Quantum Computing Benchmarks

Surface code quantum error correction · other

Google Quantum AI's metric for measuring progress toward fault-tolerant quantum computing, based on the error suppression factor Lambda achieved by surface code logical qubits. On the Willow processor, a distance-7 surface code using 101 qubits achieved a logical error rate of 0.143% per cycle, with errors suppressed by a factor of 2.14 for each increase in code distance by 2, operating below the surface code threshold.

Parameterized model circuits (Quantum Volume-type) · other

CLOPS is a hardware-agnostic speed metric developed by IBM that measures how fast a quantum processor can execute parameterized circuit layers. It captures the full hardware-software stack including gate execution speed, classical compilation time, control instruction generation, and data transfer rates. IBM updated the metric in 2023 to CLOPSh, replacing Quantum Volume with layer fidelity for a more comprehensive assessment.

Linear cross-entropy on random quantum circuits · other

Cross-entropy benchmarking (XEB) is a statistical method for evaluating quantum processor performance by measuring how well experimental output distributions of random circuits correlate with ideal distributions. Google used linear XEB to demonstrate quantum computational advantage on the 53-qubit Sycamore processor in 2019. The XEB fidelity decays exponentially with circuit depth, providing a measure of cumulative gate quality.

VQE with Hamiltonian Variational Ansatz · simulation

Comprehensive benchmark of 30 classical optimizers on 372 instances of VQE for the Fermi-Hubbard system using the Hamiltonian variational ansatz. Best-performing optimizers are gradient-descent variants (Momentum, ADAM with finite differences), SPSA, CMA-ES, and BayesMGD. The study demonstrates that finite-difference step size has a very significant impact on VQE convergence quality.

Grover's search algorithm · cryptography

Benchmark measuring the scaling behavior of Grover's quantum search algorithm on NISQ hardware. Better-than-classical success probabilities have been demonstrated up to five qubits on IBM superconducting platforms using robust dynamical decoupling. The GRADE benchmarking toolkit provides standardized evaluation of quantum hardware reliability using generalized Grover circuits with customizable search configurations.

Variational Quantum Eigensolver (VQE) · chemistry

Benchmark for quantum simulation of the water molecule (H2O), computing ground-state energies and potential energy surfaces using VQE. As a triatomic molecule with a non-trivial electronic structure, H2O serves as a step up in complexity from diatomic benchmarks like H2 and LiH. VQE implementations can approximate the ground-state energy with high chemical accuracy on both simulators and NISQ hardware.

Variational Quantum Eigensolver (VQE) · simulation

Benchmark for quantum simulation of the Hubbard model, a fundamental model of strongly correlated electrons in condensed matter physics. The relative simplicity of the Hubbard Hamiltonian allows efficient implementation on quantum hardware. Classical benchmarks show that even the most accurate VQE ansatze exhibit plateauing ground-state energy errors for larger lattices, with stronger electronic correlations magnifying the issue.

Variational Quantum Eigensolver (VQE) · chemistry

Benchmark for computing the ground-state energy of lithium hydride (LiH) using the Variational Quantum Eigensolver. LiH is a standard test molecule for quantum chemistry on NISQ devices, requiring 6-12 qubits depending on the active space and basis set. The benchmark tests VQE accuracy against Full Configuration Interaction (FCI) results across a range of bond distances.

Randomized mirror circuits (Clifford-based) · other

Scalable randomized benchmarking method using mirror circuits that estimates the average infidelity of many-qubit logic layers. Unlike standard randomized benchmarking, which becomes infeasible beyond roughly five qubits, mirror circuits scale to hundreds of qubits. The technique can reveal and quantify crosstalk errors, demonstrated with simulations of up to 225 qubits and experiments on up to 16 physical qubits.

70+ quantum algorithms across multiple domains · other

MQT Bench is a comprehensive quantum computing benchmark suite containing over 70 algorithm implementations across various application domains. Developed by the Technical University of Munich, it provides quantum circuits at different abstraction levels (2 to 130 qubits) and in multiple formats. MQT Bench is designed for evaluating quantum compilers, simulators, and hardware platforms.

VQE with UCCSD / Contextual Subspace VQE · chemistry

Benchmark of quantum chemistry methods for computing the potential energy curve of molecular nitrogen (N2) dissociation. The N2 triple bond makes this a challenging test of strongly correlated electron methods. Contextual Subspace VQE has been demonstrated on superconducting quantum hardware for the entire N2 potential energy curve, while cyclic VQE variants achieve chemical accuracy even where standard UCCSD errors increase significantly.

QAOA MaxCut · optimization

Q-Score is a quantum computing benchmark metric developed by Atos that evaluates quantum processors using QAOA applied to the MaxCut problem. It provides a standardized, hardware-agnostic way to compare quantum computing platforms. Q-Score measures the maximum number of variables in a MaxCut problem that can be solved with acceptable quality.

Quantum Approximate Optimization Algorithm (QAOA) · optimization

Benchmark of the Quantum Approximate Optimization Algorithm for solving the MaxCut problem on random graphs. Tests the quality of approximate solutions as a function of circuit depth (p-levels) and graph size. A standard benchmark for hybrid quantum-classical optimization.

Multiple algorithms (BV, QFT, Hidden Shift, etc.) · other

QED-C (Quantum Economic Development Consortium) Application-Oriented Benchmarks is a comprehensive suite of quantum algorithms designed to evaluate quantum computing platforms. The suite includes implementations across multiple problem domains and is designed to be scalable across varying qubit counts. These benchmarks focus on real-world applications rather than abstract metrics.

Volumetric benchmarking framework · other

Extension of the QED-C benchmark framework focused on application-specific benchmark profiles that measure quantum hardware capability across problem-relevant circuit shapes and depths. The volumetric benchmarking methodology plots result quality against circuit width and depth to produce capability maps, enabling comparison across different quantum platforms for specific application domains such as optimization, chemistry, and machine learning.

BB84 quantum key distribution protocol · cryptography

Benchmark evaluating quantum key distribution throughput and security using the BB84 protocol, the foundational QKD scheme proposed by Bennett and Brassard in 1984. The secure bound of the quantum bit error rate (QBER) for BB84 is approximately 11%, extendable to 12.6% with six-state variants. Benchmarks assess secret key rates, QBER tolerance, and distance limits under coherent attacks with decoy-state methods.

Quantum Amplitude Estimation (BHMT algorithm) · optimization

Benchmark based on the Brassard-Hoyer-Mosca-Tapp (BHMT) quantum counting algorithm, which estimates the number of marked items in an unstructured database with quadratic speedup over classical counting. Given N items with K marked, the algorithm estimates K to within relative error epsilon using O(1/epsilon * sqrt(N/K)) queries. A simplified variant achieves the same query complexity using only Grover iterations.

Quantum Generative Adversarial Network (QGAN) · machine-learning

Benchmark for quantum generative adversarial networks that learn probability distributions and load them into quantum states. Through the interplay of quantum channels and classical neural networks, QGANs achieve polynomial gate complexity for distribution loading. Implementations on IBM Quantum hardware demonstrate competitive image generation on MNIST and Fashion-MNIST, with the MosaiQ framework achieving significant improvements in Frechet Inception Distance scores.

Quantum Support Vector Machine (QSVM) with quantum kernels · machine-learning

Benchmark evaluating quantum kernel estimation (QKE) and quantum kernel training (QKT) for classification tasks using quantum support vector machines. A large-scale study examined fidelity-based and projected quantum kernels across 64 datasets and five dataset families. While quantum kernels outperform classical methods on specifically constructed ad-hoc datasets, results on standard classical benchmarks are mixed.

QAOA / Copula-QAOA / AAM-QAOA · optimization

Benchmark of quantum optimization algorithms applied to the knapsack problem, a fundamental NP-hard constrained combinatorial optimization problem. Recent advances include copula-QAOA for hardware-efficient constraint handling and Amplitude Amplification-mixer QAOA (AAM-QAOA), which uses a quantum tree generator for feasible-solution state preparation. Benchmark sets test instances with up to 20 items.

Quantum Convolutional Neural Network (QCNN) · machine-learning

Benchmark comparing quantum neural network architectures for image classification tasks on standard datasets including MNIST and Fashion-MNIST. Fully parameterized QCNNs achieve excellent classification accuracy despite a small number of free parameters. Comparative studies evaluate noise robustness across different QCNN model structures, data encoding methods, and optimizers under realistic quantum noise channels.

QAOA / VQE / Quantum Annealing · optimization

Finance-specific benchmark evaluating quantum optimization algorithms for Markowitz mean-variance portfolio selection. Compares QAOA, VQE, and quantum annealing against classical solvers (MIP, meta-heuristics) on asset selection and allocation problems. Recent studies highlight that while quantum methods minimize cost functions effectively, resulting portfolios may violate practical financial constraints.

Random quantum circuits · other

Quantum Volume is a hardware-agnostic metric developed by IBM to measure the capability of quantum computers. It uses random quantum circuits to test the combination of qubit count, gate fidelity, connectivity, and error rates. Quantum Volume provides a single-number metric that captures overall quantum processor performance and has become an industry-standard benchmark.

Quantum Annealing / QAOA / Simulated Bifurcation · optimization

Benchmark suite for Quadratic Unconstrained Binary Optimization (QUBO), the standard formulation for mapping combinatorial problems onto quantum hardware. Compares quantum solvers (D-Wave Hybrid Solver, gate-based QAOA) against quantum-inspired solvers (Toshiba SBM, Fujitsu Digital Annealer) and classical heuristics on established instance libraries including MQLib, random NAE 3-SAT, and Sherrington-Kirkpatrick models.

Clifford randomized benchmarking protocol · other

The standard protocol for estimating average gate error rates in quantum processors, developed by Magesan, Gambetta, and Emerson. Random sequences of Clifford gates followed by an inversion gate are applied, and the survival probability decays exponentially with sequence length at a rate determined by the average gate infidelity. Interleaved randomized benchmarking extends the protocol to characterize individual gate errors.

Shor's factoring algorithm · cryptography

Benchmark of Shor's quantum factoring algorithm on small semiprime instances. Current NISQ hardware can only factor very small numbers (N=15, N=21) with low success rates due to high error rates. GPU-based supercomputer simulations have assessed Shor's algorithm for larger numbers, finding average success probabilities above 50% due to high frequency of 'lucky' cases, with the largest simulated factorization reaching 549,755,813,701.

QAOA, VQE, Hamiltonian Simulation · optimization

SupermarQ is an application-oriented quantum benchmark suite developed by the SuperTech group. It focuses on practical quantum algorithms including QAOA, VQE, and Hamiltonian simulation with scalable problem sizes. SupermarQ provides both algorithm implementations and standardized metrics for evaluating quantum hardware performance on real-world applications.

VQE / Hamiltonian Simulation / Exact Diagonalisation · simulation

Benchmark for quantum simulation of the transverse-field Ising model (TFIM), a paradigmatic model for studying quantum phase transitions. Implementations range from VQE ground-state calculations on NISQ hardware (tested on IQM Garnet, IBM Quantum) to Hamiltonian simulation of dynamics on trapped-ion platforms. Studies cover 1D chains up to 25 qubits and 2D lattices up to 6x6 sites.

QAOA / Variational Quantum Eigensolver (VQE) · optimization

Benchmark of quantum approaches to the Travelling Salesman Problem, a canonical NP-hard combinatorial optimization task. Instances are encoded as QUBO/Ising Hamiltonians and solved using QAOA or VQE on gate-based hardware, or via quantum annealing. Current NISQ-era implementations handle instances of up to roughly six cities, making TSP a demanding test of quantum optimizer quality.

Variational Quantum Classifier (VQC) · machine-learning

Benchmark of variational quantum classifiers that use parameterized quantum circuits for supervised learning. Circuit-centric quantum classifiers encode classical data into quantum states via feature maps, then apply trainable unitaries and measure a readout qubit for binary classification. Benchmarks evaluate different ansatz architectures, encoding strategies, and circuit depths on standard datasets including Iris, MNIST subsets, and domain-specific applications.

Variational Quantum Eigensolver (VQE) · chemistry

Benchmark of the Variational Quantum Eigensolver algorithm for computing the ground state energy of the hydrogen molecule (H2). This is a standard benchmark for hybrid quantum-classical chemistry simulations, testing the interplay between quantum circuit evaluation and classical optimization.

QAOA / Quantum Walk Optimisation Algorithm (QWOA) · optimization

Extended Max-Cut benchmark using weighted graph instances, which present a harder optimization landscape than the standard unweighted variant. Weighted instances introduce a proliferation of poor local optima and exacerbate barren-plateau issues. Recent work compares QAOA with non-variational quantum walk algorithms on weighted instances up to 31 nodes.