Simulating Special but Natural Quantum Circuits

TL;DR

This paper identifies a specific subclass of quantum algorithms that can be efficiently simulated classically if they measure few qubits, revealing insights into the structure of quantum algorithms and potential cryptographic implications.

Contribution

It introduces a new subclass of BQP algorithms that are classically simulatable under certain measurement constraints and provides a novel characterization of Fourier-type sums.

Findings

Classical simulation is possible for certain quantum algorithms measuring O(log n) qubits.

The subclass includes algorithms with structural similarities to some known quantum algorithms.

Highlights a potentially hard function relevant for cryptography.

Abstract

We identify a sub-class of BQP that captures certain structural commonalities among many quantum algorithms including Shor's algorithms. This class does not contain all of BQP (e.g. Grover's algorithm does not fall into this class). Our main result is that any algorithm in this class that measures at most O(log n) qubits can be simulated by classical randomized polynomial time algorithms. This does not dequantize Shor's algorithm (as the latter measures n qubits) but our work also highlights a new potentially hard function for cryptographic applications. Our main technical contribution is (to the best of our knowledge) a new exact characterization of certain sums of Fourier-type coefficients (with exponentially many summands).