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Near-optimal decomposition of unitary matrices using phase masks and the discrete Fourier transform

Vincent Girouard, Nicolás Quesada

TL;DR

This work tackles the design of universal multiport interferometers (UMIs) by proposing a constructive analytic decomposition of any $N imes N$ unitary into a sequence of $2N+5$ diagonal phase masks interleaved with $2N+4$ discrete Fourier transforms (DFTs). Grounded on Bell and Walmsley’s symmetric MZI unit cell, and enhanced by phase-relocation and circulant-diagonal simplifications, the method yields a near-optimal, low-depth architecture with fixed mixing layers and a programmable phase-mask sequence. It achieves approximately a 3x reduction in analytic-depth relative to the previous best analytic method, equivalent to a 66% depth reduction for large $N$, and is complemented by an open-source Python implementation. The approach promises robust, scalable UMIs with lower losses and fabrication complexity, applicable to both classical and quantum photonic processing and potentially to other linear-wave systems.

Abstract

Universal multiport interferometers (UMIs) have emerged as a key tool for performing arbitrary linear transformations on optical modes, enabling precise control over the state of light in essential applications of classical and quantum information processing such as neural networks and boson sampling. While UMI architectures based on Mach-Zehnder interferometer networks are well established, alternative approaches that involve interleaving fixed multichannel mixing layers and phase masks have recently gained interest due to their high robustness to losses and fabrication errors. However, these approaches currently lack optimal analytical methods to compute design parameters with low optical depth. In this work, we introduce a constructive decomposition of unitary matrices using a sequence of $2N+5$ phase masks interleaved with $2N+4$ discrete Fourier transform matrices. This decomposition can be leveraged to design universal interferometers based on phase masks and multimode interference couplers, implementing a discrete Fourier transform, offering an analytical alternative to conventional numerical optimization-based designs and reducing by a factor of 3 the previous best known analytical methods.

Near-optimal decomposition of unitary matrices using phase masks and the discrete Fourier transform

TL;DR

This work tackles the design of universal multiport interferometers (UMIs) by proposing a constructive analytic decomposition of any unitary into a sequence of diagonal phase masks interleaved with discrete Fourier transforms (DFTs). Grounded on Bell and Walmsley’s symmetric MZI unit cell, and enhanced by phase-relocation and circulant-diagonal simplifications, the method yields a near-optimal, low-depth architecture with fixed mixing layers and a programmable phase-mask sequence. It achieves approximately a 3x reduction in analytic-depth relative to the previous best analytic method, equivalent to a 66% depth reduction for large , and is complemented by an open-source Python implementation. The approach promises robust, scalable UMIs with lower losses and fabrication complexity, applicable to both classical and quantum photonic processing and potentially to other linear-wave systems.

Abstract

Universal multiport interferometers (UMIs) have emerged as a key tool for performing arbitrary linear transformations on optical modes, enabling precise control over the state of light in essential applications of classical and quantum information processing such as neural networks and boson sampling. While UMI architectures based on Mach-Zehnder interferometer networks are well established, alternative approaches that involve interleaving fixed multichannel mixing layers and phase masks have recently gained interest due to their high robustness to losses and fabrication errors. However, these approaches currently lack optimal analytical methods to compute design parameters with low optical depth. In this work, we introduce a constructive decomposition of unitary matrices using a sequence of phase masks interleaved with discrete Fourier transform matrices. This decomposition can be leveraged to design universal interferometers based on phase masks and multimode interference couplers, implementing a discrete Fourier transform, offering an analytical alternative to conventional numerical optimization-based designs and reducing by a factor of 3 the previous best known analytical methods.

Paper Structure

This paper contains 3 sections, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Decomposition Method
  3. Discussion

Figures (4)

  • Figure 1: Different configurations of universal multiport interferometers. (a) Triangular mesh which implements the decomposition of Reck et al.reck1994experimental in an integrated photonics setting carolan2015universal made of MZIs (blue rounded rectangles) and phase shifters (orange squares). (b) Rectangular mesh of Clements et al.clements2016optimal. (c) Sequence of $L=N+1$ diagonal phase masks interleaved with a fixed multichannel mixing layer (green rounded rectangles).
  • Figure 2: Diagram of the Bell and Walmsley universal multiport interferometer bell2021further. (a) Layout of the interferometer for $N=6$ modes, where blue rounded rectangles correspond to symmetric MZIs and orange squares correspond to single-mode phase shifters. A distinctive feature of this design is the presence of additional edge phase shifters in even MZI layers. (b) Unit cell of the interferometer. The unit cell consists of a sMZI made from two 50:50 beam splitters and two internal phase shifters.
  • Figure 3: Modified multiport interferometer used to derive the mask sequence. (a) Interactions are added between the first and last channel to simulate a periodic system. Channels are also relabelled using the permutation matrix of Eq. (\ref{['eq:k_permutation_matrix']}). (b) Structure of one bi-layer. In the odd layer, sMZIs generate interactions between channels $j$ ($j \geq N/2$) and $j-N/2$. In even layers, sMZIs generate interactions between channels $j$ ($j \geq N/2$) and $j-N/2+1$. Note that the sMZIs connecting modes 0 and $N-1$ are set to the bar (identity) configuration.
  • Figure 4: Relocation of edge phase shifters in the interferometer. (a - b) In a sMZI network, phase shifters can be moved to adjacent modes by changing their sign and adjusting the global phase of the neighboring sMZI. (c) The edge phase shifter can be divided into $N$ equal partial phases that can be distributed to all channels of the interferometer.