Group Sparse Matrix Optimization for Efficient Quantum State Transformation

TL;DR

The paper tackles efficient quantum state transformation by optimizing a sparse, unitary transformation $\mathbf{X}$ that maps an initial state $\mathbf{A}$ to a target state $\mathbf{C}$ under $\mathbf{X}^{\dagger}\mathbf{X} = \mathbf{I}$ and row/column-sum constraints. It introduces an ADMM-based framework with dual variables and an auxiliary matrix to handle unitary and equality constraints, and compares $\ell_1$ and $\ell_{2,1}$ sparsity penalties within this non-convex setting. The authors demonstrate convergence behaviors, analyze sparsity trade-offs, and provide a practical 3-qubit example where the optimized $\mathbf{X}$ is decomposed into a quantum circuit via cosine-sine (two-level) decomposition, obtaining gate-level representations and potential reductions in gate counts. The work highlights a pathway to more efficient quantum circuit synthesis by leveraging group sparsity, while noting scalability challenges and the need for tighter enforcement of row/column-sum constraints. Overall, the approach offers a principled, sparse-optimization route to design unitary transformations for quantum information processing with concrete implications for circuit depth and fault tolerance.

Abstract

Finding ways to transform a quantum state to another is fundamental to quantum information processing. In this paper, we apply the sparse matrix approach to the quantum state transformation problem. In particular, we present a new approach for searching for unitary matrices for quantum state transformation by directly optimizing the objective problem using the Alternating Direction Method of Multipliers (ADMM). Moreover, we consider the use of group sparsity as an alternative sparsity choice in quantum state transformation problems. Our approach incorporates sparsity constraints into quantum state transformation by formulating it as a non-convex problem. It establishes a useful framework for efficiently handling complex quantum systems and achieving precise state transformations.

Figures (3)

• Figure 1: Magnitude of the resulted matrix (a) $\mathbf{X}_{\ell_1}$ and (b) $\mathbf{X}_{\ell_{2,1}}$.
• Figure 2: Magnitude of sparsed values in each row of $\mathbf{X}$ by (a) $\ell_1$-norm and (b) ${\ell_{2,1}}$-norm.
• Figure 3: Quantum circuit for transformation of a 3-qubit system.