Decomposing large unitaries into multimode devices of arbitrary size

Letter
Open Access

Decomposing large unitaries into multimode devices of arbitrary size

Christian Arends, Lasse Wolf, Jasmin Meinecke, Sonja Barkhofen, Tobias Weich, and Tim J. Bartley

Phys. Rev. Research 6, L012043 – Published 29 February 2024

Abstract

Decomposing complex unitary evolution into a series of constituent components is a cornerstone of practical quantum information processing. While the decomposition of an $n \times n$ unitary into a product of $2 \times 2$ subunitaries (which can for example be realized by beam splitters and phase shifters in linear optics) is well established, we show how for any $m > 2$ this decomposition can be generalized into a product of $m \times m$ subunitaries (which can then be realized by a more complex device acting on $m$ modes). If the cost associated with building each $m \times m$ multimode device is less than constructing with $\frac{m (m - 1)}{2}$ individual $2 \times 2$ devices, we show that the decomposition of large unitaries into $m \times m$ submatrices is more resource efficient and exhibits a higher tolerance to errors, than its $2 \times 2$ counterpart. This allows larger-scale unitaries to be constructed with lower errors, which is necessary for various tasks, not least boson sampling, the quantum Fourier transform, and quantum simulations.

Received 29 September 2023
Accepted 23 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.L012043

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum algorithms & computation Quantum circuits Quantum computation Quantum information processing Quantum protocols Quantum simulation

Quantum Information, Science & Technology

Authors & Affiliations

Christian Arends ^1,2, Lasse Wolf ³, Jasmin Meinecke^4,5,6,7, Sonja Barkhofen^2,8, Tobias Weich ^3,2, and Tim J. Bartley^2,8

¹Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
²PhoQS, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
³Institute of Mathematics, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
⁴Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
⁵Department für Physik, Ludwig-Maximilians-Universität, 80539 München, Germany
⁶Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany
⁷Institute of Solid State Physics, Technische Universität Berlin, 10623 Berlin, Germany
⁸Department of Physics, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany

Article Text

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Issue

Vol. 6, Iss. 1 — February - April 2024

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Images

Figure 1
Example of the matrix embedding for $m = 3$ (left). The matrix $\tilde{Q}$ only affects the input modes $j_{1}, ..., j_{m}$ . This is illustrated on the circuit diagram (right).
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Figure 2
Decomposition for $m = 3$ and $n = 5$ (left) with an experimental circuit (right). In a first step we can create an upper triangular matrix in the lower left $3 \times 3$ submatrix, which means that we produce zeros on the blue $L$ -shaped region. In the next step the submatrix $Q_{2}^{- 1}$ creates zeros in the yellow $L$ -shaped region. This operation is followed by a two-mode device that only acts on modes 2 and 4 and creates a zero in the red box. Finally, the submatrix $Q_{4}^{- 1}$ creates further zeros in the green $L$ -shaped region. The final phase shifts $φ_{1}, ..., φ_{5}$ transform the matrix to the identity matrix. Continuous lines that bypass a colored box in the circuit diagram (e.g., mode 3 in $Q_{3}^{- 1}$ ) denote modes that are not affected by the submatrix operation.
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Figure 3
(a) Illustration of the algorithm using triangle blocks for $m = 4$ and $n = 13$ and (b) corresponding circuit diagram. Each colored region corresponds to the zeros created by the action of one multimode device. Note that in the last line all four-mode devices act on adjacent modes and thus the colored regions are connected step-shaped regions. By contrast, the red region in the penultimate line is split over modes 3, 6 and 9, which is possible due to a matrix embedding as described in Fig. 1. Note that as in the example presented in Fig. 2, at some places one may also use subunitaries of smaller size, e.g., the pink box on the top that stems from a $2 \times 2$ unitary.
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Figure 4
Scaling behavior of our algorithm for the number of elements to construct a unitary of size $n$ according to $\frac{n (n - 1)}{m (m - 1)} + n - 1$ .
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Figure 5
Numerical simulations of the fidelity for $n = 50$ . For each data point we reconstructed 100 random unitary matrices $U$ , each with 20 different perturbations and averaged over all 2000 reconstructions. The size of the calculated submatrices is indicated in the legend. The error bars of the statistical fluctuations are smaller than the symbol size.
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Figure 6
Numerical simulations of the fidelity for increasing $n \geq m$ at a fixed component quality $F_{Q} = 0.95 \pm 0.0005$ . For each data point we reconstructed 100 random unitary matrices $U$ , each with 20 different perturbations. The size of the submatrices is indicated in the legend. The error bars of the statistical fluctuations are often smaller than the symbol size.
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