Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

TL;DR

This paper introduces the HC1Q quantum model, proves its classical intractability, and explores classical verification of quantum computations within the second Fourier hierarchy level.

Contribution

It defines the HC1Q model, demonstrates its classical hardness, and establishes a Merlin-Arthur verification system for certain Fourier hierarchy quantum circuits.

Findings

HC1Q model cannot be classically efficiently sampled unless the polynomial hierarchy collapses.

PDD-Max is BQP-complete, but admits Merlin-Arthur protocols for specific Fourier hierarchy circuits.

The proof techniques differ from previous models, potentially aiding the discovery of other hard-to-simulate quantum models.

Abstract

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.