Download ADIABATIC QUANTUM COMPUTATION

University of Ljubljana
Faculty for Mathematics and Physics
Department for Physics
SEMINAR - 4.letnik
ADIABATIC QUANTUM COMPUTATION
Avtor: Enej Ilievski
email: [email protected]
Mentor: dr. Marko Žnidarič
May 17, 2010
Abstract
In this seminar we present adiabatic quantum algorithm – a quantum computational method
for solving hard computational problems which relies on the adiabatic theorem. Such approach is
particularly interesting because it may offer possibilities to reduce the effects of quantum decoherence.
We are mainly focusing on the conceptual background.
1
INTRODUCTION
In adiabatic quantum computation (AQC) one encodes a computational problem in a suitable physical
system in such way that the structure of the ground (lowest energy) state reveals the answer to the
problem. In order to find the ground state one starts with engineering some simple Hamiltonian in
its ground state and gradually deforms it into complex Hamiltonian whose ground state encodes the
solution to the problem. If this deformation is sufficiently gradual, then the transformation of the state
is adiabatic, and the system remains in its ground state throughout the evolution. AQC is known to be
an universal model of quantum computation.
We begin this seminar by briefly describing some basic concepts in quantum computation, especially related to the standard (logic gate) quantum computer, following by a short introduction to the
possibilities of implementing quantum computers and comparing their power with respect to classical
computers.
1
Before focusing on the adiabatic quantum computation and quantum annealing principles, we present
some alternative approaches in quantum computation.
At the end we present some recent theoretical and practical developements together with practical
experimental implementation achievements concerning AQC.
Although we give a short introduction to quantum computing some basic knowledge of quantum
mechanics is required for proper understanding of this seminar.
2
QUANTUM COMPUTER
Quantum computer is a device for computation that uses quantum mechanics phenomena (such as
superposition of states and quantum entanglement) to perform operations on data. A theoretical model
is an abstract machine, called quantum Turing machine (or universal quantum computer), which has the
same relation to quantum computation that normal Turing machines have to classical computation [1].
1
2.1
BITS VS. QUBITS
A memory of classical computer consists of well-known bits, where each bit represents either a one or a
zero. Quantum computer, on the other hand, performs operations on qubits (Fig. 1). Qubit is formally
a two-level quantum mechanical system where one eigenstate refers to value 0 (or |0i in Dirac notation)
and other to value 1. Quantum systems can be found in superposition of states, thus a qubit is given by
|ψi = α|0i + β|1i.
Figure 1: Qubit, a fundamental entity in quantum informatics, can be represented as a point on Bloch
sphere; |ψi = cos (θ/2)|0i + eiφ sin (θ/2)|1i. A state of qubit is a vector in two-dimensional complex
vector (Hilbert) space. Special states |0i and |1i are known as computational basis states and form an
orthogonal basis of this vector space.
The crucial fact is, that we cannot examine a qubit to determine its quantum state, i.e. coefficients
α and β. In fact, when we measure a qubit we get either the result 0, with probability |α|2 , or result
1, with probability |β|2 . 2 After the measurement a qubit is found in a state that corresponds to the
measurement result, e.g. if we get 0, |ψi “collapses” to state |0i.
Moreover, a pair of qubits can be found in a superposition of 4 states (|00i, |01i, |10i, |11i), three
qubits in 8 states, or in general, n qubits in 2n states 3 (n classical bits represent only one of possible
2n states). The computational basis vectors of n-qubit system are of the form |x1 x2 . . . xn i, xi ∈ {0, 1},
1 It
means that any quantum algorithm can be represented as particular quantum Turing machine.
course, |α|2 + |β|2 = 1 holds.
3 Unfortunately there is no simple generalization of Bloch sphere for multiple qubits.
2 Of
2
so a quantum state of such a system is specified with 2n amplitudes. For n = 300 this number is larger
than the estimated number of atoms in the Universe! Trying to store all this information on classical
computer would be impossible. However, it seems Nature is capable of manipulating such enormous
quantities of data during evolution of quantum systems and this huge computational power is really
something we would like to take advantage of.
Many physical systems can be used to represent qubits, for example two different polarizations of a
photon, alignment of a nuclear spin in uniform magnetic field, two states of an electron orbiting a single
atom etc.
2.2
QUANTUM COMPUTER USING QUANTUM LOGIC GATES
Undoubtly the most popular model for quantum computation is a quantum circuit model in which
a computation is a sequence of reversible transformations on n-qubit (quantum) register. 4 These
transformations are unitary (probability preserving) and are called quantum gates [2, 3].
An arbitrary quantum computation (reversible operation) on any number of qubits can be generated
by a finite set of quantum gates. Such set is said to be universal for quantum computation. An universal
gates for classical computation are NAND and NOR gates. 5 In quantum computation, any multiple
qubit logic gate can be composed from CNOT (2-qubit gate) and single qubit gates (described by 2 × 2
unitary matrices) such as Hadamard gate and phase gate. 6
As we intuitively expect, quantum computers can simulate classical computations (non-deterministic
as well), but there would be little point in going to all the trouble with quantum effects if we would not
be able to solve some problems much more efficiently than classical computers.
Broadly speaking, there are three classes of quantum algorithms which provide an advantage over
known classical algorithms.
1. A class of algorithms based upon quantum version of the Fourier transform (Shor’s algorithm for
factoring and discrete logarithm).
2. Quantum search algorithms (Grover’s algorithm 7 ).
3. Quantum simulations (algorithms for simulatiing quantum systems 8 ).
2.3
QUANTUM DECOHERENCE
One of the greatest challenges in quantum computation is controlling or removing quantum decoherence
which usually means isolating system from its environment. Quantum decoherence is a consequence of
interaction of quantum systems with their environments resulting in their probabilistic behaviour. 9
This is a non-unitary effect (irreversible) and can be viewed as the loss of infomation from the system
to environment.
4 Note
that classical logic gates are irreversible (non-invertible)!
classical computation can also be done combining only 3-bit Toffoli gates.
6 A single-gate universal quantum gates can also be formulated using 3-qubit Deutsch gate.
7 Given a search space of size N (with no prior knowledge about the structure of information in it) we want to find the
element satisfying a known
√ property. Classicaly, this problem can be solved in O(N ), but the quantum search algorithm
can do it using only O( N ) operations.
8 The main difficulty of simulating quantum systems on classical computers is exponentially growing size of the system
(number of complex numbers needed to describe it). By contrast, a quantum computer can perform the simulation using
O(n) qubits, however the question how to efficiently extract desired information from the quantum system still remains.
9 Note that the combined system (system + environment togehter) is always in a pure quantum state and desribed by
unitary evolution. However, due to interacions of with external (unknown) degrees of freedom, system behaves as statistical
ensemble of different states, rather than quantum superposition of them.
5 Universal
3
2.4
PHYSICAL REALIZATION OF QUANTUM COMPUTER
To realize a quantum computer, we must not only give qubits some robust physical representation, but
also select a system in which they can be made to evolve as desired. Furthermore, we must be able to
prepare qubits in some specified set of initial states, and to measure the final output.
A single nuclear spin can be a good choice for a qubit, because superpositions of being aligned with or
against an external magnetic field can last a long time. Unfortunately it is difficult to build a quantum
computer from nuclear spins because their coupling to the world is so small that it is hard to measure
the orientation of a single nuclei.
Physical support
Single photon
Electrons
Nucleus
Optical lattices
Josephson junction
Singly charged
quantum dot pair
Quantum dot
Current
Energy
|0i
Horizontal
Vacuum
Up
No electron
Up
Up
Uncharged SC island
(Q = 0)
CW current
Ground state
|1i
Vertical
Single photon state
Down
One electron
Down
Down
Charged SC island
(Q = 2e, extra Cooper pair)
CCW current
First exicited state
Charge
Spin
Electron on left dot
Down
Electron on right dot
Up
Name
Polarization encoding
Photon number
Electronic spin
Electron number
Nuclear spin (NMR)
Atomic spin
SC charge qubit
Information support
Polarization of light
Photon number
Spin
Charge
Spin
Spin
Charge
SC flux qubit
SC phase qubit
Electron localization
Dot spin
Table 1: Incomplete list of physical implementations of qubits (choices of basis are by convention)[4].
Despite huge number of entries in the table above, only three fundamentally different qubit representations exist: spin, charge and photon.
A key concept in understanding the merit of a particular realization is the notion of quantum noise
(sometimes called decoherence). This is because the length of the longest possible quantum computation
is given by the ratio of decoherence time 10 to operation time 11 . These two times are determined by the
strength of coupling of the system to the external world.
Still, if the error rate (due to decoherence) is small enough, it is thought to be possible to use quantum
error correction, thereby allowing the total calculation time to be longer than decoherence time (the cost
is greatly increased number of required qubits).
The key idea is that if we wish to protect a system against the effects of noise we have to incorporate
some redundant information. Protecting bits against the effects of noise is somehow trivial in classical
world, where for instance one needs to simply replace each bit with three copies of itself. Similar
procedure is not applicable to quantum systems due to no-cloning theorem, however there are two
additional important issues:
1. Errors are continuous - different types of continuous errors may affect the state on a single qubit.
Determining which error has occured in order to correct it would require infinite precision and
resources.
2. Destructive measurements - observation in quantum mechanics destroys the quantum state under
observation, thus making it’s recovery impossible.
Fortunately these problems are not always fatal. Basic principles of error-correcting procedures are
presented in [5].
10 The
11 The
time for which the system remains quantum-mechanically coherent.
time it takes to perform elementary unitary transformations (at least two qubits).
4
2.5
THE POWER OF QUANTUM COMPUTATION
How powerful quantum computers really are? Nobody yet knows the answer to this question, despite
some examples (such as factoring) suggest that quantum computers are more powerful than classical
computers. It is still possible that quantum computers are no more powerful than classical ones, in the
sense that any problem which can be efficiently solved on a quantum computer can be also efficiently
solved on a classical computer.
Computational complexity theory is the subject of classifying the difficulty of various computational
problems (both classical and quantum). The basic concept is a complexity class, which can be thought
as a collection of computational problems that share some common feature(s) with respect to the computational resources needed to solve those problems.
The most important classes are P and NP. The former is the class of computational problems that
can be solved efficiently (in polynomial time) on classical computer and the later is the class of problems
which have solutions that can be quickly verified (again in polynomial time) on classical computer. It is
clear that P is a subset of NP, since the ability to solve problem implies the ability to check potential for
solutions. Perhaps the most important problem of theoretical computer science is to determine whether
?
these two classes are different: P6=NP.
12
There is an important subclass of NP problems, called NP-complete problems (NPC). Any NPC
problem is at least as hard as all other problems in NP. It means that an algorithm to solve a specific
NPC problem can be adapted 13 to solve any other problem in NP. If P6=NP, then it follows that no
NP-complete problem can be efficiently solved on classical computer.
It is not known wheter quantum computers can be used to quickly solve all the problems in NP
(although they can be used to solve some of them, e.g. factoring, which is believed not to be in P). 14
Another important class in PSPACE. It consist of problems which can be solved using resources
which are few in spatial size, but not necessary in time. PSPACE is believed to be strictly larger than
P and NP (see Fig. 2), although this has never been proved.
Finally we mention BPP complexity class containing problems that can be solved using randomized
algorithms in polynomial time, if a bounded probability of error is allowed.
What about quantum complexity classes? We can define BQP (acronym stands for bounded error,
quantum, polynomial time) to be the class of all computational problems which can be solved efficiently
on a quantum computer, allowing a bounded probability of error. Quantum computers run only probabilistic algorithms 15 , so BQP on quantum computer is the counterpart of BPP on classical computers.
Exactly where BQP fits with respect to P, NP and PSPACE is not known. What is known is that
quantum computers can solve all the problems in P efficiently, but there are no problems outside of
PSPACE which can be solved efficiently, therefore BQP probably lies somewhere between them. An
important implication is that if it is proved that quantum computers are strictly more powerful than
classical computers, then it will follow that P is not equal to PSPACE.
Although quantum computers may be faster than classical computers they can’t solve any problems
the classical computers can’t (given enough time and memory). Since a probabilistic Turing machine can
simulate quantum computers, they could never solve an undecidable problem like the halting problem.
12 Most
scientists believe there are problems in NP that are not included in P.
precisely, any NP problem is polynomial time reducible to NPC problem.
14 Note that factoring is not known to be NP-complete, otherwise we would already know how to efficiently solve all the
problems in NP on a quantum computer.
15 However we can significally reduce the probability of error by repeating an algorithm several times.
13 More
5
Figure 2: The suspected relationship between classical and quantum complexity classes. Where quantum
computers fit between P and PSPACE is not known, in part of because we do not even know whether
PSPACE is bigger than P!
2.6
OTHER QUANTUM COMPUTER TYPES
It is important to stress that quantum computer based on quantum logical gates (also known as standard
quantum computer) mentioned above is not the only possible way of performing quantum computations!
Several other approaches have been proposed so far:
• One-way quantum computer
In 2001, Robert Raussendorf and Hans J. Briegel presented a scheme of quantum computation
that consist entirely of one-qubit measurements on a particular class of entangled states called
cluster states 16 . The measurements are used to imprint a quantum logic circuit on the state (see
Fig. 3). As the computation proceeds, the entanglement in the resource cluster state is progressively
destroyed. Cluster states are thus one-way quantum computers and the measurements form the
program (they replace the unitary evolution) [6, 7].
Figure 3: After initially creating a multiparticle entangled cluster state, a sequence of adaptive singleparticle measurements is carried out. In each step of the computation, the measurement basis of the
next qubit depends on the specific program and on the outcome of previous measurement results [6].
• Quantum cellular automata
It refers to any of several models of quantum computation, which have been devised in analogy
16 In quantum computation, a cluster state is a type of highly entangled state of multiple qubits. Cluster states can be
created efficiently in any system with a quantum Ising-type interaction between two-state particles in a lattice configuration.
6
to conventional models of cellular automata 17 introduced by von Neumann. It may also refer to
quantum dot cellular automata, which is a proposed implementation of classical cellular automata
exploiting quantum mechanical phenomena.
The computation is considered to come about by parallel operation of multiple cells. These are
usually taken to be identical quantum systems, e.g. qubits. Cells togheter form a (usually regular)
network. The evolution of the system has several symmetries. The most important are locality
(next state of the cell depends only on current state and that of its neighbours) and homogeneity
(the evolution is indepentendt of time and acts the same everywhere). The state space of the cells,
as well as the operations performed on them, should be motivated by the principles of quantum
mechanics [8].
• Topological quantum computer
It is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons
18
, whose world lines 19 cross over one another to form braids in a 3D spacetime (one temporal
plus two spatial dimensions). These braids act like the logic gates that make up the computer and
are described in terms of braid group [10]. The main advantage of a quantum computer based on
quantum braids is its stability. While the smallest perturbations can cause a quantum particle do
decohere (introducing errors in the computation) they do not change the topological properties of
the braids [11].
In a key developement for topological quantum computers, in 2005 Vladimir J. Goldman et.al.
were said to have created the first experimental evidence of using fractional quantum Hall effect
to create actual anyons [12].
Topologial quantum computers are equivalent in computational power to other standard models
of quantum computation.
To learn more about quantum topological computer we refer the reader to [13].
• Adiabatic quanum computer
Finally we mention adiabatic quantum computation which was initiated in 2001 by Edward Farhi
et.al. Authors suggested a novel quantum algorithm for solving classical optimization problems
such as satisfiability 20 (SAT) based on adiabatic theorem.
In the remaining of this seminar we are focusing on the adiabatic quantum computer and presenting
it in more detail.
3
ADIABATIC QUANTUM COMPUTATION
Adiabatic quantum computation relies on adiabatic theorem to do calculations. The goal is to find a
Hamiltonian whose ground state corresponds to the solution of the problem of interest. First, a system
with a simple Hamiltonian is taken and initialized to its ground state. Finally, the simple Hamiltonian
is adiabatically evolved to the desired Hamiltonian. By the adiabatic theorem, the system remains in
the ground state, so that the final state of the system describes the solution to the problem.
17 The cellular automata is an abstract system consisting of uniform (finite or infinite) grid of cells. Each of these cells can
only be in one of finite number of states, which are determined by its adjacent cells (neighbourhood). The most popular
example is known as “The Game of Life”[9].
18 Anyons are neither fermions nor bosons, but they share characteristics of fermions that they cannot occupy the same
state. In the real world anyons emerge from the excitations in an electron gas in a very strong magnetic field, and carry
fractional units of magnetic flux in a particle-like manner (fractional quantum Hall effect).
19 In physics, the world line of an object in the unique path of that object as it travels through 4D timespace.
20 Satisfiability is the problem of determining if variables of a given Boolean formula can be assigned in such a way as to
make the formula evaluate to TRUE.
7
Adiabatic quantum computation offers possibility to avoid the problem of quantum decoherece. Since
the system is in the ground state, interference with the outside world cannot make it move to a lower
state. The only thing we need to ensure is to keep the temperature of the bath (energy of envoronment)
lower than the energy gap between ground and first exited state of the system. If this condition is
fulfilled, the system has a very low probability of going to a higher energy state. Thus the system can
in principe stay coherent as long as needed. 21
3.1
ADIABATIC THEOREM
Adiabatic theorem simply states that a physical system remains in its instantaneous eigenstate if a given
perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of
Hamiltonian spectrum.
Note that the term adiabatic is traditionally used in thermodynamics to describe processes without
the exchange of heat between the system and environment. The quantum mechanical definition is
somehow closer to the thermodynamical concept of quasistatic process 22 , and has no direct relation
with heat exchange. A true analogy comes when entropy (TD system) and quantum number (QM
system) are both considered to remain unchanged in adiabatic processes.
Suppose an energy of the system H(t0 ) is given at some initial time t0 , with the corresponding
eigenstate labelled as ψ(x, t0 ). Changing conditions in a continuous manner we end up with a final
Hamiltonian H(t1 ) at some later time t1 in a final state ψ(x, t1 ). The adiabatic theorem states that the
modification of the system critically depends on time difference τ = t1 − t0 .
Diabatic process Rapidly changing conditions prevent the system from adapting its configuration
during the process, hence the probability density remains unchanged. Diabatic passage (infinitely rapid)
is reached in the limit τ → 0: |ψ(x, t1 )|2 = |ψ(x, t0 )|2 .
Adiabatic process For a trully adiabatic process we require τ → ∞; in this case ψ(x, t1 ) will be an
eigenstate of the final Hamiltonian H(t1 ), with a modified configuration: |ψ(x, t1 )|2 6= |ψ(x, t0 )|2 .
Example: Quantum harmonic oscillator Let us quickly explain the effects of the adiabatic theorem
on the simple example. Consider a quantum harmonic oscillator as the spring constant k is increased
(resulting in a narrowing of the potential well in the Hamiltonian).
If k is increased adiabatically ( dk
dt → 0) then the system at time t will be in an instantaneous eigenstate
of the current Hamiltonian H(t), corresponding to the initial eigenstate of H(0). In particular case, this
means the quantum number associated with quantum harmonic oscilator will remain unchanged (if
system is initially in its ground state n = 1, remains in the ground state as potential is compressed).
For a rapidly increased spring constant, the system undergoes a diabatic process ( dk
dt → ∞) in which
the system has no time to adapt its functional form to the changing conditions. While the final state
must look identical to the initial state (|ψ(t)|2 = |ψ(0)|2 ), there is no eigenstate of the new Hamiltonian
H(t) that resembles the initial state, thus the final state is composed of a linear superposition of different
eigenstates of H(t) that reproduce the form of the initial state.
3.2
QUANTUM ANNEALING
Quantum annealing (abbrev. QA) is a general method for finding the global minimum of a given
objective (cost) function over a given set of candidate solutions (the search space). It is used mainly for
21 In
reality problems with decoherence cannot be completely avoided as we shall see later on.
quasistatic process is a TD process that happens infinitely slowly. In practice, it must be carried out on a time-scale
that is much longer than the relaxation time of the corresponding system.
22 A
8
the problems where the search space is discrete (combinatorial optimization problems) with many local
minima.
In quantum annealing, a current state is replaced by randomly selected “neighbour” state if the
latter has lower energy (value of the objective function). The process is controlled by the tunneling
field strength, the parameter that determines the extent of the neighborhood of states explored by the
method. The tunneling field starts high (so that the neighborhood extends over the whole search space)
and is slowly reduced through the computation.
Method is actually derived from its classical analogue called simulated annealing, where “temperature”
parameter plays similar role to QA’s tunneling field strength. However in simulated annealing the
neighborhood stays the same throughout the search and the temperature determines the probability
(given by Boltzmann distribution) of moving to a higher energy state, while in QA the tunneling field
strength determines the neighborhood radius. 23
It has been demonstrated experimentally as well as theoretically, that QA can out rate thermal
annealing in certain cases, specially, where the potential energy landscape consist of very high but thin
barriers, surrounding shallow local minima - it is very unlikely for the thermal fluctuations to get the
system out of such local minima, while quantum tunneling probabilities depends not only on the height
of barrier, but also on its width.
Figure 4: Schematic view on quantum annealing - while optimizing the cost function of computationally
hard problem one has to get out of a shallower minimum in order to reach a deeper minimum. Classically
one has to jump over the energy of the cost barriers separating them, while quantum mechanically one
can tunnel thought the same. If the barrier is high enough, thermal jump becomes very difficult, however,
if the barrier is narrow enough, quantum tunneling becomes quite easy [14].
In practice, the optimization problem is encoded in Hamiltonian HP . The algorithm starts by
introducing strong quantum fluctuations by adding a disordering Hamiltonian H 0 that does not commute
with H,
H = HP + ΓH 0 ,
where Γ changes from one to zero during the evolution, thus slowly removing the disordering part. Γ
indeed plays a role of so called tunneling field strength. If the process is slow enough, the system will
settle in a local minima close to exact solution (the slower the process the better the solution will be
achieved). The performance of the computation is conditioned by the residual energy (the distance from
23 In more elaborate simulated annealing variants (such as adaptive simulated annealing), the neighborhood radius is
also varied using temperature value.
9
exact solution) versus evolution time. The computation time is the time required to generate a residual
energy below some acceptable threshold value.
The main difference between QA and AQC is that in the latter, the system is constrained to its
ground state at all times, starting from the ground state of initial Hamiltonian and ending in the ground
state of HP . In other words, AQC is an exact algorithm, while QA is heuristic.
3.3
ADIABATIC QUANTUM EVOLUTION AS A COMPUTATIONAL
TOOL
Quantum system evolves in time according to Schröedinger equation
i
d
|ψ(t)i = H(t)|ψ(t)i,
dt
where |ψ(t)i is the time-dependent state vector and H(t) is the time-dependent Hamiltonian operator
(here we set ~ = 1). A quantum algorithm can be viewed as a specification of a Hamiltonian H(t) and
initial state |ψ(0)i. These are chosen so that the state at time T (|ψ(T )i) encodes the answer to the
problem at hand.
In designing our quantum algorithm we rely on the above introduced adiabatic quantum theorem.
Adiabatic evolution refers to the situation where H(t) is slowly varying. Suppose evolution starts at
time t = 0 in |ψg (0)i which is the ground state of Hamiltonian H(0). The adiabatic theorem guarantees
that evolving state vector |ψ(t)i remains close to instantaneous ground state |ψg (t)i if H(t) varies slowly
enough.
To specify our algorithm we have to provide H(t) for 0 ≤ t ≤ T , where T denotes the running
time of the algorithm. We choose H(t) so that the ground state of H(0) is known in advance and easy
to construct. For any particular instance of the problem, there is a Hamiltonian HP , whose ground
state encodes the solution. Although it is easy to construct HP , finding its ground state might be
computationally difficult. Thus we take HP = H(T ), which implies that |ψg (T )i encodes our solution.
For intermediate times, H(t) smoothly interpolates between H(0) and H(T ) = HP . If time T is large
enough, H(t) will indeed be slowly varying and the final state |ψ(T )i will be close to the solution of the
problem encoded in |ψg (T )i. The cruicial question is how to estimate the appropriate time T .
3.3.1
MINIMUM SPECTRAL GAP
Adiabatic quantum evolution of Hamiltonian can be generally expressed as
H(t) = [1 − s(t)]Hinit + s(t)HP ,
with s(t) changing from 0 to 1.
The performance of AQC is determined by the minimal gap
gm = min (E1 (s) − E0 (s)).
0≤s≤1
The adiabatic theorem imposes the minimum time it takes for the switching from Hinit to HP to be
adiabatic. This time can be thought of as the algorithm complexity! If H(s) has an exponentially small
minimum gap (with respect to the number of qubits used in computation) then the corresponding algorithm is inefficient, whereas minimum gap which scales inverse polynomially gives an efficient quantum
adiabatic algorithm whose running time is also polynomial.
10
3.3.2
LOCAL VS. GLOBAL EVOLUTION
In the global adiabatic evolution scheme, s is changed uniformly with time (ṡ = const.) and the compu−2
tation time scales as τglobal ∝ gm
. On the other hand, in the local adiabatic scheme s in a non-linear
−1
function of time chosen in such a way to optimize the computation time. 24 In this case τlocal ∝ gm
!
Global evolution After evolution under H(s) for a time T , the system is found in the ground state
of HP with probability (1 − 2 )2 , provided the evolution rate satisfies
|h dH
dt i1,0 |
≤ ,
2
gmin
dH dH
= E1 ; t|
|E0 ; t ,
1,0
dt
dt
(1)
where 1. The above formula (Eq. (1)) follows directly from adiabatic theorem by applying firstorder perturbation theory on a two-level system of relevant states. In particular, Eq. (1) implies that
the minimum gap cannot be smaller than a certain value if we require the state at time t to differ from
instantaneous ground state by a negligible amount (a smaller gap implies a higher transition probability
to the first excited state). As long as the gap is finite, for any finite and positive , the time of evolution
can be finite.
Local evolution Adiabatic evolution scheme can be improved, since we have applied Eq. (1) to the
entire time interval T , hence imposing the limit on the evolution rate during the whole computation
while this limit is only severe in the vicinity of gmin . Thus, by dividing T into infinitesimal time intervals
dt and applying adiabaticity condition locally to each of these intervals, we can vary the evolution rate
continuously in time and thereby speeding up the computation. New condition would be
|
g 2 (t)
ds
| ≤ dH
,
dt
|h ds i1,0 |
(2)
for all times t. 25
The local and global schemes of AQC are also different in their response to decoherence. The global
scheme is robust against environmental noise, on the contrary, local adiabatic scheme is very sensitive
−2
to decoherence. It was shown that in order for the local scheme to change the scaling time from ∝ gm
26
−1
to ∝ gm , the computation time should be smaller than global dephasing time.
Beside that, local
adiabatic evolution requires knowledge of the spectrum which is not feasible for general Hamiltonians.
3.3.3
LANDAU-ZENER FORMULA
Landau-Zener formula [15] is an analytic solution to the equations of motions governing the transition
dynamics of a 2-level quantum mechanical system, with a time-dependent Hamiltonian varying such
that the energy separation of the two states is a linear function of time.
If the system starts in the lower energy eigenstate we are interested in the probability of finding
the system in the upper energy eigenstate in infinite future (so called Landau-Zener transition). For
infinitely slow variation of the energy difference, the adiabatic theorem tells us that no such transition
will take place, as the system will always be in an instantaneous eigenstate of the Hamiltonian at some
24 This
is done by spending the majority of time in the vicinity of anticrossing.
instance, local evolution applied to adiabatic Grover’s search algorithm
provides quadratic speed up over global
√
evolution, enabling to solve the problem in total running time of order N (same as logic gate quantum computer)[18].
26 Characteristic time over which the mutual phases between qubits are destroyed. This implies loss of information from
the system.
25 For
11
time. At non-zero velocities, transitions (diabatic) occur with probability given by the Landau-Zener
formula
a2 /~
a2 ~
PD = e−2πΓ ,
Γ= ∂
=
,
dq ∂
| ∂t (E2 − E1 )|
| ( ) ∂q (E2 − E1 )|
dt }
| {z
vLZ
where q is the perturbation variable (e.g. electric or magnetic field), E1,2 are the energies of the crossing
states and vLZ denotes Landau-Zener velocity which is inversely proportional to time. The quantity a
is the off-diagonal element of the two-level Hamiltonian coupling given eigenstates.
Since computational time is always finite, there is a nonzero probability that system would undergo
Landau-Zener transitions and end up in an exited state. Hence, Landau-Zener transitions are important
in estimating the sufficient computation time.
4
WHAT KIND OF PROBLEMS CAN AQC SOLVE?
AQC is not restricted only to optimization problems (unlike QA) - an universal AQC can run any
quantum algorithm, and has been shown to be computationally equivalent to the gate model of quatnum
computation, as both can be efficiently mapped into each other [16].
Quantum adiabatic algorithms have been aplied to solve various optimization problems, for instance
finding cliques 27 in random graphs [17]. There is no know classical algorithm that finds the largest
clique in a random graph with high probability and runs in polynomial time.
In these algorithms, the condition for adiabaticity is fulfilled globaly by using only the minimum
energy gap between ground and first exited state to determine the computation time. This method
(global evolution) is not efficient in some cases, such as adiabatic Grover’s search algorithm [18] and
adiabatic Deutsch-Jozsa algorithm [19] as they result in a complexity τglobal = O(N ) (which is complexity
of classical algorithms). However these algorithms can be improved by application
of local adiabatic
√
evolution yielding an optimal performance of a quantum algorithm, τlocal = O( N ).
On the other hand, the universal AQC can provide solution to a problem in polynomial time if
the same problem can be solved in polynomial time using logic gate quantum computer. Evidently,
the polynomial advantage does not depend on the local evolution (instead it only provides a quadratic
enhancement).
4.1
RECENT DEVELOPEMENT
An important question is what kind of problems can benefit from AQC without requiring local adiabatic
evolution and therefore phase coherence? Using a perturbative approach to estimate the gap size of
adiabatic quantum optimization it has been found that the gap is inversely proportional to the square
root of the number of states that have energies close to global minimum, which means if the number of
low energy local minima becomes exponentially large, then the gap will be exponentially small. In such
cases, only a local adiabatic evolution scheme can provide advantage over classical computation. Local
AQC however requires phase coherence during the evolution and knowledge of the energy spectrum,
which limits its practicality [20].
These problems, although unsuitable for AQC, could still be suitable for heuristic algorithm (such
is quantum annealing) if approximate solutions are acceptable, because the chance of finding a solution
within the acceptance range will be large.
Equally important question is whether the interactions between the computer and its environment
can spoil the computation. It is clear that AQC has fundamental advantages over the gate model in
27 A
clique in an undirected graph is a subset of its vertices such that every two vertices are connected by an edge.
12
regards to robustness against decoherence, however there does not yet exist an equivalent of the threshold
theorem [5] that describes under what conditions AQC coupled to an environment will succeed [21].
In an isolated system with no decoherence the limitation is due to the usual Landau-Zener tunneling
at the anticrossing. Probability to leave out of the ground state is given by the a adiabatic theorem.
The main assumption here is that there exist a well-defined energy gap between the two lowest energy
states. In reality, the energy levels of qubit register are broadened by the coupling to an environment.
Figure 5: Broadening of the energy levels of a closed system (a) due to coupling to the environment
made of a single two-state system (b) or infinitely many degrees of freedom with a continuous energy
spectrum (c). In latter case the anticrossing turns into a continuous transition region [22].
Since the broadening W typically increases with the number of qubits (Fig. 5), while the minimum
gap gm decreases, the realistic large-scale system will eventually fall in the (incoherent) regime W gm
[22].
5
PRACTICAL IMPLEMENTATION OF ADIABATIC QUANTUM COMPUTER
First experimental implementation of Shor’s algorithm was demonstrated by Vandersypen et. al. [23]
in 2001 using nuclear spins to find the prime factors of number 15. More recent experiments by Lu et.
al. [24] and Lanyon et. al. [25] used photons as qubits and found the same factors. In 2005, Mitra
et. al demonstrated the experimental implementations of local adiabatic evolution algorithms (Grover’s
search and Deutsch-Jozsa algorithm) on a 2-qubit quantum information processor using NMR [26, 27].
Chuang et. al. [28] have demonstrated the implementation of a quantum adiabatic algorithm by
solving MAXCUT 28 problem on a 3-qubit system by NMR.
Note that in actual implementation, the Hamiltonian of the system is discretized in order to recast
adiabatic evolution in terms of unitary operators,
U=
M
Y
m=0
Um ,
Um = exp (−i[(1 −
m
m
)Hinit +
HP ]∆t),
M
M
∆t = T /(M + 1).
Discretizing a continuous Hamiltonian is straightforward process and changes the total run time T
28 One wants a subset S of vertex set such that the number of edges between S and complementary subset is as large as
possible.
13
of the adiabatic evolution only polynomially. The required adiabatic limit is achieved when both T and
number of discrete steps M approach infinity.
5.1
D-WAVE SYSTEMS (The Quantum Computing Company)
D-Wave’s core focus is the developement of superconducting processors capable of running adiabatic
quantum algorithms for solving quadratic unconstrained binary optimization problems (NP-hard optimization problem) 29 . Many important scientific and commercial problems require the solution of
QUBO.
D-Wave processors actually perform quantum annealing. Combinatorial optimization is represented
by a disordered Ising spin Hamiltonian, e.g.
H0 (t) =
X
hi Siz +
i=1
N
X
Jij Siz Sjz +
i,j
X
∆i (t)Siy
i
Example is taken from a Hydra processor manufactured by D-Wave. Si are representing Pauli
matrices, hi is the local bias on qubit i, Jij is the coupling strength between qubits i and j and ∆i (t)
tunneling matrix element. A problem instance is encoded in the h and J values. The traverse term is
used to control the quantum annealing schedule [29].
Many artificial intelligence problems can be mapped to NP-hard optimization problems, particulary
QUBO is found to be very useful in pattern matching, common in machine learning applications [30].
D-Wave processors are fabricated using superconducting metals instead of semiconductors and are
operated at ultra-low temperatures.
A circuit consisting of a network of coupled compound Josephson junction rf-SQUID flux qubits has
been used to impelement an adiabatic quantum optimization algorithm [31].
Flux qubit Flux qubits (also known as persistent current qubits, are micro-metre sized loops of
superconducting metal interrupted by a number of Josephson junctions. The junction parameters are
engineered during fabrication so that a persistent current will flow continuously when an external flux is
applied. The computational basis states of the qubit are defined by the circulating currents which can
flow either clockwise or counter-clockwise. These currents screen the applied flux limiting it to multiples
of the flux quantum and give the qubit its name. When the applied flux through the loop area is close
to a half integer number of flux quanta the two energy levels corresponding to the two directions of
circulating current are brought close together and the loop may be operated as a qubit.
Computational operations are performed by pulsing the qubit with microwave frequency radiation
which has an energy comparable to that of the gap between the energy of the two basis states.
29 QUBO
is given by the formula E(X1 , . . . , XN ) =
PN
i≤j
Qij Xi Xj , Xi ∈ {0, 1}
14
Figure 6: Large loop interrupted by two Josephson junctions (the SQUID) merged with the smaller
loop on the right side comprising three in-line Josephson junctions (the flux qubit). Arrows indicate the
direction of the persistent current for each qubit state and the corresponding measured Rabi oscillations
(cyclic behaviour of a two-state quantum system in the presence of an oscillatory driving field) are shown
below. Image is taken from qsd.magnet.fsu.edu.
Conclusions
We have seen that the efficiency of AQC approach is fundamentally limited by the small spectral gaps
between ground and exited states. It was shown that these gaps can become exponentially small under
specific conditions, such as bad choice of initial Hamiltonian [32] or specifically designed hard instances.
As it unfortunately seems, exponentially small gaps appear close to the end of the adiabatic algorithm
for large random instances of NPC problems, which indicates the failure of the adiabatic quantum optimization (the sistem gets trapped in one of the numerous local minima) [33, 36]. However it is important
to point out that we are still lacking the rigorous analytical result characterizing the performance of AQC
on random instances of NPC problems. Anyway, by assuming worst case scenario AQC could still be
found more efficient in comparisson to classical algorithms on average case.
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Computing
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