Methods for Reducing Ancilla-Overhead in Block Encodings

TL;DR

This paper introduces new techniques to reduce ancilla qubit overhead in quantum block encodings by establishing tradeoffs between space and time, and between space and accuracy, optimizing quantum resource usage.

Contribution

It presents novel algorithms demonstrating space-time and space-accuracy tradeoffs, significantly reducing ancilla requirements for block encoding operations.

Findings

Approximate uncomputation of ancillae enables reuse in quantum algorithms.

Logarithmic ancillae are necessary for exact multiplication of block encodings.

High-precision approximate multiplication can be achieved with just one ancilla in certain regimes.

Abstract

Block encodings are a fundamental primitive in quantum algorithms, but can often have large ancilla overhead. In this work, we introduce novel techniques for reducing this overhead in two distinct ways. In Part I, we prove the existence of a "space-time tradeoff" by deriving an algorithm that, for any block encoding, approximately uncomputes all but one of its ancilla (freeing up those ancillae for reuse in later parts of a quantum algorithm). In Part II, we evaluate the minimum number of ancillae required to perform coherent multiplication of block encodings and introduce a "space-accuracy tradeoff". Specfically, we prove that logarithmic ancillae is optimal for exact multiplication of block encodings, but show that (in certain block encoding regimes) approximate multiplication of block encodings can be achieved to high-precision with just one ancilla.