Noisy Intermediate-scale Quantum Computers Research Articles

Warm-Starting and Quantum Computing: A Systematic Mapping Study

Due to low numbers of qubits and their error-proneness, Noisy Intermediate-Scale Quantum (NISQ) computers impose constraints on the size of quantum algorithms they can successfully execute. State-of-the-art research introduces various techniques addressing these limitations by utilizing known or inexpensively generated approximations, solutions, or models as a starting point to approach a task instead of starting from scratch. These so-called warm-starting techniques aim to reduce quantum resource consumption, thus facilitating the design of algorithms suiting the capabilities of NISQ computers. In this work, we collect and analyze scientific literature on warm-starting techniques in the quantum computing domain. In particular, we (i) create a systematic map of state-of-the-art research on warm-starting techniques using established guidelines for systematic mapping studies, (ii) identify relevant properties of such techniques, and (iii) based on these properties classify the techniques identified in the literature in an extensible classification scheme. Our results provide insights into the research field and aim to help quantum software engineers to categorize warm-starting techniques and apply them in practice. Moreover, our contributions may serve as a starting point for further research on the warm-starting topic since they provide an overview of existing work and facilitate the identification of research gaps.

Efficient quantum simulation of open quantum system dynamics on noisy quantum computers

Quantum simulation represents the most promising quantum application to demonstrate quantum advantage on near-term noisy intermediate-scale quantum (NISQ) computers, yet available quantum simulation algorithms are prone to errors and thus difficult to realize with limited circuit depth on nowadays quantum devices. Herein, we propose a novel scheme to utilize intrinsic gate errors of NISQ devices to enable controllable simulation of open quantum system dynamics without ancillary qubits or explicit bath engineering, thus turning unwanted quantum noises into useful quantum resources. Specifically, we simulate the energy transfer process in a photosynthetic dimer system on IBM-Q cloud. By employing tailored decoherence-inducing gates, we show that quantum dissipative dynamics can be simulated efficiently across coherent-to-incoherent regimes with results comparable to those of the numerically exact classical method. Moreover, we demonstrate a calibration routine that enables consistent and predictive simulations of open-quantum system dynamics in the intermediate coupling regime. This work provides a new direction for quantum advantage in the NISQ era.

Systematic study on the dependence of the warm-start quantum approximate optimization algorithm on approximate solutions

Quantum approximate optimization algorithm (QAOA) is a promising hybrid quantum-classical algorithm to solve combinatorial optimization problems in the era of noisy intermediate-scale quantum computers. Recently it has been revealed that warm-start approaches can improve the performance of QAOA, where approximate solutions are obtained by classical algorithms in advance and incorporated into the initial state and/or unitary ansatz. In this work, we study in detail how the accuracy of approximate solutions affects the performance of the warm-start QAOA (WS-QAOA). We numerically find that in typical MAX-CUT problems, WS-QAOA achieves higher fidelity (probability that exact solutions are observed) and approximation ratio than QAOA as the Hamming distance of approximate solutions to the exact ones becomes smaller. We reveal that this could be quantitatively attributed to the initial state of the ansatz. We also solve MAX-CUT problems by WS-QAOA with approximate solutions obtained via QAOA, having higher fidelity and approximation ratio than QAOA especially when the circuit is relatively shallow. We believe that our study may deepen understanding of the performance of WS-QAOA and also provide a guide as to the necessary quality of approximate solutions.

Layerwise Quantum Deep Reinforcement Learning for Joint Optimization of UAV Trajectory and Resource Allocation

This study proposes a layerwise quantum-based deep reinforcement learning (LQ-DRL) method for optimizing continuous large space and time series problems using deep layer training. The actions in LQ-DRL are optimized using a layerwise quantum embedding that leverages the advantages of quantum computing to maximize reward and reduce training loss. Moreover, this study employs a local loss to minimize the occurrence of barren plateaus phenomena and further enhance performance. As a particular case, the proposed scheme is employed to jointly optimize: (1) UAV trajectory planning, (2) user grouping, and (3) power allocation for higher energy efficiency of a UAV as the reward. The combination of these optimized factors is referred to as action space in the presented LQ-DRL. The LQ-DRL is employed to solve the optimization problem due to its non-convexity, continuous and large action space, and time-series domain. In a practical view, LQ-DRL aims to solve the issue of energy consumption related to limited-battery energy of a UAV base station while maintaining quality-of-service (QoS) for users, by gaining maximum energy efficiency as the reward. One of real applications, as an example, LQ-DRL can be employed to maximize the energy efficiency of a UAV base station in UAV empowered disaster recovery networks scenario. The quantum circuits of layerwise quantum embedding are presented to show the practical implementation in noisy intermediate-scale quantum computers. Based on the results, LQ-DRL outperformed the classical DRL by achieving higher effective dimension, rewards, and lower learning losses. In addition, better performances were achieved using more layers.

A universal variational quantum eigensolver for non-Hermitian systems

Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional O(log2N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(log_{2}{N})$$\\end{document} quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.

Quantum computing for solid mechanics and structural engineering – A demonstration with Variational Quantum Eigensolver

Variational quantum algorithms exploit the features of superposition and entanglement to optimize a cost function efficiently by manipulating the quantum states. They are suitable for noisy intermediate-scale quantum (NISQ) computers that recently became accessible to the worldwide research community. Here, we implement and demonstrate the numerical processes on the 5-qubit and 7-qubit quantum processors on the IBM Qiskit Runtime platform. We combine the commercial finite-element-method (FEM) software ABAQUS with the implementation of Variational Quantum Eigensolver (VQE) to establish an integrated pipeline. Three examples are used to investigate the performance: a hexagonal truss, a Timoshenko beam, and a plane-strain continuum. We conduct parametric studies on the convergence of fundamental natural frequency estimation using this hybrid quantum-classical approach. Our findings can be extended to problems with many more degrees of freedom when quantum computers with hundreds of qubits become available in the near future.

Two-qubit entangling gates for superconducting quantum computers

Noisy intermediate-scale quantum computers (NISQ) have demonstrated superior performance compared to conventional computing schemes. Nonetheless, errors associated with two-qubit entangling gates remain a substantial obstacle in fully harnessing the potential of current superconducting quantum computers. The achievement of quantum computation and quantum information processing critically depends on the presence of entangled states. These states are indispensable for quantum computers to surpass classical computers in terms of computational power. This paper presents a comprehensive characterization of a two-qubit entangling gate, known as Dagwood Bumstead (DB) gate, to investigate the challenges posed by errors in the DB gate on NISQ devices. Firstly, we propose a scheme for the realization of the DB gate using two capacitively coupled Josephson Superconducting qubits, based on the experimental setup carried out in Bialczak et al. (2010). Additionally, we introduce and characterize a novel two-qubit quantum gate. Our proposed gate showcases an impressive average gate fidelity of 0.85450. The DB gate’s performance is fully described through quantum process tomography (QPT) and quantum state tomography (QST) experiments conducted in three different environments: noisy, noiseless, and on a real IBM quantum computer. The process fidelity of the DB gate is measured to be 0.8069705, and the state fidelity is determined to be 0.8858750. This study offers valuable insights into the performance of the DB gate on a NISQ device, showcasing its potential for executing intricate algorithms and operations on an authentic quantum computer.

The cross-sectional stock return predictions via quantum neural network and tensor network

In this paper, we investigate the application of quantum and quantum-inspired machine learning algorithms to stock return predictions. Specifically, we evaluate the performance of quantum neural network, an algorithm suited for noisy intermediate-scale quantum computers, and tensor network, a quantum-inspired machine learning algorithm, against classical models such as linear regression and neural networks. To evaluate their abilities, we construct portfolios based on their predictions and measure investment performances. The empirical study on the Japanese stock market shows the tensor network model achieves superior performance compared to classical benchmark models, including linear and neural network models. Though the quantum neural network model attains the lowered risk-adjusted excess return than the classical neural network models over the whole period, both the quantum neural network and tensor network models have superior performances in the latest market environment, which suggests capability of model’s capturing non-linearity between input features.

Maximizing Local Rewards on Multi-Agent Quantum Games through Gradient-Based Learning Strategies.

This article delves into the complex world of quantum games in multi-agent settings, proposing a model wherein agents utilize gradient-based strategies to optimize local rewards. A learning model is introduced to focus on the learning efficacy of agents in various games and the impact of quantum circuit noise on the performance of the algorithm. The research uncovers a non-trivial relationship between quantum circuit noise and algorithm performance. While generally an increase in quantum noise leads to performance decline, we show that low noise can unexpectedly enhance performance in games with large numbers of agents under some specific circumstances. This insight not only bears theoretical interest, but also might have practical implications given the inherent limitations of contemporary noisy intermediate-scale quantum (NISQ) computers. The results presented in this paper offer new perspectives on quantum games and enrich our understanding of the interplay between multi-agent learning and quantum computation. Both challenges and opportunities are highlighted, suggesting promising directions for future research in the intersection of quantum computing, game theory and reinforcement learning.

Boosting quantum amplitude exponentially in variational quantum algorithms

We introduce a family of variational quantum algorithms, which we coin as quantum iterative power algorithms (QIPAs), and demonstrate their capabilities as applied to global-optimization numerical experiments. Specifically, we demonstrate the QIPA based on a double exponential oracle as applied to ground state optimization of the H 2 molecule, search for the transmon qubit ground-state, and biprime factorization. Our results indicate that QIPA outperforms quantum imaginary time evolution (QITE) and requires a polynomial number of queries to reach convergence even with exponentially small overlap between an initial quantum state and the final desired quantum state, under some circumstances. We analytically show that there exists an exponential amplitude amplification at every step of the variational quantum algorithm, provided the initial wavefunction has non-vanishing probability with the desired state and that the unique maximum of the oracle is given by , while all other values are given by the same value (here λ can be taken as eigenvalues of the problem Hamiltonian). The generality of the global-optimization method presented here invites further application to other problems that currently have not been explored with QITE-based near-term quantum computing algorithms. Such approaches could facilitate identification of reaction pathways and transition states in chemical physics, as well as optimization in a broad range of machine learning applications. The method also provides a general framework for adaptation of a class of classical optimization algorithms to quantum computers to further broaden the range of algorithms amenable to implementation on current noisy intermediate-scale quantum computers.

An improved QFT-based quantum comparator and extended modular arithmetic using one ancilla qubit

Quantum comparators and modular arithmetic are fundamental in many quantum algorithms. Current research mainly focuses on operations between two quantum states. However, various applications, such as integer factorization, optimization, and financial risk analysis, commonly require one of the inputs to be classical. It requires many ancillary qubits, especially when subsequent computations are involved. In this paper, we propose a quantum–classical comparator based on the quantum Fourier transform. Then we extend it to compare two quantum integers and modular arithmetic. Proposed operators only require up to one ancilla qubit, which is optimal for qubit resources. We analyze limitations in the current modular addition circuit and develop it to process arbitrary quantum states in the entire n-qubit space. The proposed algorithms reduce computing resources and make them valuable for noisy intermediate-scale quantum computers.

Classical Surrogates for Quantum Learning Models.

The advent of noisy intermediate-scale quantum computers has put the search for possible applications to the forefront of quantum information science. One area where hopes for an advantage through near-term quantum computers are high is quantum machine learning, where variational quantum learning models based on parametrized quantum circuits are discussed. In this work, we introduce the concept of a classical surrogate, a classical model which can be efficiently obtained from a trained quantum learning model and reproduces its input-output relations. As inference can be performed classically, the existence of a classical surrogate greatly enhances the applicability of a quantum learning strategy. However, the classical surrogate also challenges possible advantages of quantum schemes. As it is possible to directly optimize the Ansatz of the classical surrogate, they create a natural benchmark the quantum model has to outperform. We show that large classes of well-analyzed reuploading models have a classical surrogate. We conducted numerical experiments and found that these quantum models show no advantage in performance or trainability in the problems we analyze. This leaves only generalization capability as a possible point of quantum advantage and emphasizes the dire need for a better understanding of inductive biases of quantum learning models.

The SWITCH test for discriminating quantum evolutions

We study different quantum circuits that can discriminate between two arbitrary quantum evolution operators. These circuits can be used to check whether two quantum operators are equal or not and to estimate a fidelity measure telling how close the operators are. This operator comparison is related to the SWAP test for discriminating two quantum states. In terms of their practical realization, we comment possible laboratory implementations with light along the same lines of recent experimental realizations of quantum superpositions of causal orders exploiting the different degrees of freedom of photons. We also discuss hardware efficient realizations for noisy intermediate scale quantum computers. Finally, we comment potential applications to the discrimination of quantum communication channels and to the search for simpler quantum circuits in quantum compilers.

Meta-learning digitized-counterdiabatic quantum optimization

The use of variational quantum algorithms for optimization tasks has emerged as a crucial application for the current noisy intermediate-scale quantum computers. However, these algorithms face significant difficulties in finding suitable ansatz and appropriate initial parameters. In this paper, we employ meta-learning using recurrent neural networks to address these issues for the recently proposed digitized-counterdiabatic quantum approximate optimization algorithm (QAOA). By combining meta-learning and counterdiabaticity, we find suitable variational parameters and reduce the number of optimization iterations required. We demonstrate the effectiveness of our approach by applying it to the MaxCut problem and the Sherrington–Kirkpatrick model. Our method offers a short-depth circuit ansatz with optimal initial parameters, thus improving the performance of the state-of-the-art QAOA.

Simulating Open Quantum System Dynamics on NISQ Computers with Generalized Quantum Master Equations.

We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.

Solving Graph Problems Using Gaussian Boson Sampling.

Gaussian boson sampling (GBS) is not only a feasible protocol for demonstrating quantum computational advantage, but also mathematically associated with certain graph-related and quantum chemistry problems. In particular, it is proposed that the generated samples from the GBS could be harnessed to enhance the classical stochastic algorithms in searching some graph features. Here, we use Jiǔzhāng, a noisy intermediate-scale quantum computer, to solve graph problems. The samples are generated from a 144-mode fully connected photonic processor, with photon click up to 80 in the quantum computational advantage regime. We investigate the open question of whether the GBS enhancement over the classical stochastic algorithms persists-and how it scales-with an increasing system size on noisy quantum devices in the computationally interesting regime. We experimentally observe the presence of GBS enhancement with a large photon-click number and a robustness of the enhancement under certain noise. Our work is a step toward testing real-world problems using the existing noisy intermediate-scale quantum computers and hopes to stimulate the development of more efficient classical and quantum-inspired algorithms.

QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer

Quantum Natural Language Processing (QNLP) deals with the design and implementation of NLP models intended to be run on quantum hardware. In this paper, we present results on the first NLP experiments conducted on Noisy Intermediate-Scale Quantum (NISQ) computers for datasets of size greater than 100 sentences. Exploiting the formal similarity of the compositional model of meaning by Coecke, Sadrzadeh, and Clark (2010) with quantum theory, we create representations for sentences that have a natural mapping to quantum circuits. We use these representations to implement and successfully train NLP models that solve simple sentence classification tasks on quantum hardware. We conduct quantum simulations that compare the syntax-sensitive model of Coecke et al. with two baselines that use less or no syntax; specifically, we implement the quantum analogues of a “bag-of-words” model, where syntax is not taken into account at all, and of a word-sequence model, where only word order is respected. We demonstrate that all models converge smoothly both in simulations and when run on quantum hardware, and that the results are the expected ones based on the nature of the tasks and the datasets used. Another important goal of this paper is to describe in a way accessible to AI and NLP researchers the main principles, process and challenges of experiments on quantum hardware. Our aim in doing this is to take the first small steps in this unexplored research territory and pave the way for practical Quantum Natural Language Processing.

Exploring the optimality of approximate state preparation quantum circuits with a genetic algorithm

We study the approximate state preparation problem on noisy intermediate-scale quantum (NISQ) computers by applying a genetic algorithm to generate quantum circuits for state preparation. The algorithm can account for the specific characteristics of the physical machine in the evaluation of circuits, such as the native gate set and qubit connectivity. We use our genetic algorithm to optimize the circuits provided by the low-rank state preparation algorithm introduced by Araujo et al., and find substantial improvements to the fidelity in preparing Haar random states with a limited number of CNOT gates. Moreover, we observe that already for a 5-qubit quantum processor with limited qubit connectivity and significant noise levels (IBM Falcon 5T), the maximal fidelity for Haar random states is achieved by a short approximate state preparation circuit instead of the exact preparation circuit. We also present a theoretical analysis of approximate state preparation circuit complexity to motivate our findings. Our genetic algorithm for quantum circuit discovery is freely available at https://github.com/beratyenilen/qc-ga.

A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Strategies for quantum-optimized construction of interpolating operators in classical simulations of lattice quantum field theories

It has recently been argued that noisy intermediate-scale quantum computers may be used to optimize interpolating operator constructions for lattice quantum field theory (LQFT) calculations on classical computers. Here, two concrete realizations of the method are developed and implemented. The first approach is to maximize the overlap, or fidelity, of the state created by an interpolating operator acting on the vacuum state to the target eigenstate. The second is to instead minimize the energy expectation value of the interpolated state. These approaches are implemented in a proof-of-concept calculation in ($1+1$)-dimensions for a single-flavor massive Schwinger model to obtain quantum-optimized interpolating operator constructions for a vector meson state in the theory. Although fidelity maximization is preferable in the absence of noise due to quantum gate errors, it is found that energy minimization is more robust to these effects in the proof-of-concept calculation. This work serves as a concrete demonstration of how quantum computers in the intermediate term might be used to accelerate classical LQFT calculations.