Eliminating Intermediate Measurements using Pseudorandom Generators

TL;DR

This paper demonstrates that quantum algorithms with intermediate measurements can be simulated by purely unitary quantum algorithms with only polynomial overhead in time and space, using pseudorandom generators to replace randomness.

Contribution

It introduces a method to simulate measurement-based quantum algorithms with unitary-only algorithms, leveraging pseudorandom generators for quantum space-bounded computation.

Findings

Quantum algorithms with intermediate measurements can be simulated unitarily.

INW pseudorandom generator fools quantum space-bounded algorithms.

Simulation achieves polynomial overhead in time and space.

Abstract

We show that quantum algorithms of time $T$ and space $S\ge \log T$ with unitary operations and intermediate measurements can be simulated by quantum algorithms of time $T \cdot \mathrm{poly} (S)$ and space $ {O}(S\cdot \log T)$ with unitary operations and without intermediate measurements. The best results prior to this work required either $\Omega(T)$ space (by the deferred measurement principle) or $\mathrm{poly}(2^S)$ time [FR21,GRZ21]. Our result is thus a time-efficient and space-efficient simulation of algorithms with unitary operations and intermediate measurements by algorithms with unitary operations and without intermediate measurements. To prove our result, we study pseudorandom generators for quantum space-bounded algorithms. We show that (an instance of) the INW pseudorandom generator for classical space-bounded algorithms [INW94] also fools quantum space-bounded algorithms. More precisely, we show that for quantum space-bounded algorithms that have access to a read-once tape consisting of random bits, the final state of the algorithm when the random bits are drawn from the uniform distribution is nearly identical to the final state when the random bits are drawn using the INW pseudorandom generator. This result applies to general quantum algorithms which can apply unitary operations, perform intermediate measurements and reset qubits.