Approximate Sparse State Preparation with the Grover-Rudolph Algorithm

TL;DR

This paper improves the Grover-Rudolph algorithm for sparse quantum state preparation by reducing gate complexity and introducing an approximate merging method with controllable error.

Contribution

It extends gate-merging techniques and proposes an approximate variant with error control, enhancing efficiency in sparse state preparation.

Findings

Reduced number of CNOTs and control qubits in state preparation

Introduced an approximate merging method with controllable error

Derived an estimate of the overlap guiding merging decisions

Abstract

Sparse quantum state preparation is a common subroutine in quantum algorithms, where classical data with few nonzero entries must be loaded into a quantum state. In this work, we consider the Grover-Rudolph algorithm, which has recently been shown to efficiently prepare sparse states, and we propose two improvements. First, we extend an existing gate-merging procedure by allowing rotations to merge with virtual zero-angle gates on unreachable branches of the preparation tree, reducing the number of CNOTs and control qubits. Second, we introduce an approximate variant in which rotations with similar but not identical angles are merged at the cost of a small, controllable error in the prepared state. We derive a classically computable estimate of the resulting overlap with the target state, which is used to guide the merging decisions.