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Matrix optimization on universal unitary photonic devices

Sunil Pai, Ben Bartlett, Olav Solgaard, David A. B. Miller

arXiv:1808.00458v3eess.SPcs.ETcs.NEphysics.optics

TL;DR

Uniform-random settings bias locally interacting photonic meshes toward banded unitaries, limiting optimization toward random unitary matrices. The paper introduces Haar initialization and tests mesh alterations; Haar initialization and the altered meshes improve convergence in the reported simulations.

  • Problem

    Uniform-random phase initialization biases rectangular meshes toward banded unitary matrices, while optimization seeks random unitaries.

    Source: A. Mesh initialization · p. 6, D. Results · p. 7

  • Method

    The paper samples phase settings from the random-unitary distribution for Haar initialization and simulates redundant and permuting rectangular meshes.

    Source: §I · p. 2

  • Results

    Haar-initialized rectangular meshes converge faster than uniformly initialized meshes, permuting meshes converge faster than rectangular meshes, and doubled-layer redundant meshes achieve up to five orders of magnitude better convergence.

    Source: D. Results · p. 7

  • Takeaways & Limitations

    The simulations show that initialization and mesh architecture can improve optimization performance, including with significant simulated fabrication errors.

    Source: §I · p. 2

    Photonic errors slightly decrease optimization performance in the 256-layer redundant mesh, although mean squared error remains around 10^-5.

    Source: Appendix E · p. 14

Abstract

from arXiv · show

Universal unitary photonic devices can apply arbitrary unitary transformations to a vector of input modes and provide a promising hardware platform for fast and energy-efficient machine learning using light. We simulate the gradient-based optimization of random unitary matrices on universal photonic devices composed of imperfect tunable interferometers. If device components are initialized uniform-randomly, the locally-interacting nature of the mesh components biases the optimization search space towards banded unitary matrices, limiting convergence to random unitary matrices. We detail a procedure for initializing the device by sampling from the distribution of random unitary matrices and show that this greatly improves convergence speed. We also explore mesh architecture improvements such as adding extra tunable beamsplitters or permuting waveguide layers to further improve the training speed and scalability of these devices.

I. INTRODUCTION

Uniform-random phase settings make locally interacting meshes favor banded unitaries, while the paper introduces Haar initialization and mesh alterations to improve optimization. The rectangular mesh implements transformations through locally interacting MZIs.

A. Photonic unitary implementation

  • A rectangular mesh applies an N-mode unitary through locally interacting MZIs, each built from two 50:50 beamsplitters and two tunable phase shifters.The mesh represents the operation with Givens rotations and input single-mode phase shifts.Source: §I · p. 1, A. Photonic unitary implementation · p. 2

I. INTRODUCTION

  • Uniform-random phases produce random-walk-like propagation; fewer paths between distant inputs and outputs make the resulting unitary increasingly banded as matrix size grows.The paper identifies random unitary matrices as a desired alternative in applications such as machine learning and quantum computing.Source: §I · pp. 1–2
  • Haar initialization lets light propagate uniformly to all outputs from any input and supports convergence despite significant simulated fabrication errors.The procedure initializes gradient-based optimization to learn unknown random unitaries from training data.Source: §I · p. 2
  • Redundant MZIs reduce convergence error by up to five orders of magnitude, while permuting interactions improve tolerances, off-diagonal errors, and convergence time.The permuting mesh retains the same number of tunable components; the redundant mesh adds MZIs.Source: §I · p. 2

B. Beamsplitter error tolerances

The section models beamsplitter fabrication errors and describes phase-shift tolerances as another source of uncertainty affecting mesh operation.

B. Beamsplitter error tolerances

  • The modeled device expressions assume perfect fabrication, motivating simulations of errors in beamsplitters and phase-shift components.The imperfect beamsplitter model introduces ϵ as twice the displacement of the split ratio from 50:50.Source: B. Beamsplitter error tolerances · p. 3
  • Imperfect beamsplitters restrict the realizable transmissivity range by up to ϵ², potentially limiting greedy progressive algorithms; simulated adaptive learning can nearly match perfect-mesh performance at ϵ = 0.1 for N = 128.With identical beamsplitter errors, realistic transmissivity is tϵ = t(1−ϵ²) and reflectivity is rϵ = r + t·ϵ².Source: B. Beamsplitter error tolerances · p. 3

C. Phase shift tolerances

  • Thermal crosstalk or environmental drift can shift phase settings from their intended values, primarily changing θ controls that set MZI split ratios.The paper frames mean behavior and sensitivity of θ as relevant to optimizing random unitaries.Source: C. Phase shift tolerances · p. 3

III. HAAR INITIALIZATION

Haar-random unitaries require position-dependent MZI settings: central components are biased toward the cross state and are more sensitive. The paper converts the resulting phase distribution into a practical initialization procedure.

A. Cross state bias and sensitivity index

  • For Haar-random unitaries, transmissive MZIs toward the mesh center are more sensitive, and cross-state bias narrows their acceptable setting range.The sensitivity index depends on the numbers of reachable input and output waveguides and increases toward the center of a rectangular mesh.Source: A. Cross state bias and sensitivity… · p. 3

B. Phase shift distributions and Haar phase

C. Haar initialization

  • Haar initialization samples a uniform Haar phase and inverts the mapping to set each MZI transmissivity, improving gradient-descent convergence speed.The initialization can be performed progressively or by directly setting transmissivities when phase shifters are characterized.Source: C. Haar initialization · p. 4
  • Under Haar initialization, light spreads quickly from an input and interacts more near mesh boundaries, unlike the random-walk-like propagation of uniform-random phases.The comparison is illustrated with field propagation from a single input.Source: C. Haar initialization · p. 4

D. Tolerance dependence on N

As mesh size N grows, phase-shifter tolerances become more restrictive for optimization of unknown random unitaries. The paper proposes redundant and permuting mesh architectures to relax transmissivity tolerances and improve optimization.

D. Tolerance dependence on N

  • Phase-shifter tolerances tighten as N increases, challenging convergence to an unknown unitary during on-chip optimization.The reported phase-shift standard deviation is on the order of 10^-2 radians for most N values in the specified range.Source: D. Tolerance dependence on N · p. 5

IV. ARCHITECTURE MODIFICATIONS

A. Redundant rectangular mesh (RRM)

  • Adding redundant tunable MZI layers can accelerate optimization of an unknown unitary, though it increases optical depth.Embedding smaller meshes with extra beamsplitter layers in a rectangular mesh of acceptable optical depth is not described as posing intrinsic waveguide-loss problems.Source: A. Redundant rectangular mesh (RRM) · p. 5

B. Permuting rectangular mesh (PRM)

  • Permuting mesh interactions shuffles outputs at intervals, relaxing component tolerances and making the number of paths more uniform across input-output transitions.Permutation blocks can use fixed-cross-state MZIs or low-loss waveguide crossings.Source: B. Permuting rectangular mesh (PRM) · p. 5

V. SIMULATIONS

The simulations examine gradient-based optimization of unknown Haar-random unitaries and show that initialization and mesh architecture affect convergence. Haar initialization avoids a banded starting configuration, while permuting meshes generally retain Haar-like bandsizes as N grows.

V. SIMULATIONS

  • The simulated framework improves convergence performance by up to five orders of magnitude, including with fabrication error.Source: §V · p. 6

A. Mesh initialization

  • Uniform-random phase initialization produces banded unitary matrices whose bandsize decreases as circuit size N increases.The simulations compare rectangular, permuting rectangular, and redundant rectangular meshes.Source: A. Mesh initialization · p. 6, Figure 5 · p. 6
  • Permuting rectangular meshes generally retain Haar-like bandsizes independent of N, unlike rectangular meshes, and this correlates with faster optimization and less initialization dependence.Haar initialization avoids starting from a banded unitary configuration and primes component average behavior; larger meshes can also converge reasonably when initialized toward higher transmissivity (θ_nℓ near 0).Source: A. Mesh initialization · pp. 6–7
  • For permuting meshes, the proposed Haar-initialization equivalent initializes each tunable block independently using its block layer count.The authors note that better initialization strategies for the nonlocal structure may exist.Source: A. Mesh initialization · p. 7

B. Optimization problem and synthetic data

  • The optimization minimizes mean-square error between the estimated and target unitary using synthetic unit-norm complex inputs and outputs generated by the target matrix.The parameterized optimization is non-convex, and the test set is the identity matrix.Source: B. Optimization problem and synthetic… · p. 7

C. Training algorithm

The simulations compare optimization methods and mesh variants, finding especially strong convergence from redundant layers. The training approach uses measured or simulated gradients with adaptive updates, while permuting meshes outperform rectangular meshes at equal parameter count.

C. Training algorithm

  • Training uses automatic differentiation, with in-situ optical backpropagation described as a physical route for measuring phase-shift gradients; the simulations find Adam outperforms standard stochastic gradient descent.The authors favor first-order adaptive updates as more physically straightforward than quasi-Newton methods such as L-BFGS.Source: C. Training algorithm · p. 7

D. Results

  • At N = 128, rectangular meshes show significant off-diagonal errors, and Haar initialization converges faster than uniform-random initialization.Uniform-random initialization can converge toward a banded configuration.Source: D. Results · p. 7
  • Permuting rectangular meshes converge faster than rectangular meshes despite having the same number of total parameters.Source: D. Results · p. 7
  • Beamsplitter imperfections slightly reduce overall optimization performance in permuting and redundant rectangular meshes, while significantly reducing performance in the rectangular mesh.Source: D. Results · p. 8
  • Up to five orders of magnitude better convergence is reported for redundant rectangular meshes when vertical layers are doubled, compared with rectangular and permuting rectangular meshes.Figure 7 compares test error for N = 128; 32 added layers outperform permuting meshes, while 16 give almost identical performance.Source: D. Results · p. 7, Figure 7 · p. 8

VI. DISCUSSION

Haar initialization spreads input light uniformly across outputs, whereas random phase initialization yields increasingly narrow bands as mesh size grows. The discussion relates this initialization and architectural changes to optimization robustness, scalability, and implementation conditions.

A. Haar initialization

  • Haar initialization spreads light from one input uniformly across waveguides regardless of N, while random phases confine it to a band that shrinks relative to mesh size.Uniform spreading matches the behavior required of Haar random matrices.Source: A. Haar initialization · pp. 8–9
  • For N = 1000, phase-shifter tolerances are just above 10^-2 radians, and thermal crosstalk may make those tolerances difficult to achieve.The average reflectivities from Haar initialization may also inform inverse design of compact tunable or passive multiport interferometers.Source: A. Haar initialization · p. 9
  • Simulations assume a linear relation between control parameter and phase shift, although thermal phase shifters can have nonlinear, bounded phase responses.The passage says phase uncertainty versus voltage needs particular attention because Haar internal-phase variance is small for large N.Source: A. Haar initialization · p. 9

B. Applications of mesh optimization

  • Gradient-based global optimization remains effective with split-ratio error variance up to σϵ = 0.1, particularly for permuting rectangular meshes.The approach is model-free and can tune photonic neural networks; the cited comparison is with perfect-mesh performance.Source: B. Applications of mesh optimization · p. 9
  • Adding tunable vertical MZI layers can improve convergence by up to five orders of magnitude, at the cost of doubling mesh size and parameter space.Waveguide permutations preserve the number of tunable components while relaxing cross-state requirements and improving optimization.Source: B. Applications of mesh optimization · p. 9
  • For globally optimized meshes, adding waveguide permutation or extra tunable vertical MZI layers relaxes cross-state requirements and accelerates optimization, achieving up to five orders of magnitude convergence improvement at the expense of doubling mesh size and parameter space.Permuting rectangular meshes are considered in the same class as permuting architectures using nonlocal waveguide interactions; extra vertical layers add tunable paths for light.Source: B. Applications of mesh optimization · p. 9
  • Experimental simulated annealing on small unitary photonic devices suggests that algorithms beyond gradient descent may support on-chip tuning.Whether such approaches can compete with backpropagation in classical applications remains open; annealing requires only output detectors, while gradient-based approaches allow continuous phase-shifter control.Source: B. Applications of mesh optimization · p. 9

VII. CONCLUSION

  • For N = 128, in situ backpropagation generally fails to converge when phase shifters are initialized randomly.The conclusion attributes scalability limits to small reflectivities and MZI sensitivities under locally interacting components, and notes that Haar initialization can sufficiently bias optimization.Source: §VII · p. 10

Appendix A: Software

The appendices provide software for reproducing the simulations, derive how beamsplitter fabrication errors enter the implemented unitary, and develop Haar measure in physical mesh coordinates. The Haar-measure derivation connects measured output powers to component transmissivities and their induced distributions.

Appendix A: Software

  • The optimizations and simulations are implemented in the open-source neurophox Python package, with the exact result-generating code in neurophox-notebooks.Source: Appendix A · p. 12

Appendix B: Derivation of beamsplitter errors

  • The beamsplitter error ϵ is twice the split-ratio displacement from 50:50, and the derivation uses it to model imperfect component amplitudes.The error definition is selected because it is measurable on-chip, and strategies to minimize ϵ are noted.Source: Appendix B · p. 12
  • Assuming equal errors for nearby fabricated structures, the derivation solves for transmissivity and reflectivity in terms of their ideal values.The resulting constraints suggest fabrication errors can limit transmissivity and inhibit progressive mesh setup, though in situ backpropagation can address this to some extent.Source: Appendix B · p. 12

Appendix C: Haar measure

  • The Haar measure for U(N) is expressed in measurement and transmissivity bases, corresponding to measured MZI outputs and component transmissivities.Source: Appendix C · p. 12
  • The derivation relates measured powers x_n to transmissivities t_n through a recursive path relation, with t_n = cos^2(θ_n/2).The relation is represented by tracing light from the top input port through the triangular mesh.Source: Appendix C · p. 13
  • In each triangular-mesh layer, the measurement-basis power x_n follows a geometric distribution because light is successively transmitted and then reflected along its path.The basis uses unit input power and omits the final output power because energy conservation determines it from the other outputs.Source: Appendix C · p. 13
  • A lower-triangular Jacobian links x_n and t_n; its diagonal yields the unbiased measure, which is normalized to obtain transmissivity and phase distributions.The phase distribution uses t_n = cos^2(θ_n/2), and the Haar phase parameter ξ is recovered as a cumulative density function.Source: Appendix C · p. 13
  • For a triangular mesh, the sensitivity index is α_nℓ = N − n, independent of layer ℓ; the Clements decomposition gives the corresponding rectangular-mesh labeling.The triangular-mesh construction uses N diagonal layers, with the operators recursively reducing the projected Hilbert-space dimension.Source: Appendix C · pp. 12–13

Appendix D: Unitary bandsizes

The appendix defines an η-bandsize measure of matrix bandedness and compares it across rectangular, permuting rectangular, and redundant meshes. At η = 0.001, permuting rectangular meshes match the bandsize of Haar random matrices.

Appendix D: Unitary bandsizes

  • At η = 0.001, permuting rectangular meshes match Haar random matrices in bandsize.The comparison includes rectangular and redundant meshes; η-bandsize counts the minimum matrix elements whose squared magnitudes sum to (1−η)N.Source: Figure 9 · p. 14, Appendix D · p. 14

Appendix E: Introducing photonic errors in a redundant mesh

The appendices examine photonic-error effects in a redundant mesh, compare rectangular and permuting meshes for SVD, and describe periodic parameter representations. The permuting mesh is more resilient to random phase initialization in the SVD simulations, while the redundant mesh retains test MSE around 10^-5 with photonic errors.

Appendix E: Introducing photonic errors in a redundant mesh

  • With photonic errors, the 256-layer redundant mesh has a slight performance decrease but still achieves mean square error around 10^-5.The authors suggest this may indicate greater robustness to photonic errors during on-chip optimization.Source: Appendix E · p. 14, Figure 10 · p. 14

Appendix F: Photonic singular value decomposition simulations

  • The SVD simulations use a mesh implementation of A = UΣV†, with equal matrix dimensions M = N = 64.The authors choose the square case because the architecture is redundant when M ≠ N.Source: Appendix F · pp. 14–15
  • The permuting rectangular mesh converges slightly faster and is more resilient to uniform-random phase initialization than the rectangular mesh in SVD simulations.Both optimizations are minimally affected by beamsplitter error; the comparison assumes a procedure like can be used with gains and optimization.Source: Figure 11 · p. 14, Appendix F · p. 15

Appendix G: Periodic parameters

  • The appendix maps absolute θnℓ values into [0, π/2] and represents periodic parameters with periods 2π for eθ and 2 for eξ.The parameters are mapped to their half-periods using symmetry arguments.Source: Appendix G · p. 15

Appendix H: Training simulation comparisons

The appendix compares rectangular and permuting rectangular mesh training conditions and reports differences in optimized reflectivities, parameter behavior, and response to beamsplitter error. It also specifies the layer ordering used for its permuting mesh.

Appendix H: Training simulation comparisons

  • The Figure 12 plots compare the unitary-network conditions described in Section V and have accompanying videos of parameter distributions, estimates, and errors during optimization.The figure conditions include rectangular and permuting rectangular meshes under ideal, beamsplitter-error, and random-initialization settings.Source: Appendix H · p. 15
  • After uniform-random phase initialization, gradient descent struggles to reach the Haar-random regime, with θnℓ values still larger than needed after 100 iterations.The authors identify this as likely evidence of a vanishing-gradient problem.Source: Appendix H · p. 15
  • With beamsplitter error, the θnℓ/2 distribution shifts slightly toward zero in the rectangular mesh, consistent with MZI reflectivity and transmissivity limits.The plots also show smaller optimized central-MZI reflectivities in rectangular meshes than in permuting rectangular meshes.Source: Appendix H · p. 15
  • The simulated permuting rectangular mesh places layers with the most Pk components at the center and those with the fewest near the inputs and outputs.For N = 128, the ordering is [P2, P4, P6, P5, P3, P1]; the authors report it as their best permuting mesh configuration so far.Source: Appendix H · p. 15

Appendix I: An equivalent definition for αnℓ

The appendix gives an equivalent reachability-based definition of the sensitivity index for triangular and rectangular meshes. It proves the identity by induction and notes that permuting meshes allow larger tolerances, which eventually results in faster optimization.

Appendix I: An equivalent definition for αnℓ

  • Permuting meshes have larger tolerance, which eventually results in faster mesh optimization.The appendix relates the sensitivity index to reachable inputs and outputs for both triangular and rectangular meshes.Source: Appendix I · p. 17
  • For triangular and rectangular meshes, the sensitivity index αnℓ equals the number of reachable inputs plus reachable outputs minus N − 1.Reachable inputs and outputs are the modes affecting the MZI's immediate inputs and outputs, respectively.Source: Appendix I · pp. 15, 17
  • The rectangular-mesh proof changes to diagonal x,y coordinates and uses induction within quadrants and across their boundaries.Figure 14 depicts the diagonal basis and the quadrant-separating boundaries.Source: Appendix I · p. 17, Figure 14 · p. 18
  • Tables I and II organize induction cases within mesh quadrants and across quadrant boundaries.The proof maintains the reachability identity in both sets of cases.Source: Appendix I · p. 17, Table I · p. 18, Table II · p. 18
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