QAOA-based Fair Sampling on NISQ Devices.

test. We find that fairness is relatively high for circuits with small and large error rates, but drops for circuits with medium error rates. This indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits. Our results provide a simple, intuitive means of quantifying fairness in quantum circuits, and show that reducing structured errors is necessary to improve fair sampling on NISQ hardware.

Related papers

Fair Sampling Error Analysis on NISQ Devices

John K. Golden, Andreas Bärtschi, Daniel O’Malley +1 more

We study the status of fair sampling on Noisy Intermediate Scale Quantum (NISQ) devices, in particular the IBM Q family of backends. Using the recently introduced Grover Mixer-QAOA algorithm for discrete optimization, we generate fair sampling circuits to solve six problems of varying difficulty, each with several optimal solutions, which we then run on twenty backends across the IBM Q system. For a given circuit evaluated on a specific set of qubits, we evaluate: how frequently the qubits return an optimal solution to the problem, the fairness with which the qubits sample from all optimal solutions, and the reported hardware error rate of the qubits. To quantify fairness, we define a novel metric based on Pearson’s χ 2 test. We find that fairness is relatively high for circuits with small and large error rates, but drops for circuits with medium error rates. This indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits. Our results show that fairness can be a powerful tool for understanding the intricate web of errors affecting current NISQ hardware.We study the status of fair sampling on Noisy Intermediate Scale Quantum (NISQ) devices, in particular the IBM Q family of backends. Using the recently introduced Grover Mixer-QAOA algorithm for discrete optimization, we generate fair sampling circuits to solve six problems of varying difficulty, each with several optimal solutions, which we then run on twenty backends across the IBM Q system. For a given circuit evaluated on a specific set of qubits, we evaluate: how frequently the qubits return an optimal solution to the problem, the fairness with which the qubits sample from all optimal solutions, and the reported hardware error rate of the qubits. To quantify fairness, we define a novel metric based on Pearson’s χ 2 test. We find that fairness is relatively high for circuits with small and large error rates, but drops for circuits with medium error rates. This indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits. Our results show that fairness can be a powerful tool for understanding the intricate web of errors affecting current NISQ hardware.

arXiv:2101.03258ACM Transactions on Quantum ComputingMar 7, 202216 citations

Sampling on NISQ Devices: "Who’s the Fairest One of All?"

Elijah Pelofske, John K. Golden, Andreas Bärtschi +2 more

Modern NISQ devices are subject to a variety of biases and sources of noise that degrade the solution quality of computations carried out on these devices. A natural question that arises in the NISQ era, is how fairly do these devices sample ground state solutions. To this end, we run five fair sampling problems (each with at least three ground state solutions) that are based both on quantum annealing and on the Grover Mixer-QAOA algorithm for gate-based NISQ hardware. In particular, we use seven IBM Q devices, the Aspen-9 Rigetti device, the IonQ device, and three D-Wave quantum annealers.For each of the fair sampling problems, we measure the ground state probability, the relative fairness of the frequency of each ground state solution with respect to the other ground state solutions, and the aggregate error as given by each hardware provider. Overall, our results show that NISQ devices do not achieve fair sampling yet. We also observe differences in the software stack with a particular focus on compilation techniques that illustrate what work will still need to be done to achieve a seamless integration of frontend (i.e. quantum circuit description) and backend compilation.Modern NISQ devices are subject to a variety of biases and sources of noise that degrade the solution quality of computations carried out on these devices. A natural question that arises in the NISQ era, is how fairly do these devices sample ground state solutions. To this end, we run five fair sampling problems (each with at least three ground state solutions) that are based both on quantum annealing and on the Grover Mixer-QAOA algorithm for gate-based NISQ hardware. In particular, we use seven IBM Q devices, the Aspen-9 Rigetti device, the IonQ device, and three D-Wave quantum annealers.For each of the fair sampling problems, we measure the ground state probability, the relative fairness of the frequency of each ground state solution with respect to the other ground state solutions, and the aggregate error as given by each hardware provider. Overall, our results show that NISQ devices do not achieve fair sampling yet. We also observe differences in the software stack with a particular focus on compilation techniques that illustrate what work will still need to be done to achieve a seamless integration of frontend (i.e. quantum circuit description) and backend compilation.

arXiv:2107.06468IEEE International Conference on Quantum Computing and Engineering (QCE)Oct 1, 202126 citations

NISQ Benchmarking

Elijah Pelofske, Andreas Bärtschi, Stephan Eidenbenz +1 more

Test suite of quantum algorithms for Noisy Intermediate Scale Quantum (NISQ) computers. The test suite includes benchmark-style code for quantum volume circuits (QV), fairness sampling circuits, quantum telecloning circuits, and other NISQ benchmark style algorithms on small problems (i.e., up to 100 qubits), such as Variational Quantum Eigensolver (VQE), Hamiltonian Simulation, and Grover unstructured search example circuits. These benchmark-style applications are implemented in quantum software packages, mostly IBM's QISKIT, but may include vendor-specific frameworks, such as PyQuil (for Rigetti) or Q\# for Microsoft, or CirQ (for Google) as the test suite grows with the vendor sample. The test suite also includes numerical simulation code for Quantum Alternating Operator Ansatz (QAOA) algorithms, VQE, Hamiltonian Simulation and search examples. Numerical simulation code simulates quantum computers on classical computers, which is only possible for small problem instances; the implementation framework of choice is typically within Python, using the numpy/scipy libraries as well as extensions to the Julia language.Test suite of quantum algorithms for Noisy Intermediate Scale Quantum (NISQ) computers. The test suite includes benchmark-style code for quantum volume circuits (QV), fairness sampling circuits, quantum telecloning circuits, and other NISQ benchmark style algorithms on small problems (i.e., up to 100 qubits), such as Variational Quantum Eigensolver (VQE), Hamiltonian Simulation, and Grover unstructured search example circuits. These benchmark-style applications are implemented in quantum software packages, mostly IBM's QISKIT, but may include vendor-specific frameworks, such as PyQuil (for Rigetti) or Q\# for Microsoft, or CirQ (for Google) as the test suite grows with the vendor sample. The test suite also includes numerical simulation code for Quantum Alternating Operator Ansatz (QAOA) algorithms, VQE, Hamiltonian Simulation and search examples. Numerical simulation code simulates quantum computers on classical computers, which is only possible for small problem instances; the implementation framework of choice is typically within Python, using the numpy/scipy libraries as well as extensions to the Julia language.

Jan 1, 20220 citations

Test Points for Online Monitoring of Quantum Circuits

Nikita Acharya, Miroslav Urbánek, Wibe Albert de Jong +1 more

Noisy Intermediate-Scale Quantum (NISQ) computers consisting of tens of inherently noisy quantum bits (qubits) suffer from reliability problems. Qubits and their gates are susceptible to various types of errors. Due to limited numbers of qubits and high error rates, quantum error correction cannot be applied. Physical constraints of quantum hardware including the error rates are used to guide the design and the layout of quantum circuits. The error rates determine the selection of qubits and their operations. The resulting circuit is executed on the quantum computer. This study explores the risk of unexpected changes in the error rates of NISQ computers post-calibration. We show that unexpected changes in error rates can alter the output state of a quantum circuit. To detect these changes, we propose the insertion of test points into the quantum circuit to enable online monitoring of the physical qubit behavior. We utilize classical, superposition, and uncompute test points. Furthermore, we use a gate error coverage metric to assess the quality of the tests. We verify the effectiveness of the proposed scheme on different IBM quantum computers (IBM Q), in addition to a noisy simulation that shows the scalability of the proposed approach.Noisy Intermediate-Scale Quantum (NISQ) computers consisting of tens of inherently noisy quantum bits (qubits) suffer from reliability problems. Qubits and their gates are susceptible to various types of errors. Due to limited numbers of qubits and high error rates, quantum error correction cannot be applied. Physical constraints of quantum hardware including the error rates are used to guide the design and the layout of quantum circuits. The error rates determine the selection of qubits and their operations. The resulting circuit is executed on the quantum computer. This study explores the risk of unexpected changes in the error rates of NISQ computers post-calibration. We show that unexpected changes in error rates can alter the output state of a quantum circuit. To detect these changes, we propose the insertion of test points into the quantum circuit to enable online monitoring of the physical qubit behavior. We utilize classical, superposition, and uncompute test points. Furthermore, we use a gate error coverage metric to assess the quality of the tests. We verify the effectiveness of the proposed scheme on different IBM quantum computers (IBM Q), in addition to a noisy simulation that shows the scalability of the proposed approach.

ACM Journal on Emerging Technologies in Computing SystemsOct 22, 20217 citations

Limits of QAOA Depth Scaling on IBM Quantum Hardware for Fault Tree QUBOs

Devin Peters

The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for near-term quantum advantage in combinatorial optimization. A central design parameter is circuit depth p, with theoretical analyses establishing asymptotic convergence under idealized conditions. This paper presents a limits study demonstrating that this theoretical relationship does not translate to measurable improvement on current noisy intermediate-scale quantum (NISQ) hardware for fault tree analysis applications. Across 120 QAOA executions on 60 QUBO-encoded fault tree instances (N ∈ {12, 16, 20, 24, 28, 32}) on IBM Quantum hardware (ibm_torino, 133-qubit Heron processor), circuit depths p = 1 and p = 2 were directly compared. The mean change in optimal state sampling probability from doubling circuit depth was Δmax_prob = 1.22 × 10⁻⁵ with a median Δ of zero. For 8,192 shots, the binomial sampling uncertainty is O(10⁻²); the observed mean delta is therefore approximately three orders of magnitude below the shot-noise scale, making the difference statistically indistinguishable from zero. Output distribution entropy approaches the theoretical maximum of log₂(8192) ≈ 13 bits, indicating hardware noise dominates algorithmic signal. These findings establish that for fault tree minimal cut set identification on current NISQ devices, computational resources are more effectively allocated to increased shot counts than to circuit depth. Complete experimental artifacts with SHA-256 verification are provided, enabling full reproducibility.The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for near-term quantum advantage in combinatorial optimization. A central design parameter is circuit depth p, with theoretical analyses establishing asymptotic convergence under idealized conditions. This paper presents a limits study demonstrating that this theoretical relationship does not translate to measurable improvement on current noisy intermediate-scale quantum (NISQ) hardware for fault tree analysis applications. Across 120 QAOA executions on 60 QUBO-encoded fault tree instances (N ∈ {12, 16, 20, 24, 28, 32}) on IBM Quantum hardware (ibm_torino, 133-qubit Heron processor), circuit depths p = 1 and p = 2 were directly compared. The mean change in optimal state sampling probability from doubling circuit depth was Δmax_prob = 1.22 × 10⁻⁵ with a median Δ of zero. For 8,192 shots, the binomial sampling uncertainty is O(10⁻²); the observed mean delta is therefore approximately three orders of magnitude below the shot-noise scale, making the difference statistically indistinguishable from zero. Output distribution entropy approaches the theoretical maximum of log₂(8192) ≈ 13 bits, indicating hardware noise dominates algorithmic signal. These findings establish that for fault tree minimal cut set identification on current NISQ devices, computational resources are more effectively allocated to increased shot counts than to circuit depth. Complete experimental artifacts with SHA-256 verification are provided, enabling full reproducibility.

Jan 18, 20260 citations

Error Mitigation–Driven Enhancement of QAOA Performance on NISQ Devices

Kondreddy Sathwika, Maka Sri Varshith, Marati Nanda Kumar +1 more

The Quantum Approximate Optimization Algorithm (QAOA) is widely regarded as a promising near-term quantum algorithm for addressing hard combinatorial optimization problems on Noisy Intermediate-Scale Quantum (NISQ) devices; however, its real-world effectiveness is severely constrained by hardware noise, including gate infidelities, decoherence, and measurement errors. This work explores an error mitigation–driven approach to enhance the performance and robustness of QAOA without altering its algorithmic structure or requiring fault-tolerant quantum resources. Measurement error mitigation and zero-noise extrapolation are incorporated into the execution and post-processing stages of QAOA circuits and evaluated using the Minimum Vertex Cover problem as a representative benchmark. Performance is assessed through approximation ratios, solution stability, and noise sensitivity across multiple circuit depths. The results indicate that error mitigation not only improves approximation accuracy but also stabilizes output distributions and delays performance degradation at increased circuit depths. Additionally, the mitigated executions demonstrate better consistency across repeated runs, highlighting improved resilience to stochastic noise. These findings emphasize that lightweight and hardware-efficient error mitigation techniques can significantly extend the practical usability of QAOA on current NISQ hardware, providing a scalable and realistic pathway for executing quantum optimization algorithms more reliably in near-term quantum computing environments.The Quantum Approximate Optimization Algorithm (QAOA) is widely regarded as a promising near-term quantum algorithm for addressing hard combinatorial optimization problems on Noisy Intermediate-Scale Quantum (NISQ) devices; however, its real-world effectiveness is severely constrained by hardware noise, including gate infidelities, decoherence, and measurement errors. This work explores an error mitigation–driven approach to enhance the performance and robustness of QAOA without altering its algorithmic structure or requiring fault-tolerant quantum resources. Measurement error mitigation and zero-noise extrapolation are incorporated into the execution and post-processing stages of QAOA circuits and evaluated using the Minimum Vertex Cover problem as a representative benchmark. Performance is assessed through approximation ratios, solution stability, and noise sensitivity across multiple circuit depths. The results indicate that error mitigation not only improves approximation accuracy but also stabilizes output distributions and delays performance degradation at increased circuit depths. Additionally, the mitigated executions demonstrate better consistency across repeated runs, highlighting improved resilience to stochastic noise. These findings emphasize that lightweight and hardware-efficient error mitigation techniques can significantly extend the practical usability of QAOA on current NISQ hardware, providing a scalable and realistic pathway for executing quantum optimization algorithms more reliably in near-term quantum computing environments.

International Conference on AI-Driven Smart Systems and Ubiquitous Computing (ICAUC)Jan 19, 20260 citations

\chi^2