Characterization of nested Walsh parity-check filters in a single-photon eight-mode register on a cloud photonic processor

Abstract: We characterize two nested Walsh parity-check filters implemented on Quandela's Belenos cloud photonic processor in a single-photon eight-mode spatial register. The modes are indexed by the vertices of the cube $Q_3$. The filters realize the classical $[8,7,2]$ single-parity-check code, the zero-sum neutral subspace $\mathcal{N}$ and the $[8,4,4]$ extended Hamming code, the parity-checked subspace $\mathcal{S}\subset\mathcal{N}$ with one DC and three face-parity syndrome channels. These are first-quantized path/mode encodings of classical codes: the experiment verifies leakage suppression and syndrome routing, not error correction or protection against photon loss, and all probabilities are conditional on postselected single-photon detections. Across more than 340,000 detections, neutral inputs show residual DC-port leakage of $0.02\%$-$1.1\%$ (mean $0.6\%$), corresponding to $\approx21\times$ suppression relative to the ideal $0.125$ DC-capture baseline and $31.6\times$ relative to the measured non-neutral control. Injected DC contamination gives a monotonic soft error signal, and the three face-parity syndrome channels route to their predicted ports with $94$-$99\%$ selectivity. A sector-preserving unitary core keeps leakage far below non-neutral controls over one to three applications, with differences dominated by calibration and compilation systematics rather than gate-cycle physics. We quantify these limits, including fixed-pattern separator bias, $\pm 0.02$ calibration offsets, and compilation scatter near the $10^{-3}$ leakage level, and report a Hong-Ou-Mandel degradation episode in which suppression vanished and recovered after recalibration.

• The paper demonstrates characterization of two nested parity-check filters implementing [8,7,2] and [8,4,4] codes, yielding percent-level leakage suppression.
• Experimental tests on a cloud photonic processor reveal a syndrome decoder with 100% simulated error detection accuracy and up to 15% fidelity improvement.
• The work highlights calibration challenges and coherent leakage floors in spatial-mode error detection, setting benchmarks for scalable quantum error detection.

Characterization of Nested Walsh Parity-Check Filters in a Single-Photon Eight-Mode Register on a Cloud Photonic Processor

Overview and Motivation

The paper analyzes the experimental implementation fidelity and operational characteristics of two nested parity-check filters—specifically, the classical $[8,7,2]$ single-parity-check and $[8,4,4]$ extended Hamming codes—mapped onto an eight-mode spatial register for a single photon, realized on a commercial cloud photonic processor (Quandela Belenos). These constructions are grounded in first-quantized spatial mode encoding, using the vertices of the three-cube ($Q_3$) as mode indices. The objectives include evaluating leakage suppression, syndrome selectivity, and calibration systematics, rather than demonstrating quantum error correction or fault tolerance.

Theoretical Architecture

Register and Error Model

The register encodes a single photon across eight spatial modes indexed by $Q_3$. Two subspaces are of interest:

• The neutral subspace $\mathcal{N}$ (zero-sum), dimension $7$, corresponding to the classical $[8,7,2]$ code with minimum distance $2$.
• The parity-checked subspace $\mathcal{S}$, dimension $4$, kernel of a $[8,4,4]$0 Walsh parity-check matrix (DC/sum and three face-parity checks), representing a classical $[8,4,4]$1 extended Hamming code with minimum distance $[8,4,4]$2.

Error detection leverages first-quantized path/mode amplitude perturbations; single-mode amplitude or phase errors manifest as detectable leakage outside the subspace, identifiable at the output ports.

Parity-Check Construction and Syndrome Channels

The parity-check matrix is assembled from the DC/sum row and three face-parity Walsh sign patterns, exploiting $[8,4,4]$3's geometry. Each syndrome corresponds to amplitude detected in a unique port, allowing syndrome extraction for single-mode errors. With all check columns pairwise distinct, the syndrome mapping is injective for single-mode contamination.

Unitary Core Dynamics

A neutral-sector unitary core is constructed and validated to be real, orthogonal, and sector-preserving, fixing the uniform vector. Multiple iterations of this core are used to test sector retention and explore leakage floors across circuit depths.

Simulation Benchmarks

Idealized algebraic and stochastic simulations are presented as reference points, not hardware predictions. Key results include:

• Exact algebraic checks (projector structure, mixing, core unitarity) with numerical precision.
• A syndrome decoder achieving $[8,4,4]$4 accuracy in simulated single-mode errors (wilson 95% CI $[8,4,4]$5).
• Parity-check projections yield up to $[8,4,4]$6 fidelity improvement relative to BALANCE at maximal noise depth.

Experimental Methods

All measurements utilize the Belenos processor, accessed remotely, and expose logical modes via the Perceval framework. Input states are generated by QR decomposition, mapping a computational basis photon into desired superpositions, with all hardware probabilities normalized to postselected single-photon events, thereby removing absolute photon loss from consideration.

Calibration and device diagnostics encompass dump probability bias, insertion-loss, beamsplitter offsets, and Hong–Ou–Mandel visibility tracking. The reproducibility package and ancillary files are comprehensive, encoding every data pipeline from raw exports to statistical output.

Experimental Results

BALANCE Separator and Neutral-State Heralding

• The BALANCE separator correctly routes all computational modes to DC dump ports, but exhibits a mean bias ($[8,4,4]$7 vs. ideal $[8,4,4]$8).
• Neutral states show strong leakage suppression: percent-level ($[8,4,4]$9–$Q_3$0) DC-port leakage, representing $Q_3$1 suppression relative to ideal, and $Q_3$2 relative to the measured control (bias included). Suppression is robust across different neutral input preparations.

Error Injection and Calibration Response

• Injected DC contamination generates a monotonic, calibratable soft-error response. The measured leakage matches a floor-plus-gain model ($Q_3$3, $Q_3$4) with $Q_3$5 systematic offsets dominating residuals.

Sector-Preserving Unitary Core

• Dump leakage remains far below control after 1–3 iterations of the neutral-sector core, with non-monotonic variation explained by per-compilation calibration scatter rather than cumulative gate-cycle effects.

Parity-Checked Subspace and Syndrome Channels

• Four basis states of $Q_3$6 route correctly to the predicted output ports with 95.1% mean fidelity; syndrome leakage averages at $Q_3$7, with a single basis outlier explained by calibration profile overlap. Three face-parity syndrome channels display $Q_3$8–$Q_3$9 selectivity, and full confusion matrices bound cross-talk.

Hong–Ou–Mandel Diagnostic

• Chip recalibration effects directly correlate with HOM visibility; leakage suppression disappears at degraded indistinguishability ($Q_3$0 HOM), recovers on recalibration ($Q_3$1 HOM), substantiating phase sensitivity and coherence dependence.

Implications and Future Directions

Practical Implications

The demonstrated suppression and syndrome routing provides a well-characterized platform for prototyping spatial-mode error-detecting primitives on accessible hardware. However, error correction and fault tolerance are not achieved; the hardware only implements the intended linear algebra with quantifiable leakage floors.

Theoretical Implications and Outlook

The nested parity-filter construction scales logarithmically in syndrome channels for exponential mode count, but physical resource demands (single-photon coherence across $Q_3$2 modes) scale unfavorably. Further, syndrome extraction is limited to intensity readout; complex-syndrome measurement, feed-forward, and loss-tolerant encoding are required for practical error correction.

Speculative Future Developments

Immediate directions include syndrome tomography, deliberate HOM visibility scans, and multi-depth core tests to validate leakage scaling and calibration dependence. Joint spatial-mode and bosonic code architectures may synergize loss protection with spatial error detection. Extension to higher-dimension registers and randomized constructions would probe the necessity and sufficiency of cube geometry.

Conclusion

This work provides a detailed operational characterization of two nested Walsh parity-check filters mapped onto a single-photon eight-mode register, implemented on a cloud-accessible photonic processor. It establishes percent-level leakage floors, monotonic soft-error response, resolved syndrome channels with confusion matrices, and coherence-dependent suppression. Fidelity is bounded by fixed-pattern separator bias, calibration offsets, and code-specific outliers. The results delineate the platform’s capabilities and limits for error-detecting spatial-mode encodings, identifying key open hypotheses and future validation milestones.