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arXiv:2508.20010 · Journal of the Optical Society of America B

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

Vincent Girouard, Nicolás Quesada

Published Dec 3, 2025• 6 citations
arXivHTMLPDF

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 a low optical depth. In this work, we introduce a constructive decomposition of unitary matrices using a sequence of 2 N +5 phase masks interleaved with 2 N +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.

DOI
10.1364/josab.577579

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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 a low optical depth. In this work, we introduce a constructive decomposition of unitary matrices using a sequence of 2 N +5 phase masks interleaved with 2 N +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.
A large number of applications in classical and quantum photonics require the capability of implementing arbitrary linear unitary transformations on a set of optical modes. In a seminal work by Reck et al. it was shown how to build such multiport universal interferometers with a mesh of beam splitters and phase shifters, and this design became the basis for most experimental implementations in the last decades. However, the design of Reck et al. is difficult to scale up to a large number of modes, which would be required for many applications. Here we present a constructive proof that it is possible to realize a multiport universal interferometer on N modes with a succession of 6N Fourier transforms and 6N+1 phase masks, for any even integer N. Furthermore, we provide an algorithm to find the correct succesion of Fourier transforms and phase masks to realize a given arbitrary unitary transformation. Since Fourier transforms and phase masks are routinely implemented in several optical setups and they do not suffer from the scalability issues associated with building extensive meshes of beam splitters, we believe that our design can be useful for many applications in photonics.
Universal multiport interferometers, which can be programmed to implement any linear transformation between multiple channels, are emerging as a powerful tool for both classical and quantum photonics. These interferometers are typically composed of a regular mesh of beam splitters and phase shifters, allowing for straightforward fabrication using integrated photonic architectures and ready scalability. The current, standard design for universal multiport interferometers is based on work by Reck et al. [Phys. Rev. Lett.73, 58 (1994)PRLTAO0031-900710.1103/PhysRevLett.73.58]. We demonstrate a new design for universal multiport interferometers based on an alternative arrangement of beam splitters and phase shifters, which outperforms that by Reck et al. Our design requires half the optical depth of the Reck design and is significantly more robust to optical losses.
Reconfigurable devices which can implement arbitrary unitary operations are crucial for photonic quantum computation, optical neural networks, and boson sampling. Here, we address the problem of, using multiport interferometers, approximating with a given infidelity any pure-state preparation and any unitary operation. By means of numerical calculations, we show that pure states, in any dimension ddd, can be prepared with infidelity $\ensuremath{\le}{10}^{\ensuremath{-}15}$ with three layers of ddd-dimensional Fourier transforms and three layers of configurable phase shifters. We also present numerical evidence that d+1d+1d+1 layers of ddd-dimensional Fourier transforms and d+2d+2d+2 layers of configurable phase shifters can produce any unitary operation with infidelity $\ensuremath{\le}{10}^{\ensuremath{-}14}$. The conclusions are achieved by numerical simulations in the range from d=3d=3d=3 to d=10d=10d=10. These results are interesting in light of the recent availability of multicore fiber-integrated multiport interferometers.
Universal multi-port interferometers that can be reconfigured to implement any unitary operation are crucial for photonic universal quantum computation, optical neural networks, and boson sampling. Recently, Saygin et al. [Phys. Rev. Lett. 124, 010501 (2020)] have presented numerical evidence that an extremely compact arrangement consisting in ddd layers of ddd-dimensional Fourier transforms and d+1d+1d+1 layers of configurable phase shifters might be universal in any dimension ddd. Here, we first present counterexamples to this statement. We show that for d=3d=3d=3 and 444, around 5% of the unitary matrices from a Haar-uniform distribution cannot be recovered with high fidelity using Saygin et al.'s scheme. We also identify a family of unitary matrices with elements that cannot be recovered with high fidelity for any 5≤d≤105\leq d \leq 105≤d≤10. Secondly, we present numerical evidence that universality is recovered in all dimensions by adding a Fourier transformation and a layer of phase shifters, which preserves the compactness of the scheme and its robustness against imperfections.
A large number of applications in classical and quantum photonics require the capability of implementing arbitrary linear unitary transformations on a set of optical modes. In a seminal work by Reck et al. [Phys. Rev. Lett.73, 58 (1994)10.1103/PhysRevLett.73.58], it was shown how to build such multiport universal interferometers with a mesh of beam splitters and phase shifters, and this design became the basis for most experimental implementations in the last decades. However, the design of Reck et al. is difficult to scale up to a large number of modes, which would be required for many applications. Here we present a deterministic algorithm that can find an exact and efficient implementation of any unitary transformation, using only Fourier transforms and phase masks. Since Fourier transforms and phase masks are routinely implemented in several optical setups and they do not suffer from some of the scalability issues associated with building extensive meshes of beam splitters, we believe that our design can be useful for many applications in photonics.
Abstract We introduce procedures for decomposing N × N unitary matrices into smaller M × M unitary matrices. Our procedures enable designing modular and optimal architectures for implementing arbitrary discrete unitary transformations on light. Such architectures rely on systematically combining the M -mode linear optical interferometers together to implement a given N -mode transformation. Thus this work enables the implementation of large linear optical transformations using smaller modules that act on the spatial or the internal degrees of freedom of light such as polarization, time or orbital angular momentum. The architectures lead to a rectangular gate structure, which is optimal in the sense that realizing arbitrary transformations on these architectures needs a minimal number of optical elements and minimal circuit depth. Moreover, the rectangular structure ensures that each of the different optical modes incurs balanced optical losses, so the architectures promise substantially enhanced process fidelities as compared to existing schemes.