Quantum Sample-to-Query Lifting

• Quantum Sample-to-Query Lifting is an information-theoretic method that connects quantum state sample complexity to query complexity.
• It leverages techniques like Hamiltonian simulation and quantum singular value transformation to establish optimal quadratic relationships between samples and queries.
• The framework unifies polynomial and adversary methods, providing compositional analysis for quantum property testing, distributed tasks, and algorithm subroutines.

Quantum sample-to-query lifting is an information-theoretic technique that establishes quantitative connections between the sample complexity of quantum state testing (i.e., the number of independent copies of a quantum state required) and the query complexity of quantum algorithms with oracle access to unitary representations or block encodings of these states. The framework systematically translates lower bounds from the sample-based setting to the query-based regime, saturates these bounds for discrimination tasks, and unifies previous methods (polynomial, adversary) via reductions rooted in quantum information theory. This paradigm provides optimal bounds for a range of quantum property testing problems and enables compositional analysis of quantum subroutines in contexts such as block encoding, Pauli sampling, and distributed quantum computation.

1. Foundational Principles and Formalisms

Quantum sample-to-query lifting is built upon precise models of access to quantum data:

• Sample complexity (S(ℙ)): Involves performing quantum tests on independent copies of an unknown state ρ to decide a property ℙ with success probability at least 2/3.
• Query complexity (Q_◇(ℙ)): Counts the number of calls to a unitary oracle U (block-encoding or purified oracle models), encoding ρ, needed to decide the corresponding unitary property ℙ^◇ with comparable success probability.
• Access abstraction (ASQ): Recent models formalize the simultaneous ability to sample (draw indices i with probability |x(i)|²/||x||²) and query vector amplitudes or matrix elements up to a specified error, permitting compositional computation (such as linear combinations, inner products) with controlled overhead factors (Murça et al., 2 Dec 2024).

The central result, the quantum sample-to-query lifting theorem, is:

$$Q_◇(ℙ) = \widetilde{\Omega}\left(\sqrt{S(ℙ)}\right), \qquad S(ℙ) = \widetilde{O}\left(Q_◇(ℙ)^2\right)$$

where tilde notation hides polylogarithmic factors. This quadratic relationship is optimal and holds for both standard block-encoding input models and for purified oracle access, and is established via reductions using sample-based Hamiltonian simulation and quantum singular value transformation (QSVT) (Wang et al., 2023, Chen et al., 1 Dec 2025).

2. Technical Derivation and Proof Structure

The lifting theorem is rooted in the capacity to simulate Q-query algorithms (unitary oracle model) using approximately $\widetilde O(Q^2)$ samples, leveraging sample-based Hamiltonian simulation and QSVT protocols. For any query, block-encoding constructions allow controlled approximation using multiple state copies, with overall quadratic cost overhead.

Information-theoretic lower bounds (e.g., Helstrom–Holevo bounds for quantum state discrimination) translate directly via this framework: any problem that is hard in the sample-based instance remains hard under oracle model, up to the quadratic relationship. Key proof techniques include:

• Density-matrix exponentiation: Approximates $e^{-i\rho t}$ using multiple applications of swap operations, allowing simulation of purified oracles (Chen et al., 1 Dec 2025).
• Reductions from discrimination: Constructing state pairs or eigenvalue spectra with trace-distance or entropy gaps forces sample lower bounds; applying lifting then dictates query lower bounds.

3. Complexity Bounds in Quantum Property Testing

Quantum sample-to-query lifting has led to comprehensive listings of complexity bounds, with 49 property testing results cataloged (41 new, 18 optimal) (Chen et al., 1 Dec 2025). These include lower and upper bounds for both classical distribution testing (hypothesis testing, $\ell₁/\ell₂$-closeness, entropy estimation) and quantum state tasks (mixedness, rank, entanglement, fidelity), as well as generalizations to matrix spectrum testing and quantum algorithms (Gibbs sampling, time evolution). Examples:

Property Testing Problem	Sample Lower Bound S(·)	Query Lower Bound Q(·)
$\ell_2$-closeness	$\Omega(1/\epsilon^2)$	$\Omega(1/\epsilon)$
Mixedness (state vs. maximally mixed)	$\Omega(d/\epsilon^2)$	$\Omega(\sqrt{d}/\epsilon)$
Shannon entropy	$\Omega(d/(\epsilon \log d))$	$\Omega(\sqrt{d}/\epsilon)$
Quantum Gibbs sampling ($\beta$)	-	$\Omega(\beta)$
Entanglement entropy (gap $\Delta$)	-	$\Omega(1/\sqrt{\Delta})$
Rank testing	$\Omega(r/\epsilon)$	$\Omega(\sqrt{r}/\epsilon^{1/2})$

The quadratic sample-to-query gap is saturated by discrimination problems such as $Dis_{\rho_+, \rho_-}$ with $S = \Theta(1/\epsilon^2)$ and $Q_◇ = \Theta(1/\epsilon)$ (Wang et al., 2023, Chen et al., 1 Dec 2025).

4. Compositionality, Access Models, and Algorithmic Applications

Recent work extends lifting to compositional models of sample-and-query access (ASQ, ASQ[$\phi$]), formalizing the computational power derived from preparations and measurements of block-encoded states, quantum circuit simulation, or Pauli sampling (Murça et al., 2 Dec 2024). Central results include:

• Composition Lemmas: Linear combinations and nested compositions of ASQ access amplify computational ability, with oversampling factors scaling as $\tau^2$ for trees of size $\tau$.
• Inner Product Estimation: Via ASQ access to vectors/states, one efficiently computes symmetric or asymmetric inner products, outperforming naive tomography approaches.
• Distributed Quantum Tasks: Pauli sampling combined with ASQ and symmetric inner product lemmas yields polynomially improved protocols for distributed inner-product estimation, with precise cost analysis in terms of entanglement and stabilizer norms (Murça et al., 2 Dec 2024).

Concrete upper bounds follow from the "samplizer" principle: any query algorithm using $Q$ queries can be simulated by $O(Q^2)$ samples, translating query-based algorithms into viable sample-based protocols.

5. Relations to Polynomial and Adversary Methods

Sample-to-query lifting provides a unified framework that subsumes the polynomial method and adversary method:

• Polynomial method: Yields lower bounds through polynomial degree arguments in amplitude estimation and Hamiltonian simulation (typically $\Omega(1/\epsilon)$, $\Omega(t)$).
• Adversary method: Applies spectral adversary matrices to track progress, providing bounds in various problems.
• Lifting framework ("lifting theorem," Editor's term): Reduces state-testing lower bounds (Helstrom bounds) to query lower bounds in a direct, information-theoretic manner, matching or improving classical methods in Gibbs sampling, entanglement entropy, amplitude/phase estimation, and Hamiltonian simulation (Wang et al., 2023, Chen et al., 1 Dec 2025).

The lifting approach is optimal in cases such as quadratic sample-to-query gaps and cannot be improved by encoding stronger oracles (e.g., encoding $\sqrt{H}$ for Gibbs sampling leaves the lower bound at $\Omega(\beta)$).

6. Catalog of Problem-Specific Quantum Bounds

Quantum sample-to-query lifting has underpinned the derivation and cataloging of new and matching complexity bounds for testing classical and quantum properties. Notable results include optimal bounds for $\ell_1$ and $\ell_2$ testing, entropy estimation (Shannon, Rényi, Tsallis), rank and mixedness testing, support-size estimation, fidelity and trace distance, entanglement characterization, and quantum algorithm subroutines (amplitude estimation, Hamiltonian simulation, Gibbs sampling). All known classical sample lower bounds feed directly as quantum sample bounds which then lift to query bounds via this method (Chen et al., 1 Dec 2025).

7. Impact, Future Directions, and Unified Perspective

Quantum sample-to-query lifting has unified and resolved complexity characterizations in property testing, provided compositional analysis of quantum measurement and query access, and closed gaps between classical, sample-based, and query-based regimes. It yields black-box conversions from sample testing to query lower bounds and vice versa, and optimally quantifies the costs for a range of distributed, compositional, and nested quantum protocols. As quantum algorithms and data access models continue to evolve, this framework promises further generalizations for hybrid quantum-classical computation, fault-tolerant circuit analysis, and subroutine dequantization (Murça et al., 2 Dec 2024, Chen et al., 1 Dec 2025).