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Hardware error correction for programmable photonics

Saumil Bandyopadhyay, Ryan Hamerly, Dirk Englund

arXiv:2103.04993v1cs.ETphysics.optics

TL;DR

Static fabrication errors accumulate in programmable photonic circuits, while existing compensation often depends on real-time optimization. The paper corrects errors locally within optical gates, and reports that sufficiently low hardware errors permit settings that yield the exact desired unitary.

  • Problem

    Static component errors accumulate in larger photonic circuits, and ideal-component programming can therefore produce the wrong operation.

    Source: §1 · p. 1

  • Method

    The approach uses one-time device calibration to compute settings that locally correct hardware errors within each optical gate.

    Source: §1 · p. 2

  • Results

    For sufficiently low hardware errors, the computed settings yield the exact desired unitary.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Because correction is local to each 2 × 2 gate, the strategy does not assume a particular circuit structure and can generalize to interferometer architectures including recirculating meshes.

    Source: §1 · p. 2

    Component errors restrict the physically realizable range of θ.

    Source: §2 · p. 3

Abstract

from arXiv · show

Programmable photonic circuits of reconfigurable interferometers can be used to implement arbitrary operations on optical modes, facilitating a flexible platform for accelerating tasks in quantum simulation, signal processing, and artificial intelligence. A major obstacle to scaling up these systems is static fabrication error, where small component errors within each device accrue to produce significant errors within the circuit computation. Mitigating this error usually requires numerical optimization dependent on real-time feedback from the circuit, which can greatly limit the scalability of the hardware. Here we present a deterministic approach to correcting circuit errors by locally correcting hardware errors within individual optical gates. We apply our approach to simulations of large scale optical neural networks and infinite impulse response filters implemented in programmable photonics, finding that they remain resilient to component error well beyond modern day process tolerances. Our results highlight a new avenue for scaling up programmable photonics to hundreds of modes within current day fabrication processes.

1. INTRODUCTION

Programmable photonic circuits offer reconfigurable optical computation, but static component errors accumulate as circuits scale. The paper proposes deterministic, gate-local correction to recover desired operations without relying on global numerical optimization.

1. INTRODUCTION

  • Static fabrication errors accumulate across larger circuits, while ideal-component decompositions can program the wrong operation; even 2% beamsplitter variation degraded MNIST classifier accuracy by nearly 50%.Scaling to hundreds or thousands of modes would require precise fabrication of tens of thousands of interferometers.Source: §1 · p. 1
  • Component variation also alters phase-sensitive filter responses in recirculating waveguide meshes, making control challenging as their mode counts grow.The cited architectures include triangular and hexagonal MZI lattices.Source: §1 · p. 1
  • Global optimization can be time-consuming and scale poorly, while progressive local-feedback algorithms require O(N^2) tap photodiodes, increasing electrical lines and power consumption.Retraining settings chip by chip can also be costly when model parameters are broadcast across many chips.Source: §1 · p. 1
  • The correction strategy addresses a photonics scaling challenge analogous to FPGA programming: it aims to compute hardware settings from an idealized software model and applies gate by gate across interferometer architectures.The paper focuses its analysis on feedforward circuits implementing arbitrary unitary matrices, while identifying redundant feedforward circuits and recirculating meshes as generalizable architectures.Source: §1 · p. 2
  • The proposed local approach corrects hardware errors within each optical gate, computing settings after one-time device calibration and yielding the exact desired unitary for sufficiently low errors.It requires neither additional interferometers nor internal detectors within every device.Source: §1 · p. 2

2. HARDWARE ERROR CORRECTION

The method corrects fabrication errors locally in each MZI, first matching the imperfect gate’s splitting magnitudes to the intended gate and then correcting phase errors. Auxiliary phases are propagated through the circuit, allowing imperfect hardware to implement the desired unitary when its required splitting settings are physically realizable.

2. HARDWARE ERROR CORRECTION

  • Once phase-shifter and splitter errors are characterized and stored in a lookup table, the same calibration supports programming arbitrary functions without recalibrating each setting.The calibration is performed once; ideal MZI settings are converted device by device.Source: §2 · p. 2
  • Local corrections first choose θ′ to match the imperfect MZI’s element magnitudes to the intended gate, then adjust phase settings to correct the remaining phase errors.The ideal decomposition supplies θ and φ; each device is converted to imperfect-device settings using its calibrated component parameters.Source: §2 · pp. 2–3
  • Auxiliary output phases are propagated forward through the circuit and incorporated into later MZIs or the output phase screen.This carries the local corrections through the network after each device is adjusted.Source: §2 · p. 4
  • Exact recovery requires each MZI to realize its required splitting; when it cannot, the method chooses the nearest feasible θ′ and minimizes gate error.The target θ may fall outside the error-dependent physically realizable range.Source: §2 · pp. 3–4
  • The method corrects each 2 × 2 optical gate individually and can be adapted to interferometer architectures including feedforward circuits and recirculating waveguide meshes.The expressions can be modified for other gate designs, such as the dual-drive tunable basic unit.Source: §2 · p. 4

A. Hardware Performance

Simulations show that local error correction improves matrix accuracy across circuit sizes and splitter variations, and improves fabrication tolerance for optical neural networks. The gains persist in circuits with hundreds of modes.

A. Hardware Performance

  • For 32-mode circuits, correction reduces matrix error ϵ significantly, sometimes by more than an order of magnitude, across tested splitter statistics.The simulations used 100 Haar-random unitaries implemented on 100 randomly generated circuits.Source: A. Hardware Performance · p. 5
  • Error correction significantly improves neural-network fabrication tolerance beyond current-day process tolerances, including systems with hundreds of modes.For the MNIST classifier, corrected circuits retain baseline performance within 1% even with 4% splitter error.Source: Figure 4 · p. 5
  • At σBS = 2%, error correction substantially improves matrix error ϵ for circuits with N = 64, 128, and 256 modes.The improvement diminishes as N increases, but remains substantial through 256 modes.Source: A. Hardware Performance · p. 5

B. Application: Optical Neural Networks on Feedforward Pro-

The paper evaluates local hardware correction in simulated two-layer photonic neural networks trained for MNIST handwritten-digit recognition. Correction extends the networks’ tolerance to splitter variation, with less than 1% accuracy loss at variations as high as 4%.

B. Application: Optical Neural Networks on Feedforward Pro-

  • The application addresses the energy-intensive cost of training, where retraining each chip for its distinct fabrication errors is impractical.The intended approach is to train a model in software once and correct hardware to restore that model’s behavior.Source: B. Application: Optical Neural… · p. 6
  • The simulated networks use two layers, N = 36, 64, 144, or 256 neurons, and a modReLU activation implemented with an electro-optic nonlinearity.They were trained to recognize low-frequency Fourier features of MNIST handwritten digits.Source: B. Application: Optical Neural… · pp. 5–6
  • With correction, MNIST neural networks show less than 1% accuracy loss at splitter variations as high as 4%, while the uncorrected accuracy cutoff below 90% occurs near 3%.The simulations covered 300 circuits and network sizes from 36 to 256 neurons.Source: B. Application: Optical Neural… · p. 6

C. Application: Tunable Dispersion Compensators on Recir-

The paper applies local hardware-error correction to a recirculating tunable dispersion compensator, restoring its group-delay profile after fabrication errors without retraining the mesh. Residual profile variation at higher splitter error is attributed to loss in bar-state TBUs.

C. Application: Tunable Dispersion Compensators on Recir-

  • For 500 simulated TDC circuits, locally correcting each TBU restores the desired group-delay profile after beamsplitter errors perturb it.The figure compares profiles and group-delay-dispersion distributions before and after correction.Source: Figure 5 · p. 6, C. Application: Tunable Dispersion… · pp. 6–7
  • The TDC uses 15 serially coupled ring resonators in a hexagonal mesh, with ideal-model settings trained for −85 ps/nm group-delay dispersion across a 50 GHz channel.Each ring uses a TBU as its coupler, and five other TBUs implement feedback.Source: C. Application: Tunable Dispersion… · p. 6
  • Fabrication errors introduce spurious phases at resonances, varying the dispersion profile even for slight component errors.The paper reports that local correction of each TBU is sufficient to restore the desired profile.Source: C. Application: Tunable Dispersion… · pp. 6–7
  • At σBS = 4%, residual dispersion variation is attributed to bar-state TBU loss that alters the critical-coupling condition.Because the simulations assume independent Gaussian α and β although single-device errors are strongly correlated, they likely overestimate this loss.Source: C. Application: Tunable Dispersion… · p. 7

D. Scalability and Outlook

Local correction reduces circuit error and can extend operation across a wider wavelength range, while its error bound depends on circuit architecture and device programmability. The paper also distinguishes infrequently reprogrammed hardware from applications requiring real-time configuration.

D. Scalability and Outlook

  • At σBS as high as 4%, correction improves error by at least 2× for circuits up to N = 500.Figure 6a plots ⟨ϵ⟩ and ⟨ϵcorrected⟩ against circuit size for σBS = 1.2%, 2%, and 4%.Source: Figure 6 · p. 7, D. Scalability and Outlook · p. 7
  • For N = 256, corrected error across 1520–1580 nm is lower than uncorrected error at the design wavelength of 1550 nm.The estimate uses optimal directional couplers with σBS = 1.2%; the paper notes MMI couplers as another route to large instantaneous bandwidth.Source: D. Scalability and Outlook · p. 7, Figure 6 · p. 7
  • When the stated condition is satisfied, local correction yields ϵcorrected = 0 in O(1) time; otherwise global optimization can sometimes reduce error further.The paper contrasts local correction’s minimal overhead with global methods that require in-device photodiodes or output measurements scaling nonlinearly with mode count.Source: D. Scalability and Outlook · p. 8
  • The error bound applies only to feedforward unitary circuits without redundant devices; additional redundant MZIs can achieve lower error.A redundant MZI with an extra phase shifter can implement desired 2 × 2 unitaries perfectly under the stated setting condition.Source: D. Scalability and Outlook · p. 8
  • For recirculating meshes, the authors expect correction’s benefit not to diminish with increasing N because phase-shifter settings are not constrained by the Haar measure.A few extra MZIs in high-accuracy unitary circuits can also be optimized once on an ideal model and ported across devices.Source: D. Scalability and Outlook · p. 8
  • The approach assumes hardware is reprogrammed infrequently; real-time mode unscrambling instead requires robust real-time configuration.The authors point to separate self-configuration approaches for the latter setting.Source: D. Scalability and Outlook · p. 8

4. CONCLUSION

The conclusion presents a one-time calibration protocol for correcting hardware errors in programmable photonic circuits. The authors report resilience beyond modern process tolerances in optical neural networks and coupled-ring systems, with improved functionality expected up to hundreds of modes.

4. CONCLUSION

  • A one-time calibration protocol corrects hardware errors and converts ideal MZI settings to those for imperfect devices, avoiding retraining each chip.The conclusion says the protocol can implement desired functionality up to hardware limits.Source: §4 · p. 8
  • Applications including optical neural networks and programmable coupled-ring systems remain resilient to fabrication errors beyond modern-day process tolerances.Source: §4 · p. 8
  • Current process tolerances suggest improved functionality for programmable photonic systems of up to hundreds of modes.Source: §4 · p. 8

Hardware error correction for programmable photonics: supplemental document

The supplied passage identifies the supplemental document’s authors and arXiv version.

Hardware error correction for programmable photonics: supplemental document

1. HARDWARE CALIBRATION

The protocol calibrates individual MZI phase shifters and splitter errors using circuit-output measurements, then extends characterization through the network. Triangular Reck geometry simplifies this procedure, while the section also defines and checks a matrix-error measure for beamsplitter errors.

1. HARDWARE CALIBRATION

  • Finite extinction ratio scatters stray light along wire paths, potentially introducing an O(ζ) calibration error comparable to the splitter errors being measured.The protocol assumes coherent detection at circuit outputs, but does not require detectors embedded at each MZI.Source: §1 · p. 10, A. Device Calibration · p. 10

A. Device Calibration

  • Averaging top-port transmission over φ calibrates θ despite stray-light contamination, while cross/bar-state interference visibilities yield |α + β| and |α − β|.The sign ambiguity is resolved interferometrically, providing α and β; coherent output-phase measurements calibrate φ.Source: A. Device Calibration · pp. 11–13

B. System Calibration

  • Calibration proceeds from the output toward the input: once the last column is characterized, homodyne measurements infer fields for successively earlier columns.The generalized procedure applies to arbitrary MZI networks, characterizing devices one step removed from the outputs at a time.Source: B. System Calibration · p. 13

C. Triangular Network Calibration

  • In triangular (Reck) circuits, the network ensures ζ = 0, and symmetry simplifies calibration: direct intensity detection, rather than homodyne detection, is sufficient.Measurements at ψ = 0, π/2 extract the in-phase and quadrature components of U1x, enabling characterization of the remainder with intensity measurements only.Source: C. Triangular Network Calibration · pp. 13–14

A. Beamsplitter Errors

  • The Frobenius-norm hardware error is unitarily invariant, and simulations show excellent agreement with the derived average beamsplitter-error expression.The metric is interpreted as average relative error per matrix entry.Source: A. Beamsplitter Errors · pp. 14–15

B. Phase Errors

The calibration removes static phase errors but does not correct dynamic phase errors. The supplied estimates place typical thermal drift below beamsplitter error, while thermal crosstalk and DAC quantization remain relevant after static correction.

B. Phase Errors

  • Dynamic phase effects contribute significantly less hardware error than the static errors considered in the main text, though they matter more after static correction.The calibration does not account for thermal drift, thermal crosstalk, or quantization error.Source: B. Phase Errors · pp. 15, 17

Thermal drift

  • A temperature gradient below 0.01°C induces a phase error of approximately 1.5 × 10^-3 at 1550 nm, an order of magnitude below expected beamsplitter error.The estimate assumes a 200 µm phase shifter and the stated silicon thermo-optic coefficient.Source: B. Phase Errors · pp. 15–16

Thermal crosstalk

  • Thermal crosstalk is largely deterministic and dominated by nearest neighbors, so it can be accounted for during phase-shifter characterization.A 135 µm spacing was measured to produce less than 0.02 rad/rad crosstalk with a neighboring MZI.Source: B. Phase Errors · p. 16

Quantization error

  • A 12-bit DAC suppresses worst-case per-phase-shifter quantization error to approximately 9 × 10^-4, while 16 bits reduce it below 6 × 10^-5.The phase setting has roughly one fewer bit of accuracy than the voltage setting because phase depends on voltage squared.Source: B. Phase Errors · p. 17

C. Calculating the corrected error

The section derives an estimate of corrected circuit error from the fraction of MZIs requiring unrealizable splittings, and describes both phase-shifter correction and redundant-MZI gates. Its application methods cover neural networks, ring filters, and wavelength-dependent couplers.

C. Calculating the corrected error

  • The expected corrected error is estimated using the Haar distribution of required θ values, with the rare upper-boundary unrealizable case neglected for moderately large N.The probability of a required splitting below the realizable range is used in the average-error calculation.Source: C. Calculating the corrected error · pp. 17–18
  • The derived expression for average corrected error shows excellent agreement with simulation results.The estimate is based on the fraction of MZIs whose required splitting cannot be realized.Source: C. Calculating the corrected error · pp. 17–18

A. Correcting the internal phase shifter

B. Perfect optical gates with redundant devices

  • A redundant MZI with a tunable splitter can realize any 2 × 2 unitary, giving zero corrected circuit error when |β| < π/4 − max[|a1 + a2|, |a1 − a2|].The tunable splitter is set to compensate the passive splitter error, with separate settings covering lower and upper target splitting ranges.Source: B. Perfect optical gates with… · p. 19

4. METHODS

  • The neural-network simulations use Fourier-preprocessed MNIST inputs, an electro-optic activation, and power levels increased with network size to maintain activation.The networks are trained to minimize mean squared error between normalized output power and the correct one-hot label.Source: §4 · pp. 20–21
  • The filter model comprises 15 serially coupled rings; individual transfer functions are multiplied, and phase settings are optimized against the desired group-delay profile.The wavelength-dependent directional-coupler design is computed with MPB and assumes λ0 = 1550 nm for Figure 6b.Source: §4 · pp. 21–22
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