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Effect of non-unital noise on random circuit sampling

Bill Fefferman, Soumik Ghosh, Michael Gullans, Kohdai Kuroiwa, Kunal Sharma

TL;DR

It is proved that the output distribution of random quantum circuits under practical non-unital noise sources with constant noise rates never anticoncentrates, meaning it is never too"flat"$\unicode{x2014}$ regardless of the depth of the circuit.

Abstract

In this work, drawing inspiration from the type of noise present in real hardware, we study the output distribution of random quantum circuits under practical non-unital noise sources with constant noise rates. We show that even in the presence of unital sources like the depolarizing channel, the distribution, under the combined noise channel, never resembles a maximally entropic distribution at any depth. To show this, we prove that the output distribution of such circuits never anticoncentrates $\unicode{x2014}$ meaning it is never too "flat" $\unicode{x2014}$ regardless of the depth of the circuit. This is in stark contrast to the behavior of noiseless random quantum circuits or those with only unital noise, both of which anticoncentrate at sufficiently large depths. As consequences, our results have interesting algorithmic implications on both the hardness and easiness of noisy random circuit sampling, since anticoncentration is a critical property exploited by both state-of-the-art classical hardness and easiness results.

Effect of non-unital noise on random circuit sampling

TL;DR

It is proved that the output distribution of random quantum circuits under practical non-unital noise sources with constant noise rates never anticoncentrates, meaning it is never too"flat" regardless of the depth of the circuit.

Abstract

In this work, drawing inspiration from the type of noise present in real hardware, we study the output distribution of random quantum circuits under practical non-unital noise sources with constant noise rates. We show that even in the presence of unital sources like the depolarizing channel, the distribution, under the combined noise channel, never resembles a maximally entropic distribution at any depth. To show this, we prove that the output distribution of such circuits never anticoncentrates meaning it is never too "flat" regardless of the depth of the circuit. This is in stark contrast to the behavior of noiseless random quantum circuits or those with only unital noise, both of which anticoncentrate at sufficiently large depths. As consequences, our results have interesting algorithmic implications on both the hardness and easiness of noisy random circuit sampling, since anticoncentration is a critical property exploited by both state-of-the-art classical hardness and easiness results.
Paper Structure (65 sections, 19 theorems, 264 equations, 7 figures)

This paper contains 65 sections, 19 theorems, 264 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Our conventions
  4. Our noise model
  5. Our results
  6. Our techniques
  7. Removing the last layer of noise
  8. Lightcone arguments
  9. Mapping to classical partition functions
  10. Putting our results in context
  11. High noise regime
  12. Modeling the high noise regime
  13. Low noise regime
  14. Implications of our results
  15. Notation and setting
  16. ...and 50 more sections

Key Result

Proposition 1

Let $\mathcal{B}$ be an ensemble of random quantum circuits which satisfy hiding and anticoncentration with respect to Definiton: anticoncentration. Then, it also satisfies anticoncentration with respect to newdef.

Figures (7)

  • Figure 1: The circuit model that we use for our analysis. Every pink rectangle is a single qubit noise channel. Every white rectangle is either a single qubit or a two--qubit Haar random gate. In the end, the final state is measured in the standard basis.
  • Figure 2: $U_1$, $U_2$, …, $U_n$ are single qubit gates.
  • Figure 3: A portion of the circuit drawn from the ensemble $\mathcal{B}$. $U_1$ and $U_2$ are two qubit Haar random gates and $\mathcal{N}_1$ and $\mathcal{N}_2$ are arbitrary noise channels.
  • Figure 4: $U_3$ and $U_4$ are single qubit Haar random gates.
  • Figure 5: If the circuit terminates with a last layer of noiseless gates, then there is no way to "sandwich" the last layer of noise between a Haar random single qubit unitary and its adjoint. So, twirling does not work.
  • ...and 2 more figures

Theorems & Definitions (77)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 1: Parallel quantum circuit
  • Definition 2: Geometrically local quantum circuit
  • Definition 3: Noisy quantum circuit
  • Definition 4: Random quantum circuit
  • Definition 5: Amplitude Damping Noise Channel
  • Definition 6: Depolarizing Noise Channel
  • ...and 67 more