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arXiv:2104.07561 · APL Photonics

Further compactifying linear optical unitaries

B. A. Bell, I. A. Walmsley

Published Jul 1, 2021• 46 citations
arXivHTMLPDF

Abstract

Quantum integrated photonics requires large-scale linear optical circuitry, and for many applications, it is desirable to have a universally programmable circuit, able to implement an arbitrary unitary transformation on a number of modes. This has been achieved using the Reck scheme, consisting of a network of Mach–Zehnder interferometers containing a variable phase shifter in one path as well as an external phase shifter after each Mach–Zehnder. It subsequently became apparent that with symmetric Mach–Zehnders containing a phase shifter in both paths, the external phase shifters are redundant, resulting in a more compact circuit. The rectangular Clements scheme improves on the Reck scheme in terms of circuit depth, but it has been thought that an external phase-shifter was necessary after each Mach–Zehnder. Here, we show that the Clements scheme can be realized using symmetric Mach–Zehnders, requiring only a small number of external phase-shifters that do not contribute to the depth of the circuit. This will result in a significant saving in the length of these devices, allowing more complex circuits to fit onto a photonic chip, and reducing the propagation losses associated with these circuits. We also discuss how similar savings can be made to alternative schemes, which have robustness to imbalanced beam-splitters.

DOI
10.1063/5.0053421

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Quantum integrated photonics requires large-scale linear optical circuitry, and for many applications, it is desirable to have a universally programmable circuit, able to implement an arbitrary unitary transformation on a number of modes. This has been achieved using the Reck scheme, consisting of a network of Mach–Zehnder interferometers containing a variable phase shifter in one path as well as an external phase shifter after each Mach–Zehnder. It subsequently became apparent that with symmetric Mach–Zehnders containing a phase shifter in both paths, the external phase shifters are redundant, resulting in a more compact circuit. The rectangular Clements scheme improves on the Reck scheme in terms of circuit depth, but it has been thought that an external phase-shifter was necessary after each Mach–Zehnder. Here, we show that the Clements scheme can be realized using symmetric Mach–Zehnders, requiring only a small number of external phase-shifters that do not contribute to the depth of the circuit. This will result in a significant saving in the length of these devices, allowing more complex circuits to fit onto a photonic chip, and reducing the propagation losses associated with these circuits. We also discuss how similar savings can be made to alternative schemes, which have robustness to imbalanced beam-splitters.
Universal linear optical (ULO) circuit is a versatile device that plays a fundamental role in quantum computation, optical machine learning, and MIMO processing for optical communications. The design of the ULO, i.e., a universal multiport interferometer, was initially proposed by Reck et al. in 1994 [1] and improved in terms of circuit depth and robustness to optical losses by Clements et al. in 2016 [2]. Recently, a further compactified circuit design was proposed Bell and Walmsley [3]. In Bell's scheme, the circuit layout consists of a network of symmetric Mach-Zehnder interferometers (MZI) with two variable phase shifters in each arm. The design significantly reduces the whole device length compared to the conventional designs, where the circuit consists of a network of asymmetric MZIs, each of which has a phase shifter in one arm and an external phase shifter outside of itself. This downsizing becomes significantly important toward the realization of large-scale ULOs in the limited chip area, particularly when each phase shifter is long or when the spacing between phase shifters cannot be made small. This time we experimentally demonstrate that Bell's scheme can express arbitrary unitary matrices as well as Clements' scheme.
Programmable photonic integrated circuits have mainly been developed based on the single wavelength channel operation of fundamental building blocks consisting of Mach-Zehnder interferometers (MZIs) with tunable phase shifters. We propose and study optical circuit models consisting of cascaded optical resonators that enable the independent operation of multiple wavelength channels in a more compact footprint than the conventional MZIs. By adopting experimental values reported for silicon micro-ring resonators, the fidelities of different types of 2×2 unitary transformations and higher-dimensional unitary transformations are examined by employing the Reck algorithm and the Clements algorithm.
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.
Programmable linear optical processors (LOPs) can have widespread applications in computing and information processing due to their capabilities of implementing reconfigurable on-chip linear transformations. A conventional LOP that uses a mesh of Mach-Zehnder interferometers (MZIs) requires 2N+32N+32N+3 stages of phase shifters for $N\ifmmode\times\else\texttimes\fi{}N$ matrices. However, it is beneficial to reduce the number of phase-shifter stages to realize a more compact and lower-loss LOP, especially when long and lossy electro-optic phase shifters are used. In this work, we propose a structure for LOPs that can implement arbitrary matrices as long as they can be realized by previous MZI-based schemes. Through numerical analysis, we further show that the number of phase-shifter stages in the proposed structure can be reduced to N+2N+2N+2 and N+3N+3N+3 for a large number of random dense matrices and sparse matrices, respectively. This work contributes to the realization of compact, low-loss, and energy-efficient programmable LOPs.
General-purpose programmable photonic processors provide a versatile platform for integrating diverse functionalities on a single chip. Leveraging a two-dimensional hexagonal waveguide mesh of Mach-Zehnder interferometers, these systems have demonstrated significant potential in microwave photonics applications. Additionally, they are a promising platform for creating unitary linear transformations, which are key elements in quantum computing and photonic neural networks. However, a general procedure for implementing these transformations on such systems has not been established yet. This work demonstrates the programming of universal unitary transformations on a general-purpose programmable photonic circuit with a hexagonal topology. We detail the steps to split the light on-chip, demonstrate that an equivalent structure to the Mach-Zehnder interferometer with one internal and one external phase shifter can be built in the hexagonal mesh, and program both the triangular and rectangular architectures for matrix multiplication. We recalibrate the system to account for passive phase deviations. Experimental programming of 3x3 and 4x4 random unitary matrices yields fidelities > 98% and bit precisions over 5 bits. To the best of our knowledge, this is the first time that random unitary matrices are demonstrated on a general-purpose photonic processor and pave the way for the implementation of programmable photonic circuits in optical computing and signal processing systems.
Linear optics underpins fundamental tests of quantum mechanics and quantum technologies. We demonstrate a single reprogrammable optical circuit that is sufficient to implement all possible linear optical protocols up to the size of that circuit. Our six-mode universal system consists of a cascade of 15 Mach-Zehnder interferometers with 30 thermo-optic phase shifters integrated into a single photonic chip that is electrically and optically interfaced for arbitrary setting of all phase shifters, input of up to six photons, and their measurement with a 12-single-photon detector system. We programmed this system to implement heralded quantum logic and entangling gates, boson sampling with verification tests, and six-dimensional complex Hadamards. We implemented 100 Haar random unitaries with an average fidelity of 0.999 ± 0.001. Our system can be rapidly reprogrammed to implement these and any other linear optical protocol, pointing the way to applications across fundamental science and quantum technologies.