Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The power of quantum circuits in sampling

Guy Blanc, Caleb Koch, Jane Lange, Carmen Strassle, Li-Yang Tan

TL;DR

This work proves a relativized, approximate separation between quantum and classical sampling: relative to a random oracle, there exists a distribution that quantum circuits can sample exactly with polynomial size, yet no subexponential-size classical circuit can approximate it in total variation distance within $1-o(1)$. The authors build on Yamakawa–Zhandry’s hardness framework for the NullCodeword problem, introduce a $k$-fold hardness amplification to force exponential decay in the success probability of any $q$-query classical strategy, and then transfer this hardness to the sampling of the solution space $\mathrm{Unif}(\mathcal{C}_{\mathcal{O}})$. A key technical contribution is showing that a large family of query algorithms, if collectively successful, must be well-spread and thus highly unlikely to all land in the zero-output set, enabling a union-bound argument over all size-$q$ circuits. The result highlights a representational separation for sampling, distinct from algorithmic separations, and opens a path for further exploration of quantum advantages in sampling tasks under relativized assumptions. The approach also clarifies the limitations of extending exact-separation strategies to approximate settings and underscores the role of hardness amplification and coding-theoretic structure in oracle-based quantum supremacy results.

Abstract

We give new evidence that quantum circuits are substantially more powerful than classical circuits. We show, relative to a random oracle, that polynomial-size quantum circuits can sample distributions that subexponential-size classical circuits cannot approximate even to TV distance $1-o(1)$. Prior work of Aaronson and Arkhipov (2011) showed such a separation for the case of exact sampling (i.e. TV distance $0$), but separations for approximate sampling were only known for uniform algorithms. A key ingredient in our proof is a new hardness amplification lemma for the classical query complexity of the Yamakawa-Zhandry (2022) search problem. We show that the probability that any family of query algorithms collectively finds $k$ distinct solutions decays exponentially in $k$.

The power of quantum circuits in sampling

TL;DR

This work proves a relativized, approximate separation between quantum and classical sampling: relative to a random oracle, there exists a distribution that quantum circuits can sample exactly with polynomial size, yet no subexponential-size classical circuit can approximate it in total variation distance within . The authors build on Yamakawa–Zhandry’s hardness framework for the NullCodeword problem, introduce a -fold hardness amplification to force exponential decay in the success probability of any -query classical strategy, and then transfer this hardness to the sampling of the solution space . A key technical contribution is showing that a large family of query algorithms, if collectively successful, must be well-spread and thus highly unlikely to all land in the zero-output set, enabling a union-bound argument over all size- circuits. The result highlights a representational separation for sampling, distinct from algorithmic separations, and opens a path for further exploration of quantum advantages in sampling tasks under relativized assumptions. The approach also clarifies the limitations of extending exact-separation strategies to approximate settings and underscores the role of hardness amplification and coding-theoretic structure in oracle-based quantum supremacy results.

Abstract

We give new evidence that quantum circuits are substantially more powerful than classical circuits. We show, relative to a random oracle, that polynomial-size quantum circuits can sample distributions that subexponential-size classical circuits cannot approximate even to TV distance . Prior work of Aaronson and Arkhipov (2011) showed such a separation for the case of exact sampling (i.e. TV distance ), but separations for approximate sampling were only known for uniform algorithms. A key ingredient in our proof is a new hardness amplification lemma for the classical query complexity of the Yamakawa-Zhandry (2022) search problem. We show that the probability that any family of query algorithms collectively finds distinct solutions decays exponentially in .

Paper Structure

This paper contains 31 sections, 7 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Exact vs. approximate sampling.
  4. Separations for constant-depth circuits.
  5. Dequantizing quantum circuits.
  6. Broader context: Representational vs. algorithmic quantum advantage
  7. Three requirements for a representational separation
  8. Technical Overview
  9. Quick overview of Yamakawa--Zhandry.
  10. Overview of our approach
  11. $k$-fold NullCodeword and its hardness
  12. From $k$-fold NullCodeword to sampling
  13. Obstacles in extending AA11's separation to the approximate setting
  14. Discussion
  15. Preliminaries
  16. ...and 16 more sections

Key Result

Theorem 1

The following holds relative to a random oracle. For every sufficiently large $n$, there is an explicit distribution $\mathcal{D}^{(n)}$ over $\{0,1\}^n$ that can be sampled exactly by a polynomial-size quantum circuit but no subexponential-size classical circuit can approximate $\mathcal{D}^{(n)}$

Figures (1)

  • Figure 1: This family of $5$ query algorithms outputs $4$ elements of $\mathcal{C}_{\mathcal{O}}$, among which $3$ are distinct. It therefore $3$-succeeds on $\mathcal{O}$.

Theorems & Definitions (36)

  • Theorem 1
  • Remark 1
  • Remark 2: Salting CDGS18
  • Remark 3: Uniform quantum easiness; non-uniform classical hardness
  • Lemma 2.1
  • Remark 4: Comparison with direct product theorems
  • Remark 5
  • Definition 1: TV distance
  • Definition 2: Codewords that are $t$-queried by $S$
  • Definition 3: List Recoverability
  • ...and 26 more