Hybrid quantum-classical computer for variational coupled cluster method
A hybrid quantum classical (HQC) computer, which includes both a classical computer component and a quantum computer component, solves linear systems. The HQC decomposes the linear system to be solved into subsystems that are small enough to be solved by the quantum computer component, under control of the classical computer component. The classical computer component synthesizes the outputs of the quantum computer component to generate the complete solution to the linear system.
1. A method performed by a classical computer for implementing, on a quantum computer, a non-unitary operation of the form I+aU, where I is the identity operator, α is a scalar, and U is a unitary operator,
the quantum computer having a plurality of qubits, including an ancilla qubit,
the classical computer including a processor, a non-transitory computer-readable medium, and computer program instructions stored in the non-transitory computer-readable medium, the computer program instructions being executable by the processor to perform the method, the method comprising:
(A) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a first quantum circuit W which, when executed by the quantum computer, probabilistically realizes the non-unitary operation by the technique of linear combination of unitaries;
(B) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a second quantum circuit, the second quantum circuit comprising a sequence of quantum gates S d =S d-1 RS d-1 † RS d-1 parametrized by an integer d, wherein S 0 =WRW † RW and wherein
R
=
(
1
0
0
e
i
π
/
3
)
is a single-qubit rotation applied to the ancilla qubit.
2. The method of claim 1 , further comprising:
(C) on the quantum computer, executing the first quantum circuit to probabilistically realize the non-unitary operation by the technique of linear combination of unitaries.
3. The method of claim 2 , further comprising:
(D) on the quantum computer, executing the second quantum circuit, comprising executing the sequence of quantum gates S d =S d-1 RS d-4 † RS d-1.
4. The method of claim 3 , wherein (D) comprises, on the quantum computer:
(D)(1) applying a single-qubit rotation R α to transform the ancilla qubit into the state
1
1
+
α
0
〉
+
α
1
+
α
1
〉
;
(D)(2) applying controlled-U operator on a subset of the plurality of qubits, not including the ancilla qubit, conditioned on the ancilla qubit being in the state |1 ; and
(D)(3) applying R a † on the ancilla qubit.
5. The method of claim 1 , further comprising generating and storing, in the non-transitory computer-readable medium, additional computer-readable data that, when executed on the quantum computer, causes the second quantum circuit to execute repeatedly, on the quantum computer, to perform a sequence of operations
∏
i
=
1
r
∏
j
=
1
k
(
1
+
t
κ
j
r
P
j
)
,
that approximates the operator e tT({right arrow over (K)}) with t being a scalar and
T
(
κ
→
)
=
∑
j
=
1
k
κ
j
P
j
being the cluster operator which is a linear combination of operators P j parametrized by weights {right arrow over (K)}.
6. The method of claim 5 , further comprising:
(C) performing mean-field approximation to generate and store, in the non-transitory computer-readable medium, computer-readable data representing a description of a third quantum circuit to prepare a reference state;
(D) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a parametrized quantum circuit for approximating e tT({right arrow over (K)}) ;
(E) on the quantum computer, executing the third quantum circuit to prepare the reference state;
(F) on the quantum computer, applying the parametrized quantum circuit to the reference state to generate the ansatz state |ψ({right arrow over (K)}) ;
(G) using the classical computer in cooperation with the quantum computer to measure an energy of the ansatz |ψ({right arrow over (K)}) ; and
(H) on the classical computer, iteratively tuning the parameters {right arrow over (K)} is to minimize the energy of the ansatz |ψ({right arrow over (K)}) .
7. A system comprising:
a classical computer the classical computer including a processor, a non-transitory computer-readable medium, and computer program instructions stored in the non-transitory computer-readable medium;
a quantum computer comprising a plurality of qubits, including an ancilla qubit;
wherein the computer program instructions, when executed by the processor, perform a method for implementing, on the quantum computer, a non-unitary operation of the form I+aU, where I is the identity operator, α is a scalar, and U is a unitary operator, the method comprising:
(A) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a first quantum circuit W which, when executed by the quantum computer, probabilistically realizes the non-unitary operation by the technique of linear combination of unitaries;
(B) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a second quantum circuit, the second quantum circuit comprising a sequence of quantum gates S d =S d-1 RS d-1 † RS d-1 parametrized by an integer d, wherein S 0 =WRW † RW and wherein
R
=
(
1
0
0
e
i
π
/
3
)
is a single-qubit rotation applied to the ancilla qubit.
8. The system of claim 7 , wherein the method further comprises:
(C) on the quantum computer, executing the first quantum circuit to probabilistically realize the non-unitary operation by the technique of linear combination of unitaries.
9. The system of claim 8 , wherein the method further comprises:
(D) on the quantum computer, executing the second quantum circuit, comprising executing the sequence of quantum gates S d =S d-1 RS d-1 † RS d-1 .
10. The system of claim 9 , wherein (D) comprises, on the quantum computer:
(D)(1) applying a single-qubit rotation R a to transform the ancilla qubit into the state
1
1
+
α
0
〉
+
α
1
+
α
1
〉
;
(D)(2) applying controlled-U operator on a subset of the plurality of qubits, not including the ancilla qubit, conditioned on the ancilla qubit being in the state |1 ; and
(D)(3) applying R α † on the ancilla qubit.
11. The system of claim 7 , wherein the method further comprises generating and storing, in the non-transitory computer-readable medium, additional computer-readable data that, when executed on the quantum computer, causes the second quantum circuit to execute repeatedly, on the quantum computer, to perform a sequence of operations
∏
i
=
1
r
∏
j
=
1
k
(
1
+
t
κ
j
r
P
j
)
,
that approximates the operator e tT({right arrow over (K)}) with t being a scalar and
T
(
κ
→
)
=
∑
j
=
1
k
κ
j
P
j
being the cluster operator which is a linear combination of operators P j parametrized by weights {right arrow over (K)}.
12. The system of claim 11 , wherein the method further comprises:
(C) performing mean-field approximation to generate and store, in the non-transitory computer-readable medium, computer-readable data representing a description of a third quantum circuit to prepare a reference state;
(D) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a parametrized quantum circuit for approximating e tT({right arrow over (K)}) ;
(E) on the quantum computer, executing the third quantum circuit to prepare the reference state;
(F) on the quantum computer, applying the parametrized quantum circuit to the reference state to generate the ansatz state |ψ({right arrow over (K)}) ;
(G) using the classical computer in cooperation with the quantum computer to measure an energy of the ansatz |ψ({right arrow over (K)}) ; and
(H) on the classical computer, iteratively tuning the parameters is to minimize the energy of the ansatz |ψ({right arrow over (K)}) .