On the Need for Large Quantum Depth

TL;DR

This paper investigates the computational power of shallow-depth quantum circuits, proving an oracle separation that challenges the conjecture that polynomial-time quantum and hybrid quantum-classical models are equivalent.

Contribution

It proves Aaronson's oracle conjecture for a related problem, demonstrating limitations of shallow quantum circuits and refuting Jozsa's conjecture in an oracle setting.

Findings

Proves an oracle separation supporting the limitations of shallow quantum circuits.

Demonstrates that BQP may not be contained within certain hybrid models.

Refutes Jozsa's conjecture relative to an oracle.

Abstract

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates, and thus a potential way to use these quantum devices is using a hybrid scheme that interleaves them with classical computers. For example, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Along the line, it seems possible that a general quantum computer may only be polynomially faster than a hybrid quantum-classical computer. Jozsa raised the question of whether $BQP = BPP^{BQNC}$ and conjectured that they are equal, where $BQNC$ means $polylog$-depth quantum circuits. Nevertheless, Aaronson conjectured an oracle separation for these two classes and gave a candidate. In this work, we prove Aaronson's conjecture for a different but related oracle problem. Our result also proves that Jozsa's conjecture fails relative to an oracle.