Download Lecture 6: QUANTUM CIRCUITS 1. Simple Quantum Circuits We`ve

Lecture 6: QUANTUM CIRCUITS
1. Simple Quantum Circuits
We’ve already mentioned the term ”quantum circuit”. Now it is the time to
provide a detailed look at quantum circuits because the term ”quantum computer”
itself is synonymous with the quantum circuit model of computation. Generally, a
quantum circuit is formed by the gates connected by lines. The simplest quantum
circuits containing the single qubit gates are shown in Figure 4.2. Each line in the
circuit represents a wire in the quantum circuit. This wire does not necessarily
correspond to a physical wire. Instead, it can correspond simply to the passage of
time, or perhaps to a particle such as a photon - a particle of light - moving from one
location to another through space. The circuit is to be read from left-to-right. It is
conventional to assume that the state input to the circuit is a computational basis
state, usually the state consisting of all |0 >s. If this rule is broken, it is necessary
to inform the user about the input state.
The multi-qubit gates themselves may be represented by some circuits. For example, Figure 4.3 shows the circuit representation for the CONTROLLED-NOT gate.
The top line represents the control qubit, the bottom line the target qubit. The
symbol ⊕ reminds about an addition modulo 2 [see: Eq. (5.76)]. It is interesting
what will happen if we change this symbol by the symbol for an arbitrary single
qubit unitary operation U ? Then we obtain the circuit drawn in Figure 4.4. The
operation represented by such a circuit is said to be CONTROLLED-U operation.
We see that this operation is again a two qubit operation, with a control and a
targed qubit. If the control qubit is set to |1 > then U is applied to the target qubit,
otherwise the targed qubit is left alone, that is
|a, b > →
|a > U a |b > .
(1)
Hence the CONTROLLED-U gate is a natural extension of the CONTROLLEDNOT gate, and the latter itself can be represented in a different way, as illustrated
in Figure 1.9.
We just consider the CONTROLLED-U operation with one target qubits. What
about n target qubits? In circuit representation, the answer is obtained immediately:
Figure 4.4 may be changed to Figure 1.8!
The circuit representation can be used in order to implement an arbitrary CONTROLLEDU operation using only single qubit operations and the CONTROLLED-NOT gate.
The example of such representation is on Figure 4.6, but we will not discuss it in
detail.
Using quantum circuits we can construct many other important operations. For
example, the circuit in Figure 1.7 accomplishes a simple but very useful task - it
swaps the states of the two qubits. To see this, let us write the sequence of effects
of the gates shown in this Figure on a computational basis state |a, b >:
|a, b > → |a, a ⊕ b >
→ |a ⊕ (a ⊕ b), a ⊕ b >= |b, (a ⊕ b) >
→ |b, (a ⊕ b) ⊕ b >= |b, a > .
(2)
Here all additions are done modulo 2. The total effect of the circuit, therefore, is to
interchange the states of the two qubits. An equivalent schematic symbol for this
important operation is shown in Figure 1.7 on the right.
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A slightly more complicated circuit, shown in Figure 1.12, has a Hadamard gate
followed by a CONTROLLED NOT and transforms the four computational basis
states according to the table given. We see that the final states are nothing but the
Bell states which example we have discussed in one of our previous lectures [see:
Eq. (4.17)]. Note the useful mnemonic rule for remembering all the Bell states (see
test 4, question 4):
|0y > +(−1)x |1, ȳ >
√
|βxy >≡
,
(3)
2
where ȳ is the negation of y.
So far we did not mention about the circuit representation of the measurement.
Meanwhile, we know that the measurement plays the role of an interface between
the quantum and classical worlds, and so it may present in any circuit designated
for the solution of a practical task. To represent a usual projective measurement,
a special ”meter” symbol is introduced, as shown in Figure 1.10. As previously
described, this operation converts a single qubit state |ψ >= c0 |0 > +c1 |1 > into a
probabilistic classical bit M which is 0 with probability |c0 |2 and 1 with probability
|c1 |2 . To distinguish bit from qubit, the bit is drawing as a double-line wire (see
Figure 1.10).
Now we are well-experienced for understanding that quantum circuits can serve
as models for all quantum processes which are in no way limited to only the specific
problem of quantum computation. We shall illustrate this intriguing guess by considering two important examples in what follows.
2. No-Cloning Theorem
As a first example, let us try to use the quantum CONTROLLED-NOT for
copying a qubit in a way analogous that classical CONTROLLED-NOT does. This
way is illustrated on the left-hand side of Figure 1.11. A classical CONTROLLEDNOT is taken with the bit to copy (in some unknown state x) as a control bit and
a ”scratchpad” bit initialized to zero as a target bit. The output is two bits, both
of which are in the same state x.
In quantum case we have to copy a qubit in the unknown state
|ψ >= a|0 > +b|1 >
(4)
and to initiate a ”scratchpad” qubit in the state |0 >. So the input state of these
two qubits may be written as
h
i
a|0 > +b|1 > |0 >= a|00 > +b|10 > .
(5)
The effect of the CONTROLLED-NOT is to negate the second qubit when the first
qubit is 1, and hence the output state is
ψ out = a|00 > +b|11 > .
(6)
Does it mean that the copying |ψ > was successful? More specifically, have we
created the state |ψ > |ψ > or not? We see that in particular cases where
|ψ >= |0 >
or |ψ >= |1 >
2
(7)
that is indeed what this circuit does. Therefore, it is possible to use the quantum
circuit shown in Figure 1.11, to copy information encoded as a |0 > or |1 >. But
with the general state |ψ > we have another story, since the output can be generally
rewritten as
|ψ > |ψ >= a2 |00 > +ab|01 > +ba|10 > +b2 |11 > .
(8)
Comparing (8) and (6), we see that unless ab = 0 the circuit above does not copy
the quantum state input! We obtained a remarkable result: it turns out to be impossible to make a copy of an unknown quantum state. This result is known as the
no-cloning theorem.
3. Quantum Teleportation
Next we will apply the quantum circuit technique to clarify something very surprising and a lot of fun - quantum teleportation! Commonly, teleportation is understood as a fictional method for transferring an object between two places by a
process of dissociation, information transmission and reconstitution. The net effect
is the destruction of the original object at the source and the creation of an exact
replica at the intended destination. A key feature of teleportation is that the actual
object does not transverse the distance between two locations. Instead, the object
is scanned to extract sufficient information to recreate the original, the information
is transmitted and an exact replica is re-assembled at the destination out of the
material is locally available.
Of course, we are hardly at the point of being able to teleport in this manner an entire person, even though it will be possible in principle. Starting from
the observation that, according to the data by the American National Institute of
Health, it requires about 10 Gigabytes to hold the information needed to describe the
three-dimensional structure of a human being to 1mm3 resolution, it can be easily
estimated that an entire human being could be described, down to the atomic level,
using roughly 1032 bits. With current communication channel capacities, it would
like take about a hundred million centuries to transmit this information down a single channel. So, I leave funny stories for beaming action heroes around the Universe
to video, and restrict myself only to the consideration of a small-scale teleportation
prototype capable of teleporting a single qubit.
Naturally, the prospects for a teleportation of such small objects as a qubit is
much better from the information point of view. Nevertheless, until recently no serious attention has been paid to the physical principles on which such teleportation
might be based. The arguments of most scientists were that the teleportation is impossible because, as the Heisenberg Uncertainty Principle stands, it is impossible to
measure all the attributes of a quantum state exactly. For example, it is impossible
to measure the position and momentum of a particle simultaneously. Consequently,
it appeared that even a scanning step of teleportation is doomed to failure because
it would never yield complete information about the original.
The situation changed in 1993 when a team of of prominent physicist and computer scientists, Charles Bennet, Gilles Brassard and others, showed that it is possible to exploit yet another aspect of modern quantum theory, namely, the notion
of entangled states and nonlocal interaction, to circumvent the limitations of the
Heisenberg Uncertainty Principle and hence to create an exact replica of a quantum
state.
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In the context of a quantum teleportation, the entangled pair of particles serve
as two ends of quantum communication channel: one particle being retained by the
person wishing to teleport the quantum state and the other by the person wishing
to receive it. Thus, in order to teleport a quantum state, the sender and recepient
must each already possess one member of a pair of entangled particles. The basic
idea is for the sender to make a measurement of the joint state of the particle whose
state is to be teleported and one of the entangled particle, and then to send the
result of the measurement as a classical message over a conventional communication
channel (such as radio). For the recepient, it is to use the information in this
classical message to determine which operation to apply to his member of the pair
of entangled particles in order to place it in a state that is an exact replica of the
state that the sender wished to teleport.
Note that in this scheme the quantum state of an object is teleported, obviously
not the object itself. Consequently, we cannot use this scheme to teleport an electron
in its entirety from one place to another, but we can teleport, say, the spin orientation
of one electron at a particular location to another electron at a different location.
However, the net effect is similar: a particle in a specific state at the source place
has its state destroyed and restored on another particle at the destination without
the original particle traversing the intermediate distance.
Now, we are able to give a more precise description of quantum teleportation
represented by the quantum circuit shown in Figure 1.13. The two top lines in this
Figure represent sender’s system, while the bottom line is the recepient’s system.
The state of the qubit to be sent is
|Ψ >= a|0 > +b|1 > .
(9)
The objective is to transmit this quantum state using classical bits and then to
reconstruct the exact quantum state at the receiver. Of course, such a statement
of the problem is surprising in light of the no-cloning principle because we want to
transmit an unknown quantum state. Nevertheless, the problem has a solution.
The key for teleportation is the use of the entangled particles in such a way
that the first and second particle of the source EPR pair is accesible for the sender
and receiver, respectively. Mathematically, the initial state of the EPR pair can be
written as the joint two-particle state that cannot be factorized as the direct product
of the states of two separate particles, say,
1 (10)
|β00 >= √ |00 > +|11 > .
2
Until a particle is transmitted, only sender can perform transformation with the
first particle and only receiver can perform transformation with the second particle
in this joint state. At this point the qubit in the state (9) is not correlated yet with
the entangled particles in the state (10), either classically or quantum mechanically,
so we can still write the combined state for the initial three-particle system as a
direct product of the states (9) and (10):
|Ψ0 >= |ψ > ⊗|β00 >= a|0 > +b|1 >
=
√1
2
√1
2
|00 > +|11 >
a|0 > ⊗ |00 > +|11 > + b|1 > ⊗ |00 > +|11 >
=
√1
2
a|000 > +a|011 > +b|100 > +b|111 > .
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(11)
We recall that at this stage the sender controls the first two bits and the receiver
the last bit in the three-qubit state (11). So, the sender is free to use first the
CONTROLLED-NOT, obtaining the state
1 ψ1 = √ a|000 > +a|011 > +b|110 > +b|101 >
2
(12)
and then the Hadamard gate, obtaining the state
ψ2 =
1
2
|00 > a|0 > +b|1 > + |01 > a|1 > +b|0 >
+|10 > a|0 > −b|1 > + |11 > a|1 > −b|0 >
!
.
(13)
After this transformation, the sender measures the first two qubits to get one of
the two-qubit states |00 >, |01 >, |10 > or|11 > with equal probability. Depending
on the result of the measurement, the quantum state of receiver’s qubit is projected
to
(14)
a|0 > +b|1 >, a|1 > +b|0 >, a|0 > −b|1 > or a|1 > −b|0 >,
respectively. That is, depending on sender’s measurement outcome, receiver’s qubit
will end up in one of the following possible states
00
01
10
11
→
→
→
→
(a|0 > +b|1 >) ≡ |ψ3 (00) >;
(a|1 > +b|0 >) ≡ |ψ3 (01) >;
(a|0 > −b|1 >) ≡ |ψ3 (10) >;
(a|1 > −b|0 >) ≡ |ψ3 (11) > .
(15)
Of course, to know which state it is in, receiver must be told the result of sender’s
measurement using an ordinary classical information channel (a telephone, say).
Once receiver has learned the measurement outcome, the state |ψ > can be recovered
by applying the appropriare quantum gate to the state of a qubit belonging to
receiver. For example, in the case where the measurement yields 00, nothing needs to
be done. If the measurement is 01, then receiver can fix up his state by applying the
Z-gate. If the measurement is 11 then receiver can fix up his state by applying first
an X and then a Z- gate. Summing up, receiver needs to apply the transformation
Z M1 X M2 to recover the state which has been send.
It seems that a teleportation creates a copy of the quantum state being teleported
- in apparent violation of the no-cloning theorem discussed in preceding section.
However, this violation is only illusory since the original state |Ψ >, which should
to be sent to the receiver, is irretrievably altered under the measurement. That is,
only target qubit is left in the state |ψ >, while the second qubit ends up in one of
the computational basis state |0 > or |1 >, depending upon the measurement result
on the first qubit. Of course, later it ”recoveres” due to the special procedures but
be recovered is not the same as be a copy, isn’t it?
Another important aspect of a quantum teleportation must be commented: doesn’t
teleportation allow one to transmit quantum states faster than light? Fortunately,
this is not the case because to complete the teleportation sender must transmit the
measurement result to receiver over a classical communication channel which work,
of course, is limited by the speed of light as it should be along the prescription of
the theory of relativity.
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It is worth noting, that if somebody can teleport a single qubit, an arbitrary
message can be teleported by decomposing it into sequence of qubits and teleporting each qubit separately. So although the ability to teleport a single qubit may
seem too modest, it actually provides the foundation for a reach new communication technology. We notice as well that quntum teleportation is closely related to
quntum computing, since it might provide an alternative way for transmission of the
primary quantum information inside a quantum computer or even between distant
quantum computers. This might be especially useful if some qubits needs to be kept
secret (as in quantum cryptography, for instance). Using quantum teleportation, a
qubit could be passed around without ever being transmitted over insecure (public)
channel.
4. Quantum dynamics: Schrödinger Equation
Although the operator level of description is adequite for representing the transformation of states in many-qubit system, it tells us nothing about the time evolution
that transforms the input state into the output one. To be able to consider this process, we need to understand a more general question of how a state vector evolves
in time. As the state of quantum memory register is described by some state vector, this amounts to asking by what rule does the memory register of a quantum
computer evolve?
Fortunately, famous physicist Erwin Schrödinger already gave us an answer to
this question back in 1926, long before quantum computers were ever imagined.
To understand this answer, we may return to the Dirac formulation of quantum
mechanics adding the time parameter to the description of quantum states:
|Ψ >≡ |Ψ(t) > .
(16)
Then the fundamental postulate of quantum mechanics will sound as:
The maximum information about the outcome of physical measurements at time
t is contained in the probability amplitudes < l, m, ...|Ψ(t) >, which correspond to a
complete set of observables L, M, ... for the system.
As expected, the only new feature here is that we now recognize formally that the
state is a function of time. Furthermore, if we take two states, |Ψ(t) > and |Ψ(t0 ) >,
with t0 < t, we shall assume that |Ψ(t) > is determined by |Ψ(t0 ) > by means of
some time-dependent operator U (t, t0 ),
|Ψ(t) >= U (t, t0 )|Ψ(t0 ) >,
(17)
which legitimizes the name time development or evolution operator for U (t, t0 ). This
assertion expresses the general principle of causality in the quantum-mechanical
form.
If Eq. (17) takes place, we can immediately construct the time-dependent composition rule for the probability amplitudes:
Ψl (t) ≡< l|Ψ(t) >=
X
< l|U (t, t0 )|m >< m|Ψ(t0 ) >≡
m
X
Slm (t, t0 ) < m|Ψ(t0 ) >,
m
(18)
where the coefficients
Slm (t, t0 ) =< l|U (t, t0 )|m >
(19)
are independent of the state |Ψ(t0 ) >. The physical interpretation of these coefficients is straightforward: they signifies the probability amplitude for finding the
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system at time t in the eigenstate |l > of the observable L, if at time t0 the system
was known to be in the eigenstate |m > of the observable M . Hereafter, we will call
these quantities the transition amplitudes.
From (17) and (19) it follows that the time evolution operator U (t, t0 ) has the
following properties:
U (t, t) = I;
U (t2 , t0 ) = U (t2 , t1 )U (t1 , t0 );
[U (t, t0 )]−1 ≡ U −1 (t, t0 ) = U (t0 , t).
(20)
Furthermore, for small we may write to first order
U (t + , t0 ) = U (t, t0 ) + dU (t, t0 )
i
≡ U (t, t0 ) − H(t)U (t, t0 ),
dt
h̄
(21)
In particular case t0 = t this takes the form
h
i
i
i
U (t + , t) = U (t, t) − H(t)U (t, t) = I − H(t) U (t, t).
h̄
h̄
(22)
In general, Eq. (22) is equivalent to the differential equation for operator U (t, t0 ),
ih̄
dU (t, t0 )
= H(t)U (t, t0 ),
dt
(23)
which should be solved with the initial condition U (t0 , t0 ) = I. By this, we introduce new operator H(t) which is the same fundamental characteristic of the system
as an original evolution operator U (t, t0 ). The factor i/h̄ in this expression could
look simply as a curious caprice if it were not be known the consequences of such
presentation.
To deduce these consequences, we consider the state
|Ψ(t + ) >= U (t + , t)|Ψ(t) >,
(24)
or, to first order in ,
"
#
d
i
|Ψ(t) > + |Ψ(t) >= I − H(t) |Ψ(t) > .
dt
h̄
(25)
Hence, we derive the differential equation for the state vector,
ih̄
d
|Ψ(t) >= H(t)|Ψ(t) > .
dt
(26)
which is the quantum-mechanical equation of motion in its most general form.
We will call Eq. (26) the Schrödinger equation though primarily this equation has
been suggested by Erwin Schrödinger not for the state vector but for the probability
amplitude (or ”wave function”) in coordinate representation, i.e. for the function
ψ(r, t) ≡< r|Ψ(t) > .
(27)
The most remarkable feature of the Schrödinger equation is the presence of a
specific operator H(t) which is called the Hamiltonian. This operator relates to
the total energy of the system which can be measured experimentally, and thus it
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can be represented by Hermitian matrix. The form of this matrix is determined
by the specific arrangement of atoms, molecules and charges that constitute the
system (in particular, the computer). In the case of computer, we can think of
the Hamiltonian as being analogous to the ”hardware” and the initial state of the
quantum memory register as being analogous to the ”data” fed into a conventional
computer. Of course, if quantum computer is specialized for performing only a single
type of computation, the ”program” is essentially folded into the definition of the
Hamiltonian.
We notice that due to the Hermitian property of the Hamiltonian H(t), the
equations adjoint to Eqs. (23) and (26) becomes
dU + (t, t0 )
= U + (t, t0 )H(t)
−ih̄
dt
(28)
and
d
< Ψ(t)| =< Ψ(t)|H(t)
(29)
dt
By multiplying (23) on the left by U + (t, t0 ) and (28) on the right by U (t, t0 ), and
subtracting the two resulting equations, we get
−ih̄
d +
[U (t, t0 )U (t, t0 )] = 0.
dt
(30)
Therefore, the product of the operators conserves in time its value at t = t0 equal
to I, i.e.
U + (t, t0 )U (t, t0 ) = I
(31)
for all t. Hence, the unitary operator is always unitary, and the norm of any state
vector remains unchanged during the motion. Say, if < Ψ(t0 )|Ψ(t0 ) >= 1, then
< Ψ(t)|Ψ(t) >= 1 for all times t.
Note that such superficially innocuous property of the evolution operator as its
inherent unitarity harbors an extremely important implication. Namely, it means
that the evolution operator of an ideal quantum computer, isolated from its enviroment, is reversible because the conjugate transpose of U (t, t0 ) is equal to the inverse
of it. Of course, quantum physicists knew all along that Schrödinger equation gives
rise to a unitary (and hence reversible) evolution. Thus if a quantum system had
any chance of serving as a computer, it had to be possible to make computers that
operated reversibly. The importance of reversible computing has been first shown
by Charles Bennett in 1973.
Using the equations of motion (26) and (29), we are able to calculate the time
derivative of the expectation value of an arbitrary operator A, which may itself vary
with time:
*
d
∂A
ih̄
< A >=< AH − HA > +ih̄
dt
∂t
+
*
∂A
≡< [A, H] > +ih̄
∂t
+
(32)
where < A >=< Ψ(t)|A|Ψ(t) >. where, as usual, the brackets signify expectation values of the operators enclosed. We see that if A commutes with H and is
independent of time, the expectation value of A is constant, and A is said to be
constant of motion.
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Finally, if the Hamiltonian of a computer is time independent, and its memory
register is in the state |Ψ(0) >, then the state of the memory register at an arbitrary
time t is
"
#
i
|Ψ(t) >= exp − Ht |Ψ(0) > .
(33)
h̄
This state determines the general solution of the Schrödinger equation with the
time-independent Hamiltonian and the initial state |Ψ(0) >.
APPENDIX: Simulation of Quantum Systems
To close out our brief consideration of some interesting and useful applications
of the quantum circuit model let cast for the simulation of the quantum systems.
Of course, some quantum systems can be simulated by a classical computer. But
their real quantum behavior might often be much more complex than its classical
simulation. So any alternative ideas for the simulation of quantum systems are
very welcome. The use of quantum computers for this purpose is one of the great
challenges in modern science and technology.
The heart of simulation of any system (classical or quantum) is the solution
of differential equations which express the physical laws governing the dynamical
behavior of a system. For quantum systems this is the Schrödinger equation (26).
In many cases this equation takes the form of a partial differential equation like the
well-known equation for a real particle moving in space in the presence of potential
V (x),
h h̄2 ∂ 2
i
∂
ih̄ ψ(x) = −
+
V
(x)
ψ(x),
(34)
∂t
2m ∂x2
where ψ(x) is the wave function in the coordinate representation,
ψ(x) =< x|ψ > .
(35)
The latter is an elliptical equation very much like the wave equation. Therefore
simulating the Schrödinger equation itself is not a special difficulty faced in simulating quantum systems since its solution can be obtain, as usual, by approximating
the state with a digital representation, then discretizing the differential equation in
space and time such that an iterative application of a procedure carries the state
from the initial to the final conditions. By this sequence of steps we achieve the
goal which is generally formulated as: what is the state at some other time and/or
position, given an initial state of the system? What is then the difficulty?
The key challenge in simulating quantum systems is the exponential number of
differential equations which must be solved. Say, for one qubit evolving according
to the the Schrödinger equation, a system of two differential equations must be
solved; for two qubits, four equations; and for n qubits, 2n equations. Sometimes,
insightful approximations can be made which reduce the effective number of equations involved, thus making classical simulation of the quantum system feasible. For
example, in many cases the Hamiltonian can be represented as a sum over many
local interactions. Specifically, for a system of n particles
H=
L
X
Hk ,
(36)
k=1
where each Hk acts on at most a constant c number of systems, and L is a polynomial
in n. Moreover, the terms Hk are often just two-body interactions. Such locality
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is quite physically reasonable, and originates in many systems from the fact that
the interactions mostly fall off with increasing distance or difference in energy. The
important point is that usually e−iHt is difficult to compute, while e−iHk t acts on
a much smaller subsystem, and is straightforward to approximate using quantum
circuits. However, it is strongly desired to have the quantum circuit which allows
one an efficient quantum simulating even Hamiltonians which are, by definition, not
the sum of local interactions.
An example of such quantum circuit is shown in Figure 4.19. This circuit performs
simulating the Hamiltonian
(37)
H = Z 1 ⊗ Z2 ⊗ Z3 ,
which acts on a three qubit system. This circuit implements e−iH∆t for arbitrary
values of ∆t. The main observation is that although the Hamiltonian involves all
the qubits in the system, it does so in a classical manner: the phase shift applied
to the system is e−i∆t if the parity of the qubits in the computational basis is even;
otherwise, the phase shift should be ei∆t . Thus simple simulation of H is achieved
by first classically computing the parity (storing the result in an ancillary qubit),
then applying the appropriate phase shift conditioned on the parity, and then uncomputing the parity (to erase the ancillary qubit). This strategy clearly works not
only for three qubits, but also for arbitrary number of qubits n. As a result, we
can use this procedure for efficient simulating more complicated Hamiltonians of
the form
1
2
n
H = σc(1)
⊗ σc(2)
... ⊗ σc(n)
(38)
,
k
where σc(k)
is a Pauli matrix acting on the kth qubit, with c(k) ∈ {0, 1, 2, 3} specifying one of the operations {I, X, Y, Z)}. Indeed, the qubits upon which the identity
operation is performed can be disregarded, and X or Y terms can be transformed,
as we early saw, to Z operations by single qubit gates. This leaves us with a Hamiltonian of the form
H = Z1 ⊗ Z2 ⊗ ... ⊗ Zn ,
(39)
which is extension of the Hamiltonian (37) for arbitrary number of operators n.
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