We show how to use parallelization to speed up sampling from an arbitrary distribution µ on a product space [q]n, given oracle access to counting queries: ℙX∼ µ[XS=σS] for any S⊆ [n] and σS ∈ [q]S. Our algorithm takes O(n2/3· polylog(n,q)) parallel time, to the best of our knowledge, the first sublinear in n runtime for arbitrary distributions. Our results have implications for sampling in autoregressive models. Our algorithm directly works with an equivalent oracle that answers conditional marginal queries ℙX∼ µ[Xi=σi | XS=σS], whose role is played by a trained neural network in autoregressive models. This suggests a roughly n1/3-factor speedup is possible for sampling in any-order autoregressive models. We complement our positive result by showing a lower bound of Ω(n1/3) for the runtime of any parallel sampling algorithm making at most poly(n) queries to the counting oracle, even for q=2.
We show how to use parallelization to speed up sampling from an arbitrary distribution µ on a product space [q]n, given oracle access to counting queries: ℙX∼ µ[XS=σS] for any S⊆ [n] and σS ∈ [q]S. Our algorithm takes O(n2/3· polylog(n,q)) parallel time, to the best of our knowledge, the first sublinear in n runtime for arbitrary distributions. Our results have implications for sampling in autoregressive models. Our algorithm directly works with an equivalent oracle that answers conditional marginal queries ℙX∼ µ[Xi=σi | XS=σS], whose role is played by a trained neural network in autoregressive models. This suggests a roughly n1/3-factor speedup is possible for sampling in any-order autoregressive models. We complement our positive result by showing a lower bound of Ω(n1/3) for the runtime of any parallel sampling algorithm making at most poly(n) queries to the counting oracle, even for q=2.
The computational equivalence between approximate counting and sampling is well established for polynomial-time algorithms. The most efficient general reduction from counting to sampling is based on simulated annealing. In this approach, the counting problem is formulated as estimating the ratio Q = Z(βmax)/Z(βmin) between partition functions Z(β) = Σx∈Ω exp(βH(x)) of Gibbs distributions μβ over Ω with Hamiltonian H, given access to a sampling oracle for μβ at any β ∈ [βmin, βmax]. The sample complexity (measured by the number of oracle calls) is typically expressed in terms of q and h, which respectively bound ln Q and H. The best upper bound achieved by known annealing algorithms with relative error ε is O(qε−2 log h). However, all known algorithms attaining this near-optimal complexity are inherently sequential, or adaptive: the queried parameters β depend on previous samples.
Aaronson and Ambainis (Theory of Computing, 2014) conjectured that quantum query algorithms admit efficient almost-everywhere classical simulation: for any T-query quantum algorithm, its acceptance probability can be approximated on a (1−δ) fraction of inputs, up to ε additive error, using poly(T,1/ε,1/δ) classical queries. At a high level, the conjecture suggests that exponential quantum speedups are possible only on sufficiently structured inputs.
In this work, we make progress on this conjecture by proving it for quantum algorithms that make massively parallel quantum queries. In contrast, Yamakawa and Zhandry (Journal of the ACM, 2024) showed that quantum algorithms restricted to parallel queries can still achieve exponential speedups over classical algorithms for sampling problems. We establish our simulation theorem by proving the stronger statement that parallel-query quantum algorithms cannot distinguish the uniform distribution over oracles from oracles drawn from so-called "dense distributions". Our main technical contribution is a coupling theorem that relates the uniform distribution over oracles to oracles drawn from dense distributions.
We further extend this approach beyond the purely parallel setting. We prove simulation theorems for quantum algorithms with constant rounds of adaptivity, and for quantum algorithms with a bounded quantum-query prefix followed by a massively parallel quantum-query stage.
The computational equivalence between approximate counting and sampling is well established for polynomial-time algorithms. The most efficient general reduction from counting to sampling is achieved via simulated annealing, where the counting problem is formulated in terms of estimating the ratio Q=Z(βmax)/Z(βmin) between partition functions Z(β)=∑x∈Ωexp(βH(x)) of Gibbs distributions μβ over Ω with Hamiltonian H, given access to a sampling oracle that produces samples from μβ for β∈[βmin,βmax]. The best bound achieved by known annealing algorithms with relative error ε is O(qlogh/ε2), where q,h are parameters which respectively bound lnQ and H. However, all known algorithms attaining this near-optimal complexity are inherently sequential, or *adaptive*: the queried parameters β depend on previous samples. We develop a simple non-adaptive algorithm for approximate counting using O(qlog2h/ε2) samples, as well as an algorithm that achieves O(qlogh/ε2) samples with just two rounds of adaptivity, matching the best sample complexity of sequential algorithms. These algorithms naturally give rise to work-efficient parallel (RNC) counting algorithms. We discuss applications to RNC counting algorithms for several classic models, including the anti-ferromagnetic 2-spin, monomer-dimer and ferromagnetic Ising models.
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.
Quantum sampling, a fundamental subroutine in numerous quantum algorithms, involves encoding a given probability distribution in the amplitudes of a pure state. Given the hefty cost of large-scale quantum storage, we initiate the study of quantum sampling in a distributed setting. Specifically, we assume that the data is distributed among multiple machines, and each machine solely maintains a basic oracle that counts the multiplicity of individual elements. Given a quantum sampling task, which is to sample from the joint database, a coordinator can make oracle queries to all machines. We focus on the oblivious communication model, where communications between the coordinator and the machines are predetermined. We present both sequential and parallel algorithms: the sequential algorithm queries the machines sequentially, while the parallel algorithm allows the coordinator to query all machines simultaneously. Furthermore, we prove that both algorithms are optimal in their respective settings.
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) queries to an NP oracle to compute a witness to a given SAT formula, quantumly 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(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(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) NP queries, then BPPNP[o(n)] contains a PNP-complete problem if the algorithm is classical and FBQPNP[o(n)] contains an FPNP-complete problem if the algorithm is quantum.