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AIP conference proceedings

Multiport universal unitary interferometer design

I. V. Kondratyev, I. V. Dyakonov, M. Yu. Saygin, Stanislav S. Straupe, S. P. Kulik

Published Jan 1, 2020• 1 citation
Publisher

Abstract

Linear optics is a promising physical architecture for the implementation of the quantum computing devices. The key element of the linear optical quantum computer is an optical circuit realizing the sequence of transformations defined by the required quantum algorithm. The most prominent approach is the reconfigurable integrated photonics which allows to miniaturize standard optical setups on the chosen integrated photonic platform. In this paper, we propose a universal linear-optical circuit design comprised of the series of multiport beamsplitters followed by the phaseshifting elements. The proposed unitary composer device can be straightforwardly implemented using standard integrated photonic technologies. We provide guidelines for appropriate multiport beamsplitter fabrication using integrated waveguide lattices. To exemplify the performance of the composer we provide the six-mode CNOT linear optical gate circuit design fitted to described interferometric circuit.

DOI
10.1063/5.0012752

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Linear optics is a promising physical architecture for the implementation of the quantum computing devices. The key element of the linear optical quantum computer is an optical circuit realizing the sequence of transformations defined by the required quantum algorithm. The most prominent approach is the reconfigurable integrated photonics which allows to miniaturize standard optical setups on the chosen integrated photonic platform. In this paper, we propose a universal linear-optical circuit design comprised of the series of multiport beamsplitters followed by the phaseshifting elements. The proposed unitary composer device can be straightforwardly implemented using standard integrated photonic technologies. We provide guidelines for appropriate multiport beamsplitter fabrication using integrated waveguide lattices. To exemplify the performance of the composer we provide the six-mode CNOT linear optical gate circuit design fitted to described interferometric circuit.
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.
We present recursive multiport schemes for implementing quantum Fourier transforms and the inversion step in Grover's algorithm on an integrated linear optics device. In particular, each scheme shows how to execute a quantum operation on 2d2d2d modes using a pair of circuits for the same operation on ddd modes. The circuits operate on path-encoded qudits and realize ddd-dimensional unitary transformations on these states using linear optical networks with O(d2)O\left({d}^{2}\right)O(d2) optical elements. To evaluate the schemes against realistic errors, we ran simulations of proof-of-principle experiments using a simple fabrication model of silicon-based photonic integrated devices that employ directional couplers and thermo-optic modulators for beam splitters and phase shifters, respectively. We find that high-fidelity performance is achievable with our multiport circuits for 2-qubit and 3-qubit quantum Fourier transforms, and for quantum search on four-item and eight-item databases.
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.
Integrated photonic circuits with many input and output modes are essential in applications ranging from conventional optical telecommunication networks, to the elaboration of photonic qubits in the integrated quantum information framework. In particular, the latter field has been object in the recent years of an increasing interest: the compactness and phase stability of integrated waveguide circuits are enabling experiments unconceivable with bulk-optics set-ups. Linear photonic devices for quantum information are based on quantum and classical interference effects: the desired circuit operation can be achieved only with tight fabrication control on both power repartition in splitting elements and phase retardance in the various paths. Here we report on a novel three-dimensional circuit architecture, made possible by the unique capabilities of femtosecond laser waveguide writing, which enables us to realize integrated multimode devices implementing arbitrary linear transformations. Networks of cascaded directional couplers can be built with independent control on the splitting ratios and the phase shifts in each branch. In detail, we show an arbitrarily designed 5×5 integrated interferometer: characterization with one- and two-photon experiments confirms the accuracy of our fabrication technique. We exploit the fabricated circuit to implement a small instance of the boson-sampling experiments with up to three photons, which is one of the most promising approaches to realize phenomena hard to simulate with classical computers. We will further show how, by studying classical and quantum interference in many random multimode circuits, we may gain deeper insight into the bosonic coalescence phenomenon.
Multiport interferometers with NNN ports are basic devices in both classical and quantum photonics. Ideally, they implement a linear unitary transformation between the input and output electric field vectors with NNN components, each associated with a spatial mode of classical coherent light or a single photon. Standard designs for a fully reconfigurable universal multiport interferometer are given by the Reck or the Clements schemes. In this work, we introduce routing schemes to implement a generic unitary transformation on classical coherent light or single photons using linear or tree geometries via multiple projective measurements on a single detector with the minimum number of components. Then, we generalize this result to the case of any multi-photon state for scattershot boson sampling experiments with multi-routing schemes. Finally, we test the robustness of routing schemes compared to universal schemes with respect to losses and phase noise.
We show that any NNN-dimensional unitary matrix can be realized using a finite sequence of concatenated identical fixed multiport beamsplitters (MBSs) and phase shifters (PSs). Our construction is based on a Lie group theorem applied to existing decompositions. Using the Bell-Walmsley-Clements framework, we prove that any NNN-dimensional unitary requires N+2N+2N+2 phase masks, N−1N-1N−1 fixed MBSs, and N−1N-1N−1 BSs. Our scheme requires only O(N)\mathcal{O}(N)O(N) fixed, identical components (MBSs and BSs) compared to the O(N2)\mathcal{O}(N^2)O(N2) fixed BSs required by conventional schemes (e.g., Clements), all while keeping the same number of PSs. Experimentally, these MBS can be realized as a monolithic element via femtosecond laser writing, offering superior performance through reduced insertion losses. As an application, we present a reconfigurable linear optical circuit that implements a three-dimensional unitary emerging in the unambiguous discrimination of two nonorthogonal qubit states.