arXiv subjects

Qisheng Wang

Publications and source records attributed to Qisheng Wang.

At least 37 records · Page 2Linked to original sources

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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arXiv2026

Quantum enhanced rare event discovery and sampling

Financial crashes, cascading failures in infrastructure, and critical errors in AI systems are frequently triggered by events that occur with extremely small probability. Efficiently discovering and sampling events with probability below a threshold is therefore of critical interest. Yet this task is highly non-trivial using existing classical or quantum methods. Being rare, such events require an immense sampling overhead to collect sufficient data samples. Moreover, because the rare events are not known in advance, they cannot be flagged for amplification using standard techniques. Here, we introduce a quantum algorithm for rare-event discovery and sampling without first learning which events are rare. The algorithm achieves the optimal quantum scaling with the rarity threshold. We further demonstrate that this can achieve a quadratic speedup for heavy-tailed systems whose tail has nonvanishing total mass, and translates into a robust polynomial speedup for stationary stochastic processes, with the exponent determined by its entropy-rate structure.

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arXiv2026

Quantum Time Lower Bounds by Permutation Invariance

Tight bounds on quantum sample complexity and quantum query complexity have been known for various computational problems in the literature, whereas tight bounds on quantum time complexity (i.e., the size of quantum circuits) remain unresolved. In this paper, we provide a framework to establish lower bounds on the quantum time complexity for testing permutation-invariant properties of quantum states, via a reduction from quantum sample complexity. As an application, we obtain a series of matching lower bounds when given sample access to the input quantum states, including: 1. The SWAP test due to Buhrman, Cleve, Watrous, and de Wolf (Phys. Rev. Lett. 2001) is time-optimal to estimate the purity $\operatorname{tr}(ρ^2)$ and the inner product $\operatorname{tr}(ρσ)$. 2. The Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki, and Kwek (Phys. Rev. Lett. 2002) is time-optimal to estimate the high-order functionals $\operatorname{tr}(ρ^k)$. 3. The productness tester for multipartite pure states due to Harrow and Montanaro (J. ACM 2013) is time-optimal. 4. The LMR protocol due to Lloyd, Mohseni, and Rebentrost (Nat. Phys. 2014) is time-optimal to implement the reflection operator about a pure state. 5. The samplizer due to Wang and Zhang (IEEE Trans. Inf. Theory 2025) is time-optimal for pure states. 6. The estimator for pure-state trace distance and fidelity due to Wang and Zhang (ICALP 2026) is time-optimal. To the best of our knowledge, this is the first method that allows us to systematically establish tight lower bounds on quantum time complexity.

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arXiv2026

Quantum Multi-Level Estimation of Functionals of Discrete Distributions

We propose a quantum multi-level estimation framework for a functional $\sum_{i=1}^n f(p_i)$ of a discrete distribution $(p_i)_{i=1}^n$. We partition the values $p_i$ into logarithmically many intervals whose length decays exponentially. For each interval, we perform non-destructive singular value discrimination to isolate the relevant $p_i$, enabling adaptive estimation of the partial sum over this interval. Unlike previous variable-time approaches, our method avoids high control overhead and requires only constant extra ancilla qubits. As an application, we present efficient quantum estimators for the $q$-Tsallis entropy of discrete distributions. Specifically: (i) For $q > 1$, we obtain a near-optimal quantum algorithm with query complexity $\tildeΘ(1/\varepsilon^{\max\{1/(2(q-1)), 1\}})$, improving the prior best $O(1/\varepsilon^{1+1/(q-1)})$ due to Liu and Wang (SODA 2025; IEEE Trans. Inf. Theory 2026). (ii) For $0 < q < 1$, we obtain a quantum algorithm with query complexity $\tilde{O}(n^{1/q-1/2}/\varepsilon^{1/q})$, exhibiting a quantum speedup over the near-optimal classical estimators due to Jiao, Venkat, Han, and Weissman (IEEE Trans. Inf. Theory 2017). Our results achieve, to our knowledge, the first near-optimal quantum estimators for parameterized $q$-entropy for non-integer $q$.

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arXiv2026

Strict Hierarchy for Quantum Channel Certification to Unitary

We consider the problem of quantum channel certification to unitary, where one is given access to an unknown $d$-dimensional channel $\mathcal{E}$, and wants to test whether $\mathcal{E}$ is equal to a target unitary channel or is $\varepsilon$-far from it in the diamond norm. We present optimal quantum algorithms for this problem, settling the query complexities in three access models with increasing power. Specifically, we show that: (i) $Θ(d/\varepsilon^2)$ queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023). (ii) $Θ(d/\varepsilon)$ queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008). (iii) $Θ(\sqrt{d}/\varepsilon)$ queries suffice for source-code access model, matching the lower bound due to Jeon and Oh (npj Quantum Inf. 2026). This demonstrates a strict hierarchy of complexities for quantum channel certification to unitary across various access models.

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arXiv2026

Trace Estimation of Quantum State Powers: Sample Complexity and Computational Hardness

As often emerges in various basic quantum properties such as Rényi and Tsallis entropies, the trace of quantum state powers $\text{tr}(ρ^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that, even for (possibly) non-integer $q>1$, $\text{tr}(ρ^q)$ can be estimated to within additive error $ε$ using a dimension-independent (and also rank-independent) sample complexity of $\widetilde O(1/ε^{3+\frac2{q-1}})$, together with a lower bound of $Ω(1/ε)$. In addition, combining this result with subsequent work of Liu (STACS 2026) shows that the corresponding promise problem is ${\sf BQP}$-complete. In this paper, we significantly improve and extend the sample complexity bounds for this problem. Furthermore, we show that for $0 2$, we settle the sample complexity with matching upper and lower bounds $\widetildeΘ(1/ε^2)$. - For $1 1$. Technically, our upper bounds are obtained by (non-plug-in) quantum estimators based on weak Schur sampling, in sharp contrast to the prior approach based on quantum singular value transformation and samplizer.

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arXiv2026

On Estimating the Quantum Tsallis Relative Entropy

The relative entropy between quantum states quantifies their distinguishability. The estimation of certain relative entropies has been investigated in the literature, e.g., the von Neumann relative entropy and sandwiched Rényi relative entropy. In this paper, we present a comprehensive study of the estimation of the quantum Tsallis relative entropy. We show that for any constant $α\in (0, 1)$, the $α$-Tsallis relative entropy between two quantum states of rank $r$ can be estimated with sample complexity $\operatorname{poly}(r)$, which can be made more efficient if we know their state-preparation circuits. As an application, we obtain an approach to tolerant quantum state certification with respect to the quantum Hellinger distance with sample complexity $\widetilde{O}(r^{3.5})$, which exponentially outperforms the folklore approach based on quantum state tomography when $r$ is polynomial in the number of qubits. In addition, we show that the quantum state distinguishability problems with respect to the quantum $α$-Tsallis relative entropy and quantum Hellinger distance are $\mathsf{QSZK}$-complete in a certain regime, and they are $\mathsf{BQP}$-complete in the low-rank case.

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arXiv2026

Quantum Data Structure for Range Minimum Query

Given an array $a[1..n]$, the Range Minimum Query (RMQ) problem is to maintain a data structure that supports RMQ queries: given a range $[l, r]$, find the index of the minimum element among $a[l..r]$, i.e., $\operatorname{argmin}_{i \in [l, r]} a[i]$. In this paper, we propose a quantum data structure that supports RMQ queries and range updates, with an optimal time complexity $\widetilde Θ(\sqrt{nq})$ for performing $q = O(n)$ operations without preprocessing, compared to the classical $\widetildeΘ(n+q)$. As an application, we obtain a time-efficient quantum algorithm for $k$-minimum finding without the use of quantum random access memory.

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arXiv2025

A List of Complexity Bounds for Property Testing by Quantum Sample-to-Query Lifting

Quantum sample-to-query lifting, a relation between quantum sample complexity and quantum query complexity presented in Wang and Zhang (SIAM J. Comput. 2025), was significantly strengthened by Tang, Wright, and Zhandry (2025) to the case of state-preparation oracles. In this paper, we compile a list of quantum lower and upper bounds for property testing that are obtained by quantum sample-to-query lifting. The problems of interest include testing properties of probability distributions and quantum states, such as entropy and closeness. This collection contains new results, as well as new proofs of known bounds. In total, we present 49 complexity bounds, where 41 are new and 18 are (near-)optimal.

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arXiv2025

Time-Efficient Quantum Entropy Estimator via Samplizer

Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropy $S(ρ)$ and Rényi entropy $S_α(ρ)$ of an $N$-dimensional quantum state $ρ$, given access to independent samples of $ρ$. Specifically, we provide the following: 1. A quantum estimator for $S(ρ)$ with time complexity $\tilde O(N^2)$, improving the prior best time complexity $\tilde O(N^6)$ by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). 2. A quantum estimator for $S_α(ρ)$ with time complexity $\tilde O(N^{4/α-2})$ for $0<α<1$ and $\tilde O(N^{4-2/α})$ for $α>1$, improving the prior best time complexity $\tilde O(N^{6/α})$ for $0<α<1$ and $\tilde O(N^6)$ for $α>1$ by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound for estimating $S_α(ρ)$. Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool called samplizer, which can "samplize" a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a unified framework for estimating quantum entropies. Specifically, when a quantum oracle $U$ block-encodes a mixed quantum state $ρ$, any quantum query algorithm using $Q$ queries to $U$ can be samplized to a $δ$-close (in the diamond norm) quantum algorithm using $\tildeΘ(Q^2/δ)$ samples of $ρ$. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.

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arXiv2025

Double-Ended Palindromic Trees in Linear Time

The palindromic tree (a.k.a. eertree) is a data structure that provides access to all palindromic substrings of a string. In this paper, we propose a dynamic version of eertree, called double-ended eertree, which supports online operations on the stored string, including double-ended queue operations, counting distinct palindromic substrings, and finding the longest palindromic prefix/suffix. At the heart of our construction, we identify a new class of substring occurrences, called surfaces, that are palindromic substring occurrences that are neither prefixes nor suffixes of any other palindromic substring occurrences, which is of independent interest. Surfaces characterize the link structure of all palindromic substrings in the eertree, thereby allowing a linear-time implementation of double-ended eertrees through a linear-time maintenance of surfaces.

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arXiv2025

Quantum algorithms for Uhlmann transformation

Uhlmann's theorem is a central result in quantum information theory, which associates the closeness of two quantum states with that of their purifications. The theorem also well characterizes a fundamental task: how close a pure quantum state can be transformed into another state via local operations acting only on its subsystem. The optimal transformation for this task is called the Uhlmann transformation, which has broad applications in various information-processing tasks. However, its quantum circuit implementation and computational cost have remained unclear, limiting the utility of the transformation. In this work, we fill this gap by proposing quantum algorithms that realize the Uhlmann transformation in query and sample access models. Notably, our Uhlmann transformation algorithms can be polynomial-time for low-rank states, exhibiting an exponential improvement over the previous approach and other naive approaches based on quantum state tomography. In addition, we derive a lower bound on the query and sample complexities of the Uhlmann transformation for a deeper understanding of its algorithmic features. We apply our Uhlmann transformation algorithms to fidelity estimation between two states, and substantially improve the previous best query and sample complexities. We further discuss other applications to several information-theoretic tasks, including entanglement transmission, quantum state merging, and the algorithmic implementation of the Petz recovery map, providing a comprehensive evaluation of their computational costs. These results, hence, contribute to the practical realization of such widely recognized and useful protocols.

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arXiv2025

Quantum Lower Bounds by Sample-to-Query Lifting

The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS 1998, J. ACM 2001), the adversary method by Ambainis (STOC 2000, J. Comput. Syst. Sci. 2002), and the compressed oracle method by Zhandry (CRYPTO 2019) have been shown to be powerful in proving quantum query lower bounds for a wide variety of problems. In this paper, we propose a new method for proving quantum query lower bounds by a quantum sample-to-query lifting theorem, which is from an information theory perspective. Using this method, we obtain the following new results: 1. A quadratic relation between quantum sample and query complexities regarding quantum property testing, which is optimal and saturated by quantum state discrimination. Here, the sample complexity is measured given sample access to the quantum state to be tested, while the query complexity is measured given query access to an oracle that block-encodes the quantum state. 2. A matching lower bound $\widetilde Ω(β)$ for quantum Gibbs sampling at inverse temperature $β$, showing that the quantum Gibbs sampler by Gilyén, Su, Low, and Wiebe (STOC 2019) is optimal. 3. A new lower bound $\widetilde Ω(1/\sqrtΔ)$ for the entanglement entropy problem with gap $Δ$, which was recently studied by She and Yuen (ITCS 2023). 4. A series of quantum query lower bounds for matrix spectrum testing, based on the sample lower bounds for quantum state spectrum testing by O'Donnell and Wright (STOC 2015, Comm. Math. Phys. 2021). In addition, we also provide unified proofs for some known lower bounds that have been proven previously via different techniques, including those for phase/amplitude estimation and Hamiltonian simulation.

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arXiv2025

Information-Theoretic Lower Bounds for Approximating Monomials via Optimal Quantum Tsallis Entropy Estimation

This paper reveals a conceptually new connection from information theory to approximation theory via quantum algorithms for entropy estimation. Specifically, we provide an information-theoretic lower bound $Ω(\sqrt{n})$ on the approximate degree of the monomial $x^n$, compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error $\varepsilon$ the Tsallis entropy of integer order $q \geq 2$ of an unknown probability distribution $p$ or an unknown quantum state $ρ$, using $\widetilde Θ(\frac{1}{\sqrt{q}\varepsilon})$ queries to the quantum oracle that produces a sample from $p$ or prepares a copy of $ρ$, improving the prior best $O(\frac{1}{\varepsilon})$ via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.

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arXiv2025

Quantum algorithms for matrix geometric means

Matrix geometric means between two positive definite matrices can be defined from distinct perspectives - as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems. We devise quantum subroutines for the matrix geometric means, and construct solutions to the algebraic Riccati equation - an important class of nonlinear systems of equations appearing in machine learning, optimal control, estimation, and filtering. Using these subroutines, we present a new class of quantum learning algorithms, for both classical and quantum data, called quantum geometric mean metric learning, for weakly supervised learning and anomaly detection. The subroutines are also useful for estimating geometric Rényi relative entropies and the Uhlmann fidelity, in particular achieving optimal dependence on precision for the Uhlmann and Matsumoto fidelities. Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines.

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arXiv2025

Space-bounded quantum interactive proof systems

We introduce two models of space-bounded quantum interactive proof systems, ${\sf QIPL}$ and ${\sf QIP_{\rm U}L}$. The ${\sf QIP_{\rm U}L}$ model, a space-bounded variant of quantum interactive proofs (${\sf QIP}$) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, ${\sf QIPL}$ allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of ${\sf QIPL}$ and ${\sf QIP_{\rm U}L}$. When the message number $m$ is polynomially bounded, ${\sf QIP_{\rm U}L} \subsetneq {\sf QIPL}$ unless ${\sf P} = {\sf NP}$: - ${\sf QIPL}^{\rm HC}$, a subclass of ${\sf QIPL}$ defined by a high-concentration condition on yes instances, exactly characterizes ${\sf NP}$. - ${\sf QIP_{\rm U}L}$ is contained in ${\sf P}$ and contains ${\sf SAC}^1 \cup {\sf BQL}$, where ${\sf SAC}^1$ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when $m$ is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where ${\sf BQL}={\sf BQ_{\rm U}L}$. We also introduce space-bounded unitary quantum statistical zero-knowledge (${\sf QSZK_{\rm U}L}$), a specific form of ${\sf QIP_{\rm U}L}$ proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (${\sf QSZK}$) defined by Watrous (SICOMP 2009). We prove that ${\sf QSZK_{\rm U}L} = {\sf BQL}$, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.

quant-ph↗

arXiv2025

Quantum Hamiltonian Certification

We formalize and study the Hamiltonian certification problem. Given access to $e^{-\mathrm{i} Ht}$ for an unknown Hamiltonian $H$, the goal of the problem is to determine whether $H$ is $\varepsilon_1$-close to or $\varepsilon_2$-far from a target Hamiltonian $H_0$. While Hamiltonian learning methods have been extensively studied, they often require restrictive assumptions and suffer from inefficiencies when adapted for certification tasks. This work introduces a direct and efficient framework for Hamiltonian certification. Our approach achieves \textit{optimal} total evolution time $Θ((\varepsilon_2-\varepsilon_1)^{-1})$ for certification under the normalized Frobenius norm, without prior structural assumptions. This approach also extends to certify Hamiltonians with respect to all Pauli norms and normalized Schatten $p$-norms for $1\leq p\leq2$ in the one-sided error setting ($\varepsilon_1=0$). Notably, the result in Pauli $1$-norm suggests a quadratic advantage of our approach over all possible Hamiltonian learning approaches. We also establish matching lower bounds to show the optimality of our approach across all the above norms. We complement our result by showing that the certification problem with respect to normalized Schatten $\infty$-norm is $\mathsf{coQMA}$-hard, and therefore unlikely to have efficient solutions. This hardness result provides strong evidence that our focus on the above metrics is not merely a technical choice but a requirement for efficient certification. To enhance practical applicability, we develop an ancilla-free certification method that maintains the inverse precision scaling while eliminating the need for auxiliary qubits, making our approach immediately accessible for near-term quantum devices with limited resources.

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