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


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)


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
  • Relational problems
  • quantum advice
  • randomized advice
  • FBQP
  • FBPP


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