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

TL;DR

The paper proposes quantum circuit constructions for three matrix operations—Hadamard product, Kronecker product, and elementary column transformations—via dedicated unitaries and measurement-based filtering. It achieves a compact set of depths: $O(1)$ for Kronecker, $O(\max(n,m))$ for Hadamard, and $O(m)$ for column transformations, with explicit success probabilities such as $1$ for Kronecker, $G^2/2^{(n+m)}$ for Hadamard, $G^2/8$ for column addition, and $1/24$ for column swapping. The methods encode input matrices into quantum registers, use auxiliary qubits to tag and extract only the relevant information, and rely on Hadamard mixing to prepare superpositions before selective measurement. The work provides a concrete, energy-aware route to quantum linear-algebra primitives that could impact quantum gate design and matrix-operation applications on quantum hardware.

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.

Figures (4)

• Figure 1: Quantum Circuit Diagram for the Hadamard Product
• Figure 2: The quantum circuit diagram for the Kronecker product
• Figure 3: The quantum circuit diagram of column addition
• Figure 4: The quantum circuit diagram of column swapping