Test of Quantumness with Small-Depth Quantum Circuits

TL;DR

This paper demonstrates that a known test of quantumness, previously thought to require complex quantum resources, can actually be implemented using very shallow quantum circuits combined with limited classical computation, highlighting their computational power.

Contribution

It shows that the test of quantumness and related cryptographic applications can be achieved with constant-depth quantum circuits and logarithmic-depth classical computation, revealing new complexity insights.

Findings

Test of quantumness implementable with shallow quantum circuits

Cryptographic applications feasible with small-depth quantum circuits

Highlights the computational superiority of small-depth quantum circuits

Abstract

Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but cannot be solved by a classical polynomial-time computer under the LWE assumption. This test has lead to several cryptographic applications. In particular, it has been applied to producing certifiable randomness from a single untrusted quantum device, self-testing a single quantum device and device-independent quantum key distribution. In this paper, we show that this test of quantumness, and essentially all the above applications, can actually be implemented by a very weak class of quantum circuits: constant-depth quantum circuits combined with logarithmic-depth classical computation. This reveals novel complexity-theoretic properties of this fundamental test of quantumness and gives new concrete evidence of the superiority of small-depth quantum circuits over classical computation.