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Guojing Tian

Publications and source records attributed to Guojing Tian.

15 recordsLinked to original sources
arXiv2025

Toward Minimum Graphic Parity Networks

Quantum circuits composed of CNOT and $R_z$ are fundamental building blocks of many quantum algorithms, so optimizing the synthesis of such quantum circuits is crucial. We address this problem from a theoretical perspective by studying the graphic parity network synthesis problem. A graphic parity network for a graph $G$ is a quantum circuit composed solely of CNOT gates where each edge of $G$ is represented in the circuit, and the final state of the wires matches the original input. We aim to synthesize graphic parity networks with the minimum number of gates, specifically for quantum algorithms addressing combinatorial optimization problems with Ising formulations. We demonstrate that a graphic parity network for a connected graph with $n$ vertices and $m$ edges requires at least $m+n-1$ gates. This lower bound can be improved to $m+\Omega(m) = m+\Omega(n^{1.5})$ when the shortest cycle in the graph has a length of at least five. We complement this result with a simple randomized algorithm that synthesizes a graphic parity network with expected $m + O(n^{1.5}\sqrt{\log n})$ gates. Additionally, we begin exploring connected graphs that allow for graphic parity networks with exactly $m+n-1$ gates. We conjecture that all such graphs belong to a newly defined graph class. Furthermore, we present a linear-time algorithm for synthesizing minimum graphic parity networks for graphs within this class. However, this graph class is not closed under taking induced subgraphs, and we show that recognizing it is $\textsf{NP}$-complete, which is complemented with a fixed-parameter tractable algorithm parameterized by the treewidth.

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arXiv2025

Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits

Quantum state preparation is a critical task in quantum computing, particularly in fields such as quantum machine learning, Hamiltonian simulation, and quantum algorithm design. The depth of preparation circuit for the most general state has been optimized to approximately optimal, but the log-depth appears only when the number of ancillary qubits reaches exponential. Actually, few log-depth preparation algorithms assisted by polynomial ancillary qubits have been come up with even for a certain kind of non-uniform state. We focus on the Hamming-Weight-preserving states, defined as $|\psi_{\text{H}}\rangle = \sum_{\text{HW}(x)=k} \alpha_x |x\rangle$, which have leveraged their strength in quantum machine learning. Especially when $k=2$, such Hamming-Weight-preserving states correspond to simple undirected graphs and will be called graph-structured states. Firstly, for the $n$-qubit general graph-structured states with $m$ edges, we propose an algorithm to build the preparation circuit of $O(\log n)$-depth with $O(m)$ ancillary qubits. Specifically for the $n$-qubit tree-structured and grid-structured states, the number of ancillary qubits in the corresponding preparation circuits can be optimized to zero. Next we move to the preparation for the HWP states with $k\geq 3$, and it can be solved in $O(\log{{n \choose k}})$-depth using $O\left({n \choose k}\right)$ ancillary qubits, while the size keeps $O\big( {n \choose k} \big)$. These depth and size complexities, for any $k \geq 2$, exactly coincide with the lower bounds of $\Omega (\log{{n \choose k}})$-depth and $\Omega ({n \choose k})$-size that we prove lastly, which confirms the near-optimal efficiency of our algorithms.

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arXiv2025

Quantum circuit synthesis with qudit phase gadget method

Current quantum devices have unutilized high-level quantum resources. More and more attention has been paid to the qudit quantum systems with larger than two dimensions to maximize the potential computing power of quantum computation. Then, a natural problem arises: How do we implement quantum algorithms on qudit quantum systems? In this work, we propose a novel qudit phase gadget method for synthesizing the qudit diagonal unitary matrices. This method is suitable for the Noisy Intermediate-Scale Quantum (NISQ) and fault-tolerant eras due to its versatility in different connectivity architectures and the optimality of its resource consumption. The method can work on any connectivity architecture with asymptotic optimal circuit depth and size. For a 10-qutrit diagonal unitary, our algorithm reduces the circuit depth form about 100000 to 500 with 300 ancillary qutrits. Further, this method can be promoted to different quantum circuit synthesis problems, such as quantum state preparation problems, general unitary synthesis problems, etc.

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arXiv2024

Towards determining the presence of barren plateaus in some chemically inspired variational quantum algorithms

In quantum chemistry, the variational quantum eigensolver (VQE) is a promising algorithm for molecular simulations on near-term quantum computers. However, VQEs using hardware-efficient circuits face scaling challenges due to the barren plateau problem. This raises the question of whether chemically inspired circuits from unitary coupled cluster (UCC) methods can avoid this issue. Here we provide theoretical evidence indicating they may not. By examining alternated dUCC ansätze and relaxed Trotterized UCC ansätze, we find that in the infinite depth limit, a separation occurs between particle-hole one- and two-body unitary operators. While one-body terms yield a polynomially concentrated energy landscape, adding two-body terms leads to exponential concentration. Numerical simulations support these findings, suggesting that popular 1-step Trotterized unitary coupled-cluster with singles and doubles (UCCSD) ansätze may not scale. Our results emphasize the link between trainability and circuit expressiveness, raising doubts about VQEs' ability to surpass classical methods.

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arXiv2024

Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices

In the field of quantum linear system algorithms, quantum computing has realized exponential computational advantages over classical computing. However, the focus has been on square coefficient matrices, with few quantum algorithms addressing non-square matrices. Towards this kind of problems defined by $ Ax = b $ where $ A $$ \in\mathbb{R}^{m \times n} $, we propose a quantum algorithm inspired by the classical multi-row iteration method and provide an explicit quantum circuit based on the quantum comparator and Quantum Random Access Memory (QRAM). The time complexity of our quantum multi-row iteration algorithm is $ O(K \log m) $, with $ K $ representing the number of iteration steps, which demonstrates an exponential speedup compared to the classical version. Based on the convergence of the classical multi-row iteration algorithm, we prove that our quantum algorithm converges faster than the quantum one-row iteration algorithm presented in [Phys. Rev. A, 101, 022322 (2020)]. Moreover, our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems and quadratic optimization problems.

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arXiv2024

Towards Optimal Circuit Size for Sparse Quantum State Preparation

Compared to general quantum states, the sparse states arise more frequently in the field of quantum computation. In this work, we consider the preparation for $n$-qubit sparse quantum states with $s$ non-zero amplitudes and propose two algorithms. The first algorithm uses $O(ns/\log n + n)$ gates, improving upon previous methods by $O(\log n)$. We further establish a matching lower bound for any algorithm which is not amplitude-aware and employs at most $\operatorname{poly}(n)$ ancillary qubits. The second algorithm is tailored for binary strings that exhibit a short Hamiltonian path. An application is the preparation of $U(1)$-invariant state with $k$ down-spins in a chain of length $n$, including Bethe states, for which our algorithm constructs a circuit of size $O\left(\binom{n}{k}\log n\right)$. This surpasses previous results by $O(n/\log n)$ and is close to the lower bound $O\left(\binom{n}{k}\right)$. Both the two algorithms shrink the existing gap theoretically and provide increasing advantages numerically.

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arXiv2023

QAOA with fewer qubits: a coupling framework to solve larger-scale Max-Cut problem

Maximum cut (Max-Cut) problem is one of the most important combinatorial optimization problems because of its various applications in real life, and recently Quantum Approximate Optimization Algorithm (QAOA) has been widely employed to solve it. However, as the size of the problem increases, the number of qubits required will become larger. With the aim of saving qubits, we propose a coupling framework for designing QAOA circuits to solve larger-scale Max-Cut problem. This framework relies on a classical algorithm that approximately solves a certain variant of Max-Cut, and we derive an approximation guarantee theoretically, assuming the approximation ratio of the classical algorithm and QAOA. Furthermore we design a heuristic approach that fits in our framework and perform sufficient numerical experiments, where we solve Max-Cut on various $24$-vertex Erdős-Rényi graphs. Our framework only consumes $18$ qubits and achieves $0.9950$ approximation ratio on average, which outperforms the previous methods showing $0.9778$ (quantum algorithm using the same number of qubits) and $0.9643$ (classical algorithm). The experimental results indicate our well-designed quantum-classical coupling framework gives satisfactory approximation ratio while reduces the qubit cost, which sheds light on more potential computing power of NISQ devices.

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arXiv2023

Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis

The Quantum State Preparation problem aims to prepare an $n$-qubit quantum state $|ψ_v\rangle =\sum_{k=0}^{2^n-1}v_k|k\rangle$ from the initial state $|0\rangle^{\otimes n}$, for a given unit vector $v=(v_0,v_1,v_2,\ldots,v_{2^n-1})^T\in \mathbb{C}^{2^n}$ with $\|v\|_2 = 1$. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth and size complexity remain open when ancillary qubits are available. In this paper, we study efficient constructions of quantum circuits with $m$ ancillary qubits that can prepare $|ψ_v\rangle$ in depth $\tilde O\left(\frac{2^n}{m+n}+n\right)$and size $O(2^n)$, achieving the optimal value for both measures simultaneously. These results also imply a depth complexity of $Θ(4^n/(m+n))$ for quantum circuits implementing a general $n$-qubit unitary using $m = O(2^n/n)$ ancillary qubits. This resolves the depth complexity for circuits without ancillary qubits, and for circuits with exponentially many ancillary qubits, this gives a quadratic saving from $O(4^n)$ to $\tilde Θ(2^n)$. Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide range of parameter regime. The results can be viewed as (optimal) time-space tradeoff bounds, which is not only theoretically interesting, but also practically relevant in the current trend that the number of qubits starts to take off, by showing a way to use a large number of qubits to compensate the short qubit lifetime.

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arXiv2023

Optimization of CNOT circuits on limited connectivity architecture

A CNOT circuit is the key gadget for entangling qubits in quantum computing systems. However, the qubit connectivity of noisy intermediate-scale quantum (NISQ) devices is constrained by their {limited connectivity architecture}. To improve the performance of CNOT circuits on NISQ devices, we investigate the optimization of the size/depth of CNOT circuits under the limited connectivity architecture. We present a method that can optimize the size of any $n$-qubit CNOT circuit $O\left(\frac{n^2}{\log δ}\right)$ on any connected graph with minimum degree $δ$, and prove this bound is optimal for the regular graph. For the near-term sparsely connected structure, we additionally present a method that can optimize the size of any $n$-qubit CNOT circuit to below $2n^2$. The numerical experiment shows that our method performs better than state-of-the-art results. Specifically, we present an example to illustrate the applicability of our algorithm. For the grid structure, which is commonly used in current quantum devices, we demonstrate that the depth of any $n$-qubit CNOT circuit can be optimized to be linear in $n$ with certain ancillary qubits (ancillas). Experimental results indicate that this method has significant improvements compared with all of the existing methods. We additionally test our algorithms on the five-qubit IBMQ devices, and the experiments show that the measurement results of the optimized circuit with our algorithm are more robust to noise compared with the IBM mapping method.

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arXiv2022

Near-Term Quantum Computing Techniques: Variational Quantum Algorithms, Error Mitigation, Circuit Compilation, Benchmarking and Classical Simulation

Quantum computing is a game-changing technology for global academia, research centers and industries including computational science, mathematics, finance, pharmaceutical, materials science, chemistry and cryptography. Although it has seen a major boost in the last decade, we are still a long way from reaching the maturity of a full-fledged quantum computer. That said, we will be in the Noisy-Intermediate Scale Quantum (NISQ) era for a long time, working on dozens or even thousands of qubits quantum computing systems. An outstanding challenge, then, is to come up with an application that can reliably carry out a nontrivial task of interest on the near-term quantum devices with non-negligible quantum noise. To address this challenge, several near-term quantum computing techniques, including variational quantum algorithms, error mitigation, quantum circuit compilation and benchmarking protocols, have been proposed to characterize and mitigate errors, and to implement algorithms with a certain resistance to noise, so as to enhance the capabilities of near-term quantum devices and explore the boundaries of their ability to realize useful applications. Besides, the development of near-term quantum devices is inseparable from the efficient classical simulation, which plays a vital role in quantum algorithm design and verification, error-tolerant verification and other applications. This review will provide a thorough introduction of these near-term quantum computing techniques, report on their progress, and finally discuss the future prospect of these techniques, which we hope will motivate researchers to undertake additional studies in this field.

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arXiv2022

BF-QC: Belief Functions on Quantum Circuits

Dempster-Shafer Theory (DST) of belief function is a basic theory of artificial intelligence, which can represent the underlying knowledge more reasonably than Probability Theory (ProbT). Because of the computation complexity exploding exponentially with the increasing number of elements, the practical application scenarios of DST are limited. In this paper, we encode Basic Belief Assignments (BBA) into quantum superposition states and propose the implementation and operation methods of BBA on quantum circuits. We decrease the computation complexity of the matrix evolution on BBA (MEoB) on quantum circuits. Based on the MEoB, we realize the quantum belief functions' implementation, the similarity measurements of BBAs, evidence Combination Rules (CR), and probability transformation on quantum circuits.

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arXiv2021

General Quantum Bernoulli Factory: Framework Analysis and Experiments

The unremitting pursuit for quantum advantages gives rise to the discovery of a quantum-enhanced randomness processing named quantum Bernoulli factory (QBF). This quantum enhanced process can show its priority over the corresponding classical process through readily available experimental resources, thus in the near term it may be capable of accelerating the applications of classical Bernoulli factories, such as the widely used sampling algorithms. In this work, we provide the framework analysis of the QBF. We thoroughly analyze the quantum state evolution in this process, discovering the field structure of the constructible quantum states. Our framework analysis shows that naturally, the previous works can be described as specific instances of this framework. Then, as a proof of principle, we experimentally demonstrate this framework via an entangled two-photon source along with a reconfigurable photonic logic, and show the advantages of the QBF over the classical model through a classically infeasible instance. These results may stimulate the discovery of advantages of the quantum randomness processing in a wider range of tasks, as well as its potential applications.

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arXiv2020

Strong quantum nonlocality for multipartite entangled states

Recently, Halder \emph{et al.} [S. Halder \emph{et al.}, Phys. Rev. Lett. \textbf{122}, 040403 (2019)] present two sets of strong nonlocality of orthogonal product states based on the local irreducibility. However, for a set of locally indistinguishable orthogonal entangled states, the remaining question is whether the states can reveal strong quantum nonlocality. Here we present a general definition of strong quantum nonlocality based on the local indistinguishability. Then, in $2 \otimes 2 \otimes 2$ quantum system, we show that a set of orthogonal entangled states is locally reducible but locally indistinguishable in all bipartitions, which means the states have strong nonlocality. Furthermore, we generalize the result in N-qubit quantum system, where $N\geqslant 3$. Finally, we also construct a class of strong nonlocality of entangled states in $d\otimes d\otimes \cdots \otimes d, d\geqslant 3$. Our results extend the phenomenon of strong nonlocality for entangled states.

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arXiv2020

Strong Quantum Nonlocality without Entanglement in Multipartite Quantum Systems

In this paper, we generalize the concept of strong quantum nonlocality from two aspects. Firstly in $\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ quantum system, we present a construction of strongly nonlocal quantum states containing $6(d-1)^2$ orthogonal product states, which is one order of magnitude less than the number of basis states $d^3$. Secondly, we give the explicit form of strongly nonlocal orthogonal product basis in $\mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3$ quantum system, where four is the largest known number of subsystems in which there exists strong quantum nonlocality up to now. Both the two results positively answer the open problems in [Halder, \textit{et al.}, PRL, 122, 040403 (2019)], that is, there do exist and even smaller number of quantum states can demonstrate strong quantum nonlocality without entanglement.

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arXiv2018

Local unitary classification for sets of generalized Bell states

In this paper, we study the local unitary classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we firstly introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all of pairs in $d\otimes d$ quantum system into $\prod_{j=1}^{n} (k_{j} + 1) $ LU-inequivalent pairs when the prime factorization of $d=\prod_{j=1}^{n}p_j^{k_j}$. Moreover, all of triples in $p^α\otimes p^α$ quantum system for prime $p$ can be partitioned into $\frac{(α+ 3)}{6}p^α + O(αp^{α-1})$ LU-inequivalent triples, especially, when $α=2$ and $p>2$, there are exactly $\lfloor \frac{5}{6}p^{2}\rfloor + \lfloor \frac{p-2}{6}+(-1)^{\lfloor\frac{p}{3}\rfloor}\frac{p}{3}\rfloor + 3$ LU-inequivalent triples.

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