Quantum Query & Sample Algorithms

• Quantum query and sample algorithms are quantum computational methods that use oracular queries and quantum samples to efficiently extract and test properties of unknown systems.
• They employ techniques such as amplitude amplification, Fourier sampling, and quantum singular value transformation to achieve provable quadratic speedups over classical methods.
• Recent advances, including the sample-to-query lifting theorem, illustrate optimal resource trade-offs and have significant implications for quantum simulation, cryptography, and machine learning.

Quantum query and sample algorithms form the algorithmic backbone of modern quantum information theory and computation. These algorithms provide procedures for learning, discrimination, sampling, property testing, and state transformation using access models that are fundamentally quantum, leveraging oracular queries or providing quantum samples for various black-box or distributional tasks. The interplay between query complexity, sample complexity, and algorithmic efficiency underlies much of the rigorous separation between quantum and classical computational models. Below, main developments and paradigms spanning quantum property testing, sampling, learning, lower bounds, and state transformation are reviewed with an emphasis on formal definitions, model-specific techniques, and resource-optimal constructions.

1. Foundations and Paradigms of Quantum Query and Sample Algorithms

Quantum query algorithms are defined in terms of black-box (oracle) access to structures such as Boolean functions, distributions, or matrix operators. Given access to an oracle (e.g., unitary $U_f$, block-encoding of an operator, or sampling from a quantum state), the task is to estimate, learn, or decide properties of an unknown object using as few queries or sample states as possible.

Quantum sample algorithms model settings where only copies of a quantum state (e.g. $|\psi\rangle$ or $\rho$) drawn from some unknown or partially known distribution are available, and the goal is to extract information (such as fidelity, support, or statistical moments) by quantum measurement strategies. Underlying these models are precise definitions of query complexity (minimum number of oracular queries needed) and sample complexity (minimum number of state samples required).

Key operational paradigms include:

• Property testing: Given query/sample access, decide whether an object possesses some property or is far from having it.
• Learning and identification: Infer the relevant structure (e.g. which variables a function depends on, as in k-junta learning (0707.3479)) or explicitly reconstruct hidden parameters (e.g. period finding, parity learning, or state identities).
• State and distributional sampling: Prepare or sample from target distributions (e.g., Gibbs, log-concave, or combinatorial distributions) either exactly or approximately via quantum routines.

These models often admit sharp quantum-classical separations in resource requirements, and lower bounds are established via polynomial method, adversary method, or lifting theorems.

2. Resource Separations and Lower Bounds: Sample-to-Query Lifting

A central recent advance is the quantum sample-to-query lifting theorem (Wang et al., 2023), which formalizes the relationship between quantum sample complexity (denoted $\mathcal{S}(\mathcal{P})$ for a testing problem $\mathcal{P}$) and quantum query complexity (denoted $\mathcal{Q}_\diamond(\mathcal{P})$) in block-encoding/unitary settings. The main result states: $$\mathcal{Q}_\diamond(\mathcal{P}) = \tilde{\mathcal{O}}(\sqrt{\mathcal{S}(\mathcal{P})}),$$ with the converse quadratic relation $$\mathcal{S}(\mathcal{P}) = \tilde{\mathcal{O}}(\mathcal{Q}_\diamond(\mathcal{P})^2)$$.

This lifting is achieved by showing that a $Q$-query quantum property testing algorithm (in the block-encoding model) can be simulated by roughly $Q^2$ samples of the underlying quantum state, via quantum singular value transformation or related block-encoding techniques.

This theorem provides a unifying lower bound technique. For instance, in quantum state discrimination (distinguishing $\rho$ vs. $\sigma$ where infidelity is $\gamma$), the Helstrom–Holevo bound shows at least $1/\gamma$ samples are required; thus, by lifting, at least $\tilde{\Omega}(1/\sqrt{\gamma})$ queries are required in the unitary model. The quadratic gap is proven to be optimal for a wide range of testing problems in this framework.

3. Quantum Property Testing and Distributional Tasks

Quantum algorithms for property testing often exploit resource-efficient amplitude amplification and estimation procedures or sampling from the Fourier spectrum (Fourier sampling). Notable results include:

• k-junta testing: Identification of Boolean functions depending on $k$ out of $n$ inputs can be done in $O(k/\epsilon)$ queries using a Fourier sampling oracle (FS), independent of $n$ (0707.3479).
• Distribution testing: L1-distance estimation between two distributions via quantum amplitude estimation achieves $O(N^{1/2})$ query complexity, while uniformity and orthogonality testing achieve $O(N^{1/3})$ queries (0907.3920).
• Polynomial separation: These quantum algorithms yield quadratic or cubic-root improvements over the best known classical query complexities, a fact that applies more generally as established by the sample-to-query lifting theorem.

In classical settings, such tasks require many more queries or samples—often linear or nearly so in the domain size.

4. Quantum Sampling, State Preparation, and Approximation

Modern quantum algorithms provide innovative primitives for state preparation and sampling:

• Quantum rejection sampling: Generalizes classical rejection sampling, allowing conversion from an initial superposition $|\psi\rangle = \sum_k T_k |k\rangle$ to a target with amplitudes $E_k$, requiring $O(1/\|E\|^2)$ queries to an oracle for state preparation (Ozols et al., 2011). This method underlies accelerated quantum procedures for linear system solving (HHL algorithm), Metropolis sampling, and the Boolean hidden shift problem.
• Quantum sampling for log-concave distributions: Algorithms that simulate underdamped Langevin diffusion using quantum evaluation (rather than gradient) oracles enable sample generation and partition function estimation with quantum query complexity matching gradient-based classical algorithms but using only zeroth-order queries; key results include achieving $\tilde{O}(1/\epsilon)$ query complexity for partition function estimation—a quadratic improvement over the classical $\tilde{O}(1/\epsilon^2)$ (Childs et al., 2022).
• Quantum algorithms for distributed data: Even when data is partitioned among multiple machines and each allows only extremely limited (counting) oracles, quantum sampling matching the form $$|\psi\rangle = (1/\sqrt{M}) \sum_i \sqrt{c_i}|i\rangle$$ can be performed with optimal query complexity $O(n\sqrt{\nu N/M})$ sequentially, or $O(\sqrt{\nu N/M})$ in parallel, matching lower bounds even in the oblivious communication model (Chen et al., 9 Jun 2025).
• Quantum D$^2$-sampling: Quantum routines can efficiently implement $D^2$-sampling crucial for $k$-means++ seeding and clustering. Using QRAM or sample-query access, the algorithm can run in $\tilde{O}(\zeta^2 k^2)$ time (with $\zeta$ the aspect ratio), enabling quantum-inspired algorithms (QI-$k$-means++) with robust classical performance (Shah et al., 22 May 2024).

5. Quantum Statistical Query Learning and Sample-Efficient Learning Algorithms

Quantum statistical query (QSQ) learning (Arunachalam et al., 2020) extends the classical SQ model, allowing a classical learner to query expectation values of general observables on quantum example states $|\psi_c\rangle$. This model allows efficient learning of classically hard concept classes (e.g., parity, O(log n)-junta, poly-size DNF), using $O(n)$ or poly$(n,1/\tau)$ queries, where each query requires only the evaluation of a measurement observable up to tolerance $\tau$.

The complexity of QSQ learning is described via weak statistical query dimension (WSQDIM), with sample complexity lower bounded by $\Omega(\log(\text{WSQDIM}(C)))$. Notably, QSQ learnability implies private quantum PAC learnability, and for differentially private learners, classical and quantum sample complexities coincide up to constants.

QSQ algorithms are also practical for near-term quantum devices, given their minimal quantum requirements.

6. State Transformation: Uhlmann Transformation and Information-Theoretic Tasks

The Uhlmann transformation is a central operation relating the purifications of two quantum states and maximizing their overlap subject to locality constraints. The most general quantum algorithms for implementing the Uhlmann transformation (Utsumi et al., 3 Sep 2025) use block-encoding, quantum singular value transformation (QSVT), and density matrix exponentiation.

• Purified query model: With unitaries preparing $|\rho\rangle$ and $|\sigma\rangle$, one can realize the Uhlmann transformation with query complexity $u = O(\min\{1/s_{\min}, r/\delta\}\log(1/\delta))$ where $s_{\min}$ is the minimal singular value of $\sqrt{\sigma}\sqrt{\rho}$ and $r$ its rank.
• Purified and mixed sample models: Using samples (copies) of $|\rho\rangle$, $|\sigma\rangle$, or $\rho$, $\sigma$, complexities depend on $1/s_{\min}^2$ and the desired accuracy, but are exponential improvements over naive tomography.
• Applications: Square root fidelity estimation is achieved with quartic improvement over previous known methods; efficient decoders for entanglement transmission, quantum state merging, and algorithmic Petz recovery map construction all benefit from the Uhlmann subroutine, with complexities dimension-independent or only polylogarithmic in the Hilbert space size.

These algorithms enable robust, efficient composition of high-level quantum information processing tasks.

7. Unified Frameworks and Algorithmic Optimization

Recent work (Te'eni et al., 23 Sep 2024) unifies quantum query algorithms with information-theoretic and communication frameworks. A quantum query algorithm can be interpreted as a communication protocol for transferring information about the hidden property of the oracle. The mutual information between the oracle's hidden property and the quantum register output is maximized by optimal pre- and post-query unitaries, with the optimum expressed in terms of the Holevo quantity minus quantum discord.

Optimal non-adaptive single- and multi-query algorithms can thus be characterized by maximizing the extracted mutual information, connecting quantum query algorithm design to optimization of accessible information in quantum communication settings.

Case studies—Deutsch–Jozsa, Bernstein–Vazirani, Hidden Subgroup Problem algorithms—illustrate that quantum query algorithms achieving maximal mutual information also "diagonalize" quantum discord, systematically extracting all available classical information.

Table: Representative Quantum Query and Sample Complexities

Task	Quantum Complexity	Classical Complexity
State discrimination ($\gamma$ inf.)	$\tilde{O}(1/\sqrt\gamma)$	$\Omega(1/\gamma)$
Gibbs sampling (inverse temp. $\beta$)	$\tilde{O}(\beta)$	[Lower bound matches]
k-junta testing ($k$ variables)	$O(k/\epsilon)$	$O((k\log k)^2/\epsilon)$
L1 distance between distributions ($N$)	$O(N^{1/2})$	$\Omega(N)$
Log-concave sampling ($\epsilon$)	$\tilde{O}(1/\epsilon)$	$\tilde{O}(1/\epsilon^2)$

All these results reflect provable, sometimes optimal, quantum speedups in key sampling, testing, learning, and property estimation tasks.

Conclusion

Quantum query and sample algorithms, as systematically developed across recent research, now constitute a principled framework for the efficient extraction, testing, and transformation of information from quantum or classical oracles and data sources. Theoretical advances—including the sample-to-query lifting theorem, optimality in distributed and property testing settings, resource-dimension independent state transformation, and performance-optimizing formulations—place these algorithms at the heart of quantum computing theory, quantum learning, and quantum information processing. Their application areas span quantum machine learning, cryptography, quantum simulation, error correction, and foundational quantum mechanics.