The Quantum Oracle Classification (QOC) problem is to classify a function, given only quantum black box access, into one of several classes without necessarily determining the entire function. Generally, QOC captures a very wide range of problems in quantum query complexity. However, relatively little is known about many of these problems. In this work, we analyze the a subclass of the QOC problems where there is a group structure. That is, suppose the range of the unknown function A is a commutative group G, which induces a commutative group law over the entire function space. Then we consider the case where A is drawn uniformly at random from some subgroup A of the function space. Moreover, there is a homomorpism f on A, and the goal is to determine f(A). This class of problems is very general, and covers several interesting cases, such as oracle evaluation; polynomial interpolation, evaluation, and extrapolation; and parity. These problems are important in the study of message authentication codes in the quantum setting, and may have other applications. We exactly characterize the quantum query complexity of every instance of QOC with group structure in terms of a particular counting problem. That is, we provide an algorithm for this general class of problems whose success probability is determined by the solution to the counting problem, and prove its exact optimality. Unfortunately, solving this counting problem in general is a non-trivial task, and we resort to analyzing special cases. Our bounds unify some existing results, such as the existing oracle evaluation and parity bounds. In the case of polynomial interpolation and evaluation, our bounds give new results for secret sharing and information theoretic message authentication codes in the quantum setting.
The Quantum Oracle Classification (QOC) problem is to classify a function, given only quantum black box access, into one of several classes without necessarily determining the entire function. Generally, QOC captures a very wide range of problems in quantum query complexity. However, relatively little is known about many of these problems. In this work, we analyze the a subclass of the QOC problems where there is a group structure. That is, suppose the range of the unknown function A is a commutative group G, which induces a commutative group law over the entire function space. Then we consider the case where A is drawn uniformly at random from some subgroup A of the function space. Moreover, there is a homomorpism f on A, and the goal is to determine f(A). This class of problems is very general, and covers several interesting cases, such as oracle evaluation; polynomial interpolation, evaluation, and extrapolation; and parity. These problems are important in the study of message authentication codes in the quantum setting, and may have other applications. We exactly characterize the quantum query complexity of every instance of QOC with group structure in terms of a particular counting problem. That is, we provide an algorithm for this general class of problems whose success probability is determined by the solution to the counting problem, and prove its exact optimality. Unfortunately, solving this counting problem in general is a non-trivial task, and we resort to analyzing special cases. Our bounds unify some existing results, such as the existing oracle evaluation and parity bounds. In the case of polynomial interpolation and evaluation, our bounds give new results for secret sharing and information theoretic message authentication codes in the quantum setting.
We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t -query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn . More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G , we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t -query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.
A central challenge in quantum algorithms and cryptography is reasoning about algorithms with oracle access to a random group element (e.g. a random function, permutation, or unitary). Can we efficiently simulate such algorithms? Can we determine what they know after t queries? A classical tool for this is lazy sampling: the oracle does not commit to the full group element upfront, but rather samples partial information about it on the fly. We study a quantum analog of lazy sampling: compressed oracles (or recording oracles). These are quantum data structures that allow on-the-fly simulation for quantum queries, originally introduced by Zhandry (CRYPTO '19) for random functions, and generalized to unitaries by Ma-Huang (STOC '25) and permutations by Carolan (STOC '26), and used to great effect in security proofs and lower bounds due to their interpretability.
We define and analyze a general-purpose and interpretable path-recording oracle, derived from first principles, that perfectly simulates random elements of any closed subgroup of U(N). Our oracle stores, in superposition, t input-output pairs, with updates described in terms of the commutant of the group's tensor power representation. This transparently records the information the algorithm has learned. Our oracle builds on recent work of Grinko-Yoshida (QIP '26), who gave a different general-purpose compressed oracle without clear interpretability.
One interesting application of our path-recording is allowing direct comparisons between compressed oracles of different groups, giving a new technique for proving pseudorandomness results. For example, comparing SN and U(N) yields what is arguably the simplest construction to date of pseudorandom unitaries: the product PC of a pseudorandom permutation and a random Clifford, improving on the prior PFC construction (Metger-Poremba-Sinha-Yuen, FOCS '24; Ma-Huang, STOC '25).
We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms.
In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. For finite abelian groups, existing Fourier-sampling algorithms use O(log(∣G∣)/ε) copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using O(log(∣G/H∣)/ε) forward and inverse queries, and prove a matching Ω(log(∣G/H∣)/ε) lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, Θ(log(∣G/H∣)/ε) copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in ε genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.
Motivated by the quantum algorithm in \cite{MN05} for testing commutativity of black-box groups, we study the following problem: Given a black-box finite ring R=∠r1,...,rk where {r1,r2,...,rk} is an additive generating set for R and a multilinear polynomial f(x1,...,xm) over R also accessed as a black-box function f:Rm→R (where we allow the indeterminates x1,...,xm to be commuting or noncommuting), we study the problem of testing if f is an \emph{identity} for the ring R. More precisely, the problem is to test if f(a1,a2,...,am)=0 for all ai∈R. We give a quantum algorithm with query complexity O(m(1+α)m/2km+1m) assuming k≥(1+1/α)m+1. Towards a lower bound, we also discuss a reduction from a version of m-collision to this problem. We also observe a randomized test with query complexity 4mmk and constant success probability and a deterministic test with km query complexity.
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 p1,…,pm and a gap parameter Δ, the goal is to estimate the fraction ρ of \emph{positive} distributions (pi≥1/2+Δ) to within additive error ε. We prove an upper bound of O~(Δερ+Δε1) queries, achieving a quadratic speedup over the classical sample complexity. % of 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)) 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 O~(ερ+ε1) and Ω(ερ), thereby characterizing the query complexity of quantum counting with bounded-error oracles.