Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Unitary matrix decompositions for optimal and modular linear optics architectures

Shreya P. Kumar, Ish Dhand

Published Dec 17, 2020• 12 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/abd4ae

Related papers

Optimal modular architectures for universal linear optics

Shreya P. Kumar, Ish Dhand

We present modular and optimal architectures for implementing arbitrary discrete unitary transformations on light. These architectures are based on systematically combining smaller M-mode linear optical interferometers together to implement a larger 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 the different optical modes incur balanced optical losses, so the architectures promise substantially enhanced process fidelities as compared to existing schemes.

arXiv:2001.02012•Jan 7, 2020•1 citation

Realization of arbitrary discrete unitary transformations using spatial and internal modes of light

Ish Dhand, Sandeep K. Goyal

Any lossless transformation on ${n}_{s}$ spatial and ${n}_{p}$ internal modes of light can be described by an ${n}_{s}{n}_{p}\ifmmode\times\else\texttimes\fi{}{n}_{s}{n}_{p}$ unitary matrix, but there is no known procedure to effect an arbitrary ${n}_{s}{n}_{p}\ifmmode\times\else\texttimes\fi{}{n}_{s}{n}_{p}$ unitary matrix on light in ${n}_{s}$ spatial and ${n}_{p}$ internal modes. We devise an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. Our realization uses beam splitters and operations on internal modes to effect arbitrary linear transformations. The number of beam splitters required to realize a unitary transformation is reduced as compared to existing realization by a factor ${n}_{p}^{2}/2$ at the cost of increasing the number of internal optical elements by a factor of 2. Our algorithm thus enables the optical implementation of higher dimensional unitary transformations.

arXiv:1508.06259•Physical Review A•Oct 12, 2015•35 citations

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

Vincent Girouard, Nicolás Quesada

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.

arXiv:2508.20010•Journal of the Optical Society of America B•Dec 3, 2025•6 citations

Collective unitary evolution with linear optics by Cartan decomposition

Wen‐Qiang Liu, Xinjie Zhou, Hai‐Rui Wei

Abstract Unitary operation is an essential step for quantum information processing. We first propose an iterative procedure for decomposing a general unitary operation without resorting to controlled-NOT gate and single-qubit rotation library. Based on the results of decomposition, we design two compact architectures to deterministically implement arbitrary two-qubit polarization-spatial and spatial-polarization collective unitary operations, respectively. The involved linear optical elements are reduced from 25 to 20 and 21 to 20, respectively. Moreover, the parameterized quantum computation can be flexibly manipulated by wave plates and phase shifters. As an application, we construct the specific quantum circuits to realize two-dimensional quantum walk and quantum Fourier transformation. Our schemes are simple and feasible with the current technology.

arXiv:2204.01984•Europhysics Letters (EPL)•Dec 1, 2021•1 citation

Hybrid spatiotemporal architectures for universal linear optics

Daiqin Su, Ish Dhand, L. G. Helt +2 more

We present two hybrid linear-optical architectures that simultaneously exploit spatial and temporal degrees of freedom of light to effect arbitrary discrete unitary transformations. Our architectures combine the benefits of spatial implementations of linear optics, namely, low loss and parallel operation, with those of temporal implementations, namely, modest resource requirements and access to transformations of potentially unbounded size. We arrive at our architectures by devising and employing decompositions of large discrete unitary transformations into smaller ones, decompositions we expect to have broad utility beyond spatiotemporal linear optics. We show that hybrid architectures promise important advantages over both spatial-only and temporal-only architectures.

arXiv:1812.07939•Physical Review A•Jun 3, 2019•23 citations

Optimized architectures for universal quantum state transformations using photonic path and polarization

Dong‐Xu Chen, Junliang Jia, Pei Zhang +1 more

Abstract An arbitrary lossless transformation in high-dimensional (HD) quantum space can be decomposed into elementary operations which are easy to implement, and an effective decomposition algorithm is important for constructing HD systems. Here, we present two optimized architectures to effectively realize an arbitrary unitary transformation by using the photonic path and polarization based on the existing decomposition algorithm. In the first architecture, the number of required interferometers is reduced by half compared with previous works. In the second architecture, by using the HD X gate, all the elementary operations are transferred to the operations which act locally on the photonic polarization in the same path. Such an architecture could be of significance in polarization-based applications. Both architectures maintain the symmetric layout. Our work facilitates the optical implementation of HD transformations and could have potential applications in HD quantum computation and quantum communication.

arXiv:2212.02703•Quantum Science and Technology•Nov 8, 2022•3 citations
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.
We present modular and optimal architectures for implementing arbitrary discrete unitary transformations on light. These architectures are based on systematically combining smaller M-mode linear optical interferometers together to implement a larger 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 the different optical modes incur balanced optical losses, so the architectures promise substantially enhanced process fidelities as compared to existing schemes.
Any lossless transformation on ns{n}_{s}ns​ spatial and np{n}_{p}np​ internal modes of light can be described by an ${n}_{s}{n}_{p}\ifmmode\times\else\texttimes\fi{}{n}_{s}{n}_{p}$ unitary matrix, but there is no known procedure to effect an arbitrary ${n}_{s}{n}_{p}\ifmmode\times\else\texttimes\fi{}{n}_{s}{n}_{p}$ unitary matrix on light in ns{n}_{s}ns​ spatial and np{n}_{p}np​ internal modes. We devise an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. Our realization uses beam splitters and operations on internal modes to effect arbitrary linear transformations. The number of beam splitters required to realize a unitary transformation is reduced as compared to existing realization by a factor np2/2{n}_{p}^{2}/2np2​/2 at the cost of increasing the number of internal optical elements by a factor of 2. Our algorithm thus enables the optical implementation of higher dimensional unitary transformations.
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.
Abstract Unitary operation is an essential step for quantum information processing. We first propose an iterative procedure for decomposing a general unitary operation without resorting to controlled-NOT gate and single-qubit rotation library. Based on the results of decomposition, we design two compact architectures to deterministically implement arbitrary two-qubit polarization-spatial and spatial-polarization collective unitary operations, respectively. The involved linear optical elements are reduced from 25 to 20 and 21 to 20, respectively. Moreover, the parameterized quantum computation can be flexibly manipulated by wave plates and phase shifters. As an application, we construct the specific quantum circuits to realize two-dimensional quantum walk and quantum Fourier transformation. Our schemes are simple and feasible with the current technology.
We present two hybrid linear-optical architectures that simultaneously exploit spatial and temporal degrees of freedom of light to effect arbitrary discrete unitary transformations. Our architectures combine the benefits of spatial implementations of linear optics, namely, low loss and parallel operation, with those of temporal implementations, namely, modest resource requirements and access to transformations of potentially unbounded size. We arrive at our architectures by devising and employing decompositions of large discrete unitary transformations into smaller ones, decompositions we expect to have broad utility beyond spatiotemporal linear optics. We show that hybrid architectures promise important advantages over both spatial-only and temporal-only architectures.
Abstract An arbitrary lossless transformation in high-dimensional (HD) quantum space can be decomposed into elementary operations which are easy to implement, and an effective decomposition algorithm is important for constructing HD systems. Here, we present two optimized architectures to effectively realize an arbitrary unitary transformation by using the photonic path and polarization based on the existing decomposition algorithm. In the first architecture, the number of required interferometers is reduced by half compared with previous works. In the second architecture, by using the HD X gate, all the elementary operations are transferred to the operations which act locally on the photonic polarization in the same path. Such an architecture could be of significance in polarization-based applications. Both architectures maintain the symmetric layout. Our work facilitates the optical implementation of HD transformations and could have potential applications in HD quantum computation and quantum communication.