On the power of interleaved low-depth quantum and classical circuits

Abstract

Low-depth quantum circuits are a well-suited model for near-term quantum devices, given short coherence times and noisy gate operations, making it pivotal to examine their computational power. It was already known as early as 2004 that simulating such low-depth quantum circuits is classically hard under complexity-theoretic assumptions. Later, it was shown that low-depth quantum circuits interleaved with low-depth classical circuits can perform approximate quantum Fourier transform, the quantum subroutine of Shor's algorithm.

Given these salient features of low-depth quantum models, Terhal and DiVincenzo, Aaronson, and Jozsa have all independently conjectured regarding the elusive power of combining low-depth quantum circuits with efficient classical computation. However, much has remained unresolved in this interleaved setting. Therefore, in this thesis, we tackle the question of characterizing the computational power of interleaved low-depth quantum and classical circuits. We first review existing separations in the low-depth setting. Then, we formally define two interleaving models based on whether the quantum device is permitted to make subset measurements (weak interleaving) or must measure all qubits together (strict interleaving).

By combining existing techniques from quantum fan-out constructions, teleportation-based quantum computation, and Clifford + T circuit synthesis, we show several results regarding the power of variants of constant-depth quantum circuits (QNC0) strictly and weakly interleaved with constant-depth classical parity circuits. Our main new result is that QNC0 with access to cat states strictly interleaved with constant-depth classical parity circuits can simulate constant-depth threshold circuits (TC0), which neither of the classes can do on their own. This strictly separates this interleaved class from constant-depth classical circuits with unbounded fan-in mod p and OR gates (AC0[p]).