Quantum Approximate Optimization Algorithm: Performance on Simulators and Quantum Hardware

Abstract: Running quantum circuits on quantum computers does not always generate "clean" results, unlike on a simulator, as noise plays a significant role in any quantum device. To explore this, we experimented with the Quantum Approximate Optimization Algorithm (QAOA) on quantum simulators and real quantum hardware. QAOA is a hybrid classical-quantum algorithm and requires hundreds or thousands of independent executions of the quantum circuit for optimization, which typically goes beyond the publicly available resources for quantum computing. We were granted access to the IBM Quantum System One at the Cleveland Clinic, the first on-premises IBM system in the U.S. This paper explores different optimization methods, techniques, error mitigation methods, and QAOA variants to observe how they react to quantum noise differently, which is helpful for other researchers to understand the complexities of running QAOA on real quantum hardware and the challenges faced in dealing with noise.

• The paper demonstrates QAOA's application to the Max-Cut problem by comparing performance across simulators and real quantum hardware.
• It employs advanced error mitigation and pulse-level optimization techniques—such as Pauli Twirling, dynamical decoupling, and Fire Opal—to combat noise.
• Results reveal that deeper QAOA circuits increase noise exposure, and classical optimizers like CG can outperform methods such as COBYLA and Powell on hardware.

Quantum Approximate Optimization Algorithm: Empirical Evaluation on Simulators and Quantum Hardware

Introduction

This paper presents a comprehensive empirical study of the Quantum Approximate Optimization Algorithm (QAOA) for the Max-Cut problem, focusing on the comparative performance of QAOA on quantum simulators and real quantum hardware. The work leverages the IBM Quantum System One (Eagle R3, 127 qubits) at the Cleveland Clinic, providing a rare opportunity to assess QAOA on a state-of-the-art superconducting platform. The study systematically investigates the impact of classical optimization strategies, circuit depth, error mitigation techniques, and advanced pulse-level optimizations, with a particular emphasis on the challenges posed by noise in Noisy Intermediate-Scale Quantum (NISQ) devices.

QAOA and Max-Cut: Algorithmic and Implementation Details

QAOA is a hybrid variational algorithm designed for combinatorial optimization, alternating between problem-specific Cost Hamiltonians and Mixer Hamiltonians. For Max-Cut, the Cost Hamiltonian encodes the cut value via parameterized $R_z$ rotations and CNOT gates for each edge, while the Mixer Hamiltonian applies $R_x$ rotations to all qubits. The ansatz is constructed with $p$ layers, each introducing two variational parameters $(\gamma_i, \beta_i)$ per layer.

The optimization loop involves:

1. Initializing parameters, typically in a uniform superposition via Hadamard gates.
2. Alternating application of Cost and Mixer Hamiltonians for $p$ layers.
3. Measurement and classical evaluation of the expected cut value.
4. Classical optimization (using COBYLA, Powell, or CG) to update parameters.

The implementation uses Qiskit for circuit construction, transpilation, and execution, with the Sampler primitive for measurement statistics. The Max-Cut value is computed by averaging over sampled bitstrings, with the objective function being the negative expected cut value.

Experimental Design and Optimization Strategies

The study evaluates QAOA on three backends:

• AerSimulator: Ideal and noisy simulation, with noise models derived from real hardware calibration data.
• Fake Providers: Static, hardware-aware simulators using historical calibration data.
• IBM Quantum Hardware: Real execution on the Eagle R3 backend.

Three classical optimizers are compared:

• COBYLA: Gradient-free, simplex-based, efficient for constrained, non-smooth landscapes.
• Powell: Gradient-free, direction-set method, but with higher iteration counts and sensitivity to noise.
• Conjugate Gradient (CG): Efficient for smooth landscapes, but more sensitive to noise; in practice, outperformed others on hardware in this study.

Circuit depth $p$ is varied ($1 \leq p \leq 5$) to assess the trade-off between expressivity and noise resilience.

Error Mitigation and Pulse-Level Techniques

The paper implements several error mitigation strategies:

• Pauli Twirling: Converts coherent errors to stochastic Pauli channels, reducing error accumulation.
• Dynamical Decoupling (XY4 sequence): Inserts idle-time pulses to refocus low-frequency noise.
• Transpiler Optimizations: optimization_level=3 in Qiskit to minimize circuit depth and gate count.
• Pulse Programming: Direct pulse-level control for native gate implementation and crosstalk mitigation (noting the deprecation of Qiskit Pulse in favor of Qiskit Dynamics).
• Q-CTRL Fire Opal: Commercial platform for automated pulse optimization, error suppression, and circuit compilation, yielding further noise reduction and symmetry breaking in degenerate solutions.

Results and Analysis

Simulator Results

• COBYLA and Powell: On AerSimulator, both optimizers reliably converge to the correct Max-Cut solutions for $p \geq 4$, with Powell requiring significantly more iterations but achieving slightly lower minimum energies.
• Noise Models: AerSimulator with live noise models produces noisier distributions, while Fake Providers yield artificially clean results due to static calibration data, sometimes outperforming ideal simulations in terms of solution sharpness due to biasing effects.

Hardware Results

• Optimizer Comparison: On real hardware, CG outperforms COBYLA and Powell in terms of solution quality, despite requiring more iterations. Powell is particularly susceptible to noise, often failing to converge to optimal solutions.
• Circuit Depth: Contrary to expectations, increasing $p$ does not consistently improve performance on hardware; higher $p$ increases circuit complexity and noise exposure, often degrading results.
• Parameter Evolution: Parameter trajectories exhibit plateauing and stabilization, with optimizers continuing to iterate due to minor cost improvements from sampling noise rather than parameter updates.

Error Mitigation

• Pauli Twirling and Dynamical Decoupling: Both techniques significantly reduce noise, yielding more prominent optimal bitstrings and improved solution quality.
• Fire Opal: Outperforms standard error mitigation, producing highly concentrated solution distributions and breaking degeneracies via symmetry-breaking effects. However, the proprietary nature of Fire Opal limits transparency and reproducibility.

Comparative Insights

• Simulators vs. Hardware: Simulators (especially with ideal or Fake Provider backends) overestimate QAOA performance relative to real hardware. AerSimulator with live noise models provides a more pessimistic but realistic assessment.
• Error Mitigation: Transparent, open-source error mitigation strategies can approach the performance of commercial solutions like Fire Opal, provided careful calibration and circuit optimization are performed.

Implications and Future Directions

This study demonstrates that QAOA, when combined with robust error mitigation and careful optimizer selection, can yield meaningful results on current NISQ hardware for small-scale Max-Cut instances. The findings challenge the assumption that deeper QAOA circuits ($p$ large) always improve performance on hardware, highlighting the critical role of noise and circuit complexity.

The results suggest several avenues for future research:

• Alternative Ansatzes: Exploring hardware-efficient or problem-inspired ansatzes to balance expressivity and noise resilience.
• Advanced Optimizers: Investigating noise-aware or quantum-aware classical optimizers tailored for NISQ landscapes.
• Benchmarking: Systematic evaluation using libraries like QOBLIB to assess QAOA performance on larger, more challenging instances.
• Hardware-Aware Compilation: Further integration of pulse-level optimizations and hardware-specific compilation strategies.
• Open-Source Error Mitigation: Development of transparent, reproducible error mitigation pipelines to match or exceed proprietary solutions.

Conclusion

The empirical evaluation of QAOA on both simulators and real quantum hardware reveals nuanced trade-offs between circuit depth, optimizer choice, and error mitigation. While simulators provide optimistic baselines, real hardware execution exposes the limitations imposed by noise and decoherence. Error mitigation techniques, both open-source and commercial, are essential for extracting meaningful results from NISQ devices. The study underscores the importance of hybrid workflows, hardware-aware optimization, and transparent benchmarking in advancing the practical utility of quantum optimization algorithms.