Optimizing Quantum Transformation Matrices: A Block Decomposition Approach for Efficient Gate Reduction

TL;DR

This work addresses the challenge of implementing quantum transformations with a limited gate budget by introducing a block-decomposition algorithm that approximates a target unitary $\mathbf{U}$ with a product $\mathbf{Y}=\prod_{k=1}^M \mathbf{X}_k$ of gate matrices under unitary and sparsity constraints. Unitarity is enforced via a penalty formulation and the subproblem for updating each gate $\mathbf{X}_w$ is solved using a KKT-based closed-form in the vectorized quadratic program, with exhaustive search over feasible positions defined by a dictionary $\mathbf{D}$. Numerical results on a 3-qubit system show that increasing gate count $M$ improves approximation accuracy (e.g., $L$ decreasing from $4.51$ to $1.83$; fidelity improving from ~0.853 to ~0.921 for a target $\mathbf{U}$ acting on $|W\rangle$), at the cost of higher computation time and complexity, notably scaling with qubit number. The authors propose indicator-variables to convert subproblems to binary quadratic programs, aiming to alleviate scaling challenges. Overall, the method offers a flexible, resource-aware framework for quantum circuit design by merging block-decomposition with unitary-gate structure constraints, enabling controlled gate-reduction in quantum transformations.

Abstract

This paper introduces an algorithm designed to approximate quantum transformation matrix with a restricted number of gates by using the block decomposition technique. Addressing challenges posed by numerous gates in handling large qubit transformations, the algorithm provides a solution by optimizing gate usage while maintaining computational accuracy. Inspired by the Block Decompose algorithm, our approach processes transformation matrices in a block-wise manner, enabling users to specify the desired gate count for flexibility in resource allocation. Simulations validate the effectiveness of the algorithm in approximating transformations with significantly fewer gates, enhancing quantum computing efficiency for complex calculations.

Figures (3)

• Figure 1: Magnitude of example unitary transformation matrix, $\mathbf{U}$ in Eq. \ref{['eq:example']}.
• Figure 2: General form of quantum circuit to represent $3$-qubit transformation matrix $\mathbf{U}$ in Eq. \ref{['eq:example']}.
• Figure 3: Magnitude of resulted transformation matrix built using $10$ gates, $\mathbf{Y}$.