arXiv subjects

Yupan Liu

Publications and source records attributed to Yupan Liu.

15 recordsLinked to original sources

arXiv2026

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

The Uhlmann fidelity ${\rm F}(\rho_0,\rho_1) = {\rm tr}|\sqrt{\rho_0}\sqrt{\rho_1}|$ 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 $\rho_0$ and $\rho_1$. 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 ${\rm F}(\rho,|\psi\rangle\!\langle\psi|)$ 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 ${\rm F}(\rho, |\psi\rangle\!\langle\psi|)$ 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 ${\rm tr}_{\sf A}(|\psi_0\rangle\!\langle\psi_1|)$ (or its inverse) to the reference register $\sf R$ of the purification $|\psi_1\rangle$ (or $|\psi_0\rangle$) 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

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 ${\rm T}_{\infty}(\rho_0,\rho_1)$, defined via the operator norm $\|A\|_{\infty} = \sigma_{\max}(A)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $\rho_0$ and $\rho_1$. 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 $\Theta(1/\epsilon)$ queries to the state-preparation circuits. Consequently, for constant additive error, say $\epsilon=1/5$, our estimator runs in ${\rm poly}(n)$ time. Since the operator norm distance ${\rm T}_{\infty}(|\psi\rangle\!\langle\psi|,\rho)$ is exactly half of the trace distance ${\rm T}(|\psi\rangle\!\langle\psi|,\rho)$, our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{\'{e}}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with ${\rm rank}(\rho)$, which can be $\exp(n)$ in general. 2. For general quantum states, we also provide a quantum estimator using $\widetilde{O}(1/\epsilon^{3/2})$ 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 $\Omega(1/\epsilon)$ 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 $|\psi\rangle$ has overlap at least $1/2$ with the top unit eigenvector of $|\psi\rangle\!\langle\psi|-\rho$, reflecting a structural feature specific to the operator norm distance.

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arXiv2026

The power of unentanglement without destructive interference

Stoquasticity, originating in sign-problem-free physical systems, gives rise to $\sf StoqMA$, introduced by Bravyi, Bessen, and Terhal (2006), a quantum-inspired intermediate class between $\sf MA$ and $\sf AM$. Unentanglement similarly gives rise to ${\sf QMA}(2)$, introduced by Kobayashi, Matsumoto, and Yamakami (CJTCS 2009), which generalizes $\sf QMA$ to two unentangled proofs and still has only the trivial $\sf NEXP$ upper bound. In this work, we initiate a systematic study of the power of unentanglement without destructive interference via ${\sf StoqMA}(2)$, the class of unentangled stoquastic Merlin--Arthur proof systems. Beyond its complexity-theoretic interest, ${\sf StoqMA}(2)$ is connected to the optimality of non-negative tensor optimization algorithms. We highlight: 1. ${\sf NP} \subseteq {\sf StoqMA}(2)$ with $\widetilde{O}(\sqrt{n})$-qubit proofs and completenes $1-2^{-{\rm polylog}(n)}$. Conversely, the Sum-of-Squares algorithm of Barak, Kelner, and Steurer (STOC 2014) gives an exponential-time upper bound for ${\sf StoqMA}(2)$. Our tightened analysis shows the optimality of our protocol and the BKS algorithm under ETH. 2. For ${\sf StoqMA}(2)_1$, the parameter dependence in the general ETH-optimal time bound can be exponentially improved, or the bound achieved simultaneously with polynomial space. 3. For logarithmic-size proofs, ${\sf NP} \subseteq {\sf StoqMA}(2)_{\log}$ with completeness $1-O(n^{-2})$ and vanishing gap, while ${\sf StoqMA}(2)_{\log} \subseteq {\sf MA}$. Consequently, quantum-inspired randomness enables \emph{exponentially} shorter unentangled proofs even under the assumption ${\sf MA}={\sf NP}$. Our lower bounds are obtained by stoquastizing the short-proof ${\sf QMA}(2)$ protocols using distribution testing techniques. Our upper bounds for the nearly perfect completeness case are proved via our rectangular closure testing framework.

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arXiv2026

Computational hardness of estimating quantum entropies via binary entropy bounds

We investigate the computational hardness of estimating the quantum $\alpha$-R\'enyi entropy ${\rm S}^{\tt R}_{\alpha}(\rho) = \frac{\ln {\rm Tr}(\rho^\alpha)}{1-\alpha}$ and the quantum $q$-Tsallis entropy ${\rm S}^{\tt T}_q(\rho) = \frac{1-{\rm Tr}(\rho^q)}{q-1}$, both of which converge to the von Neumann entropy as the order approaches $1$. The promise problems Quantum $\alpha$-R\'enyi Entropy Approximation (R\'enyiQEA$_\alpha$) and Quantum $q$-Tsallis Entropy Approximation (TsallisQEA$_q$) ask whether $ {\rm S}^ {\tt R}_{\alpha}(\rho)$ or ${\rm S}^{\tt T}_q(\rho)$, is at least $\tau_1$ or at most $\tau_2$, where $\tau_1 - \tau_2$ is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order $1$) and some cases of the quantum $q$-Tsallis entropy, while existing approaches do not readily extend to other orders. We establish that for all positive real $\alpha$ and $q$, and also for $\alpha=\infty$, the rank-$2$ variants Rank2R\'enyiQEA$_\alpha$ and Rank2TsallisQEA$_q$ are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply: - For all real orders $\alpha > 0$ or $\alpha=\infty$, and for all real orders $0 < q \leq 1$, LowRankR\'enyiQEA$_\alpha$ and LowRankTsallisQEA$_q$ are BQP-complete, where both are restricted versions of R\'enyiQEA$_\alpha$ and TsallisQEA$_q$ with $\rho$ of polynomial rank. - For all real order $q>1$, TsallisQEA$_q$ is BQP-complete. Our hardness results stem from reductions based on new inequalities relating the $\alpha$-R\'{e}nyi or $q$-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.

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arXiv2025

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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arXiv2025

StoqMA vs. MA: the power of error reduction

StoqMA characterizes the computational hardness of stoquastic local Hamiltonians, which is a family of Hamiltonians that does not suffer from the sign problem. Although error reduction is commonplace for many complexity classes, such as BPP, BQP, MA, QMA, etc.,this property remains open for StoqMA since Bravyi, Bessen and Terhal defined this class in 2006. In this note, we show that error reduction forStoqMA will imply that StoqMA = MA.

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arXiv2025

Quantum state testing beyond the polarizing regime and quantum triangular discrimination

The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$) captures computational difficulties of the time-bounded quantum state testing problem with respect to the trace distance, deciding whether $\mathrm{T}(ρ_0,ρ_1)$ is at least $α$ or at most $β$, known as the Quantum State Distinguishability Problem ($\mathrm{QSDP}$) introduced by Watrous (FOCS 2002). However, $\mathrm{QSDP}[α,β]$ is in $\mathsf{QSZK}$ only within the constant polarizing regime, where $α$ and $β$ are constants satisfying $α^2 > β$ (rather than $α> β$), similar to its classical counterpart shown by Sahai and Vadhan (JACM 2003) due to the polarization lemma (error reduction for $\mathrm{SDP}$). Recently, Berman, Degwekar, Rothblum, and Vasudevan (TCC 2019) extended the $\mathsf{SZK}$ containment of $\mathrm{SDP}$ beyond the polarizing regime via the time-bounded distribution testing problems with respect to the triangular discrimination and the Jensen-Shannon divergence. Our work introduces proper quantum analogs for these problems by defining quantum counterparts for triangular discrimination. We investigate whether the quantum analogs behave similarly to their classical counterparts and examine the limitations of existing approaches to polarization regarding quantum distances. These new $\mathsf{QSZK}$-complete problems improve $\mathsf{QSZK}$ containments of $\mathrm{QSDP}$ beyond the polarizing regime and establish a simple $\mathsf{QSZK}$-hardness for the quantum entropy difference problem ($\mathrm{QEDP}$) defined by Ben-Aroya, Schwartz, and Ta-Shma (ToC 2010). Furthermore, we prove that $\mathrm{QSDP}$ with some exponentially small errors is in $\mathsf{PP}$, while the same problem without error is in $\mathsf{NQP}$.

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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.

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arXiv2025

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

As often emerges in various basic quantum properties such as R\'enyi and Tsallis entropies, the trace of quantum state powers $\text{tr}(\rho^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that, even for (possibly) non-integer $q>1$, $\text{tr}(\rho^q)$ can be estimated to within additive error $\epsilon$ using a dimension-independent (and also rank-independent) sample complexity of $\widetilde O(1/\epsilon^{3+\frac2{q-1}})$, together with a lower bound of $\Omega(1/\epsilon)$. 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 $\widetilde\Theta(1/\epsilon^2)$. - 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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arXiv2025

On estimating Schatten norm and power distances between quantum states

We study the computational complexity of estimating the quantum Schatten $\alpha$-norm distance ${\rm T}_\alpha(\rho_0,\rho_1)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $\rho_0$ and $\rho_1$. This quantity serves as a lower bound on the trace distance and, for $\alpha > 1$, is interchangeable with its powered version $\Lambda_\alpha(\rho_0,\rho_1)$. For any constant $\alpha > 1$, we develop an efficient rank-independent quantum estimator for ${\rm T}_\alpha(\rho_0,\rho_1)$ 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 $0<\alpha<1$ is a constant, the quantum Schatten $\alpha$-power distance $\Lambda_\alpha(\rho_0,\rho_1)$ 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 $\alpha$-norm (QSD $_\alpha$), which involves deciding whether ${\rm T}_\alpha(\rho_0,\rho_1)$ is at least $2/5$ or at most $1/5$. This dichotomy arises between the cases of $\alpha > 1$ and $0 < \alpha\leq 1$: 1. For any constant $\alpha>1$, QSD$_{\alpha}$ is $\sf BQP$-complete. 2. For any $1 \leq \alpha(n) \leq 1+{\rm negl}(n)$, QSD$_\alpha$ is $\sf QSZK$-complete, implying that no efficient quantum estimator for ${\rm T}_\alpha(\rho_0,\rho_1)$ exists unless ${\sf BQP}={\sf QSZK}$. This $\sf QSZK$-hardness result also extends to the promise problem defined by $\Lambda_\alpha(\rho_0,\rho_1)$ for constant $0<\alpha<1$. The hardness results follow from reductions based on new rank-dependent inequalities for ${\rm T}_\alpha(\rho_0,\rho_1)$ when $1\leq \alpha \leq \infty$ and for $\Lambda_\alpha(\rho_0,\rho_1)$ when $0<\alpha<1$, which are of independent interest.

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arXiv2025

Quantum Merlin-Arthur proof systems for synthesizing quantum states

Complexity theory typically focuses on the difficulty of solving computational problems using classical inputs and outputs, even with a quantum computer. In the quantum world, it is natural to apply a different notion of complexity, namely the complexity of synthesizing quantum states. We investigate a state-synthesizing counterpart of the class NP, referred to as stateQMA, which is concerned with preparing certain quantum states through a polynomial-time quantum verifier with the aid of a single quantum message from an all-powerful but untrusted prover. This is a subclass of the class stateQIP recently introduced by Rosenthal and Yuen (ITCS 2022), which permits polynomially many interactions between the prover and the verifier. Our main result consists of error reduction of this class and its variants with an exponentially small gap or bounded space, as well as how this class relates to other fundamental state synthesizing classes, i.e., states generated by uniform polynomial-time quantum circuits (stateBQP) and space-uniform polynomial-space quantum circuits (statePSPACE). Furthermore, we establish that the family of UQMA witnesses, considered as one of the most natural candidates for stateQMA containments, is in stateQMA. Additionally, we demonstrate that stateQCMA achieves perfect completeness.

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arXiv2024

On estimating the trace of quantum state powers

We investigate the computational complexity of estimating the trace of quantum state powers $\text{tr}(\rho^q)$ for an $n$-qubit mixed quantum state $\rho$, given its state-preparation circuit of size $\text{poly}(n)$. This quantity is closely related to and often interchangeable with the Tsallis entropy $\text{S}_q(\rho) = \frac{1-\text{tr}(\rho^q)}{q-1}$, where $q = 1$ corresponds to the von Neumann entropy. For any non-integer $q \geq 1 + \Omega(1)$, we provide a quantum estimator for $\text{S}_q(\rho)$ 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 $\text{S}_q(\rho_0) - \text{S}_q(\rho_1)$ is at least $0.001$ or at most $-0.001$: - For any $1+\Omega(1) \leq q \leq 2$, 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 $\text{S}_q(\rho) \leq \text{S}(\rho)$, 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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arXiv2023

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 $\rho_0$ and $\rho_1$, respectively, with access to their ``source code''. Our goal is to decide whether $\rho_0$ is $\epsilon_1$-close to or $\epsilon_2$-far from $\rho_1$ 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 Gily\'en, 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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arXiv2021

Towards a quantum-inspired proof for IP = PSPACE

We explore quantum-inspired interactive proof systems where the prover is limited. Namely, we improve on a result by [AG17] showing a quantum-inspired interactive protocol ($\sf IP$) for $\sf PreciseBQP$ where the prover is only assumed to be a $\sf PreciseBQP$ machine, and show that the result can be strengthened to show an $\sf IP$ for $\sf NP^{PP}$ with a prover which is only assumed to be an $\sf NP^{PP}$ machine - which was not known before. We also show how the protocol can be used to directly verify $\sf QMA$ computations, thus connecting the sum-check protocol by [AAV13] with the result of [AG17, LFKN90]. Our results shed light on a quantum-inspired proof for ${\sf IP} = {\sf PSPACE}$, as $\sf PreciseQMA$ captures the full $\sf PSPACE$ power.

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arXiv2021

StoqMA meets distribution testing

$\mathsf{StoqMA}$ captures the computational hardness of approximating the ground energy of local Hamiltonians that do not suffer the so-called sign problem. We provide a novel connection between $\mathsf{StoqMA}$ and distribution testing via reversible circuits. First, we prove that easy-witness $\mathsf{StoqMA}$ (viz. $\mathsf{eStoqMA}$, a sub-class of $\mathsf{StoqMA}$) is contained in $\mathsf{MA}$. Easy witness is a generalization of a subset state such that the associated set's membership can be efficiently verifiable, and all non-zero coordinates are not necessarily uniform. This sub-class $\mathsf{eStoqMA}$ contains $\mathsf{StoqMA}$ with perfect completeness ($\mathsf{StoqMA}_1$), which further signifies a simplified proof for $\mathsf{StoqMA}_1 \subseteq \mathsf{MA}$ [BBT06, BT10]. Second, by showing distinguishing reversible circuits with ancillary random bits is $\mathsf{StoqMA}$-complete (as a comparison, distinguishing quantum circuits is $\mathsf{QMA}$-complete [JWB05]), we construct soundness error reduction of $\mathsf{StoqMA}$. Additionally, we show that both variants of $\mathsf{StoqMA}$ that without any ancillary random bit and with perfect soundness are contained in $\mathsf{NP}$. Our results make a step towards collapsing the hierarchy $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{SBP}$ [BBT06], in which all classes are contained in $\mathsf{AM}$ and collapse to $\mathsf{NP}$ under derandomization assumptions.

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