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Unconditional Quantum Advantage for Sampling with Shallow Circuits

Adam Bene Watts, Natalie Parham

TL;DR

The answer is yes when the number of random input bits given to the classical circuit is bounded, and an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits.

Abstract

Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: Can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes when the number of random input bits given to the classical circuit is bounded. We introduce a distribution $D_{n}$ over $\{0,1\}^n$ and construct a constant-depth uniform quantum circuit family $\{C_n\}_n$ such that $C_n$ samples from a distribution close to $D_{n}$ in total variation distance. For any $δ< 1$ we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input $kn + n^δ$ i.i.d. Bernouli random variables with entropy $1/k$ and produces output close to $D_{n}$ in total variation distance has depth $Ω(\log \log n)$. This gives an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. We also show a similar separation between constant-depth quantum circuits with advice and classical circuits with bounded fan-in and fan-out, but access to an unbounded number of i.i.d random inputs. The distribution $D_n$ and classical circuit lower bounds are inspired by work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.

Unconditional Quantum Advantage for Sampling with Shallow Circuits

TL;DR

The answer is yes when the number of random input bits given to the classical circuit is bounded, and an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits.

Abstract

Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: Can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes when the number of random input bits given to the classical circuit is bounded. We introduce a distribution over and construct a constant-depth uniform quantum circuit family such that samples from a distribution close to in total variation distance. For any we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input i.i.d. Bernouli random variables with entropy and produces output close to in total variation distance has depth . This gives an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. We also show a similar separation between constant-depth quantum circuits with advice and classical circuits with bounded fan-in and fan-out, but access to an unbounded number of i.i.d random inputs. The distribution and classical circuit lower bounds are inspired by work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.
Paper Structure (27 sections, 25 theorems, 194 equations, 12 figures)

This paper contains 27 sections, 25 theorems, 194 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Technical Overview
  4. Non-unitary gates
  5. Non-unitary circuit construction with advice
  6. Converting to Unitary
  7. Classical circuit lower bound
  8. Discussion and Open Problems
  9. Acknowledgements
  10. Reader's Guide
  11. Sampling from (X, majmod(X) + parity(X)) using a GHZ state
  12. Review of GHZ States
  13. Sampling with non-unitary operations
  14. Removing non-unitary operations
  15. Introducing multi-qubit non-unitary operations
  16. ...and 12 more sections

Key Result

Theorem 3

For each $\delta \in [0,1)$, there exists a family of distributions $\{D_n\}$ such that for each $n\in \mathbb{N}$, $D_n$ is a distribution over $\{0,1\}^n$ and The distributions $D_n$ constructed are of the form $(X, f(X))$ for a uniformly random bitstring $X$ and function $f: \{0,1\}^{n-1} \rightarrow \{0,1\}$.

Figures (12)

  • Figure 1: Table comparing a few different computational problems with either conditional or unconditional proof of quantum advantage.
  • Figure 2: A circuit constructing the state $H^{\otimes n} \ket{\text{GHZ}\xspace_n}$, as described in \ref{['eq:Hadarmarded_GHZ']}.
  • Figure 3: A diagrammatic proof of \ref{['lem:NUrot_identity']}. The equivalence between each line is explained in the proof of the lemma.
  • Figure 4: Constant-depth circuit producing approximate samples from the distribution $(X, \text{majmod}_{p}\xspace(X) \oplus \text{parity}\xspace(X))$.
  • Figure 5: Diagrammatic analysis of the circuit presented in the proof of \ref{['thm:NUrot_majmod_sampling']}. The first line follows from \ref{['eq:Hadarmarded_GHZ']}, while the second follows from \ref{['lem:NUrot_identity']}.
  • ...and 7 more figures

Theorems & Definitions (94)

  • Definition 2: Total Variation Distance, $\Delta$
  • Theorem 3
  • Corollary 4
  • Theorem 5: Separation with GHZ advice
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • Lemma 8
  • proof
  • ...and 84 more