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arXiv:2511.02413

Quantum Circuit Implementation of Two Matrix Product Operations and Elementary Column Transformations

Yu-Hang Liu, Yuan-Hong Tao, Jing-Run Lan, Shao-Ming Fei

quant-phSubmitted Nov 4, 2025• Announced Nov 5, 2025• 0 citations
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Abstract

This paper focuses on quantum algorithms for three key matrix operations: Hadamard (Schur) product, Kronecker (tensor) product, and elementary column transformations each. By designing specific unitary transformations and auxiliary quantum measurement, efficient quantum schemes with circuit diagrams are proposed. Their computational depths are: O(1) for Kronecker product; O(max(m,n)) for Hadamard product (linked to matrix dimensions); and O(m) for elementary column transformations of (2^n X 2^m) matrices (dependent only on column count).Notably, compared to traditional column transformation via matrix transposition and row transformations, this scheme reduces computation steps and quantum gate usage, lowering quantum computing energy costs.

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This paper focuses on quantum algorithms for three key matrix operations: Hadamard (Schur) product, Kronecker (tensor) product, and elementary column transformations each. By designing specific unitary transformations and auxiliary quantum measurement, efficient quantum schemes with circuit diagrams are proposed. Their computational depths are: O(1) for Kronecker product; O(max(m,n)) for Hadamard product (linked to matrix dimensions); and O(m) for elementary column transformations of (2^n X 2^m) matrices (dependent only on column count).Notably, compared to traditional column transformation via matrix transposition and row transformations, this scheme reduces computation steps and quantum gate usage, lowering quantum computing energy costs.
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D(4)D^{(4)}D(4) transforms. Our approach is to factor the operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
Quantum circuits naturally implement unitary operations on input quantum states. However, non-unitary operations can also be implemented through "block encodings", where additional ancilla qubits are introduced and later measured. While block encoding underlies a host of well-established theoretical applications, its implementation in practice has been prohibitively expensive for current quantum hardware. In this paper, we present practical and explicit block encoding circuits implementing matrix polynomial transformations of a target matrix. With standard approaches, block-encoding a degree-d matrix polynomial requires a circuit depth scaling as d times the depth for block-encoding the original matrix alone. By leveraging the recently introduced Fast One-Qubit Controlled Select Linear Combination of Unitaries (FOQCS-LCU) framework, we show that the additional circuit-depth overhead required for encoding matrix polynomials can be reduced to scale linearly in d with no dependence on system size or the cost of block encoding the original matrix. Moreover, we demonstrate that the FOQCS-LCU circuits and their associated matrix polynomial transformations can be controlled with negligible overhead, enabling efficient applications such as Hadamard tests. Finally, we provide explicit hardware-efficient circuits for representative non-unitary operators, namely quantum spin models, together with detailed non-asymptotic gate counts and circuit depths.
Quantum circuits are divided into two classes,and unitary operator expressions for two kinds of circuits in each class are offered.Transformation of unitary operators between two classes is presented according to tensor product transposition similarity theory.Besides,direct sum operation of matrices is adopted and expanded to vectors to elegantly represent the functions of quantum circuits on input vectors.On the basis of abstracting quantum circuits as controlled unitary operations,divide and conquer algorithms are proposed to simulate quantum circuits.In contrast to other algorithms based on state vector,our algorithms avoid generating matrix operators by tensor product,thus are considerably economy of memory space;and we also show that they are more time-efficient while simulating nontrivial quantum circuits.
The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.
Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum error-correction is also given.
Until today, we have not found a general and effective algorithm for synthesis, especially to multi-variables quantum circuit.This is an important question needed to be resolved, because it can not only reduce the cost of manufacture quantum circuit, but also optimize many quantum algorithms.For a quantum circuit which contains N quantum bits, the latitude of the matrix is 𝟐 𝒏 to realize its function.If we directly design the circuits of the matrix, the workload is huge.Want to reduce the dimensions of the matrix, but also to ensure the unitary of matrix after decomposing, so it is need to use Kronecker product.