QUANTUM COMPILATION DEVICE, QUANTUM COMPILATION METHOD, AND PROGRAM
A quantum compilation device obtains a probability p(k) that minimizes an error between a distribution of first observed values obtained by observing, with any observation method, a first quantum state obtained by causing the quantum circuit to be compiled represented by a unitary matrix U to act on any input quantum state, and a distribution of second observed values obtained by observing, with the observation method, a second quantum state obtained by causing a quantum circuit represented by each of a plurality of elements U k ∈{U 1 , . . . , U K } of a set {U 1 , . . . , U K } to act on the input quantum state with the probability p(k), for the set {U 1 , . . . , U K } in which a unitary matrix representing elementary gates and/or a unitary matrix representing a product of unitary matrices each representing an elementary gate are the elements U 1 , . . . , and U K , and the unitary matrix U representing a quantum circuit to be compiled, and outputs an element U k with the probability p(k). Here, K is an integer of 2 or more, and k=1, . . . , and K.
1 . A quantum compilation device comprising processing circuitry configured to:
obtain a probability p(k) that minimizes an error between
a distribution of first observed values obtained by observing, with any observation method, a first quantum state obtained by causing the quantum circuit to be compiled represented by a unitary matrix U to act on any input quantum state, and
a distribution of second observed values obtained by observing, with the observation method, a second quantum state obtained by causing a quantum circuit represented by each of a plurality of elements U k ∈{U 1 , . . . , U K } of a set {U 1 , . . . , U K } to act on the input quantum state with the probability p(k),
for the set {U 1 , . . . , U K } in which a unitary matrix representing elementary gates and/or a unitary matrix representing a product of unitary matrices each representing an elementary gate are the elements U 1 , . . . , and U K , and the unitary matrix U representing a quantum circuit to be compiled, where K is an integer of 2 or more, and k=1, . . . , and K; and
output an element U k with the probability p(k).
2 . The quantum compilation device according to claim 1 , wherein
when the unitary matrix U is compiled by any one element U k′ ∈{U 1 , . . . , U K } deterministically selected from the set {U 1 , . . . , U K }, the unitary matrix U can be approximated by the set {U 1 , . . . , U K } with an approximation accuracy E=ε, where k′∈{1, . . . , K}, and
when the unitary matrix U is compiled by the element U k stochastically selected with the probability p(k) from the set {U 1 , . . . , U K }, the unitary matrix U can be approximated by the set {U 1 , . . . , U K } with an approximation accuracy E=ε 2 or can be approximated by approximately E=ε 2 .
3 . The quantum compilation device according to claim 1 , wherein
the unitary matrix U is a 2 N ×2 N matrix, and N is an integer of 1 or more,
{σ 1 , . . . , σ J } is an orthonormal basis of a 2 2N ×2 2N Hermitian matrix, J=2 4N , and j=1, . . . , J,
e L → represents a 2 N -dimensional vertical vector, the L-th element from a head of the vertical vector e L → is 1, the other 2 N -1 elements are 0, and L=1, . . . , 2 N ,
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represents the Kronecker product of α 1 and α 2 ,
A represents a real matrix, a k-th column vector from a head of the real matrix A is ((U k → ) + σ 1 U k → , (U k → ) + σ 2 U k → , . . . , (U k → ) + σ J U k → ) T , β + represents an adjoint matrix of β, and γ T represents transposition of γ,
b → represents a real vector, and b → =((U → ) + σ 1 U → , (U → ) + σ 2 U → , . . . , (U → ) + σ J U → ) T ,
Δ represents a set of real vectors, and Δ={(p(1), p(2), . . . , p(k)) T |Σ k=1 , . . . , k p(k)=1, p(k)≥0}, and
R represents a set of real vectors, and R={(tr[σ 1 Φ], tr[σ 2 Φ], . . . , tr[σ J Φ]) T : ∃ ρ≥0, (tr [ρ]=1)∧(0≤Φ≤ρ(×)I)}, tr[κ] represents a trace of κ, I represents a unit matrix of 2 N ×2 N , ρ represents a positive semi-definite matrix of 2 N ×2 N , γ≥0 and 0≤γ in the matrix γ represent that γ is a positive semi-definite matrix, that is, a Hermitian matrix of which an eigenvalue is non-negative, γ 1 ≤γ 2 in the matrices γ 1 and γ 2 of η×η represents that γ 2 −γ 1 ≥0, that is, 65 2 −γ 1 is a positive semi-definite matrix, η is an integer of 1 or more, μ is an integer of 1 or more, and Φ represents a positive semi-definite matrix, wherein
the quantum compilation device obtains p → ={p(1), . . . , p(k)}∈Δ that achieves the following expression.
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4 . A quantum compilation method performed by a quantum compilation device comprising:
obtaining a probability p(k) that minimizes an error between
a distribution of first observed values obtained by observing, with any observation method, a first quantum state obtained by causing the quantum circuit to be compiled represented by a unitary matrix U to act on any input quantum state, and
a distribution of second observed values obtained by observing, with the observation method, a second quantum state obtained by causing a quantum circuit represented by each of a plurality of elements U k ∈{U 1 , . . . , U K } of a set {U 1 , . . . , U K } to act on the input quantum state with the probability p(k),
for the set {U 1 , . . . , U K } in which a unitary matrix representing elementary gates and/or a unitary matrix representing a product of unitary matrices each representing an elementary gate are the elements U 1 , . . . , and U K , and the unitary matrix U representing a quantum circuit to be compiled
where K is an integer of 2 or more, and k=1, . . . , and K; and
an output step of outputting an element U k with the probability p(k) in an output unit.
5 . A non-transitory computer-readable recording medium storing a program for causing a computer to function as the quantum compilation device according to claim 1 .