Article

Adiabatic quantum state generation and statistical zero knowledge

Authors:

Dorit Aharonov,

Amnon Ta-ShmaAuthors Info & Claims

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

Pages 20 - 29

https://doi.org/10.1145/780542.780546

Published: 09 June 2003 Publication History

Abstract

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'.We first show that any problem in Statistical Zero Knowledge (including eg. discrete log, quadratic residuosity and gap closest vector in a lattice) can be reduced to an instance of the quantum state generation problem. Having shown the generality of the state generation problem, we set the foundations for a new paradigm for quantum state generation. We define 'Adiabatic State Generation' (ASG), which is based on Hamiltonians instead of unitary gates. We develop tools for ASG including a very general method for implementing Hamiltonians (The sparse Hamiltonian lemma), and ways to guarantee non negligible spectral gaps (The jagged adiabatic path lemma). We also prove that ASG is equivalent in power to state generation in the standard quantum model. After setting the foundations for ASG, we show how to apply our techniques to generate interesting superpositions related to Markov chains.The ASG approach to quantum algorithms provides intriguing links between quantum computation and many different areas: the analysis of spectral gaps and groundstates of Hamiltonians in physics, rapidly mixing Markov chains, statistical zero knowledge, and quantum random walks. We hope that these links will bring new insights and methods into quantum algorithms.

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Index Terms

Adiabatic quantum state generation and statistical zero knowledge
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes
  2. Design and analysis of algorithms

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cover image ACM Conferences

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

June 2003

740 pages

ISBN:1581136749

DOI:10.1145/780542

Conference Chair:
Lawrence L. Larmore
University of Nevada Las Vegas, Las Vegas, NV
,
Program Chair:
Michel X. Goemans
Massachusetts Institute of Technology, Cambridge, MA

Copyright © 2003 ACM.

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Published: 09 June 2003

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STOC03: The 35th Annual ACM Symposium on Theory of Computing

June 9 - 11, 2003

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STOC '03 Paper Acceptance Rate 80 of 270 submissions, 30%;

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

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https://doi.org/10.3934/fods.2024013
Kirby W(2024)Analysis of quantum Krylov algorithms with errorsQuantum10.22331/q-2024-08-29-14578(1457)Online publication date: 29-Aug-2024
https://doi.org/10.22331/q-2024-08-29-1457
Low GKliuchnikov VSchaeffer L(2024)Trading T gates for dirty qubits in state preparation and unitary synthesisQuantum10.22331/q-2024-06-17-13758(1375)Online publication date: 17-Jun-2024
https://doi.org/10.22331/q-2024-06-17-1375
Ghysels EMorgan J(2024)On Quantum Ambiguity and Potential Exponential Computational Speed-Ups to Solving Dynamic Asset Pricing ModelsSSRN Electronic Journal10.2139/ssrn.4815156Online publication date: 2024
https://doi.org/10.2139/ssrn.4815156
Pande M(2024)A Comprehensive Review of Data Encoding Techniques for Quantum Machine Learning Problems2024 Second International Conference on Emerging Trends in Information Technology and Engineering (ICETITE)10.1109/ic-ETITE58242.2024.10493306(1-7)Online publication date: 22-Feb-2024
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Leadbeater CFitzpatrick NMuñoz Ramo DThom A(2024)Non-unitary Trotter circuits for imaginary time evolutionQuantum Science and Technology10.1088/2058-9565/ad53fb9:4(045007)Online publication date: 9-Jul-2024
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Granet EDreyer H(2024)Hamiltonian dynamics on digital quantum computers without discretization errornpj Quantum Information10.1038/s41534-024-00877-y10:1Online publication date: 7-Sep-2024
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