# Block Encoding of Sparse Matrices via Coherent Permutation

## TL;DR

This paper presents a unified framework for efficient block encoding of sparse matrices in quantum computing, addressing key implementation challenges and enabling hardware-friendly quantum circuits.

## Contribution

It introduces a systematic approach linking combinatorial optimization and coherent permutation operators to improve gate efficiency and connectivity in quantum matrix encoding.

## Key findings

- Reduces control overhead in quantum block encoding.
- Achieves structured amplitude reordering with coherent permutations.
- Bridges theoretical encoding methods with hardware-efficient circuits.

## Abstract

Block encoding of sparse matrices underpins powerful quantum algorithms such as quantum singular value transformation, Hamiltonian simulation, and quantum linear solvers, yet its efficient gate-level realization for general sparse matrices remains a major challenge. We introduce a unified framework that addresses key obstacles including the overhead of multi-controlled X (MCX) gates, amplitude reordering, and hardware connectivity, enabling simplified block encoding constructions with explicit gate-level implementations. Central to our approach is a connection to combinatorial optimization, which enables systematic assignment of control qubits to satisfy nearest-neighbor connectivity constraints, along with coherent permutation operators that preserve superposition while enabling structured amplitude reordering. We demonstrate our methods on structured sparse matrices, achieving systematic reductions in control overhead and circuit depth. Our framework bridges the gap between theoretical formulations and hardware-efficient quantum circuit implementations.