Quantum Lower Bounds for Approximate Counting via Laurent Polynomials

Authors Scott Aaronson, Robin Kothari, William Kretschmer, Justin Thaler

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Scott Aaronson
  • University of Texas at Austin, TX, USA
Robin Kothari
  • Microsoft Quantum, Redmond, WA, USA
  • Microsoft Research, Redmond, WA, USA
William Kretschmer
  • University of Texas at Austin, TX, USA
Justin Thaler
  • Georgetown University, Washington, D.C., USA


We are grateful to many people: Paul Burchard, for suggesting the problem of approximate counting with queries and QSamples; MathOverflow user quotedblleft fedjaquotedblright for letting us include Lemma 22 and Lemma 23; Ashwin Nayak, for extremely helpful discussions, and for suggesting the transformation of linear programs used in our extension of the method of dual polynomials to the Laurent polynomial setting; Thomas Watson, for suggesting the intersection approach to proving an SBP vs. QMA oracle separation; and Patrick Rall, for helpful feedback on writing. JT would particularly like to thank Ashwin Nayak for his warm hospitality and deeply informative discussions during a visit to Waterloo.

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Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 7:1-7:47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ⊆ [N], in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin - Arthur (QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes T quantum queries to S, and also receives an (untrusted) m-qubit quantum witness, then either m = Ω(|S|) or T = Ω(√{N/|S|}). This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of S. As a corollary, this resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA. In our second result, we ask what if, in addition to a membership oracle for S, a quantum algorithm is also given "QSamples" - i.e., copies of the state |S⟩ = 1/√|S| ∑_{i ∈ S} |i⟩ - or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either Θ(√{N/|S|}) queries or else Θ(min{|S|^{1/3},√{N/|S|}}) QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Oracles and decision trees
  • Approximate counting
  • Laurent polynomials
  • QSampling
  • query complexity


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