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Quantum Algorithms for Calculating Determinant and Inverse of Matrix and Solving Linear Algebraic Systems

Alexander I. Zenchuk

Alexander I. Zenchuk

Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RASChernogolovka, Russia
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Zenchuk, Alexander I.
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Georgii A. Bochkin

Georgii A. Bochkin

Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RASChernogolovka, Russia
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Bochkin, Georgii A.
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Wentao Qi

Wentao Qi

Institute of Quantum Computing and Computer Theory, School of Computer Science and Engineering, Sun Yat-sen UniversityGuangzhou, China
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Qi, Wentao
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Asutosh Kumar

Asutosh Kumar

  • Department of Physics, KSS CollegeLakhisarai, India
  • P.G. Department of Physics, Munger UniversityMunger, India
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Kumar, Asutosh
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Junde Wu

Junde Wu

School of Mathematical Sciences, Zhejiang UniversityHangzhou, China
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Wu, Junde
  
26 mag 2025
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Categoria dell'articolo: Article
Pubblicato online: 26 mag 2025
Pagine: 195 - 215
Ricevuto: 29 dic 2024
Accettato: 02 apr 2025
DOI: https://doi.org/10.2478/qic-2025-0010
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© 2025 Alexander I. Zenchuk et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1.

V and U operators for (a) the general case and (b) the 4 × 4-matrix case. In the latter case, ñ0 = ñ1 = 2, ñ2 = 1, ñk < n. Hence, A1 is a two-qubit ancilla.
V and U operators for (a) the general case and (b) the 4 × 4-matrix case. In the latter case, ñ0 = ñ1 = 2, ñ2 = 1, ñk < n. Hence, A1 is a two-qubit ancilla.

Figure 2.

Quantum circuit illustrating computation of determinant of a 4 × 4 matrix, see notations in Figure 1. We omit superscript (D) in |Φk〉(k = 1, …,4) and |Ψout〉 for brevity. Operators HA and XS can be applied in parallel since they affect different qubits. The lower circuit is the notation for multi-qubit controlled σ(x) operation.
Quantum circuit illustrating computation of determinant of a 4 × 4 matrix, see notations in Figure 1. We omit superscript (D) in |Φk〉(k = 1, …,4) and |Ψout〉 for brevity. Operators HA and XS can be applied in parallel since they affect different qubits. The lower circuit is the notation for multi-qubit controlled σ(x) operation.

Figure 3.

(Color online) (a) Structure of quantum circuit for matrix inversion (ancillae are shown in green). (b) The structure of the block P.
(Color online) (a) Structure of quantum circuit for matrix inversion (ancillae are shown in green). (b) The structure of the block P.

Figure 4.

(Color online) (a) Structure of quantum circuit for solving a system of linear algebraic equations. Both blocks share the same ancillae (not shown in figure). (b) Circuit for quantum algorithm solving a system of linear algebraic equations Ax = b. Circuit of matrix inversion (up to the operator WSAB(2) ) is denoted by a block A−1. All input qubits of the matrix-inversion algorithm (system S) become ancillae qubits. The output of the matrix-inversion algorithm (systems R and C) together with system b encoding the vector b becomes the input of the matrix-multiplication block. The output of the whole algorithm is encoded in the state |x〉R of the system R. All ancillae qubits are shown in green color. We omit the superscript (L) in |Φk〉(k = 5,6,7) and |Ψout〉 for brevity.
(Color online) (a) Structure of quantum circuit for solving a system of linear algebraic equations. Both blocks share the same ancillae (not shown in figure). (b) Circuit for quantum algorithm solving a system of linear algebraic equations Ax = b. Circuit of matrix inversion (up to the operator WSAB(2) ) is denoted by a block A−1. All input qubits of the matrix-inversion algorithm (system S) become ancillae qubits. The output of the matrix-inversion algorithm (systems R and C) together with system b encoding the vector b becomes the input of the matrix-multiplication block. The output of the whole algorithm is encoded in the state |x〉R of the system R. All ancillae qubits are shown in green color. We omit the superscript (L) in |Φk〉(k = 5,6,7) and |Ψout〉 for brevity.

Figure 5.

(Color online) Replacement of ordinary measurement of ancilla state (left circuit) with the subroutine of controlled measurement (right circuit).
(Color online) Replacement of ordinary measurement of ancilla state (left circuit) with the subroutine of controlled measurement (right circuit).

Figure A1.

(a) Circuit for the matrix multiplication algorithm. We omit superscript (m) in |Φk〉 (k = 0, …, 4) and |Ψout for brevity. (b) Notation for multiqubit CNOT.
(a) Circuit for the matrix multiplication algorithm. We omit superscript (m) in |Φk〉 (k = 0, …, 4) and |Ψout for brevity. (b) Notation for multiqubit CNOT.
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