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arXiv:2512.11938

Quantum circuits for permutation matrices

Jason Hanson

quant-phcs.CCSubmitted Dec 12, 2025• Announced Dec 15, 2025• 0 citations
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Abstract

Two different algorithms are presented for generating a quantum circuit realization of a matrix representing a permutation on $2^n$ letters. All circuits involve $n$ qubits and only use multi--controlled Toffoli gates. The first algorithm constructs a circuit from any decomposition of the permutation into a product of transpositions, but uses one ancilla line. The second, which uses no ancillae, constructs a circuit from a decomposition into a product of transpositions that have a Hamming distance of one. We show that any permutation admits such a decomposition, and we give a strategy for reducing the number of transpositions involved.

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Two different algorithms are presented for generating a quantum circuit realization of a matrix representing a permutation on 2n2^n2n letters. All circuits involve nnn qubits and only use multi--controlled Toffoli gates. The first algorithm constructs a circuit from any decomposition of the permutation into a product of transpositions, but uses one ancilla line. The second, which uses no ancillae, constructs a circuit from a decomposition into a product of transpositions that have a Hamming distance of one. We show that any permutation admits such a decomposition, and we give a strategy for reducing the number of transpositions involved.
We present an explicit quantum circuit that constructs a block encoding of a generator-representable semiseparable (one-pair) matrix directly from its two generators. For an N×NN\times NN×N semiseparable matrix with N=2nN=2^{n}N=2n, the circuit uses 2n+22n+22n+2 ancilla qubits and has O(polylogN)O({\rm polylog} N)O(polylogN) depth, and it realizes a (2N, 2n+2, 22 ε)(2\sqrt{N},\,2n+2,\,2 \sqrt{2}\ \varepsilon)(2N​,2n+2,22​ ε)-block encoding, where ε\varepsilonε is controlled by the accuracy of the state-preparation routines that load the generators. The building blocks of the circuit are cyclic shifts, swaps, and Hadamard gates, each admitting an efficient implementation through the binary decomposition of the relevant indices.
We present a quantum variational algorithm based on a novel circuit that generates all permutations that can be spanned by one- and two-qubits permutation gates. The construction of the circuits follows from group-theoretical results, most importantly the Bruhat decomposition of the group generated by the cx gates. These circuits require a number of qubits that scale logarithmically with the permutation dimension, and are therefore employable in near-term applications. We further augment the circuits with ancilla qubits to enlarge their span, and with these we build ansatze to tackle permutation-based optimization problems such as quadratic assignment problems, and graph isomorphisms. The resulting quantum algorithm, QuPer, is competitive with respect to classical heuristics and we could simulate its behavior up to a problem with 256 variables, requiring 20 qubits.
Abstract Boolean matrices and operations on Boolean matrices have become increasingly popular in quantum computing and quantum information because they can be used to represent quantum states and the corresponding operations. In this paper, we first give a quantum circuit direct synthesis of general Boolean matrix multiplication using Toffoli gates. We assign a qubit to each element of the matrix and add Toffoli gates at the corresponding positions where the matrix elements are multiplied according to the arithmetic of matrix multiplication. For n × p dimensional matrix A multiplied by p × m dimensional matrix B, np + pm + nm qubits and at most pmn Toffoli gates are used in total. Then, an improved heuristic algorithm based on matrix Hamming distance selection of CNOT gate is proposed to synthesize the quantum circuit of a matrix, without using wire switching and SWAP gates, using only CNOT gates. The results show that our method is as good as or better than the classical synthesis. Finally, the quantum circuit of Boolean matrix multiplication with at least one invertible or full-rank matrix using only CNOT gates is given, which is divided into five cases for discussion and analysis. For n × n dimensional invertible matrix A multiplied by n × m dimensional matrix B, nm qubits and at most mn 2 CNOT gates are used in total. For n × p dimensional column full-rank matrix A multiplied by p × m dimensional matrix B, nm qubits and at most pmn CNOT gates are used in total. The results show that this method uses less quantum resources to synthesize the quantum circuit of Boolean matrix multiplication. The correctness of the quantum circuits was verified by the Aer simulator of the IBM Quantum platform.
Given an arbitrary ${2}^{w}\ifmmode\times\else\texttimes\fi{}{2}^{w}$ unitary matrix UUU, a powerful matrix decomposition can be applied, leading to four different syntheses of a www-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. F\"uhr and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates (w=1w=1w=1) to gates with arbitrary value of www. The synthesized circuit consists of controlled one-qubit gates, such as negator gates and phasor gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled not gates in the case that UUU is a permutation matrix.
Multi-bit Toffoli gates form an essential quantum gate class for quantum algorithms. They should be efficiently decomposed into elementary single- or multi-qubit quantum gates, such as CNOT, T, and Hadarmard, for a scalable implementation of a quantum algorithm. We propose an engineering method for the practical synthesis of a multi-bit Toffoli gate. Two optimization models and their closed-form solutions are presented for optimal decomposition of the multi-bit Toffoli gate. These models are based on linearized multi-objective integer programming with parameters such as the number of target ancillae, ancillae states, and basis gates. The proposed method supports the systematic handling of quantum circuit constraints, including the total number of available qubits and maximum circuit depth, which depend on various quantum hardware specifications. Our approach exhibits promise in the noisy intermediate-scale quantum environment by providing a rapid and optimal method for synthesizing multi-bit Toffoli gates in diverse and unpredictable quantum hardware specifications.
The article presents the quantum signal-induced heap transform (QsiHT) method of the QR-decomposition of multi-qubit operations. This transform can be generated by a given signal, by using different paths, or orders, of processing the data. We propose using the concept of the fast path of calculation of the QsiHT and applying such transforms on each stage of the matrix decomposition. This allows us to build quantum circuits for multi-qubit unitary operation without permutations. Unitary operations with real and complex matrices are considered. The cases of 3- and 4-qubit operations are described in detail with quantum circuits. These circuits use a maximum of 28 and 120 Givens rotation gates for 3- and 4-qubit real operations, respectively. All rotations are performing only on adjacent bit planes. For complex unitary operation, each of the Givens gates is used in pairs with two Z-rotation gates. These two types of rotations and the global phase gate are the universal gate set for multi-qubit operations. The presented approach can be used for implementing quantum circuits for n-qubits when n≥2, with a maximum of (4n/2−2n−1) Givens rotations and no permutations.