A Gradient‐Cost Multiobjective Alternate Framework for Variational Quantum Eigensolver with Variable Ansatz

Quantum volume (QV) has become the de-facto standard benchmark to quantify the capability of Noisy Intermediate-Scale Quantum (NISQ) devices. While QV values are often reported by NISQ providers for their systems, we perform our own series of QV calculations on 24 NISQ devices currently offered by IBM Q, IonQ, Rigetti, Oxford Quantum Circuits, and Quantinuum (formerly Honeywell). Our approach characterizes the performances that an advanced user of these NISQ devices can expect to achieve with a reasonable amount of optimization, but without white-box access to the device. In particular, we compile QV circuits to standard gate sets of the vendor using compiler optimization routines where available, and we perform experiments across different qubit subsets. We find that running QV tests requires very significant compilation cycles, QV values achieved in our tests typically lag behind officially reported results and also depend significantly on the classical compilation effort invested.

We are in the noisy intermediate-scale quantum (NISQ) devices' era, in which quantum hardware has become available for application in real-world problems. However, demonstrations of the usefulness of such NISQ devices are still rare. In this work, we consider a practical railway dispatching problem: delay and conflict management on single-track railway lines. We examine the train dispatching consequences of the arrival of an already delayed train to a given network segment. This problem is computationally hard and needs to be solved almost in real time. We introduce a quadratic unconstrained binary optimization (QUBO) model of this problem, which is compatible with the emerging quantum annealing technology. The model's instances can be executed on present-day quantum annealers. As a proof-of-concept, we solve selected real-life problems from the Polish railway network using D-Wave quantum annealers. As a reference, we also provide solutions calculated with classical methods, including the conventional solution of a linear integer version of the model as well as the solution of the QUBO model using a tensor network-based algorithm. Our preliminary results illustrate the degree of difficulty of real-life railway instances for the current quantum annealing technology. Moreover, our analysis shows that the new generation of quantum annealers (the advantage system) does not perform well on those instances, either.

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 $O\left(\frac{n^2}{\log \delta}\right)$ on any connected graph with minimum degree $\delta$, 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.

Noisy, intermediate-scale quantum (NISQ) computing devices have become an industrial reality in the last few years, and cloud-based interfaces to these devices are enabling exploration of near-term quantum computing on a range of problems. As NISQ devices are too noisy for many of the algorithms with a known quantum advantage, discovering impactful applications for near-term devices is the subject of intense research interest. We explore quantum-assisted machine learning (QAML) on NISQ devices through the perspective of tensor networks (TNs), which offer a robust platform for designing resource-efficient and expressive machine learning models to be dispatched on quantum devices. In particular, we lay out a framework for designing and optimizing TN-based QAML models using classical techniques, and then compiling these models to be run on quantum hardware, with demonstrations for generative matrix product state (MPS) models. We put forth a generalized canonical form for MPS models that aids in compilation to quantum devices, and demonstrate greedy heuristics for compiling with a given topology and gate set that outperforms known generic methods in terms of the number of entangling gates, e.g., CNOTs, in some cases by an order of magnitude. We present an exactly solvable benchmark problem for assessing the performance of MPS QAML models, and also present an application for the canonical MNIST handwritten digit dataset. The impacts of hardware topology and day-to-day experimental noise fluctuations on model performance are explored by analyzing both raw experimental counts and statistical divergences of inferred distributions. We also present parametric studies of depolarization and readout noise impacts on model performance using hardware simulators.

Advancements in noisy intermediate-scale quantum (NISQ) computing are steadily pushing these systems toward outperforming classical supercomputers on specific well-defined computational tasks. In this work we explore and control quantum chaos in NISQ systems using discrete-time quantum walks (DTQWs) on cyclic graphs. To efficiently implement quantum walks on NISQ hardware, we employ the quantum Fourier transform to diagonalize the conditional shift operator, optimizing circuit depth and fidelity. We experimentally realize the transition from quantum chaos to order via DTQW dynamics on both odd and even cyclic graphs, specifically 3- and 4-cycle graphs, using the counterintuitive Parrondo paradox strategy across three different NISQ devices. While the 4-cycle graphs exhibit high-fidelity quantum evolution, the 3-cycle implementation shows significant fidelity improvement when augmented with dynamical decoupling pulses. Our results demonstrate a practical approach to probing and harnessing controlled chaotic dynamics on real quantum hardware, laying the groundwork for future quantum algorithms and cryptographic protocols based on quantum walks.

The noisy intermediate-scale quantum (NISQ) devices that have just emerged in recent years provide an opportunity for the physical realization of quantum circuits. As the first step of mapping quantum circuits to these devices, qubit allocation imposes a great impact on the number of additional quantum operations required throughout the mapping. The global optimization of qubit allocation is NP-hard, but since many current NISQ devices have very few qubits, it is possible to solve this problem exactly. In this paper, we propose an exact approach for qubit allocation, which takes advantage of the branch and bound algorithm as the basic framework. In order to further prune the considerably huge state space, three techniques have been proposed and integrated into the exact approach, including an optimized lower bound for quadratic assignment problem, prioritization of logical qubits, and branch pruning by structural symmetry of physical qubits. Moreover, based on the multiple optimal solutions obtained by the exact approach, we give an error-aware qubit allocation method. For problems that are too large to be solved by the exact approach, we also give a heuristic qubit allocation approach with polynomial time complexity. Experimental evaluations show that the exact approach proposed in this paper is feasible for the qubit allocation of current NISQ architectures. For all benchmarks considered, this exact approach can find multiple optimal solutions to the qubit allocation problem on IBM Melbourne, a 16-qubit NISQ architecture, in no more than 20 min. It can serve as a evaluation baseline of any heuristic approach. Experimental evaluations also show that the proposed heuristic approach is near-optimal in terms of routing cost. Both the exact approach and the heuristic approach can be integrated into existing quantum circuit mapping approaches.

In the era of noisy intermediate-scale quantum (NISQ) computing, the synergistic collaboration between quantum and classical computing models has emerged as a promising solution for tackling complex computational challenges. Long short-term memory (LSTM), as a popular network for modeling sequential data, has been widely acknowledged for its effectiveness. However, with the increasing demand for data and spatial feature extraction, the training cost of LSTM exhibits exponential growth. In this study, we propose the quantum convolutional long short-term memory (QConvLSTM) model. By ingeniously integrating classical convolutional LSTM (ConvLSTM) networks and quantum variational algorithms, we leverage the variational quantum properties and the accelerating characteristics of quantum states to optimize the model training process. Experimental validation demonstrates that, compared to various LSTM variants, our proposed QConvLSTM model outperforms in terms of performance. Additionally, we adopt a hierarchical tree-like circuit design philosophy to enhance the model’s parallel computing capabilities while reducing dependence on quantum bit counts and circuit depth. Moreover, the inherent noise resilience in variational quantum algorithms makes this model more suitable for spatiotemporal sequence modeling tasks on NISQ devices.

Many existing quantum supervised learning (SL) schemes consider data given a priori in a classical description. With only noisy intermediate-scale quantum (NISQ) devices available in the near future, their quantum speedup awaits the development of quantum random access memories (qRAMs) and fault-tolerant quantum computing. There, however, also exist a multitude of SL tasks whose data are acquired by sensors, e.g., pattern classification based on data produced by imaging sensors. Solving such SL tasks naturally requires an integrated approach harnessing tools from both quantum sensing and quantum computing. We introduce supervised learning assisted by an entangled sensor network (SLAEN) as a means to carry out SL tasks at the physical layer. The entanglement shared by the sensors in SLAEN boosts the performance of extracting global features of the object under investigation. We leverage SLAEN to construct an entanglement-assisted support-vector machine for data classification and entanglement-assisted principal component analyzer for data compression. In both schemes, variational circuits are employed to seek the optimum entangled probe states and measurement settings to maximize the entanglement-enabled {enhancement}. We observe that SLAEN enjoys an appreciable entanglement-enabled performance gain, even in the presence of loss, over conventional strategies in which classical data are acquired by separable sensors and subsequently processed by classical SL algorithms. SLAEN is realizable with available technology, opening a viable route toward building NISQ devices that offer unmatched performance beyond what the optimum classical device is able to afford.

In recent times, a variety of hybrid quantum-classical algorithms have been developed that aim to calculate the ground state energies of molecular systems on Noisy Intermediate-Scale Quantum (NISQ) devices. Albeit the utilization of shallow depth circuits in these algorithms, the optimization of ansatz parameters necessitates a substantial number of quantum measurements, leading to prolonged runtimes on the scantly available quantum hardware. Through our work, we lay the general foundation for an interdisciplinary approach that can be used to drastically reduce the dependency of these algorithms on quantum infrastructure. We showcase these pertinent concepts within the framework of the recently developed Projective Quantum Eigensolver (PQE) that involves iterative optimization of the nonlinearly coupled parameters through repeated residue measurements on quantum hardware. We demonstrate that one may perceive such a nonlinear optimization problem as a collective dynamic interplay of fast and slow relaxing modes. As such, the synergy among the parameters is exploited using an on-the-fly supervised machine learning protocol that dynamically casts the PQE optimization into a smaller subspace by reducing the effective degrees of freedom. We demonstrate analytically and numerically that our proposed methodology ensures a drastic reduction in the number of quantum measurements necessary for the parameter updates without compromising the characteristic accuracy. Furthermore, the machine learning model may be tuned to capture the noisy data of NISQ devices, and thus the predicted energy is shown to be resilient under a given noise model.

We have developed two quantum classifier models for the ttH classification problem, both of which fall into the category of hybrid quantumclassical algorithms for Noisy Intermediate Scale Quantum devices (NISQ). Our results, along with other studies, serve as a proof of concept that Quantum Machine Learning (QML) methods can have similar or better performance, in specific cases of low number of training samples, with respect to conventional ML methods even with a limited number of qubits available in current hardware. To utilise algorithms with a low number of qubits — to accommodate for limitations in both simulation hardware and real quantum hardware — we investigated different feature reduction methods. Their impact on the performance of both the classical and quantum models was assessed. We addressed different implementations of two QML models, representative of the two main approaches to supervised quantum machine learning today: a Quantum Support Vector Machine (QSVM), a kernel-based method, and a Variational Quantum Circuit (VQC), a variational approach.

Quantum machine learning is where nowadays machine learning (ML) meets quantum information science. In order to implement this new paradigm for novel quantum technologies, we still need a much deeper understanding of its underlying mechanisms, before proposing new algorithms to feasibly address real problems. In this context, quantum generative adversarial learning is a promising strategy to use quantum devices for quantum estimation or generative ML tasks. However, the convergence behaviours of its training process, which is crucial for its practical implementation on quantum processors, have not been investigated in detail yet. Indeed here we show how different training problems may occur during the optimization process, such as the emergence of limit cycles. The latter may remarkably extend the convergence time in the scenario of mixed quantum states playing a crucial role in the already available noisy intermediate scale quantum devices. Then, we propose new strategies to achieve a faster convergence in any operating regime. Our results pave the way for new experimental demonstrations of such hybrid classical-quantum protocols allowing to evaluate the potential advantages over their classical counterparts.

Quantum computing is gaining popularity across a wide range of scientific disciplines due to its potential to solve long-standing computational problems that are considered intractable with classical computers. One promising area where quantum computing has potential is in the speed-up of NP-hard optimisation problems that are common in industrial areas such as logistics and finance. Newcomers to the field of quantum computing who are interested in using this technology to solve optimisation problems do not have an easily accessible source of information on the current capabilities of quantum computers and algorithms. This paper aims to provide a comprehensive overview of the theory of quantum optimisation techniques and their practical application, focusing on their near-term potential for noisy intermediate scale quantum devices. The paper starts by drawing parallels between classical and quantum optimisation problems, highlighting their conceptual similarities and differences. Two main paradigms for quantum hardware are then discussed: analogue and gate-based quantum computers. While analog devices such as quantum annealers are effective for some optimisation problems, they have limitations and cannot be used for universal quantum computation. In contrast, gate-based quantum computers offer the potential for universal quantum computation, but they face challenges with hardware limitations and accurate gate implementation. The paper provides a detailed mathematical discussion with references to key works in the field, as well as a more practical discussion with relevant examples. The most popular techniques for quantum optimisation on gate-based quantum computers, the quantum approximate optimisation algorithm and the quantum alternating operator ansatz framework, are discussed in detail. However, it is still unclear whether these techniques will yield quantum advantage, even with advancements in hardware and noise reduction. The paper concludes with a discussion of the challenges facing quantum optimisation techniques and the need for further research and development to identify new, effective methods for achieving quantum advantage.

Currently available quantum processors are noisy intermediate-scale quantum (NISQ) devices. The variational quantum eigensolver (VQE) is an algorithm that is closer to near-term applicability due to lower quantum hardware requirements. In VQE, trial states with variational parameters are prepared by quantum computers, and the optimal parameters are determined by a classical optimizer. This optimization is known to be an NP-hard problem. In this work, we make a comparison between various gradient-free optimizers in terms of approximation ratio and function evaluations. As a result, we find that COBYLA method is the best to find the approximated solution with a lower number of executions of quantum computers.

The contextual subspace variational quantum eigensolver (CS-VQE) is a hybrid quantum-classical algorithm that approximates the ground-state energy of a given qubit Hamiltonian. It achieves this by separating the Hamiltonian into contextual and noncontextual parts. The ground-state energy is approximated by classically solving the noncontextual problem, followed by solving the contextual problem using VQE, constrained by the noncontextual solution. In general, computation of the contextual correction needs fewer qubits and measurements compared with solving the full Hamiltonian via traditional VQE. We simulate CS-VQE on different tapered molecular Hamiltonians and apply the unitary partitioning measurement reduction strategy to further reduce the number of measurements required to obtain the contextual correction. Our results indicate that CS-VQE combined with measurement reduction is a promising approach to allow feasible eigenvalue computations on noisy intermediate-scale quantum devices. We also provide a modification to the CS-VQE algorithm; the CS-VQE algorithm previously could cause an exponential increase in Hamiltonian terms, but with this modification now at worst will scale quadratically.

The variational quantum eigensolver (VQE) is an algorithm for finding the ground states of a given Hamiltonian. Its application to binary-formulated combinatorial optimization (CO) has been widely studied in recent years. However, typical VQE approaches for CO problems often suffer from local minima or barren plateaus, limiting their ability to achieve optimal solutions. The quantum imaginary time evolution (QITE) offers an alternative approach for effective ground-state preparation but requires large circuits to approximate non-unitary operations. Although compressed QITE (cQITE) reduces circuit depth, accumulated errors eventually cause energy increases. To address these challenges, we propose an Adaptive Weighted QITE-VQE (AWQV) algorithm that integrates the VQE gradients with the cQITE updates through an adaptive weighting scheme during optimization. In numerical simulations for MaxCut on unweighted regular graphs, AWQV achieves near-optimal approximation ratios, while for weighted Erdős-Rényi instances, it outperforms the classical Goemans-Williamson algorithm.