Universal low-depth two-unitary design of programmable photonic circuits

TL;DR

The paper addresses the challenge of scalable, programmable photonic matrix-vector multiplication (MVM) by reducing circuit depth while preserving analytical programmability. It introduces a universal two-unitary architecture that realizes any target matrix $W$ as a sum of two unitaries, $W=\frac{1}{2}(U^{(1)}+U^{(2)})$ with $U^{(1)}=UDV^\dagger$ and $U^{(2)}=UD^\ast V^\dagger$, implemented via two parallel unitary meshes fed through balanced beam splitters, achieving a depth of $\mathfrak{D}_{2U}=N+1$, i.e., half of the conventional SVD-based depth $\mathfrak{D}_{SVD}=2N+2$. The work develops an analytical programming approach, analyzes error tolerance through static corrections and a beam-splitter error model, and demonstrates that scaled error cases (RMSE_s) can be mitigated with small global factors. The architecture is architecture-agnostic, compatible with common Reck/Clements meshes and multilayer photonic platforms, and offers a practical path toward low-loss, scalable photonic MVM for both classical photonic neural networks and quantum photonic processors. Overall, the two-unitary design provides a drop-in, low-depth, analytically programmable alternative to SVD-based MVM with potential for significant gains in scalability and robustness.

Abstract

The development of large-scale, programmable photonic circuits capable of performing generic matrix-vector multiplication is essential for both classical and quantum information processing. However, this goal is hindered by high losses, hardware errors, and difficulties in programmability. We propose an enhanced architecture for programmable photonic circuits that minimizes circuit depth and offers analytical programmability, properties that have not been simultaneously achieved in previous circuit designs. Our proposal exploits a previously overlooked representation of general nonunitary matrices as sums of two unitaries. Furthermore, similar to the traditional singular value decomposition-based circuits, the circuits in our unitary-sum-based architecture inherit the advantages of the constituent unitary circuits. Overall, our proposal provides a significantly improved solution for matrix-vector multiplication compared to the established approaches.

Figures (3)

• Figure 1: Traditional method of the universal MVM operations: (a) the SVD-based programmable circuit consisting of two programmable unitary circuits and programmable amplitude modulators connected in sequence, (b) the universal planar unitary mesh built from two-mode blocks $U^{(\text{mesh})}$ and a phases-shift layer $\Phi^{(\text{out})}$, and (c) example of two-mode block---Mach-Zehnder interferometer.
• Figure 2: Proposal: The programmable circuit implementing the sum of two unitaries $U^{(1)}$ and $U^{(2)}$ proposed to perform multiplication by $N\times{}N$ nonunitary matrices. Here, the BSs before and after unitary circuits are balanced.
• Figure 3: Effect of beam-splitter errors on the quality of photonic MVM operations: The distributions of $\mathrm{RMSE}$ and $\mathrm{RMSE_s}$ (drawn in orange and blue) are shown (1) correlated errors as a function of the BS power reflectivity $R$ at $N=10$ [panel (a)] and $N=20$ [panel (b)], and (2) independently sampled random errors as a function of the distribution width $\sigma$ at $N=10$ [panel (c)] and $N=20$ [panel (d)]. Each $\mathrm{RMSE}$, $\mathrm{RMSE_s}$ distribution is based on $100$ randomly sampled target matrices (left axes). For the distributions corresponding to independently sampled random errors, the minimum, maximum, and median bars are plotted. The distributions of $|s|$ corresponding to $\mathrm{RMSE_s}$ are also plotted in the figures (right axes): The black curves are the mean values, and the shaded regions are specified by the minimum and maximum values. The kinks in the shaded regions are an intrinsic feature of the underlying statistical randomness.