Quantum information processing method for computing transition amplitude, classical computer, quantum computer, and hybrid system
A quantum computer executes quantum measurement of <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 < below based on a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2 , and outputs measurement results of the quantum measurement. A classical computer computes a transition amplitude |<ψ 1 |A|ψ 2 >| 2 based on measurement results for <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 >, wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.
1. A quantum information processing method for computing a transition amplitude by processing executed on a hybrid system including a classical computer and a quantum computer, the processing comprising:
the quantum computer executing quantum measurement based on a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2 so as to measure <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > in Equation (1) below, and outputting measurement results from the quantum measurement; and
the classical computer computing a transition amplitude |<ψ 1 |A|ψ 2 >| 2 according to Equation (1) below, based on the measurement results for <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 >
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ1|ψ2>=0.
2. The quantum information processing method for computing a transition amplitude of claim 1 , wherein the quantum computer executes the quantum measurement based on a set of parameters θ for a quantum circuit corresponding to the quantum state pair of the first quantum state ψ 1 and the second quantum state ψ 2 , obtained by computation employing variational quantum deflation.
3. The quantum information processing method for computing a transition amplitude of claim 1 , wherein:
the classical computer and the quantum computer are connected to each other over a computer network; and
the classical computer and the quantum computer exchange information with each other over the computer network.
4. A method of quantum information processing for computing a transition amplitude by processing executed by a classical processor of a classical computer including a memory and the classical processor coupled to the memory, the processing comprising:
computing a transition amplitude |<ψ 1 |A|ψ 2 > 2 according to Equation (1) below based on measurement results for <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >,
<ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > from quantum measurement executed by a quantum computer for a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.
5. A method of quantum information processing for computing a transition amplitude by processing executed by a quantum processor of a quantum computer including the quantum processor, the processing comprising:
executing quantum measurement of <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > of Equation (1) below based on a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2 , and outputting measurement results of the quantum measurement
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.
6. A classical computer including a memory and a classical processor coupled to the memory, the classical processor executing processing comprising:
computing a transition amplitude |<ψ 1 |A|ψ 2 >| 2 according to Equation (1) below based on measurement results for <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > from quantum measurement executed by a quantum computer for a quantum state pair configured by a first quantum state xvi and a second quantum state ψ 2
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.
7. A quantum computer including a quantum processor, the quantum processor executing processing comprising:
executing quantum measurement of <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > of Equation (1) below based on a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2 , and outputting measurement results of the quantum measurement
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.
8. A hybrid system including the quantum computer of claim 7 and a classical computer including a memory and a classical processor coupled to the memory, the classical processor executing processing comprising:
computing a transition amplitude |<ψ 1 |A|ψ 2 >| 2 according to Equation (1) below based on measurement results for <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > from quantum measurement executed by a quantum computer for a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.
9. A non-transitory recording medium storing a quantum information processing program executable by a classical processor to perform processing comprising:
computing a transition amplitude |<ψ 1 |A|ψ 2 | 2 according to Equation (1) below based on measurement results for <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > from quantum measurement executed by a quantum computer for a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.
10. A non-transitory recording medium storing a quantum information processing program executable by a quantum processor to perform processing comprising:
executing quantum measurement of <ψ 1 |P i |ψ 2 >, <ψ 1 |U ij,+ |ψ 2 >, <ψ 1 |U ij,− |ψ 2 >, <ψ 1 |P j |ψ 2 >, and <ψ 1 |P i P j |ψ 2 > of Equation (1) below based on a quantum state pair configured by a first quantum state ψ 1 and a second quantum state ψ 2 , and outputting measurement results of the quantum measurement
〈
ψ
1
A
ψ
2
〉
2
=
∑
i
a
i
2
〈
ψ
1
P
i
ψ
2
〉
2
+
∑
i
<
j
a
i
a
j
[
2
〈
ψ
1
U
ij
,
+
ψ
2
〉
2
+
2
〈
ψ
1
U
ij
,
-
ψ
2
〉
2
-
〈
ψ
1
P
i
ψ
2
〉
2
-
〈
ψ
1
P
j
ψ
2
〉
2
-
〈
ψ
1
P
i
P
j
ψ
2
〉
2
]
Equation
(
1
)
wherein A is a physical quantity for computation of transition amplitude, i and j are indices for identifying a and P, a is a real number, P is a tensor product of a Pauli matrix, U is a unitary gate, and <ψ 1 |ψ 2 >=0.