(Pseudo) Random Quantum States with Binary Phase

Zvika Brakerski, Omri Shmueli

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We prove a quantum information-theoretic conjecture due to Ji, Liu and Song (CRYPTO 2018) which suggested that a uniform superposition with random binary phase is statistically indistinguishable from a Haar random state. That is, any polynomial number of copies of the aforementioned state is within exponentially small trace distance from the same number of copies of a Haar random state. As a consequence, we get a provable elementary construction of pseudorandom quantum states from post-quantum pseudorandom functions. Generating pseudorandom quantum states is desirable for physical applications as well as for computational tasks such as quantum money. We observe that replacing the pseudorandom function with a (2t)-wise independent function (either in our construction or in previous work), results in an explicit construction for quantum state t-designs for all t. In fact, we show that the circuit complexity (in terms of both circuit size and depth) of constructing t-designs is bounded by that of (2t)-wise independent functions. Explicitly, while in prior literature t-designs required linear depth (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > 2$$\end{document}), this observation shows that polylogarithmic depth suffices for all t. We note that our constructions yield pseudorandom states and state designs with only real-valued amplitudes, which was not previously known. Furthermore, generating these states require quantum circuit of restricted form: applying one layer of Hadamard gates, followed by a sequence of Toffoli gates. This structure may be useful for efficiency and simplicity of implementation.
Original languageEnglish
Title of host publicationTheory of Cryptography - 17th International Conference, TCC 2019, Proceedings
EditorsDennis Hofheinz, Alon Rosen
Pages229-250
Number of pages22
Volume11891
ISBN (Electronic)978-3-030-36030-6
DOIs
StatePublished - 22 Nov 2019
EventTheory of Cryptography: 17th International Conference, TCC 2019 - Nuremberg, German
Duration: 1 Dec 20195 Dec 2019

Publication series

NameLecture Notes in Computer Science
Volume11891
ISSN (Print)0302-9743

Conference

ConferenceTheory of Cryptography: 17th International Conference, TCC 2019
Period1/12/195/12/19

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

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