arXiv subjects

Qisheng Wang

Publications and source records attributed to Qisheng Wang.

At least 19 recordsLinked to original sources

arXiv2026

Local Test for Unitarily Invariant Properties of Bipartite Quantum States

We study the power of local test for bipartite quantum states. Our central result is that, for properties of bipartite pure states, unitary invariance on one part implies an \textit{optimal} (over all global testers) local tester acting only on the other part. As an application, we demonstrate - Purified samples offer no advantage in property testing of mixed states. - A matching lower bound $Ω(r^2/\varepsilon^2)$ for testing the Schmidt rank of bipartite states with perfect completeness, settling an open question raised in the survey of Montanaro and de Wolf (ToC 2016). - A lower bound $Ω((\sqrt{n}+\sqrt{r})\cdot\sqrt{r}/\varepsilon^2)$ for testing whether an $n$-partite state is a matrix product state of bond dimension $r$ or $\varepsilon$-far, improving the prior lower bounds $Ω(\sqrt{n}/\varepsilon^2)$ by Soleimanifar and Wright (SODA 2022) and $Ω(\sqrt{r})$ by Aaronson et al. (ITCS 2024). - A matching lower bound $Ω(d/\varepsilon^2)$ for testing whether a $d$-dimensional bipartite state is maximally entangled or $\varepsilon$-far, showing that the algorithm of O'Donnell and Wright (STOC 2015) is optimal for this task. - A query lower bound $\widetildeΩ(\sqrt{d/Δ})$ for the $d$-dimensional entanglement entropy problem with gap $Δ$, improving the prior lower bounds $Ω(\sqrt[4]{d})$ by She and Yuen (ITCS 2023) and $\widetildeΩ(1/\sqrtΔ)$ by Wang and Zhang (SICOMP 2025) and Weggemans (Quantum 2025). Moreover, we extend our central result to a robust version where the tested states are subject to noise and are not guaranteed to be pure: in this case, one-way LOCC is sufficient to realize the optimal tester.

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arXiv2026

Simultaneous Estimation of Nonlinear Functionals of a Quantum State

We consider a fundamental task in quantum information theory, estimating the values of $tr(Oρ)$, $tr(Oρ^2)$, $\ldots$, $tr(Oρ^k)$ for an observable $O$ and a quantum state $ρ$. We show that $\widetildeΘ(k)$ samples of $ρ$ are sufficient and necessary to simultaneously estimate all the $k$ values. This means that estimating all the $k$ values is almost as easy as estimating only one of them, $tr(Oρ^k)$. As an application, our approach advances the sample complexity of entanglement spectroscopy and the virtual cooling for quantum many-body systems. Moreover, we extend our approach to estimating general functionals by polynomial approximation.

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arXiv2026

A Unified Complexity Framework for Quantum Property Testing

We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_ϕ(ρ) = \operatorname{tr}(ϕ(dρ))/d$, where $ρ$ is an unknown $d$-dimensional quantum state and $ϕ$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_ϕ(ρ)$ from properties of $ϕ$, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields nearly tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). Combined with known upper bounds, these results resolve several open problems and establish the optimality of 31 quantum algorithms since 2015, up to polylogarithmic factors.

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arXiv2026

Towards Optimal Quantum Estimators for State Frame Potential

The state frame potential is a standard diagnostic of how closely a quantum state ensemble approximates Haar randomness. In this work, we study the problem of estimating the state frame potential of order $t$ to within additive error $\varepsilon$ under three progressively weaker access models: (i) query access to a multi-state-preparation oracle, (ii) general sample access, and (iii) single-copy sample access. In the query model, we establish a near-optimal query complexity of $\widetildeΘ(\sqrt{t}/\varepsilon)$, yielding a quadratic improvement in the dependence on $t$ over the previous best result of Nakata, Takeuchi, Kliesch, and Darmawan (PRX Quantum 2025). In the general sample model, we establish the optimal sample complexity $Θ(t/\varepsilon^2)$. In the single-copy sample model, we present a store-and-estimate approach whose sample complexity depends on the Rényi entropy of the ensemble weights. As an application, we use the single-copy algorithm to assess the randomness of projected state ensembles, where the entropy term becomes the observational Rényi entropy associated with measuring one subsystem.

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arXiv2026

Nearly Sample-Optimal Estimators for Quantum Rényi and Tsallis Entropies

In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order $α$, dimension $d$, and additive error $\varepsilon$, 1. For $0 < α< 1$, the sample complexity is $O(d^{1+1/α}/\varepsilon^{1/α} + d^{1/α-1}/\varepsilon^{2})$ for Rényi entropy and $O(d^{1+1/α}/\varepsilon^{1/α} + d^{2-2α}/\varepsilon^2)$ for Tsallis entropy. In particular, for $0 < α\leq 1/2$, the sample complexity for both entropies is $O(d^{1+1/α}/\varepsilon^{1/α})$. 2. For non-integer $α> 1$, the sample complexity is $O(d^2/\varepsilon^{1/α} + d^{1-1/α}/\varepsilon^2)$ for Rényi entropy. Our upper bounds improve the quantum Rényi entropy estimators due to Acharya, Issa, Shende, and Wagner (2017) and the quantum Tsallis entropy estimators due to Chen, Liu, and Wang (2026), and match the lower bounds recently established by Wang (2026).

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arXiv2026

Quantum Speedups for Log-Concave Sampling from Local Structure

For a convex function $f \colon \mathbb{R}^d \to \mathbb{R}$, the problem of sampling from a distribution proportional to $e^{-f(x)}$ is called log-concave sampling. In many practical scenarios, the function $f(x)$ turns out to admit a local decomposition $f(x) = \sum_{a=1}^R ψ_a(x_{S_a})$. In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause $ψ_a(\cdot)$, which can be computationally much cheaper than the queries to $f(x)$ itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using $\widetilde{O}(\sqrtκd)$ local queries, where $κ$ is the condition number. This improves the prior best classical result $\widetilde{O}(κd)$ due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result $\widetilde{O}(\sqrtκ d^2)$ implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.

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arXiv2026

Optimal fidelity estimation when one state is pure via algorithmic Uhlmann transform

The Uhlmann fidelity ${\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_1}|$ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error $\varepsilon$ requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of $ρ_0$ and $ρ_1$. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on $1/\varepsilon$. However, the known optimal estimator for ${\rm F}(ρ,|ψ\rangle\!\langleψ|)$ due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure. In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for ${\rm F}(ρ, |ψ\rangle\!\langleψ|)$ under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of ${\rm tr}_{\sf A}(|ψ_0\rangle\!\langleψ_1|)$ (or its inverse) to the reference register $\sf R$ of the purification $|ψ_1\rangle$ (or $|ψ_0\rangle$) on the registers $\sf A$ and $\sf R$, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.

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arXiv2026

Breaking the Quadratic Barrier for von Neumann Entropy Estimation

We study the sample complexity of estimating the von Neumann entropy of an unknown $d$-dimensional quantum state. All previously known estimators require $Ω(d^2)$ samples, and plug-in estimators are known to face a quadratic barrier. We give the first subquadratic-sample estimator: for additive error $\varepsilon$, our estimator uses \[ O\!\left(\frac{d^2 \log^2(\log(d)) \log(1/\varepsilon)}{\varepsilon^2 \log^2(d)} + \frac{\log^2(d/\varepsilon)}{\varepsilon^2}\right) \] samples. In particular, for constant $\varepsilon$, the complexity is $O_\varepsilon(d^2\log^2(\log(d))/\log^2(d))=o(d^2)$. Our analysis introduces a new pinching inequality that bounds the entropy loss under a space direct-sum decomposition, together with a bias-corrected estimator for large eigenvalues and a new bounded-coefficient polynomial estimator for small eigenvalues.

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arXiv2026

Unitary Synthesis with Near-Optimal T-Count for Near-Clifford Unitaries

We present an approach to unitary synthesis that implements an arbitrary $n$-qubit unitary operator $U$ by a Clifford+T circuit with T-count $\widetilde{O}(2^n d_F^{\mathcal{C}}(U))$, where $d_F^{\mathcal{C}}(U)$ is the Frobenius norm distance of $U$ to the Clifford group. The T-count is shown to be near-optimal when $d_F^{\mathcal{C}}(U)$ is a constant. Our approach improves the previous best upper bound $\widetilde{O}(2^{4n/3})$ due to Tan (2025) for a large class of unitary operators $U$ as long as $d_F^{\mathcal{C}}(U) \ll 2^{n/3}$.

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arXiv2026

Towards Minimax Estimation of High-Order Functionals by Quantum Arguments

We propose a novel approach to the minimax estimation of high-order functionals from the perspective of quantum computing. Specifically, for any real number $α\gg 1$, we present two estimators, one for the classical functional $\mathrm{F}_α(P) = \sum_{i=1}^S p_i^α$ of a discrete distribution $P$ and the other for the quantum functional $\mathrm{F}_α(ρ) = \operatorname{tr}(ρ^α)$ of a mixed state $ρ$. These functionals have close connections with the Rényi entropy and the Tsallis entropy. We show that both estimators achieve the minimax optimal $L_2$ rate $α\mathsf{n}^{-1}$ in the range $α\lesssim \mathsf{n} \lesssim α^{3-o(1)}$, where the support size $S$ of $P$ or the dimension of $ρ$ can be much larger than the number of samples $\mathsf{n}$. As a result, both estimators achieve the \textit{optimal} sample complexity $\mathsf{n} \asymp α$, improving upon the prior best upper bounds $O(α^2)$ established by Jiao, Venkat, Han, and Weissman (IEEE Trans. Inf. Theory 2017) for classical functionals and Chen and Wang (COLT 2025) for quantum functionals. Our estimators are constructed under a unified framework using quantum primitives and run in linear time on a quantum computer. This work reveals an unexpected path from quantum computing to statistics, suggesting a conceptually new methodology for functional estimation. It adds to the growing list of quantum proofs for classical theorems.

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arXiv2026

Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer

We settle the problem of estimating the trace distance and (square root) fidelity between $n$-qubit pure quantum states to within additive error $\varepsilon$, given their independent samples, which was raised as an open question by Wang (IEEE Trans. Inf. Theory 2024). This is achieved by a quantum algorithm with optimal sample complexity $Θ(1/\varepsilon^2)$, improving the long-standing folklore with sample complexity $O(1/\varepsilon^4)$. At the heart of our algorithm is a samplized phase estimation of the product of two Householder reflections. This is realized by an improved (multi-)samplizer for pure states, through which any quantum query algorithm using $Q$ queries to the reflection operator $I - 2|ψ\rangle\!\langleψ|$ can be converted to a $δ$-close (in the diamond norm distance) quantum sample algorithm using $Θ(Q^2/δ)$ samples of the state $|ψ\rangle$. This samplizer for pure states is also shown to be optimal.

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arXiv2026

On estimating operator norm distance, with optimal trace distance estimation when one state is pure

We investigate the computational complexity of estimating the operator norm distance ${\rm T}_{\infty}(ρ_0,ρ_1)$, defined via the operator norm $\|A\|_{\infty} = σ_{\max}(A)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $ρ_0$ and $ρ_1$. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states: 1. When one state is pure, we establish an optimal quantum estimator using $Θ(1/ε)$ queries to the state-preparation circuits. Consequently, for constant additive error, say $ε=1/5$, our estimator runs in ${\rm poly}(n)$ time. Since the operator norm distance ${\rm T}_{\infty}(|ψ\rangle\!\langleψ|,ρ)$ is exactly half of the trace distance ${\rm T}(|ψ\rangle\!\langleψ|,ρ)$, our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{é}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with ${\rm rank}(ρ)$, which can be $\exp(n)$ in general. 2. For general quantum states, we also provide a quantum estimator using $\widetilde{O}(1/ε^{3/2})$ queries to the state-preparation circuits, which shows that the corresponding promise problem is ${\sf BQP}$-complete and improves the ${\sf QMA}$ upper bound sketched by Liu and Wang (ESA 2025). Together with an $Ω(1/ε)$ quantum query complexity lower bound, this leaves only square-root room for improvement. The key intuition behind our estimators is that, when one state is pure, the pure state $|ψ\rangle$ has overlap at least $1/2$ with the top unit eigenvector of $|ψ\rangle\!\langleψ|-ρ$, reflecting a structural feature specific to the operator norm distance.

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arXiv2026

Query-Optimal and Sample-Optimal Quantum Algorithms for Estimating Fidelity to a Pure State

We present two optimal quantum algorithms that estimate the (square root) fidelity of a mixed state to a pure state to within additive error $\varepsilon$: - Given query access to the state-preparation circuits of the input states, the query complexity is shown to be $Θ(1/\varepsilon)$, achieving a quadratic speedup over the folklore $O(1/\varepsilon^2)$. - Given sample access to the input states, the sample complexity is shown to be $Θ(1/\varepsilon^2)$, achieving a quadratic speedup over the folklore $O(1/\varepsilon^4)$. Our results generalize the previous approaches to pure-state fidelity estimation, and, to the best of our knowledge, are the first optimal approaches to fidelity estimation involving mixed states. Our approach is technically simple, and can be extended to estimating the uncommon quantity $\sqrt{\operatorname{tr}(ρσ^2)}$ that is of independent interest.

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arXiv2026

Space-bounded quantum state testing via space-efficient quantum singular value transformation

Driven by exploring the power of quantum computation with a limited number of qubits, we present a novel complete characterization for space-bounded quantum computation, which encompasses settings with one-sided error (unitary $\sf coRQL$) and two-sided error ($\sf BQL$), approached from a quantum state testing perspective: - The first family of natural complete problems for unitary $\sf coRQL$, namely space-bounded quantum state certification for trace distance and Hilbert-Schmidt distance; - A new family of natural complete problems for $\sf BQL$, namely space-bounded quantum state testing for trace distance, Hilbert-Schmidt distance, and (von Neumann) entropy difference. In the space-bounded quantum state testing problem, we consider two logarithmic-qubit quantum circuits (devices) denoted as $Q_0$ and $Q_1$, which prepare quantum states $ρ_0$ and $ρ_1$, respectively, with access to their ``source code''. Our goal is to decide whether $ρ_0$ is $ε_1$-close to or $ε_2$-far from $ρ_1$ with respect to a specified distance-like measure. Interestingly, unlike time-bounded state testing problems, which exhibit computational hardness depending on the chosen distance-like measure, our results reveal that the space-bounded state testing problems, considering all three measures, are computationally as easy as preparing quantum states. Our results primarily build upon a space-efficient variant of the quantum singular value transformation (QSVT) introduced by Gilyén, Su, Low, and Wiebe (STOC 2019), which is of independent interest. Our technique provides a unified approach for designing space-bounded quantum algorithms. Specifically, we show that implementing QSVT for any bounded polynomial that approximates a piecewise-smooth function incurs only a constant overhead in terms of the space required for special forms of the projected unitary encoding.

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arXiv2026

A slightly improved upper bound for quantum statistical zero-knowledge

The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound $\mathsf{QIP(2)} \cap \text{co-}\mathsf{QIP(2)}$, which was simplified following the inclusion $\mathsf{QIP(2)} \subseteq \mathsf{PSPACE}$ established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, $\mathsf{QIP(2)}$ denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically computationally unbounded, and $\text{co-}\mathsf{QIP(2)}$ denotes the complement of $\mathsf{QIP(2)}$. We slightly improve this upper bound to $\mathsf{QIP(2)} \cap \text{co-}\mathsf{QIP(2)}$ with a quantum linear-space honest prover. Specifically, the honest prover uses space linear in the size of the transcript of the original $\mathsf{QSZK}$ proof system. A similar improvement also applies to the upper bound for the non-interactive variant $\mathsf{NIQSZK}$. Our main techniques are algorithmic versions of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).

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arXiv2026

Estimating Fidelity to a Reference Quantum State

We consider the problem of estimating the fidelity of an unknown quantum state to a known reference state to within additive error $\varepsilon$. We show that the sample complexity is $O(r^2/\varepsilon^2)$ with optimal $\varepsilon$-dependence when the reference state is of rank $r$, improving the previous best $O(r^2\log^2(1/\varepsilon)/\varepsilon^4)$ due to Utsumi, Nakata, Wang, and Takagi (QIP 2026). We also provide a lower bound of $Ω(r/\varepsilon^2)$, improving the previous best $Ω(r/\varepsilon+1/\varepsilon^2)$, with implications to quantum query complexity. Moreover, we further consider the case where the unknown state is of rank at most $r$ while the reference state can be arbitrary, for which the sample complexity is shown to be $O(r^2/\varepsilon^4)$. As an application, we present an approach to tolerant quantum state certification, generalizing the exact certification studied in Bădescu, O'Donnell, and Wright (STOC 2019).

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arXiv2026

On estimating Schatten norm and power distances between quantum states

We study the computational complexity of estimating the quantum Schatten $α$-norm distance ${\rm T}_α(ρ_0,ρ_1)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $ρ_0$ and $ρ_1$. This quantity serves as a lower bound on the trace distance and, for $α> 1$, is interchangeable with its powered version $Λ_α(ρ_0,ρ_1)$. For any constant $α> 1$, we develop an efficient rank-independent quantum estimator for ${\rm T}_α(ρ_0,ρ_1)$ with time complexity ${\rm poly}(n)$, achieving an exponential speedup over the prior best results of $\exp(n)$ due to Wang, Guan, Liu, Zhang, and Ying (TIT 2024). When $0<α<1$ is a constant, the quantum Schatten $α$-power distance $Λ_α(ρ_0,ρ_1)$ becomes a distance metric. Accordingly, we provide a rank-efficient quantum estimator for this quantity. Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten $α$-norm (QSD $_α$), which involves deciding whether ${\rm T}_α(ρ_0,ρ_1)$ is at least $2/5$ or at most $1/5$. This dichotomy arises between the cases of $α> 1$ and $0 < α\leq 1$: 1. For any constant $α>1$, QSD$_α$ is $\sf BQP$-complete. 2. For any $1 \leq α(n) \leq 1+{\rm negl}(n)$, QSD$_α$ is $\sf QSZK$-complete, implying that no efficient quantum estimator for ${\rm T}_α(ρ_0,ρ_1)$ exists unless ${\sf BQP}={\sf QSZK}$. This $\sf QSZK$-hardness result also extends to the promise problem defined by $Λ_α(ρ_0,ρ_1)$ for constant $0<α<1$. The hardness results follow from reductions based on new rank-dependent inequalities for ${\rm T}_α(ρ_0,ρ_1)$ when $1\leq α\leq \infty$ and for $Λ_α(ρ_0,ρ_1)$ when $0<α<1$, which are of independent interest.

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arXiv2026

On estimating the trace of quantum state powers

We investigate the computational complexity of estimating the trace of quantum state powers $\text{tr}(ρ^q)$ for an $n$-qubit mixed quantum state $ρ$, given its state-preparation circuit of size $\text{poly}(n)$. This quantity is closely related to and often interchangeable with the Tsallis entropy $\text{S}_q(ρ) = \frac{1-\text{tr}(ρ^q)}{q-1}$, where $q = 1$ corresponds to the von Neumann entropy. For any non-integer $q \geq 1 + Ω(1)$, we provide a quantum estimator for $\text{S}_q(ρ)$ with time complexity $\text{poly}(n)$, exponentially improving the prior best results of $\exp(n)$ due to Acharya, Issa, Shende, and Wagner (ISIT 2019), Wang, Guan, Liu, Zhang, and Ying (TIT 2024), and Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (ESA 2024). Our speedup is achieved by introducing efficiently computable uniform approximations of positive power functions into quantum singular value transformation. Our quantum algorithm reveals a sharp phase transition between the case of $q=1$ and constant $q>1$ in the computational complexity of the Quantum $q$-Tsallis Entropy Difference Problem (TsallisQED$_q$), particularly deciding whether the difference $\text{S}_q(ρ_0) - \text{S}_q(ρ_1)$ is at least $0.001$ or at most $-0.001$: - For any $1+Ω(1) \leq q \leq 2$, TsallisQED$_q$ is $\mathsf{BQP}$-complete, which implies that Purity Estimation is also $\mathsf{BQP}$-complete. - For any $1 \leq q \leq 1 + \frac{1}{n-1}$, TsallisQED$_q$ is $\mathsf{QSZK}$-hard, leading to hardness of approximating the von Neumann entropy because $\text{S}_q(ρ) \leq \text{S}(ρ)$, as long as $\mathsf{BQP} \subsetneq \mathsf{QSZK}$. The hardness results are derived from reductions based on new inequalities for the quantum $q$-Jensen-(Shannon-)Tsallis divergence with $1\leq q \leq 2$, which are of independent interest.

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