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arXiv:1904.08914

Quantum Lower Bounds for Approximate Counting via Laurent Polynomials

Scott Aaronson, Robin Kothari, William Kretschmer, Justin Thaler

Published Apr 18, 2019• 12 citations
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Abstract

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 $S \subseteq [N]$, in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin--Arthur ($\mathsf{QMA}$) 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 $T=Ω\bigl(\sqrt{N/\left| S\right| } \bigr)$. 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 $\mathsf{SBP}$, the complexity class that captures approximate counting, and $\mathsf{QMA}$. 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 $\left| S\right\rangle = \frac{1}{\sqrt{\left| S\right| }} \sum_{i\in S}|i\rangle$ -- or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either $Θ\bigl(\sqrt{N/\left| S\right| }\bigr)$ queries or else $Θ\bigl(\min \bigl\{\left| S\right| ^{1/3}, \sqrt{N/\left| S\right| }\bigr\}\bigr)$ QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.

DOI
10.48550/arxiv.1904.08914

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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]S \subseteq [N]S⊆[N], in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin--Arthur (QMA\mathsf{QMA}QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes TTT quantum queries to SSS, and also receives an (untrusted) mmm-qubit quantum witness, then either m=Ω(∣S∣)m = Ω(|S|)m=Ω(∣S∣) or T=Ω(N/∣S∣)T=Ω\bigl(\sqrt{N/\left| S\right| } \bigr)T=Ω(N/∣S∣​). This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of SSS. As a corollary, this resolves the open problem of giving an oracle separation between SBP\mathsf{SBP}SBP, the complexity class that captures approximate counting, and QMA\mathsf{QMA}QMA. In our second result, we ask what if, in addition to a membership oracle for SSS, a quantum algorithm is also given "QSamples" -- i.e., copies of the state ∣S⟩=1∣S∣∑i∈S∣i⟩\left| S\right\rangle = \frac{1}{\sqrt{\left| S\right| }} \sum_{i\in S}|i\rangle∣S⟩=∣S∣​1​∑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∣)Θ\bigl(\sqrt{N/\left| S\right| }\bigr)Θ(N/∣S∣​) queries or else Θ(min⁡{∣S∣1/3,N/∣S∣})Θ\bigl(\min \bigl\{\left| S\right| ^{1/3}, \sqrt{N/\left| S\right| }\bigr\}\bigr)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.
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.
Quantum counting is a fundamental quantum algorithm that estimates the fraction of marked elements using a membership oracle, achieving a quadratic speedup over classical sampling. The membership oracle, however, assumes exact labeling of each element, but this assumption fails when the labels are inherently probabilistic. We study quantum counting with \emph{Bernoulli oracles}, where given mmm Bernoulli distributions with unknown biases p1,…,pmp_1,\dots,p_mp1​,…,pm​ and a gap parameter ΔΔΔ, the goal is to estimate the fraction ρρρ of \emph{positive} distributions (pi≥1/2+Δp_i\ge1/2+Δpi​≥1/2+Δ) to within additive error εεε. We prove an upper bound of O~ ⁣(ρΔε+1Δε)\tilde{O}\!\big(\frac{\sqrtρ}{Δε}+\frac{1}{Δ\sqrtε}\big)O~(Δερ​​+Δε​1​) queries, achieving a quadratic speedup over the classical sample complexity. % of Θ(ρ/Δ2ε2)Θ(ρ/Δ^2ε^2)Θ(ρ/Δ2ε2). Our algorithm first uses the Quantum Singular Value Transformation (QSVT) to coherently amplify the bias gap without collapsing the superposition over distributions, and then applies two-stage adaptive amplitude estimation. We complement this upper bound with a near-matching lower bound of Ω(ρ/(Δε))Ω(\sqrtρ/(Δε))Ω(ρ​/(Δε)) via a new composition theorem for the quantum adversary method in the Boolean-over-average-case direction. For the special case of a constant gap Δ=Θ(1)Δ=Θ(1)Δ=Θ(1), which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to O~(ρε+1ε)\tilde{O}\big(\frac{\sqrtρ}ε+\frac{1}{\sqrtε}\big)O~(ερ​​+ε​1​) and Ω(ρε)Ω(\frac{\sqrtρ}ε)Ω(ερ​​), thereby characterizing the query complexity of quantum counting with bounded-error oracles.
QMA\mathsf{QMA}QMA Lower Bounds for Approximate Counting
We prove a query complexity lower bound for QMA\mathsf{QMA}QMA protocols that solve approximate counting: estimating the size of a set given a membership oracle. This gives rise to an oracle AAA such that SBPA⊄QMAA\mathsf{SBP}^A \not\subset \mathsf{QMA}^ASBPA⊂QMAA, resolving an open problem of Aaronson [2]. Our proof uses the polynomial method to derive a lower bound for the SBQP\mathsf{SBQP}SBQP query complexity of the AND\mathsf{AND}AND of two approximate counting instances. We use Laurent polynomials as a tool in our proof, showing that the "Laurent polynomial method" can be useful even for problems involving ordinary polynomials.
We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial f(x)=∑j=1sajxuj∈Fq[x1±1,…,xn±1],f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], f(x)=∑j=1s​aj​xuj​∈Fq​[x1±1​,…,xn±1​], let UUU be its augmented support matrix whose columns are (1,uj)(1,u_j)(1,uj​) with rank ρρρ and N(f):=#{x∈(Fq×)n∣f(x)=0}N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}N(f):=#{x∈(Fq×​)n∣f(x)=0} be its torus point count. Our first main result is a quantum algorithm that outputs N^(f)\widehat{N}(f)N(f) satisfying ∣N^(f)−N(f)∣≤εqn+s/2−ρ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-ρ} ∣N(f)−N(f)∣≤εqn+s/2−ρ with success probability 1−δ1-δ1−δ. Provided that ρρρ and ∥U∥∞\|U\|_\infty∥U∥∞​ are bounded, the algorithm runs in both classical bit and quantum gate complexity poly(n,s,log⁡q,1/ε,log⁡(1/δ))\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/δ))poly(n,s,logq,1/ε,log(1/δ)). It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in log⁡q\log qlogq. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes #\##P-hard under randomized polynomial-time Turing reductions when the support matrix UUU varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.
What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n)Θ(n)Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(log⁡n)Θ(\log n)Θ(logn) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(log⁡n)Ω(\log n)Ω(logn) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(log⁡n)Ω(\log n)Ω(logn) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(logn)o(log n)o(logn) NP queries, then BPPNP[o(n)]\text{BPP}^{\text{NP}[o(n)]}BPPNP[o(n)] contains a PNP\text{P}^{\text{NP}}PNP-complete problem if the algorithm is classical and FBQPNP[o(n)]\text{FBQP}^{\text{NP}[o(n)]}FBQPNP[o(n)] contains an FPNP\text{FP}^{\text{NP}}FPNP-complete problem if the algorithm is quantum.
Given oracle access to an unknown unitary U=eiHU=e^{iH}U=eiH , the fractional query problem asks how many queries are required to implement a noninteger power Ut=eitHU^t=e^{itH}Ut=eitH, 0<t<10<t<10<t<1, when the spectrum is separated from the branch cut by a gap δδδ. Quantum singular value transformation gives an upper bound of O ⁣(1δlog⁡1ε)O\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)O(δ1​logε1​) queries for approximation error ε\varepsilonε. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any NNN-query circuit to the approximation of eitθe^{itθ}eitθ by a trigonometric polynomial with degree bounded by O(N)O(N)O(N), together with Remez inequality. This allows us to establish the lower bound of Ωτ ⁣(1δlog⁡1ε)Ω_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)Ωτ​(δ1​logε1​). Consequently, the optimal query complexity for fractional query problem is Θτ ⁣(1δlog⁡1ε)Θ_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)Θτ​(δ1​logε1​), showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a Ωτ ⁣(log⁡1ε)Ω_τ\!\left(\log\frac{1}{\varepsilon}\right)Ωτ​(logε1​) bound uniform to δδδ.