Bohr's complementarity principle tested on a real quantum computer via interferometer experiments

Technical Summary: Bohr's Complementarity Principle Tested on a Real Quantum Computer via Interferometer Experiments

Problem and Context
Bohr's Complementarity Principle (BCP) posits that quantum systems exhibit wave and particle natures that are mutually exclusive in observation yet equally real. While the qualitative concept is foundational, quantitative formulations have evolved, notably the wave-particle duality relation $P^2 + V^2 \leq 1$ (where $P$ is predictability and $V$ is visibility). This paper addresses the need to test updated complementarity relations on real, noisy quantum hardware. Specifically, it investigates whether the trade-off between coherence (the ondulatory aspect) and predictability (the corpuscular aspect) holds when experimental conditions force the system to modify its state through superposition and subsequent measurement collapse. The authors aim to move beyond theoretical simulations to empirical verification on a physical quantum computer, accounting for device noise and the reconstruction of final states.

Methodology
The study implements two distinct interferometric experiments on the 32-qubit quantum computer QMIO (available at the Centro de Supercomputación de Galicia, CESGA):

1. Biased Mach-Zehnder Interferometer (BMZI): Implemented on a single qubit. The circuit simulates optical elements (biased beam splitters, mirrors, and phase shifters) using unitary gates ($RX$, $iX$, $P$). The system starts in state $|0\rangle$, undergoes superposition, and is measured.
2. Partial Quantum Eraser (PQE): Implemented on two qubits. This circuit introduces polarization modes (encoded via a second qubit) and simulates half-wave plates, quarter-wave plates, and polarizing beam splitters using controlled gates (CX, C-H, C-I).

Key Technical Procedures:

• State Reconstruction: Due to the probabilistic nature of quantum measurement, the authors employ Quantum State Tomography to reconstruct the final state density matrix ($\rho$). This required 3 circuits for the BMZI and 15 circuits for the PQE per experimental configuration.
• Quantification: The authors utilize the $l_1$-norm quantum coherence ($C_{l1}$) to quantify the wave aspect and a derived predictability metric ($P_{l1}$) for the particle aspect. For pure states, the theoretical expectation is the equality $C_{l1}(\rho) + P_{l1}(\rho) = d - 1$ (where $d=2$ for qubits, summing to 1).
• Error Analysis: Instead of simple deviation analysis, the paper introduces a Deconstructed Mean Squared Error (MSE) approach. The total MSE of the sum ($C_{l1} + P_{l1}$) is decomposed into the MSE of individual terms and a correlation term ($corr$). The authors argue that a low total MSE can be misleading if it results from a high negative correlation between the errors of coherence and predictability, which masks the individual inaccuracies of the hardware.

Key Contributions and Results

• Experimental Verification: The experiments were executed over multiple runs (128 for BMZI, varying blocks for PQE) between December 2025 and January 2026. The results were compared against theoretical curves for pure states.
• Hardware Characterization: The study identified significant variability in performance across different qubits on the QMIO machine.
  • In the BMZI experiment, qubits 4, 13, 31, and 24 showed poor performance (flat curves), while qubit 20 demonstrated the best overall performance.
  • In the PQE experiment, certain qubit pairs exhibited "flat" behaviors or values exceeding the theoretical limit of 1, indicating noise artifacts.
• The Role of Correlation: A primary finding is that analyzing the sum $C_{l1} + P_{l1}$ without considering the correlation between the two variables is insufficient. The authors demonstrate that some qubits (e.g., qubit 17 in BMZI) appeared to have low MSE (suggesting good performance) only because of a strong negative correlation that artificially canceled out errors. When the correlation was accounted for, the true deviation from the theoretical pure state was revealed.
• Data Visualization: The paper utilizes "sparklines" (small, high-resolution histograms) to visualize the distribution of MSE and correlation values, allowing for a rapid assessment of reproducibility and experimental fidelity across the 32 qubits.

Significance and Claims
The paper modestly claims to contribute to the open debate on interpreting the nature of quantum theory by providing empirical data on complementarity relations in a real-world, noisy environment.

• Methodological Refinement: The authors emphasize that their deconstructed MSE analysis offers a more rigorous way to assess qubit purity and experimental fidelity than looking at the sum of coherence and predictability alone. They assert that low MSE combined with low correlation is the true indicator of a final state close to being pure.
• Broadening Analysis: The work extends the analysis of qubit state purity beyond standard metrics, suggesting that future statistical analyses (such as ANOVA) could further benchmark device performance.
• Conclusion: The interferometric tests on the QMIO computer successfully demonstrate the wave-particle duality trade-off while highlighting the specific noise characteristics and reproducibility challenges of current quantum hardware. The results confirm that while the theoretical complementarity relation holds for ideal pure states, real hardware executions require careful statistical deconstruction to distinguish between genuine physical behavior and noise-induced artifacts.