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Noise Threshold of Quantum Supremacy

Keisuke Fujii

TL;DR

The paper tackles whether quantum supremacy can be established with noisy, pre-threshold quantum circuits by proving a postselected threshold theorem that virtually simulates quantum error correction. It shows that, below a noise threshold, a noisy circuit's sampling distribution cannot be efficiently reproduced by a classical computer unless the polynomial hierarchy collapses, using postselection to link to postBQP=PP. Applied to surface-code and concatenated codes, the results yield practical supremacy thresholds (e.g., about $2.84\%$ under the studied noise model), significantly higher than the standard fault-tolerance thresholds, with the origin of the advantage traced to magic-state distillation. The work also discusses extensions to non-universal models and experimental feasibility, outlining open questions on error models and approximation notions.

Abstract

Demonstrating quantum supremacy, a complexity-guaranteed quantum advantage against over the best classical algorithms by using less universal quantum devices, is an important near-term milestone for quantum information processing. Here we develop a threshold theorem for quantum supremacy with noisy quantum circuits in the pre-threshold region, where quantum error correction does not work directly. We show that, even in such a region, we can virtually simulate quantum error correction by postselection. This allows us to show that the output sampled from the noisy quantum circuits (without postselection) cannot be simulated efficiently by classical computers based on a stable complexity theoretical conjecture, i.e., non-collapse of the polynomial hierarchy. By applying this to fault-tolerant quantum computation with the surface codes, we obtain the threshold value $2.84\%$ for quantum supremacy, which is much higher than the standard threshold $0.75\%$ for universal fault-tolerant quantum computation with the same circuit-level noise model. Moreover, contrast to the standard noise threshold, the origin of quantum supremacy in noisy quantum circuits is quite clear; the threshold is determined purely by the threshold of magic state distillation, which is essential to gain a quantum advantage.

Noise Threshold of Quantum Supremacy

TL;DR

The paper tackles whether quantum supremacy can be established with noisy, pre-threshold quantum circuits by proving a postselected threshold theorem that virtually simulates quantum error correction. It shows that, below a noise threshold, a noisy circuit's sampling distribution cannot be efficiently reproduced by a classical computer unless the polynomial hierarchy collapses, using postselection to link to postBQP=PP. Applied to surface-code and concatenated codes, the results yield practical supremacy thresholds (e.g., about under the studied noise model), significantly higher than the standard fault-tolerance thresholds, with the origin of the advantage traced to magic-state distillation. The work also discusses extensions to non-universal models and experimental feasibility, outlining open questions on error models and approximation notions.

Abstract

Demonstrating quantum supremacy, a complexity-guaranteed quantum advantage against over the best classical algorithms by using less universal quantum devices, is an important near-term milestone for quantum information processing. Here we develop a threshold theorem for quantum supremacy with noisy quantum circuits in the pre-threshold region, where quantum error correction does not work directly. We show that, even in such a region, we can virtually simulate quantum error correction by postselection. This allows us to show that the output sampled from the noisy quantum circuits (without postselection) cannot be simulated efficiently by classical computers based on a stable complexity theoretical conjecture, i.e., non-collapse of the polynomial hierarchy. By applying this to fault-tolerant quantum computation with the surface codes, we obtain the threshold value for quantum supremacy, which is much higher than the standard threshold for universal fault-tolerant quantum computation with the same circuit-level noise model. Moreover, contrast to the standard noise threshold, the origin of quantum supremacy in noisy quantum circuits is quite clear; the threshold is determined purely by the threshold of magic state distillation, which is essential to gain a quantum advantage.

Paper Structure

This paper contains 6 sections, 3 theorems, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Standard threshold theorem
  3. Postselected threshold theorem
  4. Case I: concatenated quantum computation
  5. Case II: fault-tolerant quantum computation with the surface codes
  6. Conclusion and discussion

Key Result

Theorem 1

Suppose noise is given as a stochastic one $\mathcal{N}_k = (1-\epsilon_k) \mathcal{I} + \mathcal{E}_k$. If the noise strength $\epsilon _k$ is smaller than a certain threshold value, we can simulate a probability distribution $\bar{p}(x,y)$ of an arbitrary universal quantum computation (uniformly g

Figures (2)

  • Figure 1: The depth-8 circuit for syndrome measurements of the surface code. The top view is also shown right. An error is assigned on each edge with probabilities $q_1$, $q_2$, and $q_3$ independently. In addition, correlated errors occur with probabilities $q_{1,2}$, $q_{2,3}$, and $q_{3,1}$ on the two connected edges.
  • Figure 2: The parameters for the error distribution, $q_1$, $q_2$, $q_3$, $q_{1,2}$ , $q_{2,3}$, and $q_{3,1}$, and the effective single-qubit error probability $\epsilon(\nu,\mu)$ are shown in the leading order (lines) and to all orders (points) as functions of $p_e$. The dashed line shows the threshold $13.4\%$ for the effective single-qubit error probability.

Theorems & Definitions (3)

  • Theorem 1: postselected threshold theorem
  • Lemma 1
  • Theorem 2: Threshold of Quantum Supremacy