State-Preparation Oracles

• State-preparation oracles are computational tools that provide identical copies of unknown quantum states for property testing and statistical inference.
• The sample-to-query lifting theorem reveals a quadratic relationship between sample and query complexities, ensuring tight bounds in quantum property testing.
• Implementations via block encoding, purified queries, and Pauli sampling enable advanced quantum algorithms and rigorous complexity analysis.

A state-preparation oracle, also known as a sample-preparation oracle, is a computational model in quantum information theory that provides access to identical copies of an unknown quantum state for the purposes of property testing, statistical inference, or algorithmic evaluation. Such oracles are central to quantum property testing, enabling algorithms to distinguish, certify, or estimate properties of states or distributions by analyzing prepared samples. Recent advances in sample-to-query lifting have clarified the relationships between state-preparation and purified query oracles, the associated complexity classes, and the optimality of bounds for a wide class of quantum property testing problems.

1. Formal Models: Sample-Preparation and Purified Query Oracles

A sample-preparation oracle exposes an unknown mixed quantum state $\rho \in \mathbb{C}^{d \times d}$ in the form of identical, uncoupled copies $\rho^{\otimes n}$, without further structured quantum access. The associated sample complexity $S(\mathcal{P})$ is the smallest $n$ for which an $n$-copy POVM-based tester solves a given promise problem $\mathcal{P}$ (partition of density operators) with error at most $1/3$ (Chen et al., 1 Dec 2025).

Purified quantum query oracles generalize this model: given a unitary $U$ on systems $\mathcal{A} \otimes \mathcal{B}$,

$$U |0\rangle_{\mathcal{AB}} = \sum_{i=1}^d \sqrt{\lambda_i} |\psi_i\rangle_{\mathcal{A}} |i\rangle_{\mathcal{B}},$$

where $$\rho = \operatorname{Tr}_{\mathcal{B}}(U |0\rangle\langle 0| U^\dagger) = \sum_i \lambda_i |\psi_i\rangle \langle \psi_i|$$. Query complexity $Q(\mathcal{P})$ is the minimal number of queries to $U$, $U^\dagger$ needed to decide $\mathcal{P}$ with comparable error.

2. Quantum Sample-to-Query Lifting: Theorems and Optimal Relations

The quantum sample-to-query lifting theorem establishes a generic lower bound connecting the sample complexity $S(\mathcal{P})$ and the query complexity $Q(\mathcal{P})$ for state-preparation oracles. Wang and Zhang showed that for any mixed-state promise problem, under block-encoding or purified query access,

$$Q(\mathcal{P}) = \Omega(\sqrt{S(\mathcal{P})}) \tag{TWZ25\,lift}$$

This result is tight up to absolute constants and holds without polylogarithmic overhead for purified access (Chen et al., 1 Dec 2025).

The converse "samplizer" statement provides an upper bound on sample complexity using query algorithms: $S(\mathcal{P}) = O(Q(\mathcal{P})^2)$ This relationship induces a quadratic gap between single-copy sample preparation and query-based property testing. The tightness of this relation is shown for quantum state discrimination, amplitude estimation, Hamiltonian simulation, and Gibbs sampling, among others (Wang et al., 2023).

3. Definitions and Realizations of State-Preparation Oracles

State-preparation oracles interface with quantum algorithms in several forms:

Sample-preparation (uncoupled copies): Direct access to $\rho^{\otimes n}$, as assumed in sample complexity studies (Chen et al., 1 Dec 2025). Block-encoding oracles: A unitary $U$ that is an $(\alpha, m, \epsilon)$-block encoding of an operator $A$ satisfies

$$\left\| A - \alpha (\langle 0^m| \otimes I) U (|0^m\rangle \otimes I) \right\| \leq \epsilon$$

and allows indirect sample and query access to matrix rows/columns or state amplitudes (Murça et al., 2 Dec 2024). Purified query access: Oracle access to a unitary $U$ with a unique ground state encoding $\rho$, relevant for practical quantum algorithms (phase estimation, quantum singular value transformation). Pauli sampling oracles: By Pauli decomposition, $$\rho = \sum_{P} \operatorname{Tr}(P \rho)/\sqrt{2^n} \cdot P$$, and sampling from the distribution $$\pi_\rho(i) \propto (\operatorname{Tr}(P_i \rho))^2$$ realizes sample-preparation in the Pauli basis—useful for distributed inner product estimation and non-stabilizer state benchmarking (Murça et al., 2 Dec 2024).

4. Complexity Bounds via Lifting: Property Testing Applications

State-preparation oracles admit a uniform methodology for establishing lower bounds on decision problems, via sample-to-query lifting. Table 1 summarizes representative bounds (all from (Chen et al., 1 Dec 2025)); for each problem, tight lower and upper bounds are derived according to sample and query access.

Problem	Sample Complexity $S$	Query Complexity $Q$
Mixedness ($\rho = I/d$)	$\Omega(d/\epsilon^2)$	$\Omega(\sqrt{d}/\epsilon)$
Rank Testing ($\le r$)	$\Omega(r/\epsilon)$	$\Omega(\sqrt{r}/\sqrt{\epsilon})$
Amplitude Estimation	$\Omega(1/\epsilon^2)$	$\Omega(1/\epsilon)$
Gibbs Sampling (temp $\beta$)	$\Omega(1/\beta^2)$	$\Omega(\beta)$
vN Entropy Estimation	$\Omega(d/\epsilon + \log^2 d/\epsilon^2)$	$\Omega(\sqrt{d}/\sqrt{\epsilon} + \ln d/\epsilon)$

The sample-to-query lifting paradigm is broadly applicable to distribution testing, quantum state spectrum testing, entanglement entropy, fidelity estimation, quantum entropy estimation, and more. For instance, the optimal lower bound $\Omega(\beta)$ for Gibbs state preparation persists under block-encoding and square-root Hamiltonian access (Wang et al., 2023, Chen et al., 1 Dec 2025).

5. Algorithmic Frameworks Realizing State-Preparation Oracles

Several quantum algorithmic primitives instantiate effective state-preparation oracles:

• Density-matrix exponentiation and QSVT: Block-encoding $\rho$ via single-copy tomography or QSVT gives quantum access to states and enables sample-query lifting [(Wang et al., 2023), GSLW19].
• Single-copy tomography: For $\ket{\psi} \in \mathbb{C}^{2^n}$, entry estimation and sampling from modulus-squared amplitudes can be performed in time $O(T n \epsilon^{-2})$ and $O(T)$, respectively (Murça et al., 2 Dec 2024).
• Low-$T$ Clifford+$T$ circuits: Classical simulation results yield ASQ access to amplitude vectors for circuit output states with resource overheads $O(2^t \,\mathrm{poly}(n) \epsilon^{-2})$ for $t$ T gates.
• Pauli sampling: Classical-Bell measurement procedures leverage entanglement and stabilizer norm properties to sample Pauli coefficients of $\rho$ in time $\tilde{O}(2^{4\chi}\mathrm{StabNorm}^2\Delta^{-4})$ (Murça et al., 2 Dec 2024).

6. Compositionality and Computational Power

The ASQ model (Approximate Sample and Query access, Editor's term) supports compositional operations essential for quantum algorithm design. Notably,

• Linear combinations: Given ASQ access to $\tau$ vectors $x_1, ..., x_\tau$ and coefficients $\lambda_j$, ASQ access to $u = \sum_j \lambda_j x_j$ can be constructed with controlled failure probability and explicit cost bounds.
• Inner product estimation: Using ASQ access, inner products $x^\dagger y$ can be estimated to polynomial accuracy in time nearly independent of vector dimension, provided favorable norms.
• Distributed protocols: Pauli sampling and symmetric inner-product estimation enable distributed overlap estimation with polynomial improvements in sample and computational complexity, under communication-constrained scenarios (Murça et al., 2 Dec 2024).

The ASQ abstraction thus encapsulates the computational advantages imparted by state-preparation oracles, facilitating polynomial speedups over naive sample-based schemes and enabling rigorous complexity analysis. Full dequantization of Quantum Singular Value Transforms to the quantum ASQ setting remains an open challenge, suggesting further avenues for the characterization of quantum-oracle-enabled computation.

7. Relationship with Other Lower Bound Techniques and Tightness

State-preparation oracles as formalized in sample-to-query lifting stand orthogonal to the polynomial and adversary methods. The sample-to-query paradigm reduces from information-theoretic sample bounds (e.g., Helstrom–Holevo limit) to quantum query bounds, yielding a unified framework for proving matching lower bounds for quantum property testing, phase estimation, amplitude estimation and Hamiltonian simulation (Wang et al., 2023). Theoretical tightness is demonstrated for matrix spectrum testing, entanglement entropy, and quantum distribution testing; nearly all new complexity bounds obtained by lifting are accompanied by matching algorithmic upper bounds (Chen et al., 1 Dec 2025).

A plausible implication is that state-preparation oracles and their lifting framework will remain fundamental tools guiding oracle model research, complexity separation, and quantum algorithm engineering for the foreseeable future.