Noisy intermediate-scale quantum (NISQ) algorithms

Abstract

A universal fault-tolerant quantum computer that can solve efficiently problems such as integer factorization and unstructured database search requires millions of qubits with low error rates and long coherence times. While the experimental advancement towards realizing such devices will potentially take decades of research, noisy intermediate-scale quantum (NISQ) computers already exist. These computers are composed of hundreds of noisy qubits, i.e. qubits that are not error-corrected, and therefore perform imperfect operations in a limited coherence time. In the search for quantum advantage with these devices, algorithms have been proposed for applications in various disciplines spanning physics, machine learning, quantum chemistry and combinatorial optimization. The goal of such algorithms is to leverage the limited available resources to perform classically challenging tasks. In this review, we provide a thorough summary of NISQ computational paradigms and algorithms. We discuss the key structure of these algorithms, their limitations, and advantages. We additionally provide a comprehensive overview of various benchmarking and software tools useful for programming and testing NISQ devices.

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• Efficient DCQO Algorithm within the Impulse Regime for Portfolio Optimization
• Demonstration of Algorithmic Quantum Speedup for an Abelian Hidden Subgroup Problem
• Using Quantum Switches to Mitigate Noise in Grover's Search Algorithm
• Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential
• Quantum computer error structure probed by quantum error correction syndrome measurements
• Unleashed from Constrained Optimization: Quantum Computing for Quantum Chemistry Employing Generator Coordinate Inspired Method
• Generating Universal Adversarial Perturbations for Quantum Classifiers
• Quantum error cancellation in photonic systems -- undoing photon losses
• Hardware-efficient entangled measurements for variational quantum algorithms
• Efficient witnessing and testing of magic in mixed quantum states
• Fermionic Adaptive Sampling Theory for Variational Quantum Eigensolvers
• Convergence and Quantum Advantage of Trotterized MERA for Strongly-Correlated Systems
• Error-Mitigated Quantum Simulation of Interacting Fermions with Trapped Ions
• Entanglement engineering of optomechanical systems by reinforcement learning
• Efficient simulation of parametrized quantum circuits under non-unital noise through Pauli backpropagation
• Information flow in parameterized quantum circuits
• Parameter-Parallel Distributed Variational Quantum Algorithm
• A Quantum Algorithmic Approach to Multiconfigurational Valence Bond Theory: Insights from Interpretable Circuit Design
• State preparation of lattice field theories using quantum optimal control
• Double-bracket quantum algorithms for quantum imaginary-time evolution
• Parametrized quantum circuit for weight-adjustable quantum loop gas
• A Hybrid Qubit Encoding: Splitting Fock Space into Fermionic and Bosonic Subspaces
• Calculating the energy profile of an enzymatic reaction on a quantum computer
• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits
• Gradients and frequency profiles of quantum re-uploading models
• Diagrammatic Analysis for Parameterized Quantum Circuits
• Quantum neural compressive sensing for ghost imaging
• Eigenvalue-invariant transformation of Ising problem for anti-crossing mitigation in quantum annealing
• Optimisation-free Classification and Density Estimation with Quantum Circuits
• Quantum search in many-body interacting system with long-range interaction
• A model of interacting quantum neurons with a dynamic synapse
• Probing the limits of variational quantum algorithms for nonlinear ground states on real quantum hardware: The effects of noise
• Non-adaptive measurement-based quantum computation on IBM Q
• Quantum Dynamic Programming
• Framework for Learning and Control in the Classical and Quantum Domains
• Time-optimal transfer of the quantum state in long qubit arrays
• Compilation of Entangling Gates for High-Dimensional Quantum Systems
• Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States
• Superconducting Quantum Simulation for Many-Body Physics beyond Equilibrium
• Assisted quantum simulation of open quantum systems
• Systematic study on the dependence of the warm-start quantum approximate optimization algorithm on approximate solutions
• Sparse Probabilistic Synthesis of Quantum Operations
• Scalable circuit depth reduction in feedback-based quantum optimization with a quadratic approximation
• Lattice Hamiltonians and Stray Interactions Within Quantum Processors
• Random Natural Gradient
• Characterization, synthesis, and optimization of quantum circuits over multiple-control $\text{Z}$-rotation gates: A systematic study
• MLQAOA: Graph Learning Accelerated Hybrid Quantum-Classical Multilevel QAOA
• Quantum Dynamics Simulation of the Advection-Diffusion Equation
• Variational quantum simulation of long-range interacting systems
• Quantum smoothed particle hydrodynamics algorithm inspired by quantum walks
• Connection between single-layer Quantum Approximate Optimization Algorithm interferometry and thermal distributions sampling
• Resource-efficient utilization of quantum computers
• Optimal depth and a novel approach to variational quantum process tomography
• Inverted-circuit zero-noise extrapolation for quantum gate error mitigation
• High-precision tomography of ion qubits based on registration of fluorescent photons
• A Compendious Review of Majorization-Based Resource Theories: Quantum Information and Quantum Thermodynamics
• State Preparation in a Jaynes-Cummings Lattice with Quantum Optimal Control
• Entanglement estimation of Werner states with a quantum extreme learning machine
• Determining the N-representability of a reduced density matrix via unitary evolution and stochastic sampling
• An example of use of Variational Methods in Quantum Machine Learning
• Energy Landscape Plummeting in Variational Quantum Eigensolver: Subspace Optimization, Non-iterative Corrections and Generator-informed Initialization for Improved Quantum Efficiency
• QSlack: A slack-variable approach for variational quantum semi-definite programming
• Shallow quantum circuits for efficient preparation of Slater determinants and correlated states on a quantum computer
• Machine learning of quantum channels on NISQ devices
• A modular entanglement-based quantum computer architecture
• Exploring the topological sector optimization on quantum computers
• Letting the tiger out of its cage: bosonic coding without concatenation
• Characterization of overparametrization in the simulation of realistic quantum systems
• Quantum Circuit Design using a Progressive Widening Enhanced Monte Carlo Tree Search
• Accurate harmonic vibrational frequencies for diatomic molecules via quantum computing
• Quantum computing quantum Monte Carlo algorithm
• Digital Quantum Simulation and Circuit Learning for the Generation of Coherent States
• Learning quantum phases via single-qubit disentanglement
• Iterative Quantum Optimization with Adaptive Problem Hamiltonian
• Molecular Properties from Quantum Krylov Subspace Diagonalization
• Limitations of variational quantum algorithms: a quantum optimal transport approach
• Scalability Challenges in Variational Quantum Optimization under Stochastic Noise
• Bringing Quantum Systems under Control: A Tutorial Invitation to Quantum Computing and Its Relation to Bilinear Control Systems
• Sequence of penalties method to study excited states using VQE
• Rigorous noise reduction with quantum autoencoders
• Parallel-in-time quantum simulation via Page and Wootters quantum time
• A Simple and Efficient Joint Measurement Strategy for Estimating Fermionic Observables and Hamiltonians
• Neutron-nucleus dynamics simulations for quantum computers
• Low-Rank Variational Quantum Algorithm for the Dynamics of Open Quantum Systems
• QCEDA: Using Quantum Computers for EDA
• Certifying nonstabilizerness in quantum processors
• Pitfalls of the sublinear QAOA-based factorization algorithm
• Enhancing qubit readout with Bayesian Learning
• Quantum contextual bandits and recommender systems for quantum data
• Non-Iterative Disentangled Unitary Coupled-Cluster based on Lie-algebraic structure
• Demonstration of Hardware Efficient Photonic Variational Quantum Algorithm
• Quantum Natural Gradient with Geodesic Corrections for Small Shallow Quantum Circuits
• High-dimensional counterdiabatic quantum computing
• Localizing genuine multiparty entanglement in noisy stabilizer states
• Quantum error mitigation for rotation symmetric bosonic codes with symmetry expansion
• Controlling quantum many-body systems using reduced-order modelling
• Testing quantum computers with the protocol of quantum state matching
• Quantum machine learning for multiclass classification beyond kernel methods
• Calculation of the moscovium ground-state energy by quantum algorithms
• Warm Start Adaptive-Bias Quantum Approximate Optimization Algorithm
• Variational Quantum Simulation of Valence-Bond Solids
• The Role of Quantum Computing in Advancing Scientific High-Performance Computing: A perspective from the ADAC Institute
• Large-scale Efficient Molecule Geometry Optimization with Hybrid Quantum-Classical Computing
• Encoding-dependent generalization bounds for parametrized quantum circuits
• Quantum mean value approximator for hard integer value problems
• Efficient quantum algorithms for testing symmetries of open quantum systems
• Hamiltonian-based graph-state ansatz for variational quantum algorithms
• Cross-platform hardware benchmark of style-based quantum GANs for data augmentation on superconducting and trapped-ion processors
• Quantum-inspired clustering with light
• Quantum median filter for Total Variation image denoising
• Automatic Structural Search of Tensor Network States including Entanglement Renormalization
• Switching Time Optimization for Binary Quantum Optimal Control
• Clifford circuit based heuristic optimization of fermion-to-qubit mappings
• Certified Random Number Generation using Quantum Computers
• Optimized numerical gradient and Hessian estimation for variational quantum algorithms
• Estimation of Quantum Fisher Information via Stein's Identity in Variational Quantum Algorithms
• A Monte Carlo Tree Search approach to QAOA: finding a needle in the haystack
• Predicting three-dimensional chaotic systems with four qubit quantum systems
• Survey on Computational Applications of Tensor Network Simulations
• Dual-VQE: A quantum algorithm to lower bound the ground-state energy
• Quantum Shortcut to Adiabaticity for State Preparation in a Finite-Sized Jaynes-Cummings Lattice
• Variational post-selection for ground states and thermal states simulation
• Quantum state tomography with disentanglement algorithm
• Pitfalls when tackling the exponential concentration of parameterized quantum models
• Machine learning non-Markovian two-level quantum noise spectroscopy
• Learning Fourier series with parametrized quantum circuits
• Inequality constraints in variational quantum circuits with qudits
• Distance-four quantum codes with combined postselection and error correction
• Leveraging commuting groups for an efficient variational Hamiltonian ansatz
• Lacing topological orders in two dimensions: exactly solvable models for Kitaev's sixteen-fold way
• Anticipative measurements in hybrid quantum-classical computation
• Achieving the volume-law entropy regime with random-sign Dicke states
• Minimizing the negativity of quantum circuits in overcomplete quasiprobability representations
• Stable many-body resonances in open quantum systems
• Out of the Loop: Structural Approximation of Optimisation Landscapes and non-Iterative Quantum Optimisation
• Accelerating Feedback-Based Quantum Algorithms through Time Rescaling
• Wasserstein Complexity of Quantum Circuits
• Hacking Cryptographic Protocols with Advanced Variational Quantum Attacks
• Proximal quantum control of spin and spin ensemble with highly localized control field from skyrmions
• Quantum correlations in general qubit-qudit axially symmetric states
• Transfer Learning Analysis of Variational Quantum Circuits
• Sampling Groups of Pauli Operators to Enhance Direct Fidelity Estimation
• Compact Multi-Threshold Quantum Information Driven Ansatz For Strongly Interactive Lattice Spin Models
• Probing many-body Bell correlation depth with superconducting qubits
• Experimental realization of nonunitary multi-qubit operations
• Recursive relations and quantum eigensolver algorithms within modified Schrieffer--Wolff transformations for the Hubbard dimer
• Continuous monitoring for noisy intermediate-scale quantum processors
• Learning quantum states prepared by shallow circuits in polynomial time
• Perturbative gadgets for gate-based quantum computing: Non-recursive constructions without subspace restrictions
• Quantum Approximation Optimization Algorithm for the Trellis based Viterbi Decoding of Classical Error Correcting Codes
• Expressivity of deterministic quantum computation with one qubit
• Effective temperature in approximate quantum many-body states
• Variational quantum compiling for three-qubit gates design in quantum dots
• Escaping Local Minima with Quantum Coherent Cooling
• Modified Conjugate Quantum Natural Gradient
• Tensor Decompositions and Adiabatic Quantum Computing for Discovering Practical Matrix Multiplication Algorithms
• Extracting the spin excitation spectrum of a fermionic system using a quantum processor
• High-precision and low-depth quantum algorithm design for eigenstate problems
• Adiabatic quantum computing with parameterized quantum circuits
• An Energy Estimation Benchmark for Quantum Computers
• Entanglement-informed Construction of Variational Quantum Circuits
• Cavity polariton blockade for non-local entangling gates with trapped atoms
• NISQ algorithm for the matrix elements of a generic observable
• qLEET: Visualizing Loss Landscapes, Expressibility, Entangling Power and Training Trajectories for Parameterized Quantum Circuits
• Quantum control without quantum states
• Quantum Strong-to-Weak Spontaneous Symmetry Breaking in Decohered One Dimensional Critical States
• Space-time tradeoff in networked virtual distillation
• Spatial correlations in the qubit properties of D-Wave 2000Q measured and simulated qubit networks
• State Specific Measurement Protocols for the Variational Quantum Eigensolver
• Path Matters: Industrial Data Meet Quantum Optimization
• Policy Gradients using Variational Quantum Circuits
• Practical limitations of quantum data propagation on noisy quantum processors
• The Lie Algebra of XY-mixer Topologies and Warm Starting QAOA for Constrained Optimization
• Predicting fermionic densities using a Projected Quantum Kernel method
• Dynamical decoupling of interacting spins through group factorization
• Exponential distillation of dominant eigenproperties
• Limitations of Quantum Hardware for Molecular Energy Estimation Using VQE
• Precision magnetometry exploiting excited state quantum phase transitions
• Resource-Efficient Hadamard Test Tailored Variational Framework for Nonlinear Dynamics on Quantum Computers
• Tangent Space Excitation Ansatz for Quantum Circuits
• Quantum State Preparation for Probability Distributions with Reflection Symmetry Using Matrix Product States
• Bridging Quantum Computing and Nuclear Structure: Atomic Nuclei on a Trapped-Ion Quantum Computer
• Learning the noise fingerprint of quantum devices
• Unsupervised Detection of Topological Phase Transitions with a Quantum Reservoir
• On the Noisy Road to Open Quantum Dynamics: The Place of Stochastic Hamiltonians
• Scalable Quantum Computational Science: A Perspective from Block-Encodings and Polynomial Transformations
• Solving the Hubbard model using density matrix embedding theory and the variational quantum eigensolver
• Topological Defects from Quantum Reset Dynamics
• Excluding false negative error in certification of quantum channels
• Tomography-assisted noisy quantum circuit simulator using matrix product density operators
• Integer Programming Using A Single Atom
• Demonstration of a variational quantum eigensolver with a solid-state spin system under ambient conditions
• Time crystal embodies chimeralike state in periodically driven quantum spin system
• Optimal Particle-Conserved Linear Encoding for Practical Fermionic Simulation
• A quantum tug of war between randomness and symmetries on homogeneous spaces
• Stabilizer configuration interaction: Finding molecular subspaces with error detection properties
• Quantum many-body simulation of finite-temperature systems with sampling a series expansion of a quantum imaginary-time evolution
• Quantum simulation of quantum mechanical system with spatial noncommutativity
• Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables
• Variational-toolbox-based separability detection of multiqubit states
• Microwave-activated high-fidelity three-qubit gate scheme for fixed-frequency superconducting qubits
• Quantum Computing Methods for Malware Detection
• A Noise-Aware Scalable Subspace Classical Optimizer for the Quantum Approximate Optimization Algorithm
• The Dual Role of Low-Weight Pauli Propagation: A Flawed Simulator but a Powerful Initializer for Variational Quantum Algorithms
• Energy-saving fast-forward scaling
• Making the cut: two methods for breaking down a quantum algorithm
• Distributing Quantum Computations, Shot-wise
• QTIS: A QAOA-Based Quantum Time Interval Scheduler
• Quantum simulation of entanglement dynamics in a quantum processor
• Unsupervised detection of decoupled subspaces: many-body scars and beyond
• Probabilistic imaginary-time evolution in state-vector-based and shot-based simulations and on quantum devices
• Phases and phase transition in Grover's algorithm with systematic noise
• Short-time simulation of quantum dynamics by Pauli measurements
• Noise tolerance via reinforcement: Learning a reinforced quantum dynamics
• Cheaper and more noise-resilient quantum state preparation using eigenvector continuation
• Determining the ensemble N-representability of Reduced Density Matrices
• A coherent approach to quantum-classical optimization
• OnionVQE Optimization Strategy for Ground State Preparation on NISQ Devices
• Estimating the degree of non-Markovianity using variational quantum circuits
• Accelerating Quantum Eigensolver Algorithms With Machine Learning
• Implementing Finite Impulse Response Filters on Quantum Computers
• Compressed sensing enhanced by quantum approximate optimization algorithm
• Correcting and extending Trotterized quantum many-body dynamics
• Digital quantum simulation of gravitational optomechanics with IBM quantum computers
• Digitized Counter-Diabatic Quantum Optimization for Bin Packing Problem
• Double descent in quantum kernel methods
• Quantum neural network autoencoder and classifier applied to an industrial case study
• A variational quantum algorithm for entanglement quantification
• Artificial intelligence for representing and characterizing quantum systems
• Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms
• Counterdiabatic ADAPT-VQE for molecular simulation
• Finding eigenvectors with a quantum variational algorithm
• Symmetry-based quantum algorithms for open-shop scheduling with hard constraints
• Symmetry-enhanced Counterdiabatic Quantum Algorithm for Qudits
• Adaptive Quantum Generative Training using an Unbounded Loss Function
• Efficient Fermi-Hubbard model ground-state preparation by coupling to a classical reservoir in the instantaneous-response limit
• Global optimization in variational quantum algorithms via dynamic tunneling method
• Quantum Annealing Algorithms for Estimating Ising Partition Functions
• Suppressing decoherence in quantum state transfer with unitary operations
• Quantum correlation generation capability of experimental processes
• Thermal masses and trapped-ion quantum spin models: a self-consistent approach to Yukawa-type interactions in the $\lambda {\varphi }^4$ model
• Quantum fidelity kernel with a trapped-ion simulation platform
• Warm-Start Variational Quantum Policy Iteration
• Weighted Approximate Quantum Natural Gradient for Variational Quantum Eigensolver
• Efficient Estimation and Sequential Optimization of Cost Functions in Variational Quantum Algorithms
• Describing Trotterized Time Evolutions on Noisy Quantum Computers via Static Effective Lindbladians
• Experimental factoring integers using fixed-point-QAOA with a trapped-ion quantum processor
• Vanishing performance of the parity-encoded quantum approximate optimization algorithm applied to spin-glass models
• Hybrid Real-Imaginary Time Evolution for Low-Depth Hamiltonian Simulation in Quantum Optimization
• Information Processing in Quantum Thermodynamic Systems: an Autonomous Hamiltonian Approach
• Moments of Quantum Channel Ensembles
• Correlated Purification for Restoring $N$-Representability in Quantum Simulation
• Noisy Monitored Quantum Circuits
• Applying Grover-mixer quantum alternating operator ansatz algorithm to higher-order unconstrained binary optimization problems
• Variational matrix product states for combinatorial optimization
• Disentangling quantum autoencoder
• A Taxonomy of Real Faults in Hybrid Quantum-Classical Architectures
• Resource-Optimized Grouping Shadow for Efficient Energy Estimation
• Quantum Encoding of Structured Data with Matrix Product States
• Quantum Alternating Operator Ansatz for the Preparation and Detection of Long-Lived Singlet States in NMR
• Hybrid Classical-Quantum Simulation of MaxCut using QAOA-in-QAOA
• Exponential Separation Criteria for Quantum Iterative Power Algorithms
• Equivalence between exponential concentration in quantum machine learning kernels and barren plateaus in variational algorithms
• Nonadiabatic Self-Healing of Trotter Errors in Digitized Counterdiabatic Dynamics
• Quantum State Preparation and Non-Unitary Evolution with Diagonal Operators
• Size-Consistent Quantum Chemistry on Quantum Computers
• Error bounds on the universal Lindblad equation in the thermodynamic limit
• Programming tools for Analogue Quantum Computing in the High-Performance Computing Context -- A Review
• Practical Quantum Reservoir Computing in Rydberg Atom Arrays
• A Useful Metric for the NISQ Era: Qubit Error Probability and Its Role in Zero Noise Extrapolation
• LArTPC hit-based topology classification with quantum machine learning and symmetry
• Toward Density Functional Theory on Quantum Computers?
• Performance evaluation of variational quantum eigensolver and quantum dynamics algorithms on the advection-diffusion equation
• Enhanced quantum hypothesis testing via the interplay between coherent evolution and noises
• A physical noise model for quantum measurements
• CDJ-Pontryagin Optimal Control for General Continuously Monitored Quantum Systems
• Limits of Clifford Disentangling in Tensor Network States
• Effective non-local parity-dependent couplings in qubit chains
• One-to-one correspondence between thermal structure factors and coupling constants of general bilinear Hamiltonians
• Circuit connectivity boosts by quantum-classical-quantum interfaces
• Information storage and transmission under Markovian noise
• Training Hybrid Classical-Quantum Classifiers via Stochastic Variational Optimization
• Hybrid quantum recurrent neural network for remaining useful life prediction of turbofan engines
• Qudit-native measurement protocol for dynamical correlations using Hadamard tests
• Velocity Verlet-based optimization for variational quantum eigensolvers
• Pulse engineering via projection of response functions
• Orders of chaoticity of unitaries
• QAISim: A Toolkit for Modeling and Simulation of AI in Quantum Cloud Computing Environments
• Variational Quantum Algorithm for Solving the Liouvillian Gap
• Unbiased observable estimation with approximate channels in fault-tolerant quantum computation
• Entropy density benchmarking of near-term quantum circuits
• Improving the trainability of VQE on NISQ computers for solving portfolio optimization using convex interpolation
• Quantum Algorithms for Causal Estimands
• From Hope to Heuristic: Realistic Runtime Estimates for Quantum Optimisation in NHEP
• Quantum Computation
• A variational quantum eigensolver tailored to multi-band tight-binding simulations of electronic structures
• Complex-valued in-medium potential between heavy impurities in ultracold atoms
• Controlling quantum chaos via Parrondo strategies on noisy intermediate-scale quantum hardware
• Double-bracket quantum algorithms for high-fidelity ground state preparation
• Masking Countermeasures Against Side-Channel Attacks on Quantum Computers
• Scalable Simulation of Quantum Many-Body Dynamics with Or-Represented Quantum Algebra
• Quantum-Classical Computing for Time-Dependent Ion-Atom Collision Dynamics: Applications to Charge Transfer Cross Section Simulations
• Measuring entanglement without local addressing in quantum many-body simulators via spiral quantum state tomography
• Many-body thermal states on a quantum computer: a variational approach
• Variational Quantum Subspace Construction via Symmetry-Preserving Cost Functions
• Robustness and classical proxy of entanglement in variants of quantum walk
• Analog Circuit-QED Simulator of Quantum Spin Dynamics Through the Extended Bose-Hubbard Model
• Universally Robust Control of Open Quantum Systems
• Inference of maximum parsimony phylogenetic trees with model-based classical and quantum methods
• Noise-Resilient Quantum Reinforcement Learning
• Geometric and holonomic quantum computation
• Scalable platform for qudit-based quantum computing using polar molecules
• Protocol for Purifying Noisy Preparation and Measurements of Qubits
• Capabilities and Limits of the Unitary Coupled-cluster Approach with Generalized Two-body Cluster Operators
• Role of overparametrization in quantum approximate optimization
• Superior resilience to poisoning and amenability to unlearning in quantum machine learning
• Quantum reservoir computing for predicting and characterizing chaotic maps
• Evidence for effectively constant shot complexity in the quantum approximate optimization algorithm without per-instance optimization
• Error-Mitigated Multi-Layer Quantum Routing
• Quantum approximate optimization of finite-state bosonic systems
• Exploring the Fidelity of Flux Qubit Measurement in Different Bases via the Quantum Flux Parametron
• Structural encoding with classical codes for computational-basis bit-flip correction in the early fault-tolerant regime
• Observation of the Quantum Zeno Effect on a NISQ Device