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

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

Quantum Information & Computation
Volume 25 (2025): Issue 2 (May 2025)
By: Alexander I. Zenchuk,  Georgii A. Bochkin,  Wentao Qi,  Asutosh Kumar and  Junde Wu  
Open Access
|May 2025

Figures & Tables

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.
DOI: https://doi.org/10.2478/qic-2025-0010 | Journal eISSN: 3106-0544 | Journal ISSN: 1533-7146
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Language: English
Page range: 195 - 215
Submitted on: Dec 29, 2024
Accepted on: Apr 2, 2025
Published on: May 26, 2025
Published by: Cerebration Science Publishing Co., Limited
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
quantum algorithm,
quantum circuit,
determinant and inverse of matrix,
linear algebraic systems,
depth of algorithm,
quantum measurement
Related subjects:
Physics,
Quantum physics

© 2025 Alexander I. Zenchuk, Georgii A. Bochkin, Wentao Qi, Asutosh Kumar, Junde Wu, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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