
具H共t兲典 ⬅ 具␺共t兲兩H共t兲兩␺共t兲典.
共93兲
The above equation is not linear but norm conserving, which makes the asymptotic expansion
easy. In terms of the dimensionless time s = t / ␶, the norm-conserving IT Schrödinger equation is
written as
−
d ˜
兩␺共s兲典 = ␶关H̃共s兲 − 具H̃共s兲典兴兩˜␺共s兲典.
ds
共94兲
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125210-20
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
2. Asymptotic expansion of the excitation probability
To prove the conjecture by Stella et al.,20 we derive the asymptotic expansion of the excitation
probability. The following theorem provides us with the IT version of the adiabatic theorem.
Theorem 4.2: Under the same hypothesis as in Theorem 2.1, the state vector following the
norm-conserving IT Schrödinger equation (94) has the asymptotic form in the limit of large ␶ as
兩˜␺共s兲典 = 兺 c j共s兲兩j共s兲典,
共95兲
c0共s兲 ⬇ 1 + O共␶−2兲,
共96兲
j
c j⫽0共s兲 ⬇
A j共s兲
+ O共␶−2兲.
␶
共97兲
Proof: Norm-conserving IT Schrödinger equation 共94兲 is rewritten as the equation of motion
for c j共s兲 as
冋
册
ck共s兲
dH̃共s兲
dc j
具j共s兲兩
=兺
兩k共s兲典 − ␶c j共s兲 ␧ j共s兲 − 兺 ␧l共s兲兩cl共s兲兩2 .
ds k⫽j ␧ j共s兲 − ␧k共s兲
ds
l
共98兲
To remove the second term on the right-hand side, we define
冉冕 冋
s
ds̃ ␧ j共s̃兲 − 兺 ␧l共s̃兲兩cl共s̃兲兩2
c̃ j共s兲 ⬅ exp ␶
0
l
册冊
c j共s兲
共99兲
and obtain the equation of motion for c̃ j共s兲 as
e␶兵␾ j共s兲−␾k共s兲其
dH̃共s兲
dc̃ j
= 兺 c̃k共s兲
具j共s兲兩
兩k共s兲典,
ds k⫽j
␧ j共s兲 − ␧k共s兲
ds
共100兲
where we defined ␾ j共s兲 ⬅ 兰s0ds⬘␧ j共s⬘兲 for convenience.
Integration of this equation yields the integral equation for c̃ j共s兲. It is useful to introduce the
following quantity:
␦共s兲 ⬅
冕
s
0
ds̃ 兺 关␧l共s̃兲 − ␧0共s̃兲兴兩cl共s̃兲兩2 .
共101兲
l⫽0
Since the norm of the wave function is conserved, 兺l兩cl共s兲兩2 = 1 and therefore
兺l ␧l共s兲兩cl共s兲2兩 = ␧0共s兲 + l⫽0
兺 关␧l共s兲 − ␧0共s兲兴兩cl共s兲2兩.
共102兲
Thus, the definition of c̃ j共s兲 is written as
c̃ j共s兲 = e−␶␦共s兲e␶关␾ j共s兲−␾0共s兲兴c j共s兲.
共103兲
Finally we obtain the integral equation for c j共s兲:
c0共s兲 = e
␶␦共s兲
+e
␶␦共s兲
冕
s
0
ds̃e
−␶␦共s̃兲
兺 cl共s̃兲
l⫽0
具0共s̃兲兩
dH̃
ds̃
兩l共s̃兲典
␧0共s̃兲 − ␧l共s̃兲
,
共104兲
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125210-21
J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
c j⫽0共s兲 = e
␶␦共s兲 −␶关␾ j共s兲−␾0共s兲兴
e
冕
s
ds̃e
−␶␦共s̃兲 ␶关␾ j共s̃兲−␾0共s̃兲兴
e
兺 ck共s̃兲
具j共s̃兲兩
k⫽j
0
dH̃
ds̃
兩k共s̃兲典
␧ j共s̃兲 − ␧k共s̃兲
,
共105兲
where we used the initial conditions c0共0兲 = 1 and c j⫽0 = 0.
The next step is the asymptotic expansion of these integral equations for large ␶. It is expected
that c0共s兲 = 1 and c j⫽0共s兲 = 0 for ␶ → ⬁ because of the following argument: Since coefficient c0共s兲
is less than unity, ␦共s兲 should be O共␶−1兲 at most and e␶␦共s兲 = O共1兲. The second factor on the
right-hand side of 共105兲 is small exponentially with ␶ because ␾ j共s兲 − ␾0共s兲 is positive and an
increasing function of s. Thus, c j⫽0共s兲 → 0 and then c0共s兲 → 1 owing to the norm conservation law.
Therefore we estimate the next term of order ␶−1 under the assumption that c0共s兲 Ⰷ c j⫽0共s兲.
Since ␦共s兲 is proportional to the square of c j⫽0共s兲, we have e␶␦共s兲 ⬇ 1. Thus, the e⫾␶␦共s兲 factors can
be ignored in the ␶−1 term estimation of 共105兲. Consequently, evaluation of the integral equations
yields
c j⫽0共s兲 ⬇ e−␶关␾ j共s兲−␾0共s兲兴
冕
s
具j共s̃兲兩
ds̃e␶关␾ j共s̃兲−␾0共s̃兲兴
0
dH̃
ds̃
兩0共s̃兲典
⌬ j共s̃兲
+ O共␶−2兲.
共106兲
The excitation amplitude is estimated by integration by parts as
1
c j⫽0共s兲 ⬇ 关A j共s兲 − e−␶共␾ j共s兲−␾0共s兲兲A j共0兲兴 + O共␶−2兲,
␶
共107兲
where A j共s兲 is defined by 共12兲. The second term in the square brackets is vanishingly small, which
is a different point from the RT dynamics. From the above expression, we find ␦共s兲 = O共␶−2兲, that
is, e␶␦共s兲 ⬇ 1 + O共␶−1兲. Therefore, we obtain 共96兲 and 共97兲, which is consistent with the assumption
䊐
c0共s兲 Ⰷ c j⫽0共s兲.
Remark: The excitation probability at the end of a QA process is proportional to ␶−2 in the
large ␶ limit:
兩具j共1兲兩˜␺共1兲典兩2 ⬇
1
兩A 共1兲兩2 + O共␶−3兲.
␶2 j
共108兲
Its difference from the upper bound for RT dynamics 共64兲 is only in the absence of A j共0兲. In the
IT dynamics, this term decreases exponentially because of the factor e−␶共␾ j共s兲−␾0共s兲兲. This result
proves the conjecture proposed by Stella et al.,20 that is,
⑀IT共␶兲 ⱕ ⑀RT共␶兲,
⑀IT共␶兲 ⬇ ⑀RT共␶兲
共␶ → ⬁兲.
共109兲
共110兲
Strictly speaking, the right-hand sides in the above equations denote the upper bound for the error
rate for RT-QA, not the error rate itself. In some systems, for example, the two-level system, the
error rate oscillates because A j共0兲 and A j共1兲 may cancel in 共11兲 and becomes smaller than that of
IT-QA at some ␶. However, QA for ordinary optimization problems has different energy levels at
initial and final times, and thus such a cancellation seldom occurs.
3. Numerical verification
We demonstrate a numerical verification of the above results by simulations of the IT and RT
Schrödinger equations. For this purpose, we consider the following annealing schedules 共Fig. 7兲:
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125210-22
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
Õ
¾
×
×
Õ
½
FIG. 7. 共Color online兲 The annealing schedules defined in 共111兲. f sq1共s兲 and f sq2共s兲 have a vanishing slope at the initial time
s = 0 and the final time s = 1, respectively.
f sq1共s兲 = s2,
f sq2共s兲 = s共2 − s兲.
共111兲
The former has a zero slope at the initial time s = 0 and the latter at s = 1. Thus, the A j共0兲 and A j共1兲
terms vanish with f sq1共s兲 and f sq2共s兲, respectively. Since the error rate for IT-QA depends only on
A j共1兲, IT-QA with f sq2共s兲 should show the ␶−4 error rate, while RT-QA with f sq2共s兲 exhibits the ␶−2
law. On the other hand, RT-QA and IT-QA with f sq1共s兲 should have the same error rate for large ␶.
Figure 8 shows the residual energy with two annealing schedules for the spin-glass model presented in Sec. IV C 2, which explicitly supports our results.
V. CONVERGENCE CONDITION OF QA: QUANTUM MONTE CARLO EVOLUTION
So far, we have discussed QA with the Schrödinger dynamics. When we solve the Schrödinger
equation on the classical computer, the computation time and memory increase exponentially with
the system size. Therefore, some approximations are necessary to simulate QA processes for
large-size problems. In most numerical studies, stochastic methods are used. In this section, we
investigate two types of quantum Monte Carlo methods and prove their convergence theorems,
following Ref. 43.
A. Inhomogeneous Markov chain
Since we prove the convergence of stochastic processes, it is useful to recall various definitions and theorems for inhomogeneous Markov processes.5 We denote the space of discrete states
by S and assume that the size of S is finite. A Monte Carlo step is characterized by the transition
probability from state x 共苸S兲 to state y 共苸S兲 at time step t:
Residual Energy
½
½
¼
IT, sq1
½
¼
IT, sq2
RT, sq1
½
RT, sq2
¼
10
½
¼
¼
½
¼
½
¼
½
¼
FIG. 8. 共Color online兲 The annealing-time dependence of the residual energy for IT- and RT-QA with annealing schedules
f sq1共s兲 and f sq2共s兲. The system is the spin-glass model presented in Sec. IV C 2. The solid lines stand for functions
proportional to ␶−2 and ␶−4. The parameters are h = 0.1 and ⌫ = 1.
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125210-23
J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
G共y,x;t兲 =
再
共x ⫽ y兲
P共y,x兲A共y,x;t兲
1 − 兺 z苸S P共z,x兲A共z,x;t兲 共x = y兲,
冎
共112兲
where P共y , x兲 and A共y , x ; t兲 are called the generation probability and the acceptance probability,
respectively. The former is the probability to generate the next candidate state y from the present
state x. We assume that this probability does not depend on time and satisfies the following
conditions:
∀x,y 苸 S:P共y,x兲 = P共x,y兲 ⱖ 0,
共113兲
∀x 苸 S:P共x,x兲 = 0,
共114兲
∀x 苸 S: 兺 P共y,x兲 = 1,
共115兲
y苸S
n−1
∀x,y 苸 S,
∃ n ⬎ 0,
∃ z1, . . . ,zn−1 苸 S: 兿 P共zk+1,zk兲 ⬎ 0,
z0 = x,
zn = y.
共116兲
k=0
The last condition represents irreducibility of S, that is, any state in S can be reached from any
other state in S.
We define Sx as the neighborhood of x, i.e., the set of states that can be reached by a single
step from x:
Sx = 兵y兩y 苸 S, P共y,x兲 ⬎ 0其.
共117兲
The acceptance probability A共y , x ; t兲 is the probability to accept the candidate y generated from
state x. The matrix G共t兲, whose 共y , x兲 component is given by 共112兲, 关G共t兲兴y,x = G共y , x ; t兲, is called
the transition matrix.
Let P denote the set of probability distributions on S. We regard a probability distribution p
共苸P兲 as the column vector with the component 关p兴x = p共x兲. The probability distribution at time t,
started from an initial distribution p0 共苸P兲 at time t0, is written as
p共t,t0兲 = Gt,t0 p0 ⬅ G共t − 1兲G共t − 2兲 ¯ G共t0兲p0 .
共118兲
A Markov chain is called inhomogeneous when the transition probability depends on time. In
the following sections, we will prove that inhomogeneous Markov chains associated with QA are
ergodic under appropriate conditions. There are two kinds of ergodicity, weak and strong. Weak
ergodicity means that the probability distribution becomes independent of the initial conditions
after a sufficiently long time:
∀t0 ⱖ 0:lim sup兵储p共t,t0兲 − p⬘共t,t0兲储兩p0,p0⬘ 苸 P其 = 0,
t→⬁
共119兲
where p共t , t0兲 and p⬘共t , t0兲 are the probability distributions with different initial distributions p0 and
p0⬘. The norm is defined by
储p储 =
兺 兩p共x兲兩.
共120兲
x苸S
Strong ergodicity is the property that the probability distribution converges to a unique distribution
irrespective of the initial state:
∃r 苸 P,
∀ t0 ⱖ 0:lim sup兵储p共t,t0兲 − r储兩p0 苸 P其 = 0.
t→⬁
共121兲
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J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
The following two theorems provide conditions for weak and strong ergodicities of an inhomogeneous Markov chain.5 For proofs see Appendix B.
Theorem 5.1 (condition for weak ergodicity): An inhomogeneous Markov chain is weakly
ergodic if and only if there exists a strictly increasing sequence of positive numbers 兵ti其 , 共i
= 0 , 1 , 2 , . . .兲 , such that
⬁
关1 − ␣共Gt
兺
i=0
i+1,ti
兲兴 → ⬁,
共122兲
where ␣共Gti+1,ti兲 is the coefficient of ergodicity defined by
再
冎
␣共Gti+1,ti兲 = 1 − min 兩 兺 min兵G共z,x兲,G共z,y兲其兩x,y 苸 S ,
z苸S
共123兲
with the notation G共z , x兲 = 关Gti+1,ti兴z,x .
The coefficient of ergodicity measures the variety of the transition probability. If G共z , x兲 is
independent of a state x, ␣共G兲 is equal to zero.
Theorem 5.2 (condition for strong ergodicity): An inhomogeneous Markov chain is strongly
ergodic if the following three conditions hold:
共1兲
共2兲
共3兲
The Markov chain is weakly ergodic.
For all t there exists a stationary state pt 苸 P such that pt = G共t兲pt .
pt satisfies
⬁
储pt − pt+1储 ⬍ ⬁.
兺
t=0
共124兲
Moreover, if p = lim pt , then p is equal to the probability distribution r in (121).
t→⬁
We note that the existence of the limit is guaranteed by 共124兲. This inequality implies that the
probability distribution pt共x兲 is a Cauchy sequence:
∀␧ ⬎ 0,
∃ t0 ⬎ 0,
∀ t,t⬘ ⬎ t0:兩pt共x兲 − pt⬘共x兲兩 ⬍ ␧.
共125兲
B. Path-integral Monte Carlo method
Let us first discuss convergence conditions for the implementation of QA by the path-integral
Monte Carlo 共PIMC兲 method.24,25 The basic idea of PIMC is to apply the Monte Carlo method to
the classical system obtained from the original quantum system by the path-integral formula. We
first consider the example of ground-state search of the Ising spin system whose quantum fluctuations are introduced by adding a transverse field. The total Hamiltonian is defined in 共20兲.
Although we only treat the two-body interaction for simplicity in this section, the existence of
arbitrarily many-body interactions between the z components of the Pauli matrix and longitudinal
random magnetic field −兺Ji␴zi in addition to the above Hamiltonian would not change the following argument.
In the path-integral method, the d-dimensional TFIM is mapped to a 共d + 1兲-dimensional
classical Ising system so that the quantum system can be simulated on the classical computer. In
numerical simulations, the Suzuki–Trotter formula23,24 is usually employed to express the partition
function of the resulting classical system,
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J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
冉
M
N
M
冊
␤
共k兲
共k兲 共k+1兲
,
Z共t兲 ⬇ 兺 exp
兺 兺 Jij␴共k兲
i ␴ j + ␥共t兲 兺 兺 ␴i ␴i
M k=1 具ij典
共k兲
i=0
k=1
兵S 其
i
共126兲
共= ⫾ 1兲 denotes a
where M is the length along the extra dimension 共Trotter number兲 and ␴共k兲
i
classical Ising spin at site i on the kth Trotter slice. The nearest-neighbor interaction between
adjacent Trotter slices,
冉
冊
1
␤⌫共t兲
␥共t兲 = log coth
,
2
M
共127兲
is ferromagnetic. Approximation 共126兲 becomes exact in the limit M → ⬁ for a fixed ␤ = 1 / kBT.
The magnitude of interaction 共127兲 increases with time t and tends to infinity as t → ⬁, reflecting
the decrease in ⌫共t兲. We fix M and ␤ to arbitrarily large values, which corresponds to the actual
situation in numerical simulations. Therefore the theorem presented below does not directly guarantee the convergence of the system to the true ground state, which is realized only after taking the
limits M → ⬁ and ␤ → ⬁. We will rather show that the system converges to the thermal equilibrium
represented by the right-hand side of 共126兲, which can be chosen arbitrarily close to the true
ground state by taking M and ␤ large enough.
With the above example of TFIM in mind, it will be convenient to treat a more general
expression than 共126兲,
Z共t兲 =
冉
兺 exp
x苸S
−
冊
F0共x兲 F1共x兲
−
.
T0
T1共t兲
共128兲
Here F0共x兲 is the cost function whose global minimum is the desired solution of the combinatorial
optimization problem. The temperature T0 is chosen to be sufficiently small. The term F1共x兲
derives from the kinetic energy, which is the transverse field in the TFIM. Quantum fluctuations
are tuned by the extra temperature factor T1共t兲, which decreases with time. The first term
−F0共x兲 / T0 corresponds to the interaction term in the exponent of 共126兲, and the second term
−F1共x兲 / T1共t兲 generalizes the transverse-field term in 共126兲.
For partition function 共128兲, we define the acceptance probability of PIMC as
A共y,x;t兲 = g
q共x;t兲 =
冉 冊
q共y;t兲
,
q共x;t兲
冉
共129兲
冊
1
F0共x兲 F1共x兲
exp −
−
.
Z共t兲
T0
T1共t兲
共130兲
This q共x ; t兲 is the equilibrium Boltzmann factor at a given fixed T1共t兲. Function g共u兲 is the
acceptance function, a monotone increasing function satisfying 0 ⱕ g共u兲 ⱕ 1 and g共1 / u兲 = g共u兲 / u
for u ⱖ 0. For instance, for the heat bath and the Metropolis methods, we have
g共u兲 =
u
,
1+u
g共u兲 = min兵1,u其,
共131兲
共132兲
respectively. The conditions mentioned above for g共u兲 guarantee that q共x ; t兲 satisfies the detailed
balance condition G共y , x ; t兲q共x ; t兲 = G共x , y ; t兲q共y ; t兲. Thus, q共x ; t兲 is the stationary distribution of
the homogeneous Markov chain defined by the transition matrix G共t兲 with a fixed t. In other
words, q共x ; t兲 is the right eigenvector of G共t兲 with eigenvalue 1.
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S. Morita and H. Nishimori
1. Convergence theorem for PIMC-QA
We first define a few quantities. The set of local maximum states of F1 is written as Sm,
Sm = 兵x兩x 苸 S, ∀ y 苸 Sx, F1共y兲 ⱕ F1共x兲其.
共133兲
We denote by d共y , x兲 the minimum number of steps necessary to make a transition from x to y.
Using this notation we define the minimum number of maximum steps needed to reach any other
state from an arbitrary state in the set S \ Sm,
R = min兵max兵d共y,x兲兩y 苸 S其兩x 苸 S \ Sm其.
共134兲
Also, L0 and L1 stand for the maximum changes in F0共x兲 and F1共x兲, respectively, in a single step,
L0 = max兵兩F0共x兲 − F0共y兲兩兩P共y,x兲 ⬎ 0, x,y 苸 S其,
共135兲
L1 = max兵兩F1共x兲 − F1共y兲兩兩P共y,x兲 ⬎ 0, x,y 苸 S其.
共136兲
Our main results are summarized in the following theorem and corollary.
Theorem 5.3 [strong ergodicity of system (128)]: The inhomogeneous Markov chain generated by (129) and (130) is strongly ergodic and converges to the equilibrium state corresponding to the first term of the right-hand side of (130), exp共−F0共x兲 / T0兲 , if
T1共t兲 ⱖ
RL1
.
log共t + 2兲
共137兲
Application of this theorem to the PIMC implementation of QA represented by 共126兲 immediately yields the following corollary.
Corollary 5.4 (strong ergodicity of QA-PIMC for TFIM): The inhomogeneous Markov
chain generated by the Boltzmann factor on the right-hand side of (126) is strongly ergodic and
converges to the equilibrium state corresponding to the first term on the right-hand side of (126)
if
⌫共t兲 ⱖ
1
M
tanh−1
.
␤
共t + 2兲2/RL1
共138兲
Remark: For sufficiently large t, the above inequality reduces to
⌫共t兲 ⱖ
M
共t + 2兲−2/RL1 .
␤
共139兲
This result implies that a power decay of the transverse field is sufficient to guarantee the convergence of QA of TFIM by the PIMC. Notice that RL1 is of O共N兲. Thus 共139兲 is qualitatively
similar to 共21兲.
To prove strong ergodicity it is necessary to prove weak ergodicity first. The following lemma
is useful for this purpose.
Lemma 5.5 (lower bound on the transition probability): The elements of the transition
matrix defined by (112), (129), and (130) have the following lower bound:
冉
P共y,x兲 ⬎ 0 ⇒ ∀ t ⬎ 0:G共y,x;t兲 ⱖ wg共1兲exp −
L0
L1
−
T0 T1共t兲
冊
共140兲
and
∃t1 ⬎ 0,
∀ x 苸 S \ S m,
冉
∀ t ⱖ t1:G共x,x;t兲 ⱖ wg共1兲exp −
冊
L0
L1
−
.
T0 T1共t兲
共141兲
Here, w stands for the minimum nonvanishing value of P共y , x兲,
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Mathematical foundation of quantum annealing
w = min兵P共y,x兲兩P共y,x兲 ⬎ 0, x,y 苸 S其.
共142兲
Proof of Lemma 5.5: The first part of Lemma 5.5 is proved straightforwardly. Equation 共140兲
follows directly from the definition of the transition probability and the property of the acceptance
function g. When q共y ; t兲 / q共x ; t兲 ⬍ 1, we have
G共y,x;t兲 ⱖ wg
冉 冊
冉
冊
q共x;t兲 q共y;t兲
L0
L1
ⱖ wg共1兲exp −
−
.
q共y;t兲 q共x;t兲
T0 T1共t兲
共143兲
On the other hand, if q共y ; t兲 / q共x ; t兲 ⱖ 1,
冉
G共y,x;t兲 ⱖ wg共1兲 ⱖ wg共1兲exp −
冊
L0
L1
−
,
T0 T1共t兲
共144兲
where we used the fact that both L0 and L1 are positive.
Next, we prove 共141兲. Since x is not a member of Sm, there exists a state y 苸 Sx such that
F1共y兲 − F1共x兲 ⬎ 0. For such a state y,
冉 冉
lim g exp −
t→⬁
F0共y兲 − F0共x兲 F1共y兲 − F1共x兲
−
T0
T1共t兲
冊冊
共145兲
=0
because T1共t兲 tends to zero as t → ⬁ and 0 ⱕ g共u兲 ⱕ u. Thus, for all ␧ ⬎ 0, there exists t1 ⬎ 0 such
that
冉 冉
∀t ⬎ t1:g exp −
F0共y兲 − F0共x兲 F1共y兲 − F1共x兲
−
T0
T1共t兲
冊冊
⬍ ␧.
共146兲
We therefore have
兺 P共z,x兲A共z,x;t兲 = P共y,x兲A共y,x;t兲 + z苸S\兵y其
兺 P共z,x兲A共z,x;t兲 ⬍ P共y,x兲␧ + z苸S\兵y其
兺 P共z,x兲
z苸S
= 1 − 共1 − ␧兲P共y,x兲,
共147兲
and consequently,
G共x,x;t兲 ⬎ 共1 − ␧兲P共y,x兲 ⬎ 0.
共148兲
Since the right-hand side of 共141兲 can be arbitrarily small for sufficiently large t, we obtain the
second part of Lemma 5.5.
䊐
Proof of weak ergodicity implied in Theorem 5.3: Let us introduce the following quantity:
xⴱ = arg min兵max兵d共y,x兲兩y 苸 S其兩x 苸 S \ Sm其.
共149兲
Comparison with the definition of R in 共134兲 shows that the state xⴱ is reachable by at most R
transitions from any state.
Now, consider the transition probability from an arbitrary state x to xⴱ. From the definitions of
R and xⴱ, there exists at least one transition route within R steps:
x ⬅ x0 ⫽ x1 ⫽ x2 ⫽ ¯ ⫽ xl = xl+1 = ¯ = xR ⬅ xⴱ .
Then Lemma 5.5 yields that, for sufficiently large t, the transition probability at each time step has
the following lower bound:
冉
G共xi+1,xi ;t − R + i兲 ⱖ wg共1兲exp −
冊
L0
L1
−
.
T0 T1共t − R + i兲
共150兲
Thus, by taking the product of 共150兲 from i = 0 to i = R − 1, we have
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S. Morita and H. Nishimori
Gt,t−R共xⴱ,x兲 ⱖ G共xⴱ,xR−1 ;t − 1兲G共xR−1,xR−2 ;t − 2兲 ¯ G共x1,x;t − R兲
冉
R−1
ⱖ
wg共1兲exp
兿
i=0
−
冊
冉
冊
L0
L1
RL0
RL1
−
ⱖ wRg共1兲R exp −
−
,
T0 T1共t − R + i兲
T0 T1共t − 1兲
共151兲
where we have used monotonicity of T1共t兲. Consequently, it is possible to find an integer k0 ⱖ 0
such that, for all k ⬎ k0, the coefficient of ergodicity satisfies
冉
1 − ␣共GkR,kR−R兲 ⱖ wRg共1兲R exp −
冊
RL0
RL1
−
,
T0 T1共kR − 1兲
共152兲
where we eliminate the sum over z in 共123兲 by replacing it with a single term for z = xⴱ. We now
substitute annealing schedule 共137兲. Then weak ergodicity is immediately proved from Theorem
5.1 because we obtain
⬁
共1 − ␣共G
兺
k=1
kR,kR−R
冉 冊兺
RL0
兲兲 ⱖ w g共1兲 exp −
T0
R
R
⬁
k=k0
1
→ ⬁.
kR + 1
共153兲
䊐
Proof of Theorem 5.3: To prove strong ergodicity, we refer to Theorem 5.2. Condition 共1兲 has
already been proved. As has been mentioned, Boltzmann factor 共130兲 satisfies q共t兲 = G共t兲q共t兲,
which is condition 共2兲. Thus the proof will be complete if we prove condition 共3兲 by setting pt
= q共t兲. For this purpose, we first prove that q共x ; t兲 is monotonic for large t:
∀t ⱖ 0,
∃t1 ⬎ 0,
∀ x 苸 Smin
1 :q共x;t + 1兲 ⱖ q共x;t兲,
∀ t ⱖ t 1,
∀ x 苸 S \ Smin
1 :q共x;t + 1兲 ⱕ q共x;t兲,
共154兲
共155兲
denotes the set of global minimum states of F1.
where Smin
1
To prove this monotonicity, we use the following notations for simplicity:
冉
A共x兲 = exp −
冊
F0共x兲
,
T0
B=
兺
A共x兲,
共156兲
x苸Smin
1
⌬共x兲 = F1共x兲 − Fmin
1 .
共157兲
If x 苸 Smin
1 , the Boltzmann distribution can be rewritten as
A共x兲
q共x;t兲 =
B+
冉
冊
⌬共y兲
兺min exp − T1共t兲 A共y兲
y苸S\S1
.
共158兲
Since ⌬共y兲 ⬎ 0 by definition, the denominator decreases with time. Thus, we obtain 共154兲.
To prove 共155兲, we consider the derivative of q共x ; t兲 with respect to T1共t兲,
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Mathematical foundation of quantum annealing
冋
兺
A共x兲 B⌬共x兲 +
⳵ q共x;t兲
=
⳵ T1共t兲
冉 冊冋
y苸S\Smin
1
⌬共x兲
T1共t兲 exp
T1共t兲
2
冉
共F1共x兲 − F1共y兲兲exp −
B+
冉
冊
⌬共y兲
A共y兲
T1共t兲
冊
⌬共y兲
兺min exp − T1共t兲 A共y兲
y苸S\S1
册
册
2
.
共159兲
Only F1共x兲 − F1共y兲 in the numerator has the possibility of being negative. However, the first term
B⌬共x兲 is larger than the second one for sufficiently large t because exp共−⌬共y兲 / T1共t兲兲 tends to zero
as T1共t兲 → ⬁. Thus there exists t1 ⬎ 0 such that ⳵q共x ; t兲 / ⳵T共t兲 ⬎ 0 for all t ⬎ t1. Since T1共t兲 is a
decreasing function of t, we have 共155兲.
Consequently, for all t ⬎ t1, we have
储q共t + 1兲 − q共t兲储 =
兺
关q共x;t + 1兲 − q共x;t兲兴 −
x苸Smin
1
=2
兺
兺
关q共x;t + 1兲 − q共x;t兲兴
x苸Smin
1
关q共x;t + 1兲 − q共x;t兲兴,
共160兲
x苸Smin
1
where we used 储q共t兲储 = 兺x苸Sminq共x ; t兲 + 兺x苸Sminq共x ; t兲 = 1. We then obtain
1
1
⬁
兺 储q共t + 1兲 − q共t兲储 = 2 兺
关q共x;⬁兲 − q共x;t1兲兴 ⱕ 2储q共x;⬁兲储 = 2.
共161兲
x苸Smin
1
t=t1
Therefore q共t兲 satisfies condition 共3兲:
⬁
兺 储q共t + 1兲 − q共t兲储 =
t=0
t1−1
兺
t=0
⬁
储q共t + 1兲 − q共t兲储 + 兺 储q共t + 1兲 − q共t兲储 ⱕ
t=t1
t1−1
关储q共t + 1兲储 + 储q共t兲储兴 + 2
兺
t=0
= 2t1 + 2 ⬍ ⬁,
共162兲
which completes the proof of strong ergodicity.
2. Generalized transition probability
In Theorem 5.3, the acceptance probability is defined by the conventional Boltzmann form,
共129兲 and 共130兲. However, we have the freedom to choose any transition 共acceptance兲 probability
as long as it is useful to achieve our objective since our goal is not to find finite-temperature
equilibrium states but to identify the optimal state. There have been attempts to accelerate the
annealing schedule in SA by modifying the transition probability. In particular, Nishimori and
Inoue34 proved weak ergodicity of the inhomogeneous Markov chain for classical SA using the
probability of Tsallis and Stariolo.44 There the property of weak ergodicity was shown to hold
under the annealing schedule of temperature inversely proportional to a power of time steps. This
annealing rate is much faster than the log-inverse law for the conventional Boltzmann factor.
A similar generalization is possible for QA-PIMC by using the following modified acceptance
probability:
共163兲
A共y,x;t兲 = g共u共y,x;t兲兲,
冋
u共y,x;t兲 = e−关F0共y兲−F0共x兲兴/T0 1 + 共q − 1兲
F1共y兲 − F1共x兲
T1共t兲
册
1/共1−q兲
,
共164兲
where q is a real number. In the limit q → 1, this acceptance probability reduces to the Boltzmann
form. Similarly to the discussions leading to Theorem 5.3, we can prove that the inhomogeneous
Markov chain with this acceptance probability is weakly ergodic if
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S. Morita and H. Nishimori
T1共t兲 ⱖ
b
,
共t + 2兲c
0⬍cⱕ
q−1
,
R
共165兲
where b is a positive constant. We have to restrict ourselves to the case q ⬎ 1 for a technical reason
as was the case previously.34 We do not reproduce the proof here because it is quite straightforward to generalize the discussions for Theorem 5.3 in combination with the argument of Ref. 34.
Result 共165兲 applied to the TFIM is that, if the annealing schedule asymptotically satisfies
⌫共t兲 ⱖ
冉
冊
2共t + 2兲c
M
exp −
,
␤
b
共166兲
the inhomogeneous Markov chain is weakly ergodic. Notice that this annealing schedule is faster
than the power law of 共139兲. We have been unable to prove strong ergodicity because we could not
identify the stationary distribution for a fixed T1共t兲 in the present case.
3. Continuous systems
In the above analyses we treated systems with discrete degrees of freedom. Theorem 5.3 does
not apply directly to a continuous system. Nevertheless, by discretization of the continuous space
we obtain the following result.
Let us consider a system of N distinguishable particles in a continuous space of finite volume
with the Hamiltonian
N
H=
1
兺 p2 + V共兵ri其兲.
2m共t兲 i=1 i
共167兲
The mass m共t兲 controls the magnitude of quantum fluctuations. The goal is to find the minimum of
the potential term, which is achieved by a gradual increase in m共t兲 to infinity according to the
prescription of QA. After discretization of the continuous space 共which is necessary anyway in any
computer simulation with finite precision兲 and an application of the Suzuki–Trotter formula, the
equilibrium partition function acquires the following expression in the representation to diagonalize spatial coordinates:
冉
Z共t兲 ⬇ Tr exp −
M
N
M
冊
␤
Mm共t兲
2
V共兵r共k兲
,
兺
兺 兺 兩r共k+1兲 − r共k兲
i 其兲 −
i 兩
M k=1
2␤ i=1 k=1 i
共168兲
with the unit ប = 1. Theorem 5.3 is applicable to this system under the identification of T1共t兲 with
m共t兲−1. We therefore conclude that a logarithmic increase in the mass suffices to guarantee strong
ergodicity of the potential-minimization problem under spatial discretization.
The coefficient corresponding to the numerator of the right-hand side of 共137兲 is estimated as
RL1 ⬇ M 2NL2/␤ ,
共169兲
− r共k兲
兩r共k+1兲
i
i 兩.
To obtain this coefficient, let us consider two
where L denotes the maximum value of
extremes. One is that any state is reachable at one step. By definition, R = 1 and L1 ⬇ M 2NL2 / ␤,
which yields 共169兲. The other case is that only one particle can move to the nearest-neighbor point
at one time step. With a 共ⰆL兲 denoting the lattice spacing, we have
L1 ⬇
M 2
MLa
关L − 共L − a兲2兴 ⬇
.
2␤
␤
共170兲
Since the number of steps to reach any configuration is estimated as R ⬇ NML / a, we again obtain
共169兲.
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125210-31
J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
C. Green’s function Monte Carlo method
The PIMC simulates only the equilibrium behavior at finite temperature because its starting
point is the equilibrium partition function. Moreover, it follows an artificial time evolution of
Monte Carlo dynamics, not the natural Schrödinger dynamics. An alternative approach to improve
these points is Green’s function Monte Carlo 共GFMC兲 method.25,45–47 The basic idea is to solve
the IT Schrödinger equation by stochastic processes. In the present section we derive sufficient
conditions for strong ergodicity in GFMC.
The evolution of states by the IT Schrödinger equation starting from an initial state 兩␺0典 is
expressed as
冉冕
t
兩␺共t兲典 = T exp −
0
冊
dt⬘H共t⬘兲 兩␺0典,
共171兲
where T is the time-ordering operator. The right-hand side can be decomposed into a product of
small-time evolutions,
兩␺共t兲典 = lim Ĝ0共tn−1兲Ĝ0共tn−2兲 ¯ Ĝ0共t1兲Ĝ0共t0兲兩␺0典,
n→⬁
共172兲
where tk = k⌬t, ⌬t = t / n, and Ĝ0共t兲 = 1 − ⌬tH共t兲. In the GFMC, one approximates the right-hand side
of this equation by a product with large but finite n and replaces Ĝ0共t兲 with Ĝ1共t兲 = 1 − ⌬t共H共t兲
− ET兲, where ET is called the reference energy to be taken approximately close to the final groundstate energy. This subtraction of the reference energy simply adjusts the standard of energy and
changes nothing physically. However, practically, this term is important to keep the matrix elements positive and to accelerate convergence to the ground state as will be explained shortly.
To realize the process of 共172兲 by a stochastic method, we rewrite this equation in a recursive
form,
␺k+1共y兲 = 兺 Ĝ1共y,x;tk兲␺k共x兲,
共173兲
x
where ␺k共x兲 = 具x 兩 ␺k典 and 兩x典 denotes a basis state. The matrix element of Green’s function is given
by
Ĝ1共y,x;t兲 = 具y兩1 − ⌬t关H共t兲 − ET兴兩x典.
共174兲
Equation 共173兲 looks similar to a Markov process but is significantly different in several ways. An
important difference is that Green’s function is not normalized, 兺yĜ1共y , x ; t兲 ⫽ 1. In order to avoid
this problem, one decomposes Green’s function into a normalized probability G1 and a weight w:
Ĝ1共y,x;t兲 = G1共y,x;t兲w共x;t兲,
共175兲
where
G1共y,x;t兲 ⬅
Ĝ1共y,x;t兲
兺y Ĝ1共y,x;t兲
,
w共x;t兲 ⬅
Ĝ1共y,x;t兲
.
G1共y,x;t兲
共176兲
Thus, using 共173兲, the wave function at time t is written as
␺n共y兲 = 兺 ␦y,xnw共xn−1 ;tn−1兲w共xn−2 ;tn−2兲 ¯ w共x0 ;t0兲G1共xn,xn−1 ;tn−1兲G1共xn−1,xn−2 ;tn−2兲
兵xk其
¯ G1共x1,x0 ;t0兲␺0共x0兲.
共177兲
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125210-32
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
The algorithm of GFMC is based on this formula and is defined by a weighted random walk
in the following sense. One first prepares an arbitrary initial wave function ␺0共x0兲, all elements of
which are non-negative. A random walker is generated, which sits initially 共t = t0兲 at the position x0
with a probability proportional to ␺0共x0兲. Then the walker moves to a new position x1 following
the transition probability G1共x1 , x0 ; t0兲. Thus this probability should be chosen non-negative by
choosing parameters appropriately as described later. Simultaneously, the weight of this walker is
updated by the rule W1 = w共x0 ; t0兲W0 with W0 = 1. This stochastic process is repeated to t = tn−1. One
actually prepares M independent walkers and let those walkers follow the above process. Then,
according to 共177兲, the wave function ␺n共y兲 is approximated by the distribution of walkers at the
final step weighted by Wn,
M
1
兺 W共i兲n ␦y,x共i兲n ,
M→⬁ M i=1
␺n共y兲 = lim
共178兲
where i is the index of a walker.
As noted above, G1共y , x ; t兲 should be non-negative, which is achieved by choosing sufficiently
small ⌬t 共i.e., sufficiently large n兲 and selecting ET within the instantaneous spectrum of the
Hamiltonian H共t兲. In particular, when ET is close to the instantaneous ground-state energy of H共t兲
for large t 共i.e., the final target energy兲, Ĝ1共x , x ; t兲 is close to unity whereas other matrix components of Ĝ1共t兲 are small. Thus, by choosing ET this way, one can accelerate convergence of GFMC
to the optimal state in the last steps of the process.
If we apply this general framework to the TFIM with the ␴z-diagonal basis, the matrix
elements of Green’s function are immediately calculated as
冦
1 − ⌬t关E0共x兲 − ET兴 共x = y兲
共x and y differ by a single-spin flip兲
Ĝ1共y,x;t兲 = ⌬t⌫共t兲
0
共otherwise兲,
冧
共179兲
E0共x兲 = 具x兩共−兺ijJij␴zi ␴zj 兲兩x典.
One should choose ⌬t and ET such that 1 − ⌬t共E0共x兲 − ET兲 ⱖ 0 for
where
all x. Since w共x , t兲 = 兺yĜ1共y , x ; t兲, the weight is given by
w共x;t兲 = 1 − ⌬t关E0共x兲 − ET兴 + N⌬t⌫共t兲.
共180兲
One can decompose this transition probability into the generation probability and the acceptance
probability as in 共112兲:
冦
1
共single-spin flip兲
P共y,x兲 = N
0 共otherwise兲,
A共y,x;t兲 =
冧
N⌬t⌫共t兲
.
1 − ⌬t关E0共x兲 − ET兴 + N⌬t⌫共t兲
共181兲
共182兲
We shall analyze the convergence properties of stochastic processes under these probabilities for
TFIM.
1. Convergence theorem for GFMC-QA
Similarly to the QA by PIMC, it is necessary to reduce the strength of quantum fluctuations
slowly enough in order to find the ground state in the GFMC. The following theorem provides a
sufficient condition in this regard.
Theorem 5.6 (strong ergodicity of QA-GFMC): The inhomogeneous Markov process of the
random walker for the QA-GFMC of TFIM, (112), (181), and (182), is strongly ergodic if
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J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
⌫共t兲 ⱖ
b
,
共t + 1兲c
0⬍cⱕ
1
.
N
共183兲
The lower bound of the transition probability given in the following lemma will be used in the
proof of Theorem 5.6.
Lemma 5.7: The transition probability of random walk in the GFMC defined by (112), (181),
and (182) has the lower bound
P共y,x兲 ⬎ 0 ⇒ ∀ t ⬎ 0:G1共y,x;t兲 ⱖ
∃t1 ⬎ 0,
∀ t ⬎ t1:G1共x,x;t兲 ⱖ
⌬t⌫共t兲
,
1 − ⌬t共Emin − ET兲 + N⌬t⌫共t兲
⌬t⌫共t兲
,
1 − ⌬t共Emin − ET兲 + N⌬t⌫共t兲
共184兲
共185兲
where Emin is the minimum value of E0共x兲 ,
Emin = min兵E0共x兲兩x 苸 S其.
共186兲
Proof of Lemma 5.7: The first part of Lemma 5.7 is trivial because the transition probability
is an increasing function with respect to E0共x兲 when P共y , x兲 ⬎ 0 as seen in 共182兲. Next, we prove
the second part of Lemma 5.7. According to 共179兲 and 共180兲, G1共x , x ; t兲 is written as
G1共x,x;t兲 = 1 −
N⌬t⌫共t兲
.
1 − ⌬t关E0共x兲 − ET兴 + N⌬t⌫共t兲
共187兲
Since the transverse field ⌫共t兲 decreases to zero with time, the second term on the right-hand side
tends to zero as t → ⬁. Thus, there exists t1 ⬎ 0 such that G1共x , x ; t兲 ⬎ 1 − ␧ for ∀␧ ⬎ 0 and ∀t
⬎ t1. On the other hand, the right-hand side of 共185兲 converges to zero as t → ⬁. We therefore have
共185兲.
䊐
Proof of Theorem 5.6: We show that condition 共183兲 is sufficient to satisfy the three conditions of Theorem 5.2.
共1兲
From Lemma 5.7, we obtain a bound on the coefficient of ergodicity for sufficiently large k
as
兲ⱖ
1 − ␣共GkN,kN−N
1
冋
⌬t⌫共kN − 1兲
1 − ⌬t共Emin − ET兲 + N⌬t⌫共kN − 1兲
册
N
共188兲
in the same manner as we derived 共152兲, where we used R = N. Substituting annealing
schedule 共183兲, we can prove weak ergodicity from Theorem 5.1 because
⬁
关1 −
兺
k=1
共2兲
⬁
␣共GkN,kN−N
兲兴
1
ⱖ
b ⬘N
兺 cN ,
k=k 共kN兲
共189兲
0
which diverges when 0 ⬍ c ⱕ 1 / N.
The stationary distribution of the instantaneous transition probability G1共y , x ; t兲 is
q共x;t兲 ⬅
w共x;t兲
兺x苸S w共x;t兲
=
1
⌬tE0共x兲
,
−
2N 2N关1 + ⌬tET + N⌬t⌫共t兲兴
共190兲
which is derived as follows. The transition probability defined by 共112兲, 共181兲, and 共182兲 is
rewritten in terms of weight 共180兲 as
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S. Morita and H. Nishimori
G1共y,x;t兲 =
冦
1−
N⌬t⌫共t兲
共x = y兲
w共x;t兲
⌬t⌫共t兲
w共x;t兲
共x 苸 Sy ; single-spin flip兲
0
共otherwise兲.
冧
共191兲
Thus, we have
兺 G1共y,x;t兲q共x;t兲 =
x苸S
冋
1−
册
⌬t⌫共t兲 w共x;t兲
N⌬t⌫共t兲 w共y;t兲
+ 兺
w共y;t兲
A
A
x苸Sy w共x;t兲
= q共y;t兲 −
N⌬t⌫共t兲 ⌬t⌫共t兲
+
兺 1,
A
A x苸Sy
共192兲
where A denotes the normalization factor,
兺 w共x;t兲 = Tr
x苸S
冋 冉
册
冊
1 − ⌬t − 兺 Jij␴zi ␴zj − ET + N⌬t⌫共t兲 = 2N关1 + ⌬tET + N⌬t⌫共t兲兴,
具ij典
共193兲
共3兲
where we used Tr 兺 Jij␴zi ␴zj = 0. Since the volume of Sy is N, 共192兲 indicates that q共x ; t兲 is the
stationary distribution of G1共y , x ; t兲. The right-hand side of 共190兲 is easily derived from the
above equation.
Since the transverse field ⌫共t兲 decreases monotonically with t, the above stationary distribution q共x ; t兲 is an increasing function of t if E0共x兲 ⬍ 0 and is decreasing if E0 ⱖ 0. Consequently, using the same procedure as in 共160兲, we have
储q共t + 1兲 − q共t兲储 = 2
兺
E0共x兲⬍0
关q共x;t + 1兲 − q共x;t兲兴,
共194兲
and thus
⬁
储q共t + 1兲 − q共t兲储 = 2 兺 关q共x;⬁兲 − q共x;0兲兴 ⱕ 2.
兺
t=0
E 共x兲⬍0
共195兲
0
⬁
储q共t + 1兲 − q共t兲储 is finite, which completes the proof of condition 共3兲. 䊐
Therefore the sum 兺t=0
Remark: Theorem 5.6 asserts convergence of the distribution of random walkers to equilibrium distribution 共190兲 with ⌫共t兲 → 0. This implies that the final distribution is not delta peaked at
the ground state with minimum E0共x兲 but is a relatively mild function of this energy. The optimality of the solution is achieved after one takes the weight factor w共x ; t兲 into account: The
repeated multiplication of weight factors as in 共177兲, in conjunction with the relatively mild
distribution coming from the product of G1 as mentioned above, leads to the asymptotically
delta-peaked wave function ␺n共y兲 because w共x ; t兲 is larger for smaller E0共x兲 as seen in 共180兲.
2. Alternative choice of Green’s function
So far we have used Green’s function defined in 共174兲, which is linear in the transverse field,
allowing single-spin flips only. It may be useful to consider another type of Green’s function
which accommodates multispin flips. Let us try the following form of Green’s function:
冉
冊 冉
冊
Ĝ2共t兲 = exp ⌬t⌫共t兲 兺 ␴xi exp ⌬t 兺 Jij␴zi ␴zj ,
i
ij
共196兲
which is equal to Ĝ0共t兲 to the order ⌬t. The matrix element of Ĝ2共t兲 in the ␴z-diagonal basis is
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J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
Ĝ2共y,x;t兲 = coshN共⌬t⌫共t兲兲tanh␦共⌬t⌫共t兲兲e−⌬tE0共x兲 ,
共197兲
where ␦ is the number of spins in different states in x and y. According to the scheme of GFMC,
we decompose Ĝ2共y , x ; t兲 into the normalized transition probability and the weight:
G2共y,x;t兲 =
冋
cosh共⌬t⌫共t兲兲
e⌬t⌫共t兲
册
N
tanh␦共⌬t⌫共t兲兲,
共198兲
w2共x;t兲 = e⌬tN⌫共t兲e−⌬tE0共x兲 .
共199兲
It is remarkable that the transition probability G2 is independent of E0共x兲, although it depends on
x through ␦. Thus, the stationary distribution of random walk is uniform. This property is lost if
one interchanges the order of the two factors in 共196兲.
The property of strong ergodicity can be shown to hold in this case as well.
Theorem 5.8 (strong ergodicity of QA-GFMC 2): The inhomogeneous Markov chain generated by (198) is strongly ergodic if
⌫共t兲 ⱖ −
1
log共1 − 2b共t + 1兲−1/N兲.
2⌬t
共200兲
Remark: For sufficiently large t, the above annealing schedule is reduced to
⌫共t兲 ⱖ
b
.
⌬t共t + 1兲1/N
共201兲
Since the proof is quite similar to the previous cases, we just outline the idea of the proof. The
transition probability G2共y , x ; t兲 becomes smallest when ␦ = N. Consequently, the coefficient of
ergodicity is estimated as
1 − ␣共Gt+1,t
2 兲ⱖ
冋
1 − e−2⌬t⌫共t兲
2
册
N
.
We note that R is equal to 1 in the present case because any state is reachable from an arbitrary
state in a single step. From Theorem 5.1, the condition
冋
1 − e−2⌬t⌫共t兲
2
册
N
ⱖ
b⬘
t+1
共202兲
is sufficient for weak ergodicity. From this, one obtains 共200兲. Since the stationary distribution of
G2共y , x ; t兲 is uniform as mentioned above, strong ergodicity readily follows from Theorem 5.2.
Similarly to the case of PIMC, we can discuss the convergence condition of QA-GFMC in
systems with continuous degrees of freedom. The resulting sufficient condition is a logarithmic
increase in the mass as will be shown now. The operator Ĝ2 generated by Hamiltonian 共167兲 is
written as
冉
N
Ĝ2共t兲 = exp −
冊
⌬t
兺 p2 e−⌬tV共兵ri其兲 .
2m共t兲 i=1 i
Thus, Green’s function is calculated in a discretized space as
冉
Ĝ2共y,x;t兲 ⬀ exp −
N
共203兲
冊
m共t兲
兺 兩r⬘ − ri兩2 − ⌬tV共兵ri其兲 ,
2⌬t i=1 i
共204兲
where x and y represent 兵ri其 and 兵ri⬘其, respectively. Summation over y, i.e., integration over 兵ri⬘其,
yields the weight w共x ; t兲, from which the transition probability is obtained:
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J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
w共x;t兲 ⬀ e−⌬tV共兵ri其兲 ,
冉
N
共205兲
冊
m共t兲
G2共y,x;t兲 ⬀ exp −
兺 兩r⬘ − ri兩2 .
2⌬t i=1 i
共206兲
The lower bound for the transition probability depends exponentially on the mass: G2共y , x ; t兲
ⱖ e−Cm共t兲. Since 1 − ␣共Gt+1,t
2 兲 has the same lower bound, the sufficient condition for weak ergodicity is e−Cm共t兲 ⱖ 共t + 1兲−1, which is rewritten as
m共t兲 ⱕ C−1 log共t + 1兲.
共207兲
Constant C is proportional to NL2 / ⌬t, where L denotes the maximum value of 兩r⬘ − r兩. The derivation of C is similar to 共169兲 because G2共t兲 allows any transition to arbitrary states at one time
step.
VI. SUMMARY AND PERSPECTIVE
In this paper we have studied the mathematical foundation of QA, in particular, the convergence conditions and the reduction in residual errors. In Sec. II, we have seen that the adiabaticity
condition of the quantum system representing QA leads to the convergence condition, i.e., the
condition for the system to reach the solution of the classical optimization problem as t → ⬁
following the RT Schrödinger equation. The result shows the asymptotic power decrease in the
transverse field as the condition for convergence. This rate in decrease of the control parameter is
faster than the logarithmic rate of temperature decrease for convergence of SA. It nevertheless
does not mean the qualitative reduction in computational complexity from classical SA to QA. Our
method deals with a very generic system that represents most of the interesting problems including
worst instances of difficult problems, for which drastic reduction in computational complexity is
hard to expect.
Section III reviews the quantum-mechanical derivation of the convergence condition of SA
using the classical-quantum mapping without an extra dimension in the quantum system. The
adiabaticity condition for the quantum system has been shown to be equivalent to the quasiequilibrium condition for the classical system at finite temperature, reproducing the well-known convergence condition of SA. The adiabaticity condition thus leads to the convergence condition of
both QA and SA. Since the studies of QAE often exploit the adiabaticity condition to derive the
computational complexity of a given problem, the adiabaticity may be seen as a versatile tool
traversing QA, SA, and QAE.
Section IV is for the reduction in residual errors after finite-time quantum evolution of RT and
IT Schrödinger equations. This is a different point of view from the usual context of QAE, where
the issue is to reduce the evolution time 共computational complexity兲 with the residual error fixed
to a given small value. It has been shown that the residual error can become significantly smaller
by the ingenious choice of the time dependence of coefficients in the quantum Hamiltonian. This
idea allows us to reduce the residual error for any given QAE-based algorithm without compromising the computational complexity apart from a possibly moderate numerical factor.
In Sec. V we have derived the convergence condition of QA implemented by quantum Monte
Carlo simulations of path-integral and Green’s function methods. These approaches bear important
practical significance because only stochastic methods allow us to treat practical large-size problems on the classical computer. A highly nontrivial result in this section is that the convergence
condition for the stochastic methods is essentially the same power-law decrease in the transversefield term as in the Schrödinger dynamics of Sec. II. This is surprising since the Monte Carlo
共stochastic兲 dynamics is completely different from the Schrödinger dynamics. Something deep
may lie behind this coincidence and it should be an interesting target of future studies.
The results presented and/or reviewed in this paper serve as the mathematical foundation of
QA. We have also stressed the similarity/equivalence of QA and QAE. Even the classical SA can
be viewed from the same framework of quantum adiabaticity as long as the convergence condi-
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125210-37
J. Math. Phys. 49, 125210 共2008兲
Mathematical foundation of quantum annealing
tions are concerned. Since the studies of very generic properties of QA seem to have been almost
completed, fruitful future developments would lie in the investigations of problems specific to
each case of optimization task by analytical and numerical methods.
ACKNOWLEDGMENTS
We thank G. E. Santoro, E. Tosatti, and S. Suzuki for discussions and G. Ortiz for useful
comments and correspondence on the details of the proofs of the theorems in Sec. III. Financial
supports by CREST 共JST兲, DEX-SMI, and JSPS are gratefully acknowledged.
APPENDIX A: HOPF’S INEQUALITY
In this appendix, we prove inequality 共22兲. Although Hopf32 originally proved this inequality
for positive linear integral operators, we concentrate on a square matrix for simplicity.
Let M be a strictly positive m ⫻ m matrix. The strict positivity means that all the elements of
M are positive, namely, M ij ⬎ 0 for all i, j, which will be denoted by M ⬎ 0. Similarly, M ⱖ 0
means that M ij ⱖ 0 for all i, j. We use the same notation for a vector, that is, v ⬎ 0 means that all
the elements vi are positive.
The product of the matrix M and an m element column vector v is denoted as usual by M v
and its ith element is
m
共M v兲i = 兺 M ijv j .
共A1兲
j=1
The strict positivity for M is equivalent to
Mv ⬎ 0
if v ⱖ 0,
v ⫽ 0,
共A2兲
where 0 denotes the zero vector. Of course, if v = 0, then M v = 0.
Any real-valued vector v and any strictly positive vector p ⬎ 0 satisfy
min
i
共M v兲i
共M v兲i
vi
vi
ⱕ min
ⱕ max
ⱕ max ,
pi
i 共Mp兲i
i 共Mp兲i
i pi
共A3兲
冋 冉 冊册
共A4兲
冋冉 冊 册
共A5兲
because
冉 冊
共M v兲i − min
i
vi
共Mp兲i =
pi
冉 冊
vi
max 共Mp兲i − 共M v兲i =
i pi
m
M ij
兺
j=1
m
M ij
兺
j=1
v j − min
i
max
i
vi
p j ⱖ 0,
pi
vi
p j − v j ⱖ 0.
pi
The above inequality implies that the difference between the maximum and minimum of
共M v兲i / 共Mp兲i is smaller than that of vi / pi. Following Ref. 32 we use the notation
osc
i
vi
vi
vi
⬅ max − min ,
pi
i pi
i pi
共A6兲
which is called the oscillation. For a complex-valued vector, we define
oscvi = sup osc Re共␩vi兲.
i
兩␩兩=1
i
共A7兲
It is easy to derive, for any complex c,
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S. Morita and H. Nishimori
osc共cvi兲 = 兩c兩oscvi .
i
共A8兲
i
We can also easily prove that, if oscivi = 0, vi does not depend on i.
We suppose that the simple ratio of matrix elements is bounded,
M ik
ⱕ␬
M jk
共A9兲
for all i, j,k.
This assumption is rewritten by the product form as
共M v兲i
ⱕ ␬,
共M v兲 j
v ⱖ 0,
v ⫽ 0,
共A10兲
for all i, j and such v. The following theorem states that inequality 共A3兲 is sharpened under
additional assumption 共A10兲.
Theorem A.1: If M satisfies conditions (A2) and (A10) for any p ⬎ 0 and any complex-valued
v,
osc
i
共Mv兲i ␬ − 1
vi
ⱕ
osc .
共Mp兲i ␬ + 1 i pi
共A11兲
Proof. We consider a real-valued vector v at first. For fixed i, j and fixed p ⬎ 0, we define Xk by
m
共M v兲i 共M v兲 j
−
= 兺 X kv k .
共Mp兲i 共Mp兲 j k=1
共A12兲
We do not have to know the exact form of Xk = Xk共i , j , p兲. When v = ap, the left-hand side of the
above equation vanishes, which implies 兺kXk pk = 0. Thus, we have
m
共M v兲i 共M v兲 j
−
= 兺 Xk共vk − apk兲.
共Mp兲i 共Mp兲 j k=1
共A13兲
Now we choose
vi
a = min ,
i pi
vi
b = max .
i pi
共A14兲
Since vk − apk = 共b − a兲pk − 共bpk − vk兲, vk − apk takes its minimum 0 at vk = apk and its maximum 共b
− a兲pk at vk = bpk. Therefore, the right-hand side of 共A13兲 with p given attains its maximum for
v = ap− − bp+ = ap + 共b − a兲p+ ,
共A15兲
where we defined
p−i =
再
pi 共Xi ⱕ 0兲
0 共Xi ⬎ 0兲,
冎 再
p+i =
0
共Xi ⱕ 0兲
pi 共Xi ⬎ 0兲.
冎
共A16兲
Consequently, we have
冋
册
共Mp+兲i 共Mp+兲 j
共M v兲i 共M v兲 j
−
ⱕ
−
共b − a兲.
共Mp兲i 共Mp兲 j
共Mp兲i
共Mp兲 j
共A17兲
Since, by assumptions, M ⬎ 0 and p ⬎ 0, we have
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Mathematical foundation of quantum annealing
Mp− ⱖ 0,
Mp+ ⱖ 0,
Mp = Mp− + Mp+ ⬎ 0.
共A18兲
Moreover, Mp− ⬎ 0 if p− ⫽ 0 and Mp+ ⬎ 0 if p+ ⫽ 0. In either case, namely, p− = 0 or p+ = 0, the
expression in the square brackets of 共A17兲 vanishes because p+ is equal to either p or 0. Thus, we
may assume that both Mp− ⬎ 0 and Mp+ ⬎ 0. Therefore the expression in inequality 共A17兲 is
rewritten as
1
1
共Mp+兲i 共Mp+兲 j
−
ⱕ
,
−
共Mp兲i
共Mp兲 j
1 + t 1 + t⬘
t⬅
共Mp−兲i
,
共Mp+兲i
t⬘ ⬅
共Mp−兲 j
.
共Mp+兲 j
共A19兲
Since, from assumption 共A10兲, t and t⬘ are bounded from ␬−1 to ␬, we find t⬘ ⱕ ␬2t, which yields
1
1
共Mp+兲i 共Mp+兲 j
−
.
−
ⱕ
共Mp兲i
共Mp兲 j
1 + t 1 + ␬ 2t
共A20兲
For t ⬎ 0, the right-hand side of the above inequality takes its maximum value 共␬ − 1兲 / 共␬ + 1兲 at
t = ␬−1. Finally, we obtain
共M v兲i 共M v兲 j ␬ − 1
vi
osc
−
ⱕ
共Mp兲i 共Mp兲 j ␬ + 1 i pi
共A21兲
for any i, j. Hence it holds for the sup of the left-hand side, which yields 共A11兲.
For a complex-valued vector v, we replace vi by Re共␩vi兲. Since M Re共␩v兲 = Re共␩ M v兲, the
same argument for the real vector case yields
冉
osc Re ␩
i
冊
冉 冊
␬−1
共M v兲i
vi
osc Re ␩
ⱕ
.
共Mp兲i
pi
␬+1 i
共A22兲
Taking the sup with respect to ␩, 兩␩兩 = 1, on both sides, we obtain 共A11兲.
We apply this theorem to the eigenvalue problem,
M v = ␭v .
共A23兲
The Perron–Frobenius theorem states that a non-negative square matrix, M ⱖ 0, has a real eigenvalue ␭0 satisfying 兩␭兩 ⱕ ␭0 for any other eigenvalue ␭. This result is sharpened for a strictly
positive matrix, M ⬎ 0, as the following theorems.
Theorem A.2: Under hypotheses (A2) and (A10), eigenvalue equation (A23) has a positive
solution ␭ = ␭0 ⬎ 0 and v = q ⬎ 0. Moreover, for any vector p 共p ⱖ 0 , p ⫽ 0兲, the sequence
qn =
M np
共M np兲k
共A24兲
with k fixed converges toward such q .
Theorem A.3: Under the same hypotheses, (A23) has no other non-negative solutions than
␭ = ␭0 and v = cq . For ␭ = ␭0 , (A23) has no other solutions than v = cq .
Theorem A.4: Under the same hypotheses, any (complex) eigenvalue ␭ ⫽ ␭0 of (A23) satisfies
兩␭兩 ⱕ
␬−1
␭ .
␬+1 0
共A25兲
Remark: We note that the factor 共␬ − 1兲 / 共␬ + 1兲 is the best possible if there is no further
condition. For example,
M=
冉 冊
␬ 1
,
1 ␬
␬ ⬎ 1,
共A26兲
has eigenvalues ␭0 = ␬ + 1 and ␭ = ␬ − 1.
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Proof of Theorem A.2: Let us consider two vectors p and p which are non-negative and
unequal to 0 and define
pn+1 = Mpn,
pn+1 = Mpn,
p0 = p,
p0 = p.
共A27兲
From hypothesis 共A2兲, both pn and pn are strictly positive for n ⬎ 0. We find by repeated applications of Theorem A.1 that, for n ⬎ 1,
osc
i
冉 冊
␬−1
p̄n,i
ⱕ
pn,i
␬+1
n−1
osc
i
p̄1,i
,
p1,i
共A28兲
where we used the notation pn,i = 共pn兲i. Consequently, there exists a finite constant ␭ ⬎ 0, such that
p̄n,i
→␭
pn,i
共n → ⬁兲
共A29兲
for every i. We normalize the vectors pn and pn as
qn =
pn
,
pn,k
qn =
pn
p̄n,k
,
共A30兲
with k fixed. Hypothesis 共A10兲 implies that
␬−1 ⱕ qn,i ⱕ ␬,
␬−1 ⱕ q̄n,i ⱕ ␬ .
共A31兲
Thus, we find that
兩q̄n,i − qn,i兩 = qn,i
冏
冏
pn,k p̄n,i p̄n,k
pn,k
p̄n,i
−
ⱕ␬
osc
.
p
p
p
p̄n,k n,i
p̄n,k i n,i
n,k
共A32兲
Now we specialize to the case that p = Mp = p1, namely,
pn = Mpn = pn+1,
qn = qn+1 .
共A33兲
Using 共A28兲 and 共A29兲, we estimate 共A32兲 for qn+1,i − qn,i, which implies that the sequence qn
converges to a limit vector q. Because of 共A31兲, we have q ⬎ 0. Now 共A29兲 reads
pn+1,i 共Mpn兲i 共Mqn兲i
=
=
→ ␭0 .
pn,i
共pn兲i
共qn兲i
共A34兲
Consequently, Mq = ␭0q. For any other initial vector p, the sequence qn converges to the same limit
䊐
as qn because of 共A28兲, 共A29兲, and 共A32兲. Theorem A.2 is thereby proved.
Proof of Theorem A.3: We assume that v ⱖ 0 and v ⫽ 0 is a solution of eigenvalue equation
共A23兲. Since hypothesis 共A2兲 implies M v ⬎ 0, we have ␭ ⬎ 0 and v ⬎ 0. We use this v as an initial
vector p in Theorem A.2 and apply the last part of this theorem to
M nv
␭ nv v
=
= .
共M nv兲k ␭nvk vk
共A35兲
Hence, the limit q is equal to v / vk, that is, v = cq and ␭ = ␭0. Therefore the first part of Theorem
A.3 is proved.
Next, we take ␭0 ⬎ 0 and q ⬎ 0 from Theorem A.2 and consider a solution of M v = ␭v. The
application of Theorem A.1 to q and v yields
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Mathematical foundation of quantum annealing
兩␭兩
␭vi
共M v兲i ␬ − 1
vi
vi
osc ,
osc = osc
= osc
ⱕ
␭0 i qi
␬ + 1 i qi
i ␭ 0q i
i 共Mq兲i
共A36兲
where we used 共A8兲. If ␭ = ␭0, the above inequality implies that osci vi / qi = 0 or v = cq, which
provides the second part of Theorem A.3.
Proof of Theorem A.4: We consider 共A36兲. If ␭ ⫽ ␭0 and v ⫽ 0, v cannot be equal to cq.
䊐
Therefore osci vi / qi ⬎ 0, and then 共A36兲 yields 共A25兲.
APPENDIX B: CONDITIONS FOR ERGODICITY
In this appendix, we prove Theorems 5.1 and 5.2 which provide conditions for weak and
strong ergodicities of an inhomogeneous Markov chain.5
1. Coefficient of ergodicity
Let us recall the definition of the coefficient of ergodicity,
␣共G兲 = 1 − min
再兺
冎
共B1兲
min兵G共z,x兲,G共z,y兲其 .
x,y苸S z苸S
First, we prove that this coefficient is rewritten as
1
␣共G兲 = max
2 x,y苸S
再兺
冎
兩G共z,x兲 − G共z,y兲兩 .
z苸S
共B2兲
Proof of 共B2兲: For fixed x , y 苸 S, we define two subsets of S by
+
= 兵z 苸 S兩G共z,x兲 − G共z,y兲 ⬎ 0其,
SG
−
= 兵z 苸 S兩G共z,x兲 − G共z,y兲 ⱕ 0其.
SG
Since the transition matrix satisfies 兺y苸SG共y , x兲 = 1, we have
兺
冋
关G共z,x兲 − G共z,y兲兴 = 1 −
+
z苸SG
兺
−
z苸SG
册冋
G共z,x兲 − 1 −
兺
册
G共z,y兲 = −
−
z苸SG
共B3兲
兺
关G共z,x兲 − G共z,y兲兴.
−
z苸SG
共B4兲
Thus, we find
1
兺 兩G共z,x兲 − G共z,y兲兩 =
2 z苸S
=
兺
+
z苸SG
关G共z,x兲 − G共z,y兲兴 =
兺 max兵0,G共z,x兲 − G共z,y兲其
z苸S
兺 关G共z,x兲 − min兵G共z,x兲,G共z,y兲其兴 = 1 − z苸S
兺 min兵G共z,x兲,G共z,y兲其
z苸S
共B5兲
for any x , y. Hence taking the maximum with respect to x , y on both sides, we obtain 共B2兲. 䊐
To derive the conditions for weak and strong ergodicities, the following lemmas are useful.
Lemma B.1: Let G be a transition matrix. Then the coefficient of ergodicity satisfies
0 ⱕ ␣共G兲 ⱕ 1.
共B6兲
Lemma B.2: Let G and H be transition matrices on S . Then the coefficient of ergodicity
satisfies
␣共GH兲 ⱕ ␣共G兲␣共H兲.
共B7兲
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Lemma B.3: Let G be a transition matrix and H be a square matrix on S such that
兺 H共z,x兲 = 0
共B8兲
储GH储 ⱕ ␣共G兲储H储,
共B9兲
z苸S
for any x 苸 S . Then we have
where the norm of a square matrix defined by
储A储 ⬅ max
x苸S
再兺
冎
兩A共z,x兲兩 .
z苸S
共B10兲
Proof of Lemma B.1: The definition of ␣共G兲 implies ␣共G兲 ⱕ 1 because G共y , x兲 ⱖ 0. From
共B2兲, ␣共G兲 ⱖ 0 is straightforward.
Proof of Lemma B.2: Let us consider a transition matrix G, a column vector a such that
兺z苸Sa共z兲 = 0, and their product b = Ga. We note that the vector b satisfies 兺z苸Sb共z兲 = 0 because
冋
兺 b共z兲 = z苸S
兺 兺 G共z,y兲a共y兲 = y苸S
兺 a共y兲 z苸S
兺 G共z,y兲
z苸S
y苸S
册
兺 a共y兲 = 0.
=
共B11兲
y苸S
We define subsets of S by
S+a = 兵z 苸 S兩a共z兲 ⬎ 0其,
S−a = 兵z 苸 S兩a共z兲 ⱕ 0其,
S+b = 兵z 苸 S兩b共z兲 ⬎ 0其,
S−b = 兵z 苸 S兩b共z兲 ⱕ 0其.
共B12兲
Since 兺z苸Sa共z兲 = 兺z苸Sb共z兲 = 0, we find
兺 兩a共z兲兩 = 兺
a共z兲 −
z苸Sz+
z苸S
兺
a共z兲 = 2
z苸Sz−
兺
兺 兩b共z兲兩 = 2 兺
b共z兲 = − 2
z苸Sz+
z苸S
a共z兲 = − 2
z苸Sz+
兺
共B13兲
a共z兲,
z苸Sz−
兺
共B14兲
b共z兲.
z苸Sz−
Therefore, we obtain
冋
兺 兩b共z兲兩 = 2 兺 u苸S
兺 G共z,u兲a共u兲 = 2 兺 兺
z苸S+b
z苸S
u苸S+a
ⱕ 2 max
v苸S
G共z, v兲
z苸S+b
再兺
z苸S
a共u兲 + 2 min
w苸S
u苸S+a
兩a共u兲兩 ⱕ max
u苸S
1
max
2 v,w苸S
冋
兺 兺
u苸S−a
册
G共z,u兲 a共u兲
z苸S+b
再兺 冎兺
再兺 冎兺
再兺
再兺
冎兺
冎兺
− G共z,w兲兴
=
册
G共z,u兲 a共u兲 + 2
z苸S+b
v,w苸S z苸S
兩G共z, v兲 − G共z,w兲兩
冎兺
u苸S
G共z,w兲
z苸S+b
a共u兲 = max
v,w苸S
u苸S−a
max兵0,G共z, v兲 − G共z,w兲其
关G共z, v兲
z苸S+b
兩a共u兲兩
u苸S
兩a共u兲兩 = ␣共G兲 兺 兩a共u兲兩,
共B15兲
u苸S
where we used 共B5兲 and 共B2兲.
Next, we consider transition matrices G, H, and F = GH. We take a共z兲 = H共z , x兲 − H共z , y兲, and
then 共B15兲 is rewritten as
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Mathematical foundation of quantum annealing
兺 兩F共z,x兲 − F共z,y兲兩 ⱕ ␣共G兲 u苸S
兺 兩H共u,x兲 − H共u,y兲兩
共B16兲
z苸S
for any x , y. Hence this inequality holds for the maximum of both sides with respect to x , y, which
yields Lemma B.2.
Proof of Lemma B.3: Let us consider F = GH. We can take a共z兲 = H共z , x兲 in 共B15兲 because of
the assumption 兺y苸SH共y , x兲 = 0. Thus we have
兺 兩F共z,x兲兩 ⱕ ␣共G兲 u苸S
兺 兩H共u,x兲兩
共B17兲
z苸S
for any x. Hence this inequality holds for the maximum of both sides with respect to x, which
provides Lemma B.3.
䊐
2. Conditions for weak ergodicity
The following theorem provides the reason why ␣共G兲 is called the coefficient of ergodicity.
Theorem B.4: An inhomogeneous Markov chain is weakly ergodic if and only if the transition
matrix satisfies
lim ␣共Gt,s兲 = 0
共B18兲
t→⬁
for any s ⬎ 0.
Proof: We assume that the inhomogeneous Markov chain generated by G共t兲 is weakly ergodic.
For fixed x , y 苸 S, we define probability distributions by
px共z兲 =
再
1 共z = x兲
0 共otherwise兲,
冎
py共z兲 =
再
1 共z = y兲
0 共otherwise兲.
冎
共B19兲
Since px共t , s ; z兲 = 兺u苸SGt,s共z , u兲px共u兲 = Gt,s共z , x兲 and py共t , s ; z兲 = Gt,s共z , y兲, we have
兺 兩Gt,s共z,x兲 − Gt,s共z,y兲兩 = z苸S
兺 兩px共t,s;z兲 − py共t,s;z兲兩 ⱕ sup兵储p共t,s兲 − p⬘共t,s兲储兩p0,p0⬘ 苸 P其.
z苸S
共B20兲
Taking the maximum with respect to x , y on the left-hand side, we obtain
2␣共Gt,s兲 ⱕ sup兵储p共t,s兲 − p⬘共t,s兲储兩p0,p0⬘ 苸 P其.
共B21兲
Therefore the definition of weak ergodicity 共119兲 yields 共B18兲.
We assume 共B18兲. For fixed p0 , q0 苸 P, we define the transition probabilities by
H = 共p0,q0, . . . ,q0兲,
共B22兲
F = Gt,sH = 共p共t,s兲,q共t,s兲, . . . ,q共t,s兲兲,
共B23兲
where p共t , s兲 = Gt,s p0 and q共t , s兲 = Gt,sq0. From 共B2兲, the coefficient of ergodicity for F is rewritten
as
␣共F兲 =
1
1
兩p共t,s;z兲 − q共t,s;z兲兩 = 储p共t,s兲 − q共t,s兲储.
兺
2 z苸S
2
共B24兲
Thus Lemmas B.1 and B.2 yield
储p共t,s兲 − q共t,s兲储 ⱕ 2␣共Gt,s兲␣共H兲 ⱕ 2␣共Gt,s兲.
共B25兲
Taking the sup with respect to p0 , q0 苸 S and the limit t → ⬁, we obtain
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lim sup兵储p共t,s兲 − q共t,s兲储兩p0,q0 苸 P其 ⱕ 2 lim ␣共Gt,s兲 = 0
t→⬁
共B26兲
t→⬁
for any s ⬎ 0. Therefore the inhomogeneous Markov chain generated by G共t兲 is weakly ergodic.
Next, we prove Theorem 5.1. For this purpose, the following lemma is useful.
Lemma B.5: Let a0 , a1 , . . . , an , . . . be a sequence such that 0 ⱕ ai ⬍ 1 for any i .
⬁
⬁
ai = ⬁ ⇒ 兿 共1 − ai兲 = 0.
兺
i=0
i=n
Proof. Since 0 ⱕ 1 − ai ⱕ e−ai, we have
m
m
0 ⱕ 兿 共1 − ai兲 ⱕ 兿 e
i=n
i=n
共B27兲
冉 冊
m
−ai
ⱕ exp − 兺 ai .
共B28兲
i=n
⬁
In the limit m → ⬁, the right-hand side converges to zero because of the assumption 兺i=0
ai = ⬁.
Therefore we obtain 共B27兲.
Proof of Theorem 5.1: We assume that the inhomogeneous Markov chain generated by G共t兲
is weakly ergodic. Theorem B.4 yields
lim 关1 − ␣共Gt,s兲兴 = 1
共B29兲
t→⬁
for any s ⬎ 0. Thus, there exists t1 such that 1 − ␣共Gt1,t0兲 ⬎ 1 / 2 with t0 = s. Similarly, there exists
tn+1 such that 1 − ␣共Gtn+1,tn兲 ⬎ 1 / 2 for any tn ⬎ 0. Therefore,
n
关1 − ␣共Gt
兺
i=0
i+1,ti
1
兲兴 ⬎ 共n + 1兲.
2
共B30兲
Taking the limit n → ⬁, we obtain 共122兲.
We assume 共122兲. Lemma B.5 yields
⬁
⬁
i=n
i=n
兿 兵1 − 关1 − ␣共Gti+1,ti兲兴其 = 兿 ␣共Gti+1,ti兲 = 0.
共B31兲
For fixed s and t such that t ⬎ s ⱖ 0, we define n and m by tn−1 ⱕ s ⬍ tn and tm ⬍ t ⱕ tm+1. Thus, from
Lemma B.2, we obtain
冋兿
m
␣共Gt,s兲 ⱕ ␣共Gt,tm兲␣共Gtm,tm−1兲 ¯ ␣共Gtn+1,tn兲␣共Gtn,s兲 = ␣共Gt,tm兲
i=n
册
␣共Gti+1,ti兲 ␣共Gtn,s兲.
共B32兲
In the limit t → ⬁, m goes to infinity and then the right-hand side converges to zero because of
共B31兲. Thus we have
lim ␣共Gt,s兲 = 0
t→⬁
共B33兲
for any s. Therefore, from Theorem B.4, the inhomogeneous Markov chain generated by G共t兲 is
weakly ergodic.
䊐
3. Conditions for strong ergodicity
The goal of this section is to give the proof of Theorem 5.2. Before that, we prove the
following theorem, which also provides the sufficient condition for strong ergodicity.
Theorem B.6: An inhomogeneous Markov chain generated by G共t兲 is strongly ergodic if there
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Mathematical foundation of quantum annealing
J. Math. Phys. 49, 125210 共2008兲
exists the transition matrix H on S such that H共z , x兲 = H共z , y兲 for any x , y , z 苸 S and
lim 储Gt,s − H储 = 0
t→⬁
共B34兲
for any s ⬎ 0 .
Proof: We consider p0 苸 P and p共t , s兲 = Gt,s p0. For fixed u 苸 S, we define a probability distribution r by r共z兲 = H共z , u兲. We find
储p共t,s兲 − r储 =
兺 兩 兺 Gt,s共z,x兲p0共x兲 − H共z,u兲兩 = z苸S
兺 兩 x苸S
兺 关Gt,s共z,x兲 − H共z,u兲兴p0共x兲兩
z苸S x苸S
ⱕ
兺 兺 兩Gt,s共z,x兲 − H共z,u兲兩 = z苸S
兺 兺 兩Gt,s共z,x兲 − H共z,x兲兩 ⱕ x苸S
兺 储Gt,s − H储
x苸S
z苸S x苸S
共B35兲
= 兩S兩储Gt,s − H储.
Taking the sup with respect to p0 苸 P and using assumption 共B34兲, we obtain
lim sup兵储p共t,s兲 − r储兩p0 苸 P其 = 0.
t→⬁
共B36兲
Therefore, the inhomogeneous Markov chain generated by G共t兲 is strongly ergodic.
䊐
Proof of Theorem 5.2: We assume that the three conditions in Theorem 5.2 hold. Since
condition 共3兲 is rewritten as
兺
⬁
⬁
兺 兩pt共x兲 − pt+1共x兲兩 = 兺 储pt − pt+1储 ⬍ ⬁,
x苸S t=0
共B37兲
t=0
we have
⬁
兩pt共x兲 − pt+1共x兲兩 ⬍ ⬁
兺
t=0
共B38兲
for any x 苸 S. Thus, the stationary state pt converges to p = limt→⬁ pt. Now, let us define transition
matrices H and H共t兲 by H共z , x兲 = p共z兲 and H共z , x ; t兲 = pt共z兲, respectively. For t ⬎ u ⬎ s ⱖ 0,
储Gt,s − H储 ⱕ 储Gt,uGu,s − Gt,uH共u兲储 + 储Gt,uH共u兲 − H共t − 1兲储 + 储H共t − 1兲 − H储.
共B39兲
Thus, we evaluate each term on the right-hand side and show that 共B34兲 holds.
First term. Lemma B.3 yields that
储Gt,uGu,s − Gt,uH共u兲储 ⱕ ␣共Gt,u兲储Gu,s − H共u兲储 ⱕ 2␣共Gt,u兲,
共B40兲
where we used 储Gu,s − H共u兲储 ⱕ 2. Since the Markov chain is weakly ergodic 关condition 共1兲兴, Theorem B.4 implies that
∀␧ ⬎ 0,
∃ t1 ⬎ 0,
␧
∀ t ⬎ t1:储Gt,uGu,s − Gt,uH共u兲储 ⬍ .
3
共B41兲
Second term. Since pt = G共t兲pt 关condition 共2兲兴, we find
H共u兲 = G共u兲H共u兲 = Gu+1,uH共u兲
共B42兲
Gt,uH共u兲 = Gt,u+1H共u兲 = Gt,u+1关H共u兲 − H共u + 1兲兴 + Gt,u+1H共u + 1兲.
共B43兲
and then
The last term on the right-hand side of the above equation is similarly rewritten as
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125210-46
J. Math. Phys. 49, 125210 共2008兲
S. Morita and H. Nishimori
Gt,u+1H共u + 1兲 = Gt,u+2关H共u + 1兲 − H共u + 2兲兴 + Gt,u+2H共u + 2兲.
共B44兲
We recursively apply these relations and obtain
t−2
G H共u兲 = 兺 G
t,u
t−2
t,v+1
关H共v兲 − H共v + 1兲兴 + G
t,t−1
v=u
H共t − 1兲 = 兺 Gt,v+1关H共v兲 − H共v + 1兲兴 + H共t − 1兲.
v=u
共B45兲
Thus the second term in 共B39兲 is rewritten as
t−2
储G H共u兲 − H共t − 1兲储 = 储 兺 G
t,u
t−2
t,v+1
v=u
关H共v兲 − H共v + 1兲兴储 ⱕ 兺 储Gt,v+1关H共v兲 − H共v + 1兲兴储.
v=u
共B46兲
Lemmas B.1 and B.3 yield that
储Gt,v+1关H共v兲 − H共v + 1兲兴储 ⱕ 储H共v兲 − H共v + 1兲储 = 储pv − pv+1储,
共B47兲
where we used the definition of H共t兲. Thus we obtain
t−2
储G H共u兲 − H共t − 1兲储 ⱕ 兺 储pv − pv+1储.
t,u
共B48兲
v=u
⬁
Since 兺t=0
储pt − pt+1储 ⬍ ⬁ 关condition 共3兲兴, for all ␧ ⬎ 0, there exists t2 ⬎ 0 such that
t−2
␧
∀t ⬎ ∀ u ⱖ t2: 兺 储pv − pv+1储 ⬍ .
3
v=u
共B49兲
Therefore
∀␧ ⬎ 0,
∃ t2 ⬎ 0,
␧
∀ t ⬎ ∀ u ⱖ t2:储Gt,uH共u兲 − H共t − 1兲储 ⬍ .
3
共B50兲
Third term. From the definitions of H and H共t兲, they clearly satisfy
lim 储H共t兲 − H储 = 0,
t→⬁
共B51兲
which implies that
∀␧ ⬎ 0,
∃ t3 ⬎ 0,
␧
∀ t ⬎ t3:储H共t − 1兲 − H储 ⬍ .
3
共B52兲
Consequently, substitution of 共B41兲, 共B50兲, and 共B52兲 into 共B39兲 yields that
储Gt,s − H储 ⬍
␧ ␧ ␧
+ + ⬍␧
3 3 3
共B53兲
for all t ⬎ max兵t1 , t2 , t3其. Since ␧ is arbitrarily small, 共B34兲 holds for any s ⬎ 0 and then the given
Markov chain is strongly ergodic from Theorem B.6, which completes the proof of the first part of
Theorem 5.2.
Next, we assume p = limt→⬁ pt. For any distribution q0, we have Hq0 = p because
兺 H共z,x兲q0共x兲 = p共z兲 x苸S
兺 q0共x兲 = p共z兲.
共B54兲
x苸S
Thus, we obtain
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125210-47
Mathematical foundation of quantum annealing
J. Math. Phys. 49, 125210 共2008兲
储q共t,t0兲 − p储 = 储共Gt,t0 − H兲q0储 ⱕ 储Gt,t0 − H储.
共B55兲
Hence it holds for the sup with respect to q0 苸 P, which yields 共121兲 in the limit of t → ⬁. Theorem
5.2 is thereby proved.
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