Pattern-Dependent Performance of the Bernstein-Vazirani Algorithm

TL;DR

The paper investigates how the structure of the Bernstein-Vazirani oracle encoded in a hidden string s affects algorithm performance on NISQ hardware. Using a hardware-aware benchmarking framework across 11 patterns and four IBM Q processors, it demonstrates a strong pattern dependence where dense, symmetric encodings suffer dramatic fidelity loss and near-zero success on hardware, despite favorable simulator results. A near-perfect correlation between pattern density and state fidelity (r ≈ 0.972) explains the observed performance cliff and reveals significant gaps in current noise models for structure-dependent errors. The results motivate structure-aware algorithm design and co-design approaches that minimize entanglement load, suggesting practical pathways and limitations for achieving quantum advantage on near-term devices.

Abstract

Quantum computers promise to redefine the boundaries of computational science, offering the potential for exponential speedups in solving complex problems across chemistry, optimization, and materials science. Yet, their practical utility remains constrained by unpredictable performance degradation under real-world noise conditions. A key question is how problem structure itself influences algorithmic resilience. In this work, we present a comprehensive, hardware-aware benchmarking study of the Bernstein-Vazirani algorithm across 11 diverse test patterns on multiple superconducting quantum processors, revealing that algorithmic performance is exquisitely sensitive to problem structure. Our results reveal average success rates of 100.0\% (ideal simulation), 85.2\% (noisy emulation), and 26.4\% (real hardware), representing a dramatic 58.8\% average performance gap between noisy emulation and real hardware execution. With quantum state tomography confirming corresponding average state fidelities of 0.993, 0.760, and a 0.234 fidelity drop to hardware. Performance degrades dramatically from 75.7\% success for sparse patterns to complete failure for high-density 10-qubit patterns. Most strikingly, quantum state tomography reveals a near-perfect correlation between pattern density and state fidelity degradation, providing the fundamental explanation for observed performance patterns. The catastrophic fidelity collapse observed in real hardware measurements -- dropping to 0.111 compared to the predicted 0.763 -- underscores severe limitations in current noise models for capturing structure-dependent error mechanisms. Our work establishes pattern-dependent performance as a critical consideration for quantum algorithm deployment and provides a quantitative framework for predicting algorithm feasibility in practical applications.

Figures (8)

• Figure 1: Bernstein-Vazirani Algorithm. Step-by-step quantum state evolution demonstrating exponential quantum advantage. The algorithm progresses through five key transformations: (1) Initialization prepares $\ket{\psi_0} = \ket{0}^{\otimes n}\ket{1}$, (2) Superposition creates $\ket{\psi_1} = \frac{1}{\sqrt{2^n}}\sum_{x\in\{0,1\}^n} \ket{x}\ket{-}$ via Hadamard gates, (3) Oracle application encodes the hidden string through phase kickback $\ket{\psi_2} = \frac{1}{\sqrt{2^n}}\sum_x (-1)^{s\cdot x}\ket{x}\ket{-}$, (4) Interference via final Hadamard gates collapses the state to $\ket{\psi_3} = \ket{s}\ket{-}$, and (5) Measurement reveals $s$ with unit probability. The complete state evolution is mathematically characterized by: $\ket{\psi_0} = \ket{0}^{\otimes n}\ket{1}, \quad \ket{\psi_1} = \frac{1}{\sqrt{2^n}}\sum_x \ket{x}\ket{-}, \quad \ket{\psi_2} = \frac{1}{\sqrt{2^n}}\sum_x (-1)^{s\cdot x}\ket{x}\ket{-}, \quad \ket{\psi_3} = \ket{s}\ket{-}$ This quantum workflow achieves $O(1)$ query complexity compared to $O(n)$ classically, providing exponential speedup for hidden string recovery.
• Figure 2: Output distributions for Bernstein-Vazirani algorithm across computational environments. (a) Baseline pattern '000000' showing $P_{\text{success}} = 100.0\%$ (QASM simulator), $87.0\%$ (noisy emulation), and $68.3\%$ (quantum hardware). (b) Sensitivity pattern '000001' with $P_{\text{success}} = 100.0\%$ (simulator), $88.4\%$ (emulation), and $75.7\%$ (hardware). Each row shows a test pattern, with columns representing QASM simulator, noisy emulation, and real quantum hardware. The sparse patterns maintain reasonable fidelity but reveal the onset of noise-induced degradation.
• Figure 3: Pattern complexity effects on output distributions. Structured patterns show increased sensitivity to noise, with symmetric encodings exhibiting particularly severe degradation on real hardware despite reasonable performance in simulation. (a) Alternating pattern '101010' shows moderate hardware degradation: $P_{\text{success}} = 100.0\%$ (simulator), $88.6\%$ (emulation), $30.7\%$ (hardware). (b) Symmetric pattern '011011' exhibits catastrophic failure: $100.0\%$ (simulator), $85.2\%$ (emulation), $4.1\%$ (hardware), demonstrating severe sensitivity to gate density and structured errors.
• Figure 4: Performance scaling with entanglement load across medium to high density patterns. (a) Medium-density pattern '011101': $P_{\text{success}} = 100.0\%$ (simulator), $83.9\%$ (emulation), $47.1\%$ (hardware). (b) Medium-density pattern '100100': $100.0\%$ (simulator), $89.0\%$ (emulation), $45.3\%$ (hardware). (c) High-density pattern '111111': $100.0\%$ (simulator), $81.7\%$ (emulation), $3.5\%$ (hardware), demonstrating catastrophic collapse and near-uniform output distribution under maximum gate density. The performance collapse becomes dramatic for high-density patterns, with real hardware output approaching uniform distribution while noisy emulation remains optimistic.
• Figure 5: Benchmarking the Bernstein-Vazirani algorithm across diverse pattern structures. (a) Average success probability by pattern category shows real hardware performance degrades dramatically for symmetric and high-density patterns. (b) Scaling analysis reveals performance collapse for high-density patterns as qubit count increases, with real hardware (red) showing significantly steeper degradation than noisy emulation (blue). (c) Hellinger distance from ideal distribution quantifies output quality, approaching 1.0 (completely random) for complex patterns on real hardware. (d) Performance gap between simulation and hardware correlates strongly with pattern complexity (qubits $\times$ density), highlighting the limitations of current noise models in capturing structure-dependent errors.