In this section we present examples showing how quantum gates and measurements are affected by several different tensor product operators applied to different pairs of qubits in the same circuit. In particular, basis-dependent gates such as CX and observables need to be coordinated, in a manner to be made precise, with tensor product operators, since separability of states and operators is tensor product operator-dependent.
4.1 An example of coordination
In this subsection we present a standard example of a teleportation circuit from
[6] but in which distinct tensor products among the three pairs of qubits are used and combined. The purpose is not to show there is a gain but that the modularity presents no difficulty and does not the change the physics (as expected). A nontrivial example of a localizable non-local gate is given in this subsection.
The formation of a Bell state, such as |β00⟩, is often given by applying a controlled-not operator to the result of applying a Hadamard operator to one of a pair of qubits initially in state
|0⟩⊗|0⟩:
|
(CX)(H⊗I)(|0⟩⊗|0⟩)=(CX)(H|0⟩⊗|0⟩)=1√2(|0⟩⊗|0⟩+|1⟩⊗|1⟩). |
|
(11) |
In the example below we will use the two tensor product operators ⊗1 and ⊗2 from (4). If ⊗1 is the tensor product operator for the pair of qubits
in the circuit below, then of course the Bell state |β00⟩⊗1 is formed via the standard gate sequence
(see, e.g., p27, [6])
|
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|
(12) |
If for the pair of qubits (Q1,Q2), the operator ⊗1 is replaced by ⊗2,
then the Bell state |β00⟩⊗2 is formed. By (4),
the state |+⟩⊗1|0⟩ is also formed because it is identical to |β00⟩⊗2. Similarly, if in addition to replacing ⊗1 by ⊗2, the CX gate is eliminated, then the CX gate is effectively built into the tensor product operator for the pair (Q1,Q2) and the state formed, |+⟩⊗2|0⟩, is |β00⟩⊗1
as given by the circuit below:
|
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|
(13) |
The point of this example is not to show there is a gain by eliminating the CX gate. Rather, it is to show that both operators (gates in the case of circuits) and observables for multipartite systems must be chosen and coordinated with the tensor product operators that combine the states of the separate parts of such systems.
It might be argued that when we construct circuits we do not explicitly build in the tensor product operators among the qubits. But
in fact we do implicitly build in the tensor product operators by the choice of gates and observables in the circuit that are tensor product operator-dependent. To change the tensor product operator in a circuit is to change the ordered computational basis for the combined circuit, by re-ordering the computational basis, or changing the computational basis set altogether, or both.
Suppose now that the state space of the circuit of both qubits is C4
with each of the
qubits having state space C2.
In modifying (12) to obtain (13), the change from tensor product operator
⊗1 to ⊗2
re-orders the computational basis so that for example
|1⟩⊗2|0⟩ is |1⟩⊗1|1⟩
and results in the overall state in circuit (13) immediately after the Hadamard gate being the same as it is in (12) after the CX gate, as shown in the circuit diagram.
Suppose Q2 is measured into its computational basis immediately following the CX gate in (12) and the Hadamard gate in (13). In (12) we obtain |0⟩2 or |1⟩2
with equal probability.
In (
13) we obtain
|0⟩2 with certainty even though we are measuring the same state of the overall state of the circuit in each case. We get different results because the (implicit) observables are
different in the two cases. Specifically,
an observable for the projective measurement into the computational basis of
Q2, considered in isolation, is
Z.
An observable
M for the measurement of the combined system of both qubits in (
12) is
given by
I⊗1Z=I⊗1(|0⟩⟨0|−|1⟩⟨1|). Let
P1 and
P−1 be the orthogonal projection operators in the spectral decomposition of
I⊗1Z.
In (
13), by lemma (
3.1) the observable is given by
I⊗2Z=(CX)(I⊗1Z)(CX)=Z⊗1Z=(|0⟩⟨0|−|1⟩⟨1|)⊗1(|0⟩⟨0|−|1⟩⟨1|) with corresponding orthogonal projection operators
Q1 and
Q2. Then for the circuit (
12) we have the two possible post-measurement states
|
P1|β00⟩⊗1√⟨β00|⊗1P1|β00⟩⊗1, P−1|β11⟩⊗1√⟨β11|⊗1P−1|β11⟩⊗1. |
|
(14) |
In both cases the denominators, being 12 show the post-measurement outcomes to be equally likely. In (13) the possible post-measurement states are
|
Q1|β00⟩⊗1√⟨β00|⊗1Q1|β00⟩⊗1, Q−1|β11⟩⊗1√⟨β11|⊗1Q−1|β11⟩⊗1. |
|
(15) |
The second of these two post-measurement states is undefined, and the first has unit probably.
The distinct observables for these two circuits show that if the projective measurement into the computational basis of H(Q1)⊗H(Q2) as part (13) is carried out with respect to the observable corresponding to ⊗2 while ⊗1 holds between Q1 and Q2, the improper coordination of the observable with the tensor product operator results in an unintended distribution on the possible results of the measurement.
Change of tensor product operator from ⊗1 to ⊗2 in either of these circuits
can result in the localization of some measurements.
Consider a joint projective measurement of the qubits into the eigenstates of observable
|
S=1√2⎡⎢
⎢
⎢⎣1 0 0−101−100−1−10−100−1⎤⎥
⎥
⎥⎦B1 |
|
(16) |
which is a local with respect to ⊗2 since
|
S′=(CX)S(CX)=(|−⟩⟨0|−|+⟩⟨1|)⊗3I. |
|
(17) |
The eigenvalues of
S and
S′ are
1 and
−1, each with multiplicity
2. The corresponding measurement only nontrivially involves
Q1 and measures
Q1 into the eigenstates of
S′. Thus, if the original eigenstates of
S are needed for further processing, they are known from the known observable and the post-measurement state that would have resulted from the original measurement.
Further, they recoverable from the known eigenstates of the observable.
The relationship between ⊗1 and ⊗2 in circuit (12) suggests that changing tensor products in “mid-circuit” can be useful.
Changing tensor product operators in this way is effectively achievable by keeping another related circuit but using a different tensor product operator in the other circuit.
One can then transfer
qubit states by swapping relative to the appropriate tensor product operators that connect the two circuits by connecting the corresponding qubits with swap gates. For example, consider the following circuit
|
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|
(18) |
The tensor product operator to connect Q1 with Q′1 and Q2 with Q′2 can be any tensor product operator one likes and could even be distinct for each of the qubit pairs to be connected with swap gates.
In practice, that operator would be determined by what unitary operator one chooses to be the swap gates, but one might as well choose, say, ⊗1.
In the following subsection, we show
how to combine these extra tensor product operators with ⊗1 and ⊗2 to produce states of the 4-qubit circuit.
4.2 Mixing tensor products: example with teleportation
This subsection presents a standard example of a teleportation circuit from
[6] but in which distinct tensor products among the three pairs of qubits are used and combined.
The purpose of this exercise is not to show there is a gain
but rather to demonstrate that the modularity presents no difficulty and does not change the physics.
In addition, a nontrivial example of a localizable non-local operator is given in this subsection.
Consider the following well-known 3-qubit quantum circuit to teleport the state of one of Alice’s qubits to Bob’s qubit where Alice and Bob share the Bell state |β00⟩ at the start of the process, [6].
|
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|
(19) |
Below, we explore the consequences for this teleportation circuit arising from using distinct tensor product operators for each of the three 2-qubit subsystems. The purpose is to exhibit in detail how potentially distinct tensor product operations among different pairs of qubits of a circuit affect the circuit’s action. It will be seen that if the gates and observables on the 2-qubit subsystems are coordinated with the tensor product operators on each of the 2-qubit subsystems, then the net action of the circuit is to produce the same 3-qubit states |φ1⟩, |φ2⟩ and |φ3⟩ as it would if it employed the same tensor product operator on all three 2-qubit subsystems.
The state spaces of each of the three qubits is C2 with the usual standard ordered orthonormal basis which we have labeled B0, above. For illustration purposes, we choose three distinct ordered orthonormal bases for the state spaces of the qubit pairs: HA1⊗HA2, HA1⊗HB and HA2⊗HB. The tensor product symbol in the expressions for each of these state spaces for the qubit pairs does not denote an operator on pairs of vectors.
Instead, the symbol ⊗ denotes the tensor product space to serve as the codomain for tensor product operators on the vectors in the spaces being joined.
For example,
C2⊗C2 denotes the tensor product space that serves as the codomain for all tensor product operators of type C2 \large× C2⟶C2⊗C2 such as the operators ⊗1 and ⊗2 that were defined above in example (2.1). ⊗1 and ⊗2 have the same type:
C2 \large× C2⟶C2⊗C2, where it is assumed the tensor product space has been constructed to be C4.
Two of the bases are given by:
B(HA2⊗HB)=B1
and
B(HA1⊗HB)=B2
and the third is given by
B(HA1⊗HA2)=B3,
where
B3 is derived from the Bell basis with respect to ⊗1, and is given by
|
|0⟩A1⊗3|0⟩A2=|β10⟩=1√2(|0⟩A1⊗1|0⟩A2−|1⟩A1⊗1|1⟩A2),|0⟩A1⊗3|1⟩A2=|β11⟩=1√2(|0⟩A1⊗1|1⟩A2−|1⟩A1⊗1|0⟩A2),|1⟩A1⊗3|0⟩A2=−|β01⟩=−1√2(|0⟩A1⊗1|1⟩A2+|1⟩A1⊗1|0⟩A2),|1⟩A1⊗3|1⟩A2=−|β00⟩=−1√2(|0⟩A1⊗1|0⟩A2+|1⟩A1⊗1|1⟩A2). |
|
(20) |
Thus, ⊗3 combines the subsystem consisting of qubits A1 and A2, ⊗2 combines A1 and B, and ⊗1 combines A2 and B.
The corresponding matrix for the similarity transformation S† –––S for a change of basis from B1 to B3 is
|
S=1√2⎡⎢
⎢
⎢⎣100−101−100−1−10−100−1⎤⎥
⎥
⎥⎦B3→B1. |
|
(21) |
B3 yields a third tensor product operator ⊗3 such that
|
B3={|0⟩A1⊗3|0⟩A2, |0⟩A1⊗3|1⟩A2, |1⟩A1⊗3|0⟩A2, |1⟩A1⊗3|1⟩A2}. |
|
Let H=C8 be the state space of the circuit, and choose any fixed ordered orthonormal basis B of H and denote it by
|
B={|bijk⟩∣i,j,k∈{0,1}}={|b000⟩,|b001⟩,…,|b111⟩}. |
|
(22) |
An expression such as
|
|ψ1⟩A1⊗1|ψ2⟩A2⊗1|ψ3⟩B |
|
(23) |
is not rigorously defined.
To be rigorously defined the operator
⊗1 would have to map, in particular, a pair of states, one of which is 4-level, and the other 2-level, to an 8-level state while also mapping a pair of 2-level states to a 4-level state.
Thus, the expression is ill-typed due to both an implicit overloading of the symbol
⊗1 and an ambiguous grouping. A rigorous means of treating this obstacle is to
define several tensor product operators as in the following. Define three tensor product operators: For each i,j,k∈{0,1},
|
⊗B(A1A2):(HA1⊗HA2) \large× HB⟶H:(|i⟩A1⊗3|j⟩A2,|k⟩B)↦|bijk⟩,⊗A2(A1B):(HA1⊗HB) \large× HA2⟶H:(|i⟩A1⊗2|k⟩B,|j⟩A2)↦|bijk⟩,⊗A1(A2B):(HA2⊗HB) \large× HA1⟶H:(|j⟩A2⊗1|k⟩B,|i⟩A1)↦|bijk⟩. |
|
(24) |
Prefix notation is used in these auxiliary tensor product operator definitions since it is necessary in the case of ⊗A2(A1B).
With these operators the successive states of (19) can be calculated. Let the initial state of A1 be a|0⟩+b|1⟩.
We then have
|
|φ1⟩=⊗A1(A2B)(|β00⟩⊗1,a|0⟩A1+b|1⟩A1)=a√2(⊗A1(A2B)(|0⟩A2⊗1|0⟩B,|0⟩A1))+a√2(⊗A1(A2B)(|1⟩A2⊗1|1⟩B,|0⟩A1))+ b√2(⊗A1(A2B)(|0⟩A2⊗1|0⟩B,|1⟩A1))+b√2(⊗A1(A2B)(|1⟩A2⊗1|1⟩B,|1⟩A1))=a√2|b000⟩+a√2|b011⟩+b√2|b100⟩+b√2|b111⟩=a√2(⊗B(A1A2)(|0⟩A1⊗3|0⟩A2,|0⟩B))+a√2(⊗B(A1A2)(|0⟩A1⊗3|1⟩A2,|1⟩B))+ b√2(⊗B(A1A2)(|1⟩A1⊗3|0⟩A2,|0⟩B))+b√2(⊗B(A1A2)(|1⟩A1⊗3|1⟩A2,|1⟩B)) . |
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(25) |
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|φ2⟩=((CX)B3⊗B(A1A2)IB)|φ1⟩=a√2(⊗B(A1A2)(|0⟩A1⊗3|0⟩A2,|0⟩B))+a√2(⊗B(A1A2)3(12)(|0⟩A1⊗3|1⟩A2,|1⟩B))+ b√2(⊗B(A1A2)(|1⟩A1⊗3|1⟩A2,|0⟩B))+b√2(⊗B(A1A2)(|1⟩A1⊗3|0⟩A2,|1⟩B))=a |
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