# Nonadaptive fault-tolerant verification of quantum supremacy with noise

## TL;DR

This paper proposes a nonadaptive, noise-resilient verification method for quantum samplers demonstrating quantum supremacy, with minimal experimental requirements and fault-tolerant enhancements to improve reliability and thresholds.

## Contribution

It introduces a trap-based verification scheme for quantum supremacy that requires only single-qubit preparations and extends it with fault-tolerant methods preserving nonadaptivity.

## Key findings

- Verification scheme works with constant overhead on a square lattice.
- Fault-tolerant schemes achieve higher thresholds, up to 1.97%.
- Classical sampling remains hard under plausible conjectures.

## Abstract

Quantum samplers are believed capable of sampling efficiently from distributions that are classically hard to sample from. We consider a sampler inspired by the classical Ising model. It is nonadaptive and therefore experimentally amenable. Under a plausible conjecture, classical sampling upto additive errors from this model is known to be hard. We present a trap-based verification scheme for quantum supremacy that only requires the verifier to prepare single-qubit states. The verification is done on the same model as the original sampler, a square lattice, with only a constant overhead. We next revamp our verification scheme in two distinct ways using fault tolerance that preserves the nonadaptivity. The first has a lower overhead based on error correction with the same threshold as universal quantum computation. The second has a higher overhead but an improved threshold (1.97\%) based on error detection. We show that classically sampling upto additive errors is likely hard in both these schemes. Our results are applicable to other sampling problems such as the Instantaneous Quantum Polynomial-time (IQP) computation model. They should also assist near-term attempts at experimentally demonstrating quantum supremacy and guide long-term ones.