Memory Compression with Quantum Random-Access Gates (Subhasree Patro)
Channel: IRIF
Date: 2022-04-26

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 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.