Solving in the quantum computer every classical NP problem in polynomial time

缓解线性化后的挑战以在量子计算机上求解非线性系统

Alleviating Post-Linearization Challenges for Solving Nonlinear Systems on a Quantum Computer

本文提出利用Sigma基分解和单位化完成方法，在量子计算机上高效求解非线性系统。

Solving systems of linear equations on a quantum computer

Systems of linear equations are used to model a wide array of problems in all fields of science and engineering. Recently, it has been shown that quantum computers could solve linear systems exponentially faster than classical computers, making for one of the most promising applications of quantum computation. Here, we demonstrate this quantum algorithm by implementing various instances on a photonic quantum computing architecture. Our implementation involves the application of two consecutive entangling gates on the same pair of polarisation-encoded qubits. We realize two separate controlled-…

Towards solving the BCS Hamiltonian gap in Near-Term Quantum Computers

In this work, using a NISQ framework, we obtain the gap of a BCS Hamiltonian. This could lead to interesting implications for superconductivity research. For such task, we choose to use the Variational Quantum Deflation and analyze the hardware restrictions that are needed to find the energy spectra on current quantum hardware. We also compare two different kinds of classical optimizers, Constrained Optimization BY Linear Approximations (COBYLA) and Simultaneous Perturbation Stochastic Approximation (SPSA), and study the effect of decoherence caused by the presence of noise when using simulati…

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6$^{\mathrm{th…

Solving Satisfiability Problems by the Ground-State Quantum Computer

A quantum algorithm is proposed to solve the Satisfiability problems by the ground-state quantum computer. The scale of the energy gap of the ground-state quantum computer is analyzed for the 3-bit Exact Cover problem. The time cost of this algorithm on the general SAT problems is discussed.

An adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer

The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA ansatz is not optimal, there is no systematic approach for finding better ansätze. We address this problem by developing an iterative version of QAOA that is problem-tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, whi…

在量子计算机上求解分数阶微分方程：一种变分方法

Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach

本研究提出一种求解Caputo时间分数阶偏微分方程的变分混合量子-经典算法，采用含重叠历史态线性组合的可迭代代价函数，在时间、内存效率及保真度等方面表现优异，并通过多类工程方程与噪声硬件实验验证了实用性。

在量子计算机上求解非线性微分方程的无发散算法

Divergence-free algorithms for solving nonlinear differential equations on quantum computers

本文提出在量子计算机上求解非线性微分方程的无发散算法，通过引入pivot状态并进行Carleman线性化，突破理论进化时间限制。

使用凸插值改进NISQ计算机上VQE解决投资组合优化的可训练性

Improving the trainability of VQE on NISQ computers for solving portfolio optimization using convex interpolation

本文利用凸插值改进变分量子本征求解器（VQE）的可训练性，通过近解初始化、正则代价函数景观和递归拟设平衡分区策略，在仅用10个超导量子比特实现40量子比特实验，并融合贪心算法实现全局与局部优化的互补。

机器学习量子实验数据以解决量子多体问题

Machine learning on quantum experimental data toward solving quantum many-body problems

该研究将经典机器学习与量子实验数据结合，在127量子比特硬件上降噪后，成功预测基态性质并分类量子相，验证了方法在44量子比特系统中的可扩展性。

Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer

For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.

Solving the quantum search problem in polynomial time on an NMR quantum computer

The quantum search problem is an important problem due to the fact that a general NP problem can be solved efficiently by an unsorted quantum search algorithm. Here it has been shown that the quantum search problem could be solved in polynomial time on an NMR quantum computer. The NMR ensemble quantum computation is based on the quantum mechanical unitary dynamics that both a closed quantum system and its ensemble obey the same quantum mechanical unitary dynamics instead of on the pseudopure state or the effective pure state of the classical NMR quantum computation. Based on the new principle …

在量子计算机上求解泊松方程的变分量子算法性能研究

A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer

本文系统研究了在真实量子设备上使用变分量子算法求解泊松方程的性能，发现无噪声模拟结果具有欺骗性，实际量子计算机表现很差，并指出振幅编码直接求解 PDE 在当前 NISQ 时代不具竞争力。

五个入门问题：在量子计算机上求解二次无约束二元优化模型

Five Starter Problems: Solving Quadratic Unconstrained Binary Optimization Models on Quantum Computers

本教程面向学生与行业从业者，介绍在IBM门电路架构与D-Wave量子退火机两类现有量子设备及模拟器上求解QUBO模型的方法，含实例、算法基础与实现指南，是量子求解复杂QUBO的入门资源。

Strategies for solving the Fermi-Hubbard model on near-term quantum computers

The Fermi-Hubbard model is of fundamental importance in condensed-matter physics, yet is extremely challenging to solve numerically. Finding the ground state of the Hubbard model using variational methods has been predicted to be one of the first applications of near-term quantum computers. Here we carry out a detailed analysis and optimisation of the complexity of variational quantum algorithms for finding the ground state of the Hubbard model, including costs associated with mapping to a real-world hardware platform. The depth complexities we find are substantially lower than previous work. …

在量子计算机上求解偏微分方程的高阶格式

High order schemes for solving partial differential equations on a quantum computer

本文提出一种将 $d$-带对角矩阵分解为对易泡利字符串集合的方法，用于优化量子计算机求解偏微分方程的门数量，并通过一维波动方程数值模拟证明高阶方法能减少量子比特需求但不减少Trotter步数。

Experimental realization of quantum algorithm for solving linear systems of equations

Quantum computers have the potential of solving certain problems exponentially faster than classical computers. Recently, Harrow, Hassidim and Lloyd proposed a quantum algorithm for solving linear systems of equations: given an $N\times{N}$ matrix $A$ and a vector $\vec b$, find the vector $\vec x$ that satisfies $A\vec x = \vec b$. It has been shown that using the algorithm one could obtain the solution encoded in a quantum state $|x$ using $O(\log{N})$ quantum operations, while classical algorithms require at least O(N) steps. If one is not interested in the solution $\vec{x}$ itself but cer…

Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer

Many eigenvalue problems arising in practice are often of the generalized form $A\x=λB\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than…

在光子量子计算机上求解组合优化问题

Solving Combinatorial Optimization Problems on a Photonic Quantum Computer

本文研究光子量子计算求解组合优化问题，概述了针对不同量子架构的算法，并在8量子模式设备上通过实验与模拟验证了其在 Max-Cut 和作业车间调度等问题上的性能。

在量子计算机上以指数级减少的量子比特数求解多种NP难问题

Solving various NP-Hard problems using exponentially fewer qubits on a Quantum Computer

本文提出一种随问题规模对数级扩展的量子优化方法，在NISQ设备上以指数级减少的量子比特数求解最大割、最大团等NP难问题，将可处理图规模提升近十倍。

一种利用噪声在量子计算机上求解开放系统动力学的量子算法

A quantum algorithm for solving open system dynamics on quantum computers using noise

本文提出一种将量子比特噪声作为资源、通过数字化调控浴谱函数来模拟有限温度开放系统动力学的量子算法，并在自旋-玻色子模型上验证了其在门错误高达1%时仍具有良好性能。

在量子计算机上通过测量解决线性方程组

Solving a Linear System of Equations on a Quantum Computer by Measurement

本文提出一种变分算法，通过测量测试直接最大化目标保真度参数，用于求解线性方程组，克服了传统变分算法的三个问题。

求根分解与求解非线性代数方程的量子加速

Quantum Speedup in Dissecting Roots and Solving Nonlinear Algebraic Equations

本文证明量子计算机在检测函数根存在性上具有近指数级加速，并能以平方级加速求解非线性代数方程，同时展示了其在稠密线性系统、非线性动力学、偏微分方程和机器人运动规划等方面的实际应用优势。

Solving machine learning optimization problems using quantum computers

Classical optimization algorithms in machine learning often take a long time to compute when applied to a multi-dimensional problem and require a huge amount of CPU and GPU resource. Quantum parallelism has a potential to speed up machine learning algorithms. We describe a generic mathematical model to leverage quantum parallelism to speed-up machine learning algorithms. We also apply quantum machine learning and quantum parallelism applied to a $3$-dimensional image that vary with time.

Experimental Realization of a One-way Quantum Computer Algorithm Solving Simon's Problem

We report an experimental demonstration of a one-way implementation of a quantum algorithm solving Simon's Problem - a black box period-finding problem which has an exponential gap between the classical and quantum runtime. Using an all-optical setup and modifying the bases of single-qubit measurements on a five-qubit cluster state, key representative functions of the logical two-qubit version's black box can be queried and solved. To the best of our knowledge, this work represents the first experimental realization of the quantum algorithm solving Simon's Problem. The experimental results are…

Solving hadron structures using the basis light-front quantization approach on quantum computers

Quantum computing has demonstrated the potential to revolutionize our understanding of nuclear, atomic, and molecular structure by obtaining forefront solutions in non-relativistic quantum many-body theory. In this work, we show that quantum computing can be used to solve for the structure of hadrons, governed by strongly-interacting relativistic quantum field theory. Following our previous work on light unflavored mesons as a relativistic bound-state problem within the nonperturbative Hamiltonian formalism, we present the numerical calculations on simulated quantum devices using the basis lig…

在量子计算机上求解线性微分方程的成本：快速推进到显式资源计数

The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts

本研究首次给出线性常微分方程解量子态编码的非渐近成本计算，证明稳定经典动力学可实现稳定性诱导的量子模拟快速推进，在多场景下实现渐近改进，二次降低查询复杂度。

Solving combinatorial problems by two D_Wave hybrid solvers: a case study of traveling salesman problems in the TSP Library

The D_Wave quantum computer is an analog device that approximates optimal solutions to optimization problems. The traveling salesman problems in the TSP Library are too large to process on the D_Wave quantum computer DW_2000Q_6. We report favorable approximations for solving the smallest, symmetric traveling salesman problems in the TSP Library by two D_Wave hybrid solvers, Kerberos and LeapHybridSampler. This is useful work about results from new quantum tools on problems that have been studied. It is expected to show a quantum way forward with larger problems when the hardware is upgraded. A…

在量子计算机上求解非线性弗拉索夫方程

Solving the Nonlinear Vlasov Equation on a Quantum Computer

本文探究基于卡尔曼线性化的量子算法求解带克鲁克型碰撞项的非线性静电弗拉索夫方程的适用性，推导其复杂度上界，发现其复杂度高于经典算法，且收敛准则对等离子体应用有严格限制。

Experimental Quantum Computing to Solve Systems of Linear Equations

Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2*2 linear equations for various input vectors on a quantum computer. We use…

Quantum simulation and circuit design for solving multidimensional Poisson equations

Many methods solve Poisson equations by using grid techniques which discretize the problem in each dimension. Most of these algorithms are subject to the curse of dimensionality, so that they need exponential runtime. In the paper "Quantum algorithm and circuit design solving the Poisson equation" a quantum algorithm is shown running in polylog time to produce a quantum state representing the solution of the Poisson equation. In this paper a quantum simulation of an extended circuit design based on this algorithm is made on a classical computer. Our purpose is to test an efficient circuit desi…

Quantum Annealing-Based Software Components: An Experimental Case Study with SAT Solving

Quantum computers have the potential of solving problems more efficiently than classical computers. While first commercial prototypes have become available, the performance of such machines in practical application is still subject to exploration. Quantum computers will not entirely replace classical machines, but serve as accelerators for specific problems. This necessitates integrating quantum computational primitives into existing applications. In this paper, we perform a case study on how to augment existing software with quantum computational primitives for the Boolean satisfiability prob…

Quantum Computers Speed Up Classical with Probability Zero

Let $f$ denote length preserving function on words. A classical algorithm can be considered as $T$ iterated applications of black box representing $f$, beginning with input word $x$ of length $n$. It is proved that if $T=O(2^{n/(7+e)}), e >0$, and $f$ is chosen randomly then with probability 1 every quantum computer requires not less than $T$ evaluations of $f$ to obtain the result of classical computation. It means that the set of classical algorithms admitting quantum speeding up has probability measure zero. The second result is that for arbitrary classical time complexity $T$ and $f$ chose…

在量子计算机上求解非线性微分方程：一种 Fokker-Planck 方法

Solving nonlinear differential equations on Quantum Computers: A Fokker-Planck approach

本文提出借助 Fokker-Planck 方程将非线性动力系统转化为线性系统，并结合三种量子积分策略求解，模拟结果与经典基准解高度一致。

Simulation of single photon dynamics in coupled cavities through IBM quantum computer

We design a quantum circuit in IBM quantum computer that mimics the dynamics of single photon in a coupled cavity system. By suitably choosing the gate parameters in the quantum circuit, we could transfer an unknown qubit state between the qubits. The condition for perfect state transfer is obtained by solving the unitary time dynamics governed by the Hamiltonian of the coupled cavity system. We then demonstrate the dynamics of entanglement between the two-qubits and show violation of Bell's inequality in IBM quantum computer.

基于Woodbury恒等式的近期量子求解线性方程组算法

A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity

针对现有量子线性方程组求解算法对容错硬件要求高或优化易陷局部最优的问题，提出基于Woodbury恒等式的近期量子算法，仅用基本量子子程序在IBM量子计算机上以2%误差求解超1600万方程。

Solving strongly correlated electron models on a quantum computer

One of the main applications of future quantum computers will be the simulation of quantum models. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground…

面向使用 QAOA 求解 Max-Cut 问题的量子电路模拟优化

Towards Optimizations of Quantum Circuit Simulation for Solving Max-Cut Problems with QAOA

针对 QAOA 在经典模拟大规模量子电路时速度慢的问题，本文提出数学压缩、按位运算和多核并行相结合的 QCS 加速方法，并以 Max-Cut 验证有效性。

Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation

Computer-aided engineering techniques are indispensable in modern engineering developments. In particular, partial differential equations are commonly used to simulate the dynamics of physical phenomena, but very large systems are often intractable within a reasonable computation time, even when using supercomputers. To overcome the inherent limit of classical computing, we present a variational quantum algorithm for solving the Poisson equation that can be implemented in noisy intermediate-scale quantum devices. The proposed method defines the total potential energy of the Poisson equation as…

Balanced k-Means Clustering on an Adiabatic Quantum Computer

Adiabatic quantum computers are a promising platform for approximately solving challenging optimization problems. We present a quantum approach to solving the balanced $k$-means clustering training problem on the D-Wave 2000Q adiabatic quantum computer. Existing classical approaches scale poorly for large datasets and only guarantee a locally optimal solution. We show that our quantum approach better targets the global solution of the training problem, while achieving better theoretic scalability on large datasets. We test our quantum approach on a number of small problems, and observe cluster…

Polynomial Lie algebra methods in solving the second-harmonic generation model: some exact and approximate calculations

We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of a polynomial Lie algebra $su_{pd}(2)$ and related spectral representations of the model evolution operator realized in algorithmic forms. It enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.

求解漂移-扩散方程的量子算法：复杂度分析

Quantum algorithms for solving a drift-diffusion equation: A complexity analysis

本文提出四种求解多维漂移-扩散方程的量子算法，并比较其复杂度，发现基于量子傅里叶变换的对角化方法在固定最终时间求解线性偏微分方程时具有量子优势，并利用多维振幅估计提取完整概率分布。

Backend compiler phases for trapped-ion quantum computers

A promising architecture for scaling up quantum computers based on trapped ions are so called Quantum Charged-Coupled Devices (QCCD). These consist of multiple ion traps, each designed for solving specific tasks, that are connected by transport links. In this paper we present the backend compiler phases needed for running quantum circuits on a QCCD architecture, while providing strategies to solve the optimization problems that occur when generating assembly instructions. We implement and test these strategies for the QVLS-Q1 chip architecture.

Simulation of multiple-quantum NMR dynamics of spin dimer on quantum computer

Dymanics of spin dimers in multiple quantum NMR experiment is studied on the 5-qubit superconducting quantum processor of IBM {Quantum Experience} for the both {pure} ground and thermodynamic equilibrium (mixed) initial states. The work can be considered as a first step towards an application of quantum computers to solving problems of magnetic resonance. This article is dedicated to Prof. Klaus Möbius and Prof. Kev Salikhov on the occasion of their 85th birthdays.

Noisy intermediate-scale quantum computing algorithm for solving an $n$-vertex MaxCut problem with log($n$) qubits

Quantum computers are devices, which allow more efficient solutions of problems as compared to their classical counterparts. As the timeline to developing a quantum-error corrected computer is unclear, the quantum computing community has dedicated much attention to developing algorithms for currently available noisy intermediate-scale quantum computers (NISQ). Thus far, within NISQ, optimization problems are one of the most commonly studied and are quite often tackled with the quantum approximate optimization algorithm (QAOA). This algorithm is best known for computing graph partitions with a …

Implementation of a Multiple Target Tracking Filter on an Adiabatic Quantum Computer

Recent work at Fraunhofer FKIE shows that Morefield's method for multiple target data association can in theory be solved on an adiabatic quantum computer. The present paper validates the theory and examines the significant limitations of currently available adiabatic quantum computers for solving the data association problem. The limitations of such architectures are both theoretical and practical in nature, and both are discussed. The data association problem is formulated as a quadratic unconstrained binary optimization (QUBO) problem; consequently, much of the discussion is relevant to oth…

Solving optimization problems with Rydberg analog quantum computers: Realistic requirements for quantum advantage using noisy simulation and classical benchmarks

Platforms of Rydberg atoms have been proposed as promising candidates to solve some combinatorial optimization problems. Here, we compute quantitative requirements on the system sizes and noise levels that these platforms must fulfill to reach quantum advantage in approximately solving the Unit-Disk Maximum Independent Set problem. Using noisy simulations of Rydberg platforms of up to 26 atoms interacting through realistic van der Waals interactions, we compute the average approximation ratio that can be attained with a simple quantum annealing-based heuristic within a fixed temporal computati…

Quantum vs. classical algorithms for solving the heat equation

Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE - the heat equation in a rectangular region - and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension $d \ge 2$, there is an at most quadratic quantum speedup using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative ap…

QuMeld：一个用于量子比特映射算法基准测试的模块化框架

QuMeld: A Modular Framework for Benchmarking Qubit Mapping Algorithms

QuMeld是一个开源框架，用于评估和比较不同量子比特映射算法，支持多种拓扑结构和评估指标，具有良好的扩展性。

Solving Quantum Ground-State Problems with Nuclear Magnetic Resonance

Quantum ground-state problems are computationally hard problems; for general many-body Hamiltonians, there is no classical or quantum algorithm known to be able to solve them efficiently. Nevertheless, if a trial wavefunction approximating the ground state is available, as often happens for many problems in physics and chemistry, a quantum computer could employ this trial wavefunction to project the ground state by means of the phase estimation algorithm (PEA). We performed an experimental realization of this idea by implementing a variational-wavefunction approach to solve the ground-state pr…

Quantum circuits based on qutrits as a tool for solving systems of linear equations

Recently, it has been presented some algorithms and physical models which give prospects for construction of quantum computers capable to solve systems of linear equations. The common feature which is shared in these works is the use of qubits which allow to solve systems with $2^n$ variables. In this work we propose a quantum circuit based on qutrits architecture which directly allows for solving systems of equations with three variables. Proposed circuit can be easily generalized to those with $3^n$ variables. We also present some numerical experiments to verify the correctness of proposed s…

使用效用级量子计算机进行mRNA二级结构预测

mRNA secondary structure prediction using utility-scale quantum computers

本文探讨了在效用级量子计算机上预测mRNA二级结构的可行性，使用基于CVaR的VQE算法在IBM量子处理器上处理最高60个核苷酸的问题，并在最小误差缓解下取得了与经典求解器CPLEX匹配的最小自由能结构预测结果。

基于量子退火求解线性方程算法的收敛速度

Convergence rate of algorithms for solving linear equations by quantum annealing

本文研究了基于量子退火原理的量子计算机求解线性方程 $ax=b$ 的迭代算法，在输出服从玻尔兹曼分布的假设下证明了算法的收敛条件并给出了收敛速度估计，同时探讨了该方法在无限和少量量子比特算法中的应用。

QUBO formulations for numerical quantum computing

With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all science and engineering. The Harrow-Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on gate model quantum computers, was invented and several advanced variations have been developed. For a given n by n matrix A and a vector b, we will find unconstrained binary optimization (QUBO) models for a vector x that satisfies Ax=b. To formulate QUBO models for a linear system solving problem, we make …