Universal multiport interferometer

In quantum mechanics, a universal multiport interferometer (or universal modal unitary) is an optical device capable of imposing general unitary transformations in the modal space of single photons or electromagnetic waves. Classically, a mode of the electromagnetic (EM) field is defined as a normalized solution to Maxwell's equations in vacuum. In general, a mode of the EM field is represented by a vector field that varies both in space and in time. In optics, the allowed (optical) modes are restricted by the boundary conditions imposed by the system in which they exist (e.g., in an optical fiber or an optical cavity) and are thus solutions to the Helmholtz equation. For example, the Hermite-Gauss optical modes are typically used to describe beams produced in spherical mirror cavities. To continue, a set of orthonormal modes forms an orthonormal basis which spans a modal space, or Hilbert space. The transformation from one modal basis to another is described by a rotation which, in quantum mechanics, is the action of a unitary operator (e.g., the transformation of Hermite-Gauss optical modes to Laguerre-Gauss optical modes). It has been shown that any discrete modal unitary operator can be realized using successive beam splitters and phase-shifters applied to an N × N {\textstyle N\times N} optical beam array. The Reck scheme provides an algorithmic approach to designing an experimental setup that uses such beam splitters and phase-shifters to implement any N × N {\textstyle N\times N} modal unitary transformation. The beam splitters and phase-shifters are arranged in a triangular interferometric mesh. Today, such setups are commonly referred to as universal multiport interferometers or universal modal unitaries. The transformation of a given optical mode into another, more desired optical mode has direct applications to quantum information, optical networking, and photonic computing. The first experimental realization of the Reck scheme was in 2015 by Carolan et al. who used it to implement various linear optical (LO) quantum computing protocols such as heralded quantum logic gates and performing various boson sampling experiments.

Overview

In general, fully determining any N {\textstyle N} -dimensional unitary requires specifying N 2 {\textstyle N^{2}} independent real parameters. For the simple case of transforming a two-beam array, a universal modal unitary can be implemented using a variable beam splitter and three phase-shifters. In 1994, Michael Reck and Anton Zeilinger generalized this well-known approach by proving that variable beam splitters and phase-shifters, when arranged in an interferometric mesh with N 2 {\textstyle N^{2}} arms, can be used to impose any N × N {\textstyle N\times N} (discrete) unitary mode transformation. Using their deterministic algorithm to decompose a given unitary into a triangular network of these two optical elements, it is possible to experimentally realize a discrete universal unitary, specifically for N × N {\textstyle N\times N} mode transformations. The resulting device is commonly referred to as a universal multiport interferometer. In 2016, Clements et al. introduced a variation of Reck and Zeilinger's decomposition, again using beam splitters and phase shifters, but arranged in a symmetrically-crossing network as opposed to a triangular network. Importantly, this variation has a smaller optical depth - the longest path through the interferometric mesh - and thus experiences lower propagation losses. The two aforementioned methods are strictly different from the universal unitary decomposition commonly used in quantum computing. That is, the universal gate, whereby any N {\textstyle N} -qubit gate can be realized by a circuit of single qubit gates and CNOT gates. The classical analog of such universality is the idea that an arbitrary Boolean function can be realized using a combination of NOT gates and any one of the two-bit gates (e.g. AND, OR).

Mathematical framework According to the Davenport rotation theorem, any three-dimensional rotation can be decomposed into three elemental rotations about non-orthogonal axes. The axes may be associated with a fixed coordinate system (i.e., extrinsic rotations) or with a rotating coordinate system (i.e., intrinsic rotations), but those associated with the first and third rotations must be in the plane orthogonal to those associated with the second rotation. If the axes associated with the first and third rotations are perpendicular to one another, the Davenport generalized rotations are called Tait-Bryan rotations. However, if the axes associated with the first and third rotations overlap, they are called Euler rotations. Mathematically, the three composed rotations are represented by a non-commutative product of three matrices. They are non-commutative as the order in which the rotations are applied affects the resulting orientation of the subject. The elemental rotations each occur within a two-dimensional subspace of the higher-dimensional Euclidean space. In numerical linear algebra, rotations of this type are commonly described by the Givens rotation matrix. They were introduced in the 1950s by Wallace Givens and are used to implement rotations within a plane spanned by two coordinate axes.

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