A Qubit, a Coin, and an Advice String Walk into a Relational Problem

Authors Scott Aaronson, Harry Buhrman, William Kretschmer



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Author Details

Scott Aaronson
  • University of Texas at Austin, TX, USA
  • OpenAI, San Francisco, CA, USA
Harry Buhrman
  • QuSoft, Amsterdam, The Netherlands
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands
William Kretschmer
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
  • University of California, Berkeley, CA, USA

Acknowledgements

We thank Daochen Wang and Alexandru Cojocaru for conversations that suggested the FBQP/poly versus FBQP/qpoly problem, and Lance Fortnow, Iordanis Kerenidis, Ashley Montanaro, Ronald de Wolf, and David Zuckerman for helpful discussions. We thank anonymous referees for constructive feedback.

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Scott Aaronson, Harry Buhrman, and William Kretschmer. A Qubit, a Coin, and an Advice String Walk into a Relational Problem. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.1

Abstract

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly ≠ FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP ̸ ⊂ FP/poly - that is, Adleman’s Theorem fails for relational problems - unless PSPACE ⊂ NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP ̸ ⊂ FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: - Unconditionally, FP ≠ FBPP and FP/poly ≠ FBPP/poly (even when these classes are carefully defined). - FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly ≠ SampBPP/rpoly (and likewise for SampBQP).

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Quantum complexity theory
Keywords
  • Relational problems
  • quantum advice
  • randomized advice
  • FBQP
  • FBPP

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