arXiv subjects

Lvzhou Li

Publications and source records attributed to Lvzhou Li.

At least 37 records · Page 2Linked to original sources

arXiv2025

Revisiting fixed-point quantum search: proof of the quasi-Chebyshev lemma

The original Grover's algorithm suffers from the souffle problem, which means that the success probability of quantum search decreases dramatically if the iteration time is too small or too large from the right time. To overcome the souffle problem, the fixed-point quantum search with an optimal number of queries was proposed [Phys. Rev. Lett. 113, 210501 (2014)], which always finds a marked state with a high probability when a lower bound of the proportion of marked states is given. The fixed-point quantum search relies on a key lemma regarding the explicit formula of recursive quasi-Chebyshev polynomials, but its proof is not given explicitly. In this work, we give a detailed proof of this lemma, thus providing a sound foundation for the correctness of the fixed-point quantum search. This lemma may be of independent interest as well, since it expands the mathematical form of the recursive relation of Chebyshev polynomials of the first kind, and it also constitutes a key component in overcoming the souffle problem of quantum walk-based search algorithms, for example, robust quantum walk search on complete bipartite graphs [Phys. Rev. A 106, 052207 (2022)]. The lemma is also central to a recently proposed quantum algorithm named quantum phase discrimination, which has become a fundamental subroutine in quantum search on graphs [arxiv: 2504.15194]. Hopefully, more applications of the lemma will be found in the future.

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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

A High-Performance List Decoding Algorithm for Surface Codes with Erroneous Syndrome

Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.

cs.IT↗

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

Quantum Algorithm for Online Exp-concave Optimization

We explore whether quantum advantages can be found for the zeroth-order feedback online exp-concave optimization problem, which is also known as bandit exp-concave optimization with multi-point feedback. We present quantum online quasi-Newton methods to tackle the problem and show that there exists quantum advantages for such problems. Our method approximates the Hessian by quantum estimated inexact gradient and can achieve $O(n\log T)$ regret with $O(1)$ queries at each round, where $n$ is the dimension of the decision set and $T$ is the total decision rounds. Such regret improves the optimal classical algorithm by a factor of $T^{2/3}$.

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arXiv2024

Implementation of Continuous-Time Quantum Walk on Sparse Graph

Continuous-time quantum walks (CTQWs) play a crucial role in quantum computing, especially for designing quantum algorithms. However, how to efficiently implement CTQWs is a challenging issue. In this paper, we study implementation of CTQWs on sparse graphs, i.e., constructing efficient quantum circuits for implementing the unitary operator $e^{-iHt}$, where $H=γA$ ($γ$ is a constant and $A$ corresponds to the adjacency matrix of a graph). Our result is, for a $d$-sparse graph with $N$ vertices and evolution time $t$, we can approximate $e^{-iHt}$ by a quantum circuit with gate complexity $(d^3 \|H\| t N \log N)^{1+o(1)}$, compared to the general Pauli decomposition, which scales like $(\|H\| t N^4 \log N)^{1+o(1)}$. For sparse graphs, for instance, $d=O(1)$, we obtain a noticeable improvement. Interestingly, our technique is related to graph decomposition. More specifically, we decompose the graph into a union of star graphs, and correspondingly, the Hamiltonian $H$ can be represented as the sum of some Hamiltonians $H_j$, where each $e^{-iH_jt}$ is a CTQW on a star graph which can be implemented efficiently.

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arXiv2024

Distributed quantum architecture search

Variational quantum algorithms, inspired by neural networks, have become a novel approach in quantum computing. However, designing efficient parameterized quantum circuits remains a challenge. Quantum architecture search tackles this by adjusting circuit structures along with gate parameters to automatically discover high-performance circuit structures. In this study, we propose an end-to-end distributed quantum architecture search framework, where we aim to automatically design distributed quantum circuit structures for interconnected quantum processing units with specific qubit connectivity. We devise a circuit generation algorithm which incorporates TeleGate and TeleData methods to enable nonlocal gate implementation across quantum processing units. While taking into account qubit connectivity, we also incorporate qubit assignment from logical to physical qubits within our quantum architecture search framework. A two-stage progressive training-free strategy is employed to evaluate extensive circuit structures without circuit training costs. Through numerical experiments on three VQE tasks, the efficacy and efficiency of our scheme is demonstrated. Our research into discovering efficient structures for distributed quantum circuits is crucial for near-term quantum computing where a single quantum processing unit has a limited number of qubits. Distributed quantum circuits allow for breaking down complex computations into manageable parts that can be processed across multiple quantum processing units.

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arXiv2024

Unifying quantum spatial search, state transfer and uniform sampling on graphs: simple and exact

This article presents a novel and succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex-transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with certainty. In contrast, existing quantum search algorithms generally has a certain probability of failure. Even if the graph is not vertex-transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides new results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs.

quant-ph↗

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.

quant-ph↗

arXiv2024

Variational learning algorithms for quantum query complexity

Quantum query complexity plays an important role in studying quantum algorithms, which captures the most known quantum algorithms, such as search and period finding. A query algorithm applies $U_tO_x\cdots U_1O_xU_0$ to some input state, where $O_x$ is the oracle dependent on some input variable $x$, and $U_i$s are unitary operations that are independent of $x$, followed by some measurements for readout. In this work, we develop variational learning algorithms to study quantum query complexity, by formulating $U_i$s as parameterized quantum circuits and introducing a loss function that is directly given by the error probability of the query algorithm. We apply our method to analyze various cases of quantum query complexity, including a new algorithm solving the Hamming modulo problem with $4$ queries for the case of $5$-bit modulo $5$, answering an open question raised in arXiv:2112.14682, and the result is further confirmed by a Semidefinite Programming (SDP) algorithm. Compared with the SDP algorithm, our method can be readily implemented on the near-term Noisy Intermediate-Scale Quantum (NISQ) devices and is more flexible to be adapted to other cases such as the fractional query models.

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arXiv2024

Small sets of genuinely nonlocal GHZ states in multipartite systems

A set of orthogonal multipartite quantum states are called (distinguishability-based) genuinely nonlocal if they are locally indistinguishable across any bipartition of the subsystems. In this work, we consider the problem of constructing small genuinely nonlocal sets consisting of generalized GHZ states in multipartite systems. For system (C^2)^(\otimes N) where N is large, using the language of group theory, we show that a tiny proportion Θ[1/2^(N/2)] of the states among the N-qubit GHZ basis suffice to exhibit genuine nonlocality. Similar arguments also hold for the canonical generalized GHZ bases in systems (C^d)^(\otimes N), wherever d is even and N is large. What is more, moving to the condition that any fixed N is given, we show that d + 1 genuinely nonlocal generalized GHZ states exist in (C^d)^(\otimes N), provided the local dimension d is sufficiently large. As an additional merit, within and beyond an asymptotic sense, the latter result also indicates some evident limitations of the "trivial othogonality-preserving local measurements" (TOPLM) technique that has been utilized frequently for detecting genuine nonlocality.

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arXiv2023

Lifting query complexity to time-space complexity for two-way finite automata

Time-space tradeoff has been studied in a variety of models, such as Turing machines, branching programs, and finite automata, etc. While communication complexity as a technique has been applied to study finite automata, it seems it has not been used to study time-space tradeoffs of finite automata. We design a new technique showing that separations of query complexity can be lifted, via communication complexity, to separations of time-space complexity of two-way finite automata. As an application, one of our main results exhibits the first example of a language $L$ such that the time-space complexity of two-way probabilistic finite automata with a bounded error (2PFA) is $\widetildeΩ(n^2)$, while of exact two-way quantum finite automata with classical states (2QCFA) is $\widetilde{O}(n^{5/3})$, that is, we demonstrate for the first time that exact quantum computing has an advantage in time-space complexity comparing to classical computing.

cs.CC↗

arXiv2023

Distinguishability-based genuine nonlocality with genuine multipartite entanglement

A set of orthogonal multipartite quantum states is said to be distinguishability-based genuinely nonlocal (also genuinely nonlocal, for abbreviation) if the states are locally indistinguishable across any bipartition of the subsystems. This form of multipartite nonlocality, although more naturally arising than the recently popular "strong nonlocality" in the context of local distinguishability, receives much less attention. In this work, we study the distinguishability-based genuine nonlocality of a typical type of genuine multipartite entangled states -- the d-dimensional GHZ states, featuring systems with local dimension not limited to 2. In the three-partite case, we find the existence of small genuinely nonlocal sets consisting of these states: we show that the cardinality can at least scale down to linear in the local dimension d, with the linear factor l = 1. Specifically, the method we use is semidefinite program and the GHZ states to construct these sets are special ones which we call "GHZ-lattices". This result might arguably suggest a significant gap between the strength of strong nonlocality and the distinguishability-based genuine nonlocality. Moreover, we put forward the notion of (s,n)-threshold distinguishability and utilizing a similar method, we successfully construct (2,3)-threshold sets consisting of GHZ states in three-partite systems.

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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.

quant-ph↗

arXiv2023

Derandomization of quantum algorithm for triangle finding

Derandomization is the process of taking a randomized algorithm and turning it into a deterministic algorithm, which has attracted great attention in classical computing. In quantum computing, it is challenging and intriguing to derandomize quantum algorithms, due to the inherent randomness of quantum mechanics. The significance of derandomizing quantum algorithms lies not only in theoretically proving that the success probability can essentially be 1 without sacrificing quantum speedups, but also in experimentally improving the success rate when the algorithm is implemented on a real quantum computer. In this paper, we focus on derandomizing quanmtum algorithms for the triangle sum problem (including the famous triangle finding problem as a special case), which asks to find a triangle in an edge-weighted graph with $n$ vertices, such that its edges sum up to a given weight.We show that when the graph is promised to contain at most one target triangle, there exists a deterministic quantum algorithm that either finds the triangle if it exists or outputs ``no triangle'' if none exists. It makes $O(n^{9/7})$ queries to the edge weight matrix oracle, and thus has the same complexity with the state-of-art bounded-error quantum algorithm. To achieve this derandomization, we make full use several techniques:nested quantum walks with quantum data structure, deterministic quantum search with adjustable parameters, and dimensional reduction of quantum walk search on Johnson graph.

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