Memory Compression with Quantum Random-Access Gates

TL;DR

This paper demonstrates that quantum algorithms with sparse memory states can be compressed to use significantly less memory while maintaining similar runtime, simplifying quantum data structure design.

Contribution

It introduces a quantum memory compression theorem for algorithms with sparse states, extending classical memory compression ideas to quantum computing.

Findings

Quantum algorithms with sparse states can be compressed to O(m log M) memory.

Compressed algorithms run in nearly the same time as original algorithms.

Simplifies the design and analysis of space-efficient quantum data structures.

Abstract

In the classical RAM, we have the following useful property. If we have an algorithm that uses $M$ memory cells throughout its execution, and in addition is sparse, in the sense that, at any point in time, only $m$ out of $M$ cells will be non-zero, then we may "compress" it into another algorithm which uses only $m \log M$ memory and runs in almost the same time. We may do so by simulating the memory using either a hash table, or a self-balancing tree. We show an analogous result for quantum algorithms equipped with quantum random-access gates. If we have a quantum algorithm that runs in time $T$ and uses $M$ qubits, such that the state of the memory, at any time step, is supported on computational-basis vectors of Hamming weight at most $m$, then it can be simulated by another algorithm which uses only $O(m \log M)$ memory, and runs in time $\tilde O(T)$. We show how this theorem can be used, in a black-box way, to simplify the presentation in several papers. Broadly speaking, when there exists a need for a space-efficient history-independent quantum data-structure, it is often possible to construct a space-inefficient, yet sparse, quantum data structure, and then appeal to our main theorem. This results in simpler and shorter arguments.