Achieving quantum supremacy with sparse and noisy commuting quantum computations

TL;DR

This work analyzes instantaneous quantum polynomial-time (IQP) circuits under realistic constraints: spatial locality on a 2D lattice and end-of-circuit depolarising noise. It shows that sparse IQP circuits with $O(n\log n)$ gates can be implemented on a $\sqrt{n}\times\sqrt{n}$ lattice in depth $O(\sqrt{n}\log n)$ and remain hard to simulate classically under a sparse Ising-conjecture, while also proving that even tiny noise enables efficient classical sampling for anticoncentrated outputs unless fault-tolerance is employed. The authors introduce a fault-tolerance-like classical error-correction approach that preserves IQP hardness under noise by encoding the diagonal part of the circuit, suggesting both a practical challenge for supremacy experiments and a path to mitigate it using classical techniques. The results illuminate the limits of near-term quantum supremacy demonstrations and provide a framework to study the interplay between anticoncentration, locality, noise, and classical simulability across IQP and related models.

Abstract

The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.

Figures (1)

• Figure 1: A standard IQP circuit, and an IQP circuit with depolarising noise. $D$ is a circuit made up of $\mathop{\mathrm{poly}}\nolimits(n)$ diagonal gates.

Theorems & Definitions (17)

• Conjecture 1
• Conjecture 2
• Theorem 3: informal
• Theorem 4
• Theorem 5
• Lemma 6
• Corollary 7
• Conjecture 8
• Lemma 9
• proof