Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.09804

Small-Bias Quantum Approximate Counting via the Multiplicative Adversary Method

Albert Lin, Han-Hsuan Lin

quant-phcs.CCSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
v1 · Sep 9, 2026v2 · Oct 2, 2026
arXivHTMLPDF

Abstract

We study the two-weight decision version of quantum approximate counting: given oracle access to $x\in\{0,1\}^N$, distinguish $|x|=M$ from $|x|=M+Δ$ with success probability $1/2+ζ$. Using the multiplicative adversary method, we prove $Ω\left(\max\left\{ζ\sqrt{(N-M)(M+Δ)}/Δ,\sqrt{ζN/Δ}\right\}\right)$. The same parameter dependence follows from the polynomial-method characterization of the two-layer symmetric function by Podder, Yao, and Ye. Our contribution is a multiplicative-adversary derivation that tracks the progress produced by individual oracle queries. For the first term, after complementing the input if necessary, we assume $M+Δ\le N-M$. We use the Hamming-layer subspaces from the eigenspace method of Ambainis, Spalek, and de Wolf and compose their adjacent-layer unitary maps to relate the two nonadjacent promise layers. After fixing the queried coordinate, the analysis block-diagonalizes into four-dimensional subspaces. An exact calculation of the one-query progress ratio gives the first lower bound. The same estimate also implies $\left\|(I-\widehatΠ_{\mathrm{bad}})\lvertΨ^T\rangle\right\|^2=O\left(T^2Δ^2/((N-M)(M+Δ))\right)$ for the coherent input superposition used in the adversary argument. For the second term, we prove directly using a three-eigenvalue multiplicative adversary that unique OR on $n$ bits with success probability $1/2+ζ$ requires $Ω(\sqrt{ζn})$ queries, and then reduce unique OR to the two-weight counting problem.

Comments
18 pages

Related papers

Quantum Approximate Counting with Bernoulli Oracles

Chengshen Gao, Yongzhen Xu, Lvzhou Li

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 $m$ Bernoulli distributions with unknown biases $p_1,\dots,p_m$ and a gap parameter $Δ$, the goal is to estimate the fraction $ρ$ of \emph{positive} distributions ($p_i\ge1/2+Δ$) to within additive error $ε$. We prove an upper bound of $\tilde{O}\!\big(\frac{\sqrtρ}{Δε}+\frac{1}{Δ\sqrtε}\big)$ queries, achieving a quadratic speedup over the classical sample complexity. % of $Θ(ρ/Δ^2ε^2)$. 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 $Ω(\sqrtρ/(Δε))$ 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 $Δ=Θ(1)$, which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to $\tilde{O}\big(\frac{\sqrtρ}ε+\frac{1}{\sqrtε}\big)$ and $Ω(\frac{\sqrtρ}ε)$, thereby characterizing the query complexity of quantum counting with bounded-error oracles.

quant-pharXiv:2609.07604•Sep 8, 2026•0 citations

The Multiplicative Quantum Adversary

Robert Špalek

We present a new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function. Adversary methods work as follows: one defines a progress function based on the state of the algorithm, and shows that for a successful algorithm there is a large gap between the initial and final value of the progress, and furthermore that the progress function cannot change by much with a single query. All known variants upper-bound the difference of the progress function, whereas our new variant upper-bounds the ratio and that is why we coin it the multiplicative adversary. Our new method is rooted in the quantum lower-bound method by Ambainis (2005, 2006), based on the analysis of eigenspaces of the density matrix. Ambainis's method is technically very complicated, it lacks intuition, and it only works for symmetric functions. Our method fits well into the adversary framework, has a simple formulation in terms of common block-diagonalization of two operators, and works for all functions. Furthermore, we prove an unconditional strong direct product theorem for the multiplicative quantum adversary bound.

arXiv:quant-ph/0703237•IEEE Conference on Computational Complexity (CCC)•Jun 1, 2008•29 citations

One Weird Trick Tightens the Quantum Adversary Bound, Especially for Success Probability Close to $1/2$

Duyal Yolcu

The textbook adversary bound for function evaluation states that to evaluate a function $f\colon D\to C$ with success probability $\frac{1}{2}+δ$ in the quantum query model, one needs at least $\left( 2δ-\sqrt{1-4δ^2} \right) Adv(f)$ queries, where $Adv(f)$ is the optimal value of a certain optimization problem. For $δ\ll 1$, this only allows for a bound of $Θ\left(δ^2 Adv(f)\right)$ even after a repetition-and-majority-voting argument. In contrast, the polynomial method can sometimes prove a bound that doesn't converge to $0$ as $δ\to 0$. We improve the $δ$-dependent prefactor and achieve a bound of $2δAdv(f)$. The proof idea is to "turn the output condition into an input condition": From an algorithm that transforms perfectly input-independent initial to imperfectly distinguishable final states, we construct one that transforms imperfectly input-independent initial to perfectly distinguishable final states in the same number of queries by projecting onto the "correct" final subspaces and uncomputing. The resulting $δ$-dependent condition on initial Gram matrices, compared to the original algorithm's condition on final Gram matrices, allows deriving the tightened prefactor.

arXiv:2303.10244•Mar 17, 2023•0 citations

Lower Bounds for Preprocessing Attacks on Quantum Cryptography

Fangqi Dong, Alex Lombardi

We prove near-optimal lower bounds for preprocessing attacks on quantum cryptography in the random oracle model. Specifically, we show that a $T$-query adversary with $S$ qubits of non-uniform advice can recover a random key $k$ from the $n$-qubit binary phase state $|ψ_k\rangle \propto \sum_{x} R(k,x) |x\rangle$ with probability at most $O(\frac{T^2 + \sqrt{ST}}{N})$ for $N=2^n$. In contrast, the best known bound for post-quantum one-way functions is $O(\frac{T^2 + ST}{N})$, with a trivial attack at $S = N$. This demonstrates a new advantage of quantum cryptography over classical cryptography: $n$ qubits of communication suffice for security against preprocessing attacks with space up to $N^2$ rather than $N$. Our methodology is simple: express the optimal preprocessing attack as the operator norm of a random matrix, and bound this value in expectation over the random oracle via the trace-moment method. These trace moments have a natural interpretation using compressed oracles [Zhandry, Crypto 2019], which we then analyze. This can be viewed as a simplification and generalization of the approach of Liu [Eurocrypt 2023] for proving the security of post-quantum cryptography against preprocessing attacks. We also prove the following results: (1) We tighten Liu's analysis of post-quantum PRGs in QROM, achieving a distinguishing advantage bound of $O(\frac{T^2}N + \sqrt{\frac{ST}N})$. (2) For unitary synthesis, we extend the one-query lower bound of Lombardi-Ma-Wright [STOC 2024] to hold against adversaries that can make one arbitrary function query along with polynomially many (adaptive) queries to the random oracle, either before or after the function query. This also interprets the original LMW24 result in terms of compressed oracles. (3) Finally, we prove a tight $O(\frac{\sqrt{S}}N)$ bound for the pseudorandomness of random binary phase states against space $S$ distinguishers.

quant-pharXiv:2610.02101•Oct 2, 2026•0 citations

Variations on Quantum Adversary

Aleksandrs Belovs

The (negative-weighted) quantum adversary bound is a tight characterisation of the quantum query complexity for any partial function. We analyse the extent to which this bound can be generalised. Ambainis et al. [arXiv:1012.2112] and Lee et al. [arXiv:1011.3020] generalised this bound to the state generation and state conversion problems, respectively. Using the ideas by Lee et al., we get even further generalisations of the bound. We obtain a version of the bound for general input oracles, which are just arbitrary unitaries. We also generalise the bound to the problem of implementing arbitrary unitary transformations. Similarly to the bound by Lee et al., our bound is a lower bound for exact transformation and an upper bound for approximate transformation. This version of the bound possesses the tight composition property. Using this construction, we also obtain lower bounds on the quantum query complexity of functions and relations with general input oracles.

arXiv:1504.06943•Apr 27, 2015•11 citations

Tight bounds for hybrid quantum-classical query algorithms

Andris Ambainis, András Gilyén, Martins Kokainis

We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most $q$ oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision $ε$ requires $Θ(\frac{1}{q ε^2})$ queries; (ii) (Unbiased) Amplitude estimation up to a precision $ε$ requires $Θ(\frac{p(1-p)}{q ε^2})$ queries; (iii) Search among $N$ items requires $Θ(\frac{N}{q})$ queries; (iv) Two level AND-OR tree (AND of $m$ ORs, with $n$ inputs to each OR) requires $Θ(\frac{nm}{q})$ queries. All of these bounds are optimal up to a constant factor, for all $q$ from 1, corresponding to the classical complexity, to $Q(f)$, the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.

quant-pharXiv:2610.06803•Oct 6, 2026•0 citations
We study the two-weight decision version of quantum approximate counting: given oracle access to x∈{0,1}Nx\in\{0,1\}^Nx∈{0,1}N, distinguish ∣x∣=M|x|=M∣x∣=M from ∣x∣=M+Δ|x|=M+Δ∣x∣=M+Δ with success probability 1/2+ζ1/2+ζ1/2+ζ. Using the multiplicative adversary method, we prove Ω(max⁡{ζ(N−M)(M+Δ)/Δ,ζN/Δ})Ω\left(\max\left\{ζ\sqrt{(N-M)(M+Δ)}/Δ,\sqrt{ζN/Δ}\right\}\right)Ω(max{ζ(N−M)(M+Δ)​/Δ,ζN/Δ​}). The same parameter dependence follows from the polynomial-method characterization of the two-layer symmetric function by Podder, Yao, and Ye. Our contribution is a multiplicative-adversary derivation that tracks the progress produced by individual oracle queries. For the first term, after complementing the input if necessary, we assume M+Δ≤N−MM+Δ\le N-MM+Δ≤N−M. We use the Hamming-layer subspaces from the eigenspace method of Ambainis, Spalek, and de Wolf and compose their adjacent-layer unitary maps to relate the two nonadjacent promise layers. After fixing the queried coordinate, the analysis block-diagonalizes into four-dimensional subspaces. An exact calculation of the one-query progress ratio gives the first lower bound. The same estimate also implies ∥(I−Π^bad)∣ΨT⟩∥2=O(T2Δ2/((N−M)(M+Δ)))\left\|(I-\widehatΠ_{\mathrm{bad}})\lvertΨ^T\rangle\right\|^2=O\left(T^2Δ^2/((N-M)(M+Δ))\right)​(I−Πbad​)∣ΨT⟩​2=O(T2Δ2/((N−M)(M+Δ))) for the coherent input superposition used in the adversary argument. For the second term, we prove directly using a three-eigenvalue multiplicative adversary that unique OR on nnn bits with success probability 1/2+ζ1/2+ζ1/2+ζ requires Ω(ζn)Ω(\sqrt{ζn})Ω(ζn​) queries, and then reduce unique OR to the two-weight counting problem.
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.
We present a new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function. Adversary methods work as follows: one defines a progress function based on the state of the algorithm, and shows that for a successful algorithm there is a large gap between the initial and final value of the progress, and furthermore that the progress function cannot change by much with a single query. All known variants upper-bound the difference of the progress function, whereas our new variant upper-bounds the ratio and that is why we coin it the multiplicative adversary. Our new method is rooted in the quantum lower-bound method by Ambainis (2005, 2006), based on the analysis of eigenspaces of the density matrix. Ambainis's method is technically very complicated, it lacks intuition, and it only works for symmetric functions. Our method fits well into the adversary framework, has a simple formulation in terms of common block-diagonalization of two operators, and works for all functions. Furthermore, we prove an unconditional strong direct product theorem for the multiplicative quantum adversary bound.
One Weird Trick Tightens the Quantum Adversary Bound, Especially for Success Probability Close to 1/21/21/2
The textbook adversary bound for function evaluation states that to evaluate a function f ⁣:D→Cf\colon D\to Cf:D→C with success probability 12+δ\frac{1}{2}+δ21​+δ in the quantum query model, one needs at least (2δ−1−4δ2)Adv(f)\left( 2δ-\sqrt{1-4δ^2} \right) Adv(f)(2δ−1−4δ2​)Adv(f) queries, where Adv(f)Adv(f)Adv(f) is the optimal value of a certain optimization problem. For δ≪1δ\ll 1δ≪1, this only allows for a bound of Θ(δ2Adv(f))Θ\left(δ^2 Adv(f)\right)Θ(δ2Adv(f)) even after a repetition-and-majority-voting argument. In contrast, the polynomial method can sometimes prove a bound that doesn't converge to 000 as δ→0δ\to 0δ→0. We improve the δδδ-dependent prefactor and achieve a bound of 2δAdv(f)2δAdv(f)2δAdv(f). The proof idea is to "turn the output condition into an input condition": From an algorithm that transforms perfectly input-independent initial to imperfectly distinguishable final states, we construct one that transforms imperfectly input-independent initial to perfectly distinguishable final states in the same number of queries by projecting onto the "correct" final subspaces and uncomputing. The resulting δδδ-dependent condition on initial Gram matrices, compared to the original algorithm's condition on final Gram matrices, allows deriving the tightened prefactor.
We prove near-optimal lower bounds for preprocessing attacks on quantum cryptography in the random oracle model. Specifically, we show that a TTT-query adversary with SSS qubits of non-uniform advice can recover a random key kkk from the nnn-qubit binary phase state ∣ψk⟩∝∑xR(k,x)∣x⟩|ψ_k\rangle \propto \sum_{x} R(k,x) |x\rangle∣ψk​⟩∝∑x​R(k,x)∣x⟩ with probability at most O(T2+STN)O(\frac{T^2 + \sqrt{ST}}{N})O(NT2+ST​​) for N=2nN=2^nN=2n. In contrast, the best known bound for post-quantum one-way functions is O(T2+STN)O(\frac{T^2 + ST}{N})O(NT2+ST​), with a trivial attack at S=NS = NS=N. This demonstrates a new advantage of quantum cryptography over classical cryptography: nnn qubits of communication suffice for security against preprocessing attacks with space up to N2N^2N2 rather than NNN. Our methodology is simple: express the optimal preprocessing attack as the operator norm of a random matrix, and bound this value in expectation over the random oracle via the trace-moment method. These trace moments have a natural interpretation using compressed oracles [Zhandry, Crypto 2019], which we then analyze. This can be viewed as a simplification and generalization of the approach of Liu [Eurocrypt 2023] for proving the security of post-quantum cryptography against preprocessing attacks. We also prove the following results: (1) We tighten Liu's analysis of post-quantum PRGs in QROM, achieving a distinguishing advantage bound of O(T2N+STN)O(\frac{T^2}N + \sqrt{\frac{ST}N})O(NT2​+NST​​). (2) For unitary synthesis, we extend the one-query lower bound of Lombardi-Ma-Wright [STOC 2024] to hold against adversaries that can make one arbitrary function query along with polynomially many (adaptive) queries to the random oracle, either before or after the function query. This also interprets the original LMW24 result in terms of compressed oracles. (3) Finally, we prove a tight O(SN)O(\frac{\sqrt{S}}N)O(NS​​) bound for the pseudorandomness of random binary phase states against space SSS distinguishers.
The (negative-weighted) quantum adversary bound is a tight characterisation of the quantum query complexity for any partial function. We analyse the extent to which this bound can be generalised. Ambainis et al. [arXiv:1012.2112] and Lee et al. [arXiv:1011.3020] generalised this bound to the state generation and state conversion problems, respectively. Using the ideas by Lee et al., we get even further generalisations of the bound. We obtain a version of the bound for general input oracles, which are just arbitrary unitaries. We also generalise the bound to the problem of implementing arbitrary unitary transformations. Similarly to the bound by Lee et al., our bound is a lower bound for exact transformation and an upper bound for approximate transformation. This version of the bound possesses the tight composition property. Using this construction, we also obtain lower bounds on the quantum query complexity of functions and relations with general input oracles.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most qqq oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision εεε requires Θ(1qε2)Θ(\frac{1}{q ε^2})Θ(qε21​) queries; (ii) (Unbiased) Amplitude estimation up to a precision εεε requires Θ(p(1−p)qε2)Θ(\frac{p(1-p)}{q ε^2})Θ(qε2p(1−p)​) queries; (iii) Search among NNN items requires Θ(Nq)Θ(\frac{N}{q})Θ(qN​) queries; (iv) Two level AND-OR tree (AND of mmm ORs, with nnn inputs to each OR) requires Θ(nmq)Θ(\frac{nm}{q})Θ(qnm​) queries. All of these bounds are optimal up to a constant factor, for all qqq from 1, corresponding to the classical complexity, to Q(f)Q(f)Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.