Noisy Intermediate-scale Quantum Research Articles

OnionVQE Optimization Strategy for Ground State Preparation on NISQ Devices

Abstract The Variational Quantum Eigensolver (VQE) is one of the most promising and widely used algorithms for exploiting the capabilities of current Noisy Intermediate-Scale Quantum (NISQ) devices. However, VQE algorithms suffer from a plethora of issues, such as barren plateaus, local minima, quantum hardware noise, and limited qubit connectivity, thus posing challenges for their successful deployment on hardware and simulators. In this work, we propose a VQE optimization strategy that builds upon recent advances in the literature, and exhibits very shallow circuit depths when applied to the specific system of interest, namely a model Hamiltonian representing a cuprate superconductor. These features make our approach a favorable candidate for generating good ground state approximations on current NISQ devices. Our findings illustrate the potential of VQE algorithmic development for leveraging the full capabilities of NISQ devices.

Exploring Electron-Phonon Coupling Using Quantum Computing Methods

Abstract Quantum computing in the noisy intermediate-scale quantum (NISQ) era has foregrounded the importance of Variational Quantum Algorithms (VQAs). These algorithms are crucial for addressing complex quantum mechanical problems that challenge classical computers. One such problem is the electron-phonon (e–ph) interaction, which is essential for determining the zero-point renormalization (ZPR) of electronic structure properties. The calculation of ZPR of fundamental gap relies on the accurate computation of ionization potential (IP) and electron affinity (EA) energy levels in molecular systems, where the VQAs offer the promising solutions. Despite the critical importance of IP, EA energies and ZPR in quantum chemistry calculations, research into the application of quantum algorithms for these calculations remains limited. To address these challenges, we propose two quantum algorithms for ZPR of fundamental gap calculation using Variational Quantum Deflation (VQD) and Quantum Equation of Motion (QEOM) algorithm for several molecular systems. This work opens up new possibilities for the accurate and efficient study of e-ph interaction in electronic structure calculations, even with NISQ-era hardware.

Comment on “Recovering noise-free quantum observables”

Zero-noise extrapolation (ZNE) stands as the most widespread quantum error mitigation technique used in the recovery of noise-free expectation values of observables of interest by means of noisy intermediate-scale quantum (NISQ) machines. Recently, Otten and Gray proposed a multidimensional generalization of polynomial ZNE for systems where there is no tunable global noise source [M. Otten , ]. Specifically, the authors refer to multiqubit systems where each of the qubits experiences several noise processes with different rates, i.e., a nonidentically distributed noise model. The authors proposed a hypersurface method for mitigating such noise, which is technically correct in the sense that if the required samples are obtained, the observable can be mitigated. However, the proposed method presents an excessive experiment repetition overhead, making it impractical, at least from the perspective of quantum computing. Specifically, we discuss that in order to perform a simple mitigation task to multinomial order 3 considering 100 qubits, there, 2.5 years or over a million quantum processors running in parallel would be required. In this Comment, we show that the traditional extrapolation techniques can be applied for such a nonidentically distributed noise setting consisting of many different noise sources, implying that the measurement overhead is practical. In fact, we discuss that the previous mitigation task can be resolved in the order of minutes using a single quantum processor by means of standard ZNE. To do so, we clarify what it is meant by a tunable global noise source in the context of ZNE, a concept that we consider important to be clarified for a correct understanding of how and why these methods work. Published by the American Physical Society 2024

Long-lived topological time-crystalline order on a quantum processor

Topologically ordered phases of matter elude Landau’s symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon—a prethermal topologically ordered time crystal—with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors.

Multi-objective simulated annealing-based quantum circuit cutting for distributed quantum computation

In the current noisy intermediate-scale quantum (NISQ) era, the number of qubits and the depth of quantum circuits in a quantum computer are limited because of complex operation among increasing number of qubits, low-fidelity quantum gates under noise, and short coherence time of physical qubits. However, with distributed quantum computation (DQC) in which multiple small-scale quantum computers cooperate, large-scale quantum circuits can be implemented. In DQC, it is a key step to decompose large-scale quantum circuits into several small-scale subcircuits equivalently. In this paper, we propose a quantum circuit cutting scheme for the circuits consisting of only single-qubit gates and two-qubit gates. In the scheme, the number of non-local gates and the rounds of subcircuits operation are minimized by using the multi-objective simulated annealing (MOSA) algorithm to cluster the gates and to choose the cutting positions whilst using non-local gates. A reconstruction process is also proposed to calculate the probability distribution of output states of the original circuit. As an example, the 7-qubit circuit of Shor algorithm factoring 15 is used to verify the algorithm. Five cutting schemes are recommended, which can be selected according to practical requirements. Compared with the results of the mixing integer programming (MIP) algorithm, the number of execution rounds is efficiently reduced by slightly increasing the number of nonlocal gates.

Benchmarking Quantum Computational Advantages on Supercomputers

AbstractThe achievement of quantum computational advantage, also known as quantum supremacy, is a major milestone at which a quantum computer can solve a problem significantly faster than the world's most powerful classical computers. Two tasks, boson sampling and random quantum circuit sampling, have experimentally exhibited quantum advantages on photonic and superconducting platforms respectively. Classical benchmarking is essential, yet challenging, because these tasks are intractable for classical computers. This study reviews models, algorithms and large‐scale simulations of these two sampling tasks. These approaches continue to hold substantial significance for research in both current noisy intermediate‐scale quantum (NISQ) systems and future fault‐tolerant quantum computing.

Trotterless Simulation of Open Quantum Systems for NISQ Quantum Devices

AbstractThe simulation of quantum systems is one of the flagship applications of near‐term NISQ (noisy intermediate‐scale quantum) computing devices. Efficiently simulating the rich, non‐unitary dynamics of open quantum systems remains challenging on NISQ hardware. Current simulation methods for open quantum systems employ time‐stepped Trotter product formulas (“Trotterization”) which can scale poorly with respect to the simulation time and system dimension. Here, a new simulation method is proposed based on the derivation of a time‐perturbative Kraus operator series representation of the system. A class of open quantum systems is identified for which this method produces circuits of time‐independent depth, which may serve as a desirable alternative to Trotterization, especially on NISQ devices.

Benchmarking Multipartite Entanglement Generation with Graph States

AbstractAs quantum computing technology slowly matures and the number of available qubits on a QPU gradually increases, interest in assessing the capabilities of quantum computing hardware in a scalable manner is growing. One of the key properties for quantum computing is the ability to generate multipartite entangled states. In this study, aspects of benchmarking entanglement generation capabilities of noisy intermediate‐scale quantum (NISQ) devices are discussed based on the preparation of graph states and the verification of entanglement in the prepared states. Thereby, entanglement witnesses that are specifically suited for a scalable experiment design are used. This choice of entanglement witnesses can detect A) bipartite entanglement and B) genuine multipartite entanglement for graph states with constant two measurement settings if the prepared graph state is based on a two‐colorable graph, e.g., a square grid graph or one of its subgraphs. With this, it is experimentally verified that a bipartite entangled state comprising all qubits can be prepared on a 127‐qubit IBM Quantum superconducting QPU, and genuine multipartite entanglement can be detected for states of up to 23 qubits with quantum readout error mitigation.

Mitigating noise in digital and digital–analog quantum computation

Noisy Intermediate-Scale Quantum (NISQ) devices lack error correction, limiting scalability for quantum algorithms. In this context, digital-analog quantum computing (DAQC) offers a more resilient alternative quantum computing paradigm that outperforms digital quantum computation by combining the flexibility of single-qubit gates with the robustness of analog simulations. This work explores the impact of noise on both digital and DAQC paradigms and demonstrates DAQC’s effectiveness in error mitigation. We compare the quantum Fourier transform and quantum phase estimation algorithms under a wide range of single and two-qubit noise sources in superconducting processors. DAQC consistently surpasses digital approaches in fidelity, particularly as processor size increases. Moreover, zero-noise extrapolation further enhances DAQC by mitigating decoherence and intrinsic errors, achieving fidelities above 0.95 for 8 qubits, and reducing computation errors to the order of 10−3. These results establish DAQC as a viable alternative for quantum computing in the NISQ era.

Power flow analysis using quantum and digital annealers: a discrete combinatorial optimization approach

Power flow (PF) analysis is a foundational computational method to study the flow of power in an electrical network. This analysis involves solving a set of non-linear and non-convex differential-algebraic equations. State-of-the-art solvers for PF analysis, therefore, face challenges with scalability and convergence, specifically for large-scale and/or ill-conditioned cases characterized by high penetration of renewable energy sources, among others. The adiabatic quantum computing paradigm has been proven to efficiently find solutions for combinatorial problems in the noisy intermediate-scale quantum (NISQ) era, and it can potentially address the limitations posed by state-of-the-art PF solvers. For the first time, we propose a novel adiabatic quantum computing approach for efficient PF analysis. Our key contributions are (i) a combinatorial PF algorithm and a modified version that aligns with the principles of PF analysis, termed the adiabatic quantum PF algorithm (AQPF), both of which use Quadratic Unconstrained Binary Optimization (QUBO) and Ising model formulations; (ii) a scalability study of the AQPF algorithm; and (iii) an extension of the AQPF algorithm to handle larger problem sizes using a partitioned approach. Numerical experiments are conducted using different test system sizes on D-Wave’s Advantage™ quantum annealer, Fujitsu’s digital annealer V3, D-Wave’s quantum-classical hybrid annealer, and two simulated annealers running on classical computer hardware. The reported results demonstrate the effectiveness and high accuracy of the proposed AQPF algorithm and its potential to speed up the PF analysis process while handling ill-conditioned cases using quantum and quantum-inspired algorithms.

Subspace-Search Quantum Imaginary Time Evolution for Excited State Computations.

Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate-scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems including H2 and LiH molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.

Robust optimal control for a systematic error in the control amplitude of transmon qubits

In the era of Noisy Intermediate-Scale Quantum computing as well as in error correcting circuits, physical qubits coherence time and high fidelity gates are essential to the functioning of quantum computers. In this paper, we demonstrate theoretically and experimentally, that pulses designed by optimization can be used to counteract the loss of fidelity due to a control amplitude error of the transmon qubit. We analyze the control landscape obtained by robust optimal control and find it to depend on the error range, namely the solutions can get trapped in the basin of attraction of sub-optimal solutions. Robust controls are found for different error values and are compared to an incoherent loss of fidelity mechanism due to a finite relaxation rate. The controls are tested on the IBMQ’s qubit and found to demonstrate resilience against significant ∼10% errors.

Quantum Tunneling: From Theory to Error‐Mitigated Quantum Simulation

AbstractEver since the discussions about a possible quantum computer arised, quantum simulations have been at the forefront of possible utilities, with the task of quantum simulations being one that promises quantum advantage. Recently, advancements have made it feasible to simulate complex molecules using Variational Quantum Eigensolvers or study the dynamics of many‐body spin Hamiltonians. These simulations have the potential to yield valuable outcomes through the application of error mitigation techniques. Simulating smaller models carries a great amount of importance as well and currently, in the Noisy Intermediate Scale Quantum era, is more feasible since it is less prone to errors. The objective of this work is to examine the theoretical background and the circuit implementation of a quantum tunneling simulation, with an emphasis on hardware considerations. This study presents the theoretical background required for such implementation and highlights the main stages of its development. By building on classic approaches of quantum tunneling simulations, this study aims at improving the result of such simulations by employing error mitigation techniques, Zero Noise Extrapolation, and Readout Error Mitigation and uses them in conjunction with multiprogramming of the quantum chip, a technique used for solving the hardware under‐utilization problem that arises in such contexts.

Hybrid quantum-classical computation for automatic guided vehicles scheduling

Motivated by recent efforts to develop quantum computing for practical, industrial-scale challenges, we demonstrate the effectiveness of state-of-the-art hybrid (not necessarily quantum) solvers in addressing the business-centric optimization problem of scheduling Automatic Guided Vehicles (AGVs). Some solvers can already leverage noisy intermediate-scale quantum (NISQ) devices. In our study, we utilize D-Wave hybrid solvers that implement classical heuristics with potential assistance from a quantum processing unit. This hybrid methodology performs comparably to existing classical solvers. However, due to the proprietary nature of the software, the precise contribution of quantum computation remains unclear. Our analysis focuses on a practical, business-oriented scenario: scheduling AGVs within a factory constrained by limited space, simulating a realistic production setting. Our approach maps a realistic AGVs problem onto one reminiscent of railway scheduling and demonstrates that the AGVs problem is better suited to quantum computing than its railway counterpart, the latter being denser in terms of the average number of constraints per variable. The main idea here is to highlight the potential usefulness of a hybrid approach for handling AGVs scheduling problems of practical sizes. We show that a scenario involving up to 21 AGVs, significant due to possible deadlocks, can be efficiently addressed by a hybrid solver in seconds.

Simulating Cavity-Modified Electron Transfer Dynamics on NISQ Computers.

We present an algorithm based on the quantum-mechanically exact tensor-train thermo-field dynamics (TT-TFD) method for simulating cavity-modified electron transfer dynamics on noisy intermediate-scale quantum (NISQ) computers. The utility and accuracy of the proposed methodology is demonstrated on a model for the photoinduced intramolecular electron transfer reaction within the carotenoid-porphyrin-C60 molecular triad in tetrahydrofuran (THF) solution. The electron transfer rate is found to increase significantly with increasing coupling strength between the molecular system and the cavity. The rate process is also seen to shift from overdamped monotonic decay to under-damped oscillatory dynamics. The electron transfer rate is seen to be highly sensitive to the cavity frequency, with the emergence of a resonance cavity frequency for which the effect of coupling to the cavity is maximal. Finally, an implementation of the algorithm on the IBM Osaka quantum computer is used to demonstrate how TT-TFD-based electron transfer dynamics can be simulated accurately on NISQ computers.

Solving linear systems on quantum hardware with hybrid HHL++

The limited capabilities of current quantum hardware significantly constrain the scale of experimental demonstrations of most quantum algorithmic primitives. This makes it challenging to perform benchmarking of the current hardware using useful quantum algorithms, i.e., application-oriented benchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is a critical quantum linear algebra primitive, but the majority of the components of HHL are far out of the reach of noisy intermediate-scale quantum devices, which has led to the proposal of hybrid classical-quantum variants. The goal of this work is to further bridge the gap between proposed near-term friendly implementations of HHL and the kinds of quantum circuits that can be executed on noisy hardware. Our proposal adds to the existing literature of hybrid quantum algorithms for linear algebra that are more compatible with the current scale of quantum devices. Specifically, we propose two modifications to the Hybrid HHL algorithm proposed by Lee et al., leading to our algorithm Hybrid HHL++\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {HHL}^{++}$$\\end{document}: (1) propose a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL, and (2) introduce a heuristic for compressing the HHL circuit. We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date.

Towards an efficient variational quantum algorithm for solving linear equations

Variational quantum algorithms are promising methods with the greatest potential to achieve quantum advantage, widely employed in the era of noisy intermediate-scale quantum computing. This study presents an advanced variational hybrid algorithm (EVQLSE) that leverages both quantum and classical computing paradigms to address the solution of linear equation systems. Initially, an innovative loss function is proposed, drawing inspiration from the similarity measure between two quantum states. This function exhibits a substantial improvement in computational complexity when benchmarked against the variational quantum linear solver. Subsequently, a specialized parameterized quantum circuit structure is presented for small-scale linear systems, which exhibits powerful expressive capabilities. Through rigorous numerical analysis, the expressiveness of this circuit structure is quantitatively assessed using a variational quantum regression algorithm, and it obtained the best score compared to the others. Moreover, the expansion in system size is accompanied by an increase in the number of parameters, placing considerable strain on the training process for the algorithm. To address this challenge, an optimization strategy known as quantum parameter sharing is introduced, which proficiently minimizes parameter volume while adhering to exacting precision standards. Finally, EVQLSE is successfully implemented on a quantum computing platform provided by IBM for the resolution of large-scale problems characterized by a dimensionality of 220.

Enhancing quantum annealing accuracy through replication-based error mitigation*

Abstract Quantum annealers like those manufactured by D-Wave Systems are designed to find high quality solutions to optimization problems that are typically hard for classical computers. They utilize quantum effects like tunneling to evolve toward low-energy states representing solutions to optimization problems. However, their analog nature and limited control functionalities present challenges to correcting or mitigating hardware errors. As quantum computing advances towards applications, effective error suppression is an important research goal. We propose a new approach called replication based mitigation (RBM) based on parallel quantum annealing (QA). In RBM, physical qubits representing the same logical qubit are dispersed across different copies of the problem embedded in the hardware. This mitigates hardware biases, is compatible with limited qubit connectivity in current annealers, and is well-suited for currently available noisy intermediate-scale quantum annealers. Our experimental analysis shows that RBM provides solution quality on par with previous methods while being more flexible and compatible with a wider range of hardware connectivity patterns. In comparisons against standard QA without error mitigation on larger problem instances that could not be handled by previous methods, RBM consistently gets better energies and ground state probabilities across parameterized problem sets.

Quantum-assisted rendezvous on graphs: explicit algorithms and quantum computer simulations

Abstract We study quantum advantage in one-step rendezvous games on simple graphs analytically, numerically, and using noisy intermediate-scale quantum (NISQ) processors. Our protocols realise the recently discovered (Mironowicz 2023 New J. Phys. 25 013023) optimal bounds for small cycle graphs and cubic graphs. In the case of cycle graphs, we generalise the protocols to arbitrary graph size. The NISQ processor experiments realise the expected quantum advantage with high accuracy for rendezvous on the complete graph K 3. In contrast, for the graph 2 K 4 , formed by two disconnected 4-vertex complete graphs, the performance of the NISQ hardware is sub-classical, consistent with the deeper circuit and known qubit decoherence and gate error rates.

Gate-based quantum neurons in hybrid neural networks

Abstract Quantum computing is conceived as a promising and powerful next-generation platform for information processing and it has been shown that it could bring significant accelerations to certain tasks, compared to its classical counterparts. With recent advances in noisy intermediate-scale quantum (NISQ) devices, we can process classical data from real-world problems using hybrid quantum systems. In this work, we investigate the critical problem of designing a gate-based hybrid quantum neuron under NISQ constraints to enable the construction of scalable hybrid quantum deep neural networks (HQDNNs). We explore and characterize diverse quantum circuits for hybrid quantum neurons and discuss related critical components of HQDNNs. We also utilize a new schema to infer multiple predictions from a single hybrid neuron. We further compose a highly customizable platform for simulating HQDNNs via Qiskit and test them on diverse classification problems including the iris and the wheat seed datasets. The results show that even HQDNNs with the simplest neurons could lead to superior performance on these tasks. Finally, we show that the HQDNNs are robust to certain levels of noise, making them preferred on NISQ devices. Our work provides a comprehensive investigation of building scalable near-term gate-based HQDNNs and paves the way for future studies of quantum deep learning via both simulations on classical computers and experiments on accessible NISQ devices.