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Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

Authors:

Adam Bene Watts,

Luke Schaeffer, and

Avishay TalAuthors Info & Claims

STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

June 2019

Pages 515 - 526

https://doi.org/10.1145/3313276.3316404

Published: 23 June 2019 Publication History

Abstract

Recently, Bravyi, Gosset, and Konig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0.

We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem.

Our results are shown by constructing a new problem in QNC^0, which we call the Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem.

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Gosset DGrier DKerzner ASchaeffer L(2024)Fast simulation of planar Clifford circuitsQuantum10.22331/q-2024-02-12-12518(1251)Online publication date: 12-Feb-2024
https://doi.org/10.22331/q-2024-02-12-1251
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Index Terms

Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits
1. Theory of computation
  1. Models of computation
    1. Quantum computation theory
      1. Quantum complexity theory

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cover image ACM Conferences

STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

June 2019

1258 pages

ISBN:9781450367059

DOI:10.1145/3313276

General Chair:
Moses Charikar
Stanford University
,
Program Chair:
Edith Cohen
Google, USA / Tel Aviv University, Israel

Copyright © 2019 ACM.

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Publication History

Published: 23 June 2019

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STOC '19

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STOC '19: 51st Annual ACM SIGACT Symposium on the Theory of Computing

June 23 - 26, 2019

AZ, Phoenix, USA

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Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Gosset DGrier DKerzner ASchaeffer L(2024)Fast simulation of planar Clifford circuitsQuantum10.22331/q-2024-02-12-12518(1251)Online publication date: 12-Feb-2024
https://doi.org/10.22331/q-2024-02-12-1251
Zhang ZWang QYing M(2024)Parallel Quantum Algorithm for Hamiltonian SimulationQuantum10.22331/q-2024-01-15-12288(1228)Online publication date: 15-Jan-2024
https://doi.org/10.22331/q-2024-01-15-1228
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Huang HLiu YBroughton MKim IAnshu ALandau ZMcClean JMohar BShinkar IO'Donnell R(2024)Learning Shallow Quantum CircuitsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649722(1343-1351)Online publication date: 10-Jun-2024
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Boghiu EHirsch FLin PQuintino MBowles J(2023)Device-independent and semi-device-independent entanglement certification in broadcast Bell scenariosSciPost Physics Core10.21468/SciPostPhysCore.6.2.0286:2Online publication date: 11-Apr-2023
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