arXiv subjects

Jingquan Luo

Publications and source records attributed to Jingquan Luo.

13 recordsLinked to original sources

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 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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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

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 $m = Θ({\frac{nd}{\log d}})$, our result recovers the optimal circuit depth $Θ(\log nd)$ 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 $θ\in(-π,π]$ of a given eigenstate $|ψ\rangle$ with eigenvalue $e^{iθ}$ 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 $Θ(\frac{1}λ\log\frac{1}δ)$ to the oracle $U$, where $λ$ is the gap between zero and non-zero eigenphases and $δ$ 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 $Ω(1)$ in total evolution time $ O(\frac{1}{λ\sqrt{\varepsilon}})$ and query complexity $ O(\frac{1}{\sqrt{\varepsilon}})$, where $λ$ 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 $Θ(n2^n)$ vertices. Besides these two applications, we argue that more quantum algorithms might benefit from QPD.

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arXiv2025

Nearly Optimal Circuit Size for Sparse Quantum State Preparation

Quantum state preparation is a fundamental and significant subroutine in quantum computing. In this paper, we conduct a systematic investigation on the circuit size (the total count of elementary gates in the circuit) for sparse quantum state preparation. A quantum state is said to be $d$-sparse if it has only $d$ non-zero amplitudes. For the task of preparing an $n$-qubit $d$-sparse quantum state, we obtain the following results: \textbf{Without ancillary qubits:} Any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{nd}{\log n} + n)$ without using ancillary qubits, which improves the previous best results. It is asymptotically optimal when $d = \mathrm{poly}(n)$, and this optimality holds for a broader scope under some reasonable assumptions. \textbf{With limited ancillary qubits:} (i) Based on the first result, we prove for the first time a trade-off between the number of ancillary qubits and the circuit size: any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{nd}{\log (n + m)} + n)$ using $m$ ancillary qubits for any $m \in O(\frac{nd}{\log nd} + n)$. (ii) We establish a matching lower bound $Ω(\frac{nd}{\log {(n + m)} }+ n)$ under some reasonable assumptions, and obtain a slightly weaker lower bound $Ω(\frac{nd}{\log {(n + m)} + \log d} + n)$ without any assumptions. \textbf{With unlimited ancillary qubits:} Given arbitrary amount of ancillary qubits available, the circuit size for preparing $n$-qubit $d$-sparse quantum states is $Θ(\frac{nd}{\log nd} + n)$.

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arXiv2024

Tolerant Quantum Junta Testing

Junta testing for Boolean functions has sparked a long line of work over recent decades in theoretical computer science, and recently has also been studied for unitary operators in quantum computing. Tolerant junta testing is more general and challenging than the standard version. While optimal tolerant junta testers have been obtained for Boolean functions, there has been no knowledge about tolerant junta testers for unitary operators, which was thus left as an open problem in [Chen, Nadimpalli, and Yuen, SODA2023]. In this paper, we settle this problem by presenting the first algorithm to decide whether a unitary is $ε_1$-close to some quantum $k$-junta or is $ε_2$-far from any quantum $k$-junta, where an $n$-qubit unitary $U$ is called a quantum $k$-junta if it only non-trivially acts on just $k$ of the $n$ qubits. More specifically, we present a tolerant tester with $ε_1 = \frac{\sqrtρ}{8} ε$, $ε_2 = ε$, and $ρ\in (0,1)$, and the query complexity is $O\left(\frac{k \log k}{ε^2 ρ(1-ρ)^k}\right)$, which demonstrates a trade-off between the amount of tolerance and the query complexity. Note that our algorithm is non-adaptive which is preferred over its adaptive counterparts, due to its simpler as well as highly parallelizable nature. At the same time, our algorithm does not need access to $U^\dagger$, whereas this is usually required in the literature.

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arXiv2024

Succinct quantum testers for closeness and $k$-wise uniformity of probability distributions

We explore potential quantum speedups for the fundamental problem of testing the properties of closeness and $k$-wise uniformity of probability distributions. Closeness testing is the problem of distinguishing whether two $n$-dimensional distributions are identical or at least $\varepsilon$-far in $\ell^1$- or $\ell^2$-distance. We show that the quantum query complexities for $\ell^1$- and $\ell^2$-closeness testing are $O(\sqrt{n}/\varepsilon)$ and $O(1/\varepsilon)$, respectively, both of which achieve optimal dependence on $\varepsilon$, improving the prior best results of Gilyén and Li (2020). $k$-wise uniformity testing is the problem of distinguishing whether a distribution over $\{0, 1\}^n$ is uniform when restricted to any $k$ coordinates or $\varepsilon$-far from any such distributions. We propose the first quantum algorithm for this problem with query complexity $O(\sqrt{n^k}/\varepsilon)$, achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity $O(n^k/\varepsilon^2)$ by O'Donnell and Zhao (2018). Moreover, when $k = 2$ our quantum algorithm outperforms any classical one because of the classical lower bound $Ω(n/\varepsilon^2)$. All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.

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arXiv2023

Recovering the original simplicity: succinct and deterministic quantum algorithm for the welded tree problem

This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC'2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm theoretically achieves zero-error, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it impossible to have a deterministic quantum algorithm with exponential speedups for the weled tree problem.

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arXiv2023

Playing Mastermind on quantum computers

From the 1970s up to now, Mastermind, a classic two-player game, has attracted plenty of attention, not only from the public as a popular game, but also from the academic community as a scientific issue. Mastermind with n positions and k colors is formally described as: the codemaker privately chooses a secret $s\in [k]^n$, and the coderbreaker want to determine $s$ in as few queries like $f_s(x)$ as possible to the codemaker, where $f_s(x)$ indicates how x is close to s. The complexity of a strategy is measured by the number of queries used. In this work we study playing Mastermind on quantum computers in both non-adaptive and adaptive settings, obtaining efficient quantum algorithms which are all exact (i.e., return the correct result with certainty) and show huge quantum speedups. Technically, we develop a three-step framework for designing quantum algorithms for the general string learning problem, which not only allows huge quantum speedups on playing Mastermind, but also may shed light on exploring quantum speedups for other string learning problems.

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arXiv2023

Concise and Efficient Quantum Algorithms for Distribution Closeness Testing

We study the impact of quantum computation on the fundamental problem of testing the property of distributions. In particular, we focus on testing whether two unknown classical distributions are close or far enough, and propose the currently best quantum algorithms for this problem under the metrics of $l^1$-distance and $l^2$-distance. Compared with the latest results given in \cite{gilyen2019distributional} which relied on the technique of quantum singular value transformation (QSVT), our algorithms not only have lower complexity, but also are more concise.

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arXiv2022

Quantum Speedup and Limitations on Matroid Property Problems

This paper initiates the study of quantum algorithms for matroid property problems. It is shown that quadratic quantum speedup is possible for the calculation problem of finding the girth or the number of circuits (bases, flats, hyperplanes) of a matroid, and for the decision problem of deciding whether a matroid is uniform or Eulerian, by giving a uniform lower bound $Ω(\sqrt{\binom{n}{\lfloor n/2\rfloor}})$ on the query complexity for all these problems. On the other hand, for the uniform matroid decision problem, an asymptotically optimal quantum algorithm is proposed which achieves the lower bound, and for the girth problem, an almost optimal quantum algorithm is given with query complexity $O(\log n\sqrt{\binom{n}{\lfloor n/2\rfloor}})$. In addition, for the paving matroid decision problem, a lower bound $Ω(\sqrt{\binom{n}{\lfloor n/2\rfloor}/n})$ on the query complexity is obtained, and an $O(\sqrt{\binom{n}{\lfloor n/2\rfloor}})$ quantum algorithm is presented.

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