arXiv subjects

Lvzhou Li

Publications and source records attributed to Lvzhou Li.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Approximate Counting with Bernoulli Oracles

Quantum counting is a fundamental quantum algorithm that estimates the fraction of marked elements using a membership oracle, achieving a quadratic speedup over classical sampling. The membership oracle, however, assumes exact labeling of each element, but this assumption fails when the labels are inherently probabilistic. We study quantum counting with \emph{Bernoulli oracles}, where given $m$ Bernoulli distributions with unknown biases $p_1,\dots,p_m$ and a gap parameter $\Delta$, the goal is to estimate the fraction $\rho$ of \emph{positive} distributions ($p_i\ge1/2+\Delta$) to within additive error $\epsilon$. We prove an upper bound of $\tilde{O}\!\big(\frac{\sqrt{\rho}}{\Delta\epsilon}+\frac{1}{\Delta\sqrt{\epsilon}}\big)$ queries, achieving a quadratic speedup over the classical sample complexity. % of $\Theta(\rho/\Delta^2\epsilon^2)$. Our algorithm first uses the Quantum Singular Value Transformation (QSVT) to coherently amplify the bias gap without collapsing the superposition over distributions, and then applies two-stage adaptive amplitude estimation. We complement this upper bound with a near-matching lower bound of $\Omega(\sqrt{\rho}/(\Delta\epsilon))$ via a new composition theorem for the quantum adversary method in the Boolean-over-average-case direction. For the special case of a constant gap $\Delta=\Theta(1)$, which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to $\tilde{O}\big(\frac{\sqrt{\rho}}{\epsilon}+\frac{1}{\sqrt{\epsilon}}\big)$ and $\Omega(\frac{\sqrt{\rho}}{\epsilon})$, thereby characterizing the query complexity of quantum counting with bounded-error oracles.

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arXiv2026

Fanout Complexity of Symmetric Boolean Functions in $\mathsf{QAC}^0$

Whether $\mathsf{QAC}^0$ can compute $\mathtt{PARITY}_n$ remains open. Computing $\mathtt{PARITY}_n$ is equivalent to implementing $\mathtt{FANOUT}_n$ under $\mathsf{QAC}^0$ reductions. This raises a more general question: for an arbitrary symmetric Boolean function $f:\{0,1\}^n\to\{0,1\}$, what fanout size is necessary and sufficient for computing $f$ in $\mathsf{QAC}^0$? We show that the answer is exactly the transition radius $\rho(f)$: computing $f$ and implementing $\mathtt{FANOUT}_{\rho(f)}$ are equivalent under $\mathsf{QAC}^0$ reductions. In particular, if $\rho(f)\ge n^\delta$ for some constant $\delta>0$, then computing $f$ is $\mathsf{QAC}^0_{\mathrm{f}}$-complete. Combined with Paturi's theorem, our characterization implies that if $\mathtt{PARITY}_n \notin \mathsf{QAC}^0$, then any Boolean function in $\mathsf{QAC}^0$ of approximate degree $n^{1/2+\Omega(1)}$ must be nonsymmetric.

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arXiv2026

A Minimum-Cardinality Genuinely Unextendible Product Basis in Three Qutrits

It has remained an open question whether a genuinely unextendible product basis (GUPB) exists. We resolve this problem by constructing an explicit three-qutrit GUPB of cardinality fourteen in the smallest tripartite Hilbert space in which a GUPB can exist. Together with the nonexistence of three-qutrit GUPBs of cardinality less than fourteen, our construction proves that fourteen is the minimum cardinality. A padding procedure further extends the construction to all tripartite systems whose local dimensions are at least three. As applications, the normalized projector onto the thirteen-dimensional orthogonal complement of the three-qutrit GUPB is positive under partial transposition and bound entangled across every bipartition, while the GUPB exhibits strong quantum nonlocality without entanglement.

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arXiv2026

Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States

We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies $d\geq 3$. Prior to this work, to the best of our knowledge, $\text{AME}(7,d)$ states were known to exist only when $d$ is a prime power other than $2$, or when $d$ can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no $\text{AME}(7,2)$ state exists, it remains to establish existence for all $d\geq 3$. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to $2$ modulo $4$. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension $d\geq 3$.

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arXiv2026

Sparse Quantum State Preparation with Sublinear T-Count

We study the fault-tolerant cost of preparing sparse quantum states, measured by $T$-count in the Clifford+$T$ model. Here an $n$-qubit state is called $s$-sparse if it is supported on at most $s$ computational-basis states. For arbitrary $n$-qubit states, the optimal $T$-count is $\Theta(\sqrt{2^n\log(1/\epsilon)}+\log(1/\epsilon))$, but for $s$-sparse states the best previous upper bounds remained linear in $s$. We show that any $n$-qubit $s$-sparse state can be prepared up to error $\epsilon$ using $\widetilde{O}(\min\{s,\ n^{3/4}\sqrt{s}\}+\sqrt{s\log(1/\epsilon)}+\log(1/\epsilon))$ $T$ gates, giving the first sublinear dependence on $s$ once the support is sufficiently large. Our approach is based on a support-aware synthesis theorem for sparse Boolean functions, which may be of independent interest. We also prove that, for every $0<\epsilon\le 1/6$ and $2\le s\le 2^{n/2}$, sparse-state preparation requires $\Omega(\min\{s,\sqrt{ns}\})$ $T$ gates, showing that linear dependence on $s$ is unavoidable in the small-support regime and substantially narrowing the gap between the known upper and lower bounds within this parameter range.

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arXiv2026

Quantum Speedups for Testing Similar Means

Property testing of distributions is a central topic in information theory, learning theory, and statistics. While quantum algorithms are known to offer significant speedups for property testing of a single distribution or a pair of distributions, it is unclear whether quantum algorithms provide speedups for property testing of $m$ ($m\geq 3$) distributions. This work focuses on quantum algorithms for testing whether $m$ distributions have similar means or are $\epsilon$-far from mean similarity under two models. In the query model, the algorithm can choose which distribution to sample from, whereas in the sampling model, the distributions are selected uniformly. We design quantum algorithms with complexities $\tilde{O}(1/\epsilon)$ (the $\tilde{O}$ notation hides poly-logarithmic factors) and $\tilde{O}(\sqrt{m}/\epsilon)$ in the query and sampling models, respectively, achieving quadratic speedups over the classical counterparts. We further establish quantum lower bounds of $\Omega\left(1/\epsilon\right)$ and $\Omega\rbra{m^{1/3}+\frac{m^{1/4}}{\epsilon}}$ for the query model and the sampling model, demonstrating the optimality of our quantum algorithms in terms of the dependence on $\epsilon$ up to logarithmic factors.

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arXiv2026

On the completeness of transformation rules in reversible logic synthesis

Transformation rules play a central role in reversible circuit optimization, template-based rewriting, and equivalence checking, and establishing their completeness is a fundamental problem in reversible logic synthesis. In this work, we investigate the completeness of transformation rules for reversible circuits both with and without ancillary bits and garbage outputs. For reversible circuits without ancillary bits and garbage outputs, we introduce a refined and complete transformation rule set $\mathcal{RC}^{r}$, obtained by replacing one rule in the rule set $\mathcal{RC}$ proposed in the previous work (TCAD, 45, pp. 3711--3724, 2026) with a simpler and widely adopted transformation rule. Since all rules in $\mathcal{RC}^{r}$ are commonly used in reversible logic synthesis, this result reveals that practically adopted transformation rules are already sufficient to establish a complete rewriting framework. Based on this result, we further propose an extended rule set, denoted by $\mathcal{RC}^+$, and prove its completeness for reversible circuits with ancillary bits and garbage outputs. To the best of our knowledge, this work presents the first complete transformation rule set for arbitrary reversible circuits, regardless of whether ancillary bits or garbage outputs are employed. The proposed framework establishes a theoretical foundation for circuit optimization, template generation, and equivalence checking, and may facilitate the development of automated design tools for reversible and quantum circuits.

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arXiv2026

Agnostic learning of qudit stabilizer states

Learning a classical description of a quantum state is a fundamental task in quantum computation. Among the most important classes of quantum states are stabilizer states, which play a central role in quantum error correction and fault-tolerant computation. To mitigate the effects of realistic noise, agnostic learning of stabilizer states has emerged as a natural and well-motivated problem. Recently, Chen \textit{et al.} [STOC'25, p. 429-438] resolved this problem for qubit systems by using a stabilizer bootstrapping framework. However, the agnostic learning of qudit stabilizer states remains largely unexplored, since the qudit setting introduces fundamental structural differences that prevent a direct generalization of existing qubit techniques. In this paper, we successfully generalize the stabilizer bootstrapping framework to qudit systems and present the first efficient quantum algorithm for agnostic learning of qudit stabilizer states. Specifically, given copies of an unknown $n$-qudit pure state $|\psi\rangle$ that has fidelity $\tau$ with some stabilizer state, our algorithm outputs a stabilizer state $|\phi\rangle$ such that $\left| \braket{\phi|\psi} \right|^2 \geq \tau - \varepsilon$ with high probability. The algorithm uses only single-copy and four-copy measurements, and its sample and time complexity scale as $(d/\tau)^{O(d^2 \log(1/\tau))} \cdot \mathrm{poly}(n, 1/\varepsilon)$, where the dimension $d$ is an odd prime. As a direct corollary, our algorithm enables efficient estimation of the magic of a quantum state, as quantified by its stabilizer fidelity. Completing the picture, we also present a streamlined algorithm for the high-fidelity regime $\tau > \cos^2(\pi/8)$, establishing a qudit analogue of the threshold-based approach in prior qubit work.

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arXiv2026

Quantum states supported by matroids

In this work, we establish a structural correspondence between quantum states and matroid theory. This connection demonstrates that key properties of quantum states, including entanglement and measurement, can be characterized in purely combinatorial terms via matroids, despite the apparent conceptual distance between these two fields. Using this framework, we show that a matroid-supported state is genuinely entangled when its underlying matroid is connected. Moreover, a uniform superposition over all bases of a matroid is genuinely entangled if and only if the matroid is connected. We also demonstrate that a local measurement in the $Z$-basis on such a state yields another matroid-supported state, whose underlying matroid is a minor of the original one. Inspired by matroid duality, we further propose a notion of quantum state duality, uncovering a deep structural symmetry in state transformations.

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arXiv2026

Quantum Algorithm for Elliptic Curve Discrete Logarithms with Space-Efficient Point Addition

The Elliptic Curve Discrete Logarithm Problem (ECDLP) is a fundamental problem in cryptography, and reducing the resource requirements of quantum algorithms for solving ECDLP is an important goal. In this work, we present a space-efficient quantum algorithm for solving the ECDLP over prime fields, achieving an implementation with only $3n+6\lfloor \log_2 n \rfloor+O(1)$ logical qubits and $1056n^3/\log_2 n+O(n^2)$ Toffoli gates, where $n$ is the bit-length of the prime. For a 256-bit prime-field curve, our construction requires only 835 logical qubits, reducing the previous best estimates of 1098 and 1175 logical qubits by Chevignard et al. [EUROCRYPT 2026] and Babbush et al. [ArXiv Preprint 2026], respectively. The key to our improvement is a new space-efficient reversible modular inversion circuit, which addresses the dominant space bottleneck in affine-coordinate point addition. Starting from the extended Euclidean algorithm (EEA), we refine the register-sharing technique of Proos and Zalka by introducing length registers and location-controlled arithmetic to compactly store and update intermediate variables. We further optimize the reversible update procedures and construct the corresponding controlled arithmetic circuits, resulting in a modular inversion circuit implemented by only $2n+6\lfloor \log_2 n \rfloor+O(1)$ logical qubits and $229n^2+O(n\log_2 n)$ Toffoli gates. This modular inversion circuit together with mid-circuit measurements and classical feed-forward operations provides a space-efficient controlled affine point-addition circuit and a complete implementation of Shor's algorithm for ECDLP.

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arXiv2026

Recent Advances in Quantum Architecture Search

Variational quantum algorithms (VQAs) constitute a prominent framework for exploring the capabilities of near-term quantum computers. As the effectiveness of VQAs depends heavily on the design of variational quantum circuits, Quantum Architecture Search (QAS) has emerged as a critical research area to automate the discovery of high-performing circuit structures. This paper reviews key advancements in current research on QAS, including its core concepts, representative methodologies, and applications. The content is structured to ensure broad accessibility for a diverse audience of researchers while preserving the core principles of complex methodologies. In addition, we discuss remaining challenges and suggest potential research directions to offer perspectives on future exploration in this rapidly evolving field.

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arXiv2026

Exponential quantum space advantage for Shannon entropy estimation in data streams

Near-term quantum devices with limited qubits motivate the study of space-bounded quantum computation in the data stream model. We show that Shannon entropy estimation exhibits an exponential separation between quantum and classical space complexity in this setting. Technically, we develop a two-stage quantum streaming algorithm based on a quantum procedure with an explicitly constructed oracle derived from the streaming input. This algorithm achieves logarithmic space complexity in the accuracy parameter over the data stream, whereas any classical streaming algorithm under the same pass complexity requires polynomial space. In sharp contrast, existing results for Shannon entropy estimation in the quantum query model achieve only a quadratic speedup. Our work establishes a natural problem with practical applications in computer networking that admits an exponential quantum space advantage, revealing a fundamental gap between quantum query complexity and streaming space complexity.

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arXiv2026

Optimal qudit overlapping tomography and optimal measurement order

Quantum state tomography is essential for characterizing quantum systems, but it becomes infeasible for large systems due to exponential resource scaling. Overlapping tomography addresses this challenge by reconstructing all $k$-body marginals using few measurement settings, enabling the efficient extraction of key information for many quantum tasks. While optimal schemes are known for qubits, the extension to higher-dimensional qudit systems remains largely unexplored. Here, we investigate optimal qudit overlapping tomography, constructing local measurement settings from generalized Gell-Mann matrices. By establishing a correspondence with combinatorial covering arrays, we present two explicit constructions of optimal measurement schemes. For $n$-qutrit systems, we prove that pairwise tomography requires at most $8 + 56\left\lceil \log_{8} n \right\rceil$ measurement settings, and provide an explicit scheme achieving this bound. Furthermore, we develop an efficient algorithm to determine the optimal order of these measurement settings, minimizing the experimental overhead associated with switching configurations. Compared to the worst-case ordering, our optimized schedule reduces switching costs by approximately 50\%. These results provide a practical pathway for efficient characterization of qudit systems, facilitating their application in quantum communication and computation.

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arXiv2025

Optimal Circuit Size for Fixed-Hamming-Weight Quantum States Preparation

We study the problem of efficiently preparing fixed-Hamming-weight (HW-$k$) quantum states, which are superpositions of $n$-qubit computational basis states with exactly $k$ ones. We present a quantum circuit construction that prepares any $n$-qubit HW-$k$ state with a circuit size of $O(\binom{n}{k})$ using at most $\max\{0, n-3\}$ ancillary qubits. This is the first construction that achieves the theoretical lower bound on circuit size while using only a small number of ancillary qubits. We believe that the techniques presented in this work can be extended to other quantum state preparation algorithms based on decision diagrams, potentially reducing the reliance on ancillary qubits or lowering the overall circuit size.

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arXiv2025

A complete set of transformation rules for reversible circuits

Reversible logic synthesis is a crucial component in quantum electronic design automation. While rule-based methodologies have gained prominence in reversible circuit optimization, the completeness of the transformation rule systems is a longstanding problem in this domain. In this work, we propose the first complete set of transformation rules for reversible circuits, comprising five fundamental rules: any two equivalent reversible circuits can be transformed into each other using the rules. To prove the completeness, a canonical circuit representation for reversible functions is introduced, and we show that every reversible function is computed by a unique reversible circuit in the canonical form, and any reversible circuit can be transformed into its canonical form by applying the rules.

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arXiv2025

Deterministic quantum search on all Laplacian integral graphs

Searching for an unknown marked vertex on a given graph (also known as spatial search) is an extensively discussed topic in the area of quantum algorithms, with a plethora of results based on different quantum walk models and targeting various types of graphs. Most of these algorithms have a non-zero probability of failure. In recent years, there have been some efforts to design quantum spatial search algorithms with $100\%$ success probability. However, these works either only work for very special graphs or only for the case where there is only one marked vertex. In this work, we propose a different and elegant approach to quantum spatial search, obtaining deterministic quantum search algorithms that can find a marked vertex with certainty on any Laplacian integral graph with any predetermined proportion of marked vertices. Thus, this work discovers the largest class of graphs so far that allow deterministic quantum search, making it easy to design deterministic quantum search algorithms for many graphs, including the different graphs discussed in previous works, in a unified framework.

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arXiv2025

Space-time tradeoff for sparse quantum state preparation

In this work, we investigate the trade-off between the circuit depth and the number of ancillary qubits for preparing sparse quantum states. We prove that any $n$-qubit $d$-spare quantum state (i.e., it has only $d$ non-zero amplitudes) can be prepared by a quantum circuit with depth $O\left(\frac{nd \log m}{m \log m/n} + \log nd\right)$ using $m\geq 6n$ ancillary qubits, which achieves the current best trade-off between depth and ancilla number. In particular, when $m = \Theta({\frac{nd}{\log d}})$, our result recovers the optimal circuit depth $\Theta(\log nd)$ given in \hyperlink{cite.zhang2022quantum}{[Phys. Rev. Lett., 129, 230504(2022)]}, but using significantly fewer gates and ancillary qubits.

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arXiv2025

Quantum phase discrimination with applications to quantum search on graphs

We study the phase discrimination problem, in which we want to decide whether the eigenphase $\theta\in(-\pi,\pi]$ of a given eigenstate $|\psi\rangle$ with eigenvalue $e^{i\theta}$ is zero or not, using applications of the unitary $U$ provided as a black box oracle.We propose a quantum algorithm named {\it quantum phase discrimination(QPD)} for this task, with optimal query complexity $\Theta(\frac{1}{\lambda}\log\frac{1}{\delta})$ to the oracle $U$, where $\lambda$ is the gap between zero and non-zero eigenphases and $\delta$ the allowed one-sided error. The quantum circuit is simple, consisting of only one ancillary qubit and a sequence of controlled-$U$ interleaved with single qubit $Y$ rotations, whose angles are given by a simple analytical formula. Quantum phase discrimination could become a fundamental subroutine in other quantum algorithms, as we present two applications to quantum search on graphs: i) Spatial search on graphs. Inspired by the structure of QPD, we propose a new quantum walk model, and based on them we tackle the spatial search problem, obtaining a novel quantum search algorithm. For any graph with any number of marked vertices, the quantum algorithm that can find a marked vertex with probability $\Omega(1)$ in total evolution time $ O(\frac{1}{\lambda \sqrt{\varepsilon}})$ and query complexity $ O(\frac{1}{\sqrt{\varepsilon}})$, where $\lambda$ is the gap between the zero and non-zero eigenvalues of the graph Laplacian and $\varepsilon$ is a lower bound on the proportion of marked vertices. ii) Path-finding on graphs.} By using QPD, we reduce the query complexity of a path-finding algorithm proposed by Li and Zur [arxiv: 2311.07372] from $\tilde{O}(n^{11})$ to $\tilde{O}(n^8)$, in a welded-tree circuit graph with $\Theta(n2^n)$ vertices. Besides these two applications, we argue that more quantum algorithms might benefit from QPD.

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