Factoring Matrices into the Product of Circulant and Diagonal Matrices

Huhtanen, Marko; Perämäki, Allan

doi:10.1007/s00041-015-9395-0

Cited by 17 publications

(24 citation statements)

References 12 publications

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“…Nevertheless, a sequence of alternating pulse shapers and EOMs is sufficient to reproduce any matrix transformation. As demonstrated by a recent constructive proof [46], any N × N complex matrix can be factored exactly as the product of no more than 2N − 1 circulant and diagonal matrices-or equivalently of 2N − 1 diagonal matrices spaced by DFT matrices. As described in the following section, such a decomposition is precisely that provided by EOMs and pulse shapers when discretized for numerical simulation, thereby implying that the number of components needed to implement an arbitrary unitary transformation on N spectral modes scales like O(N ).…”

Section: Protocol Componentsmentioning

confidence: 99%

Frequency-encoded photonic qubits for scalable quantum information processing

Lukens

Lougovski

2016

Optica

269

232

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Among the objectives toward large-scale quantum computation is the quantum interconnect: a device which uses photons to interface qubits that otherwise could not interact. However, current approaches require photons indistinguishable in frequency-a major challenge for systems experiencing different local environments or of different physical compositions altogether. Here we develop an entirely new platform which actually exploits such frequency mismatch for processing quantum information. Labeled "spectral linear optical quantum computation" (spectral LOQC), our protocol offers favorable linear scaling of optical resources and enjoys an unprecedented degree of parallelism, as an arbitrary N -qubit quantum gate may be performed in parallel on multiple N -qubit sets in the same linear optical device. Not only does spectral LOQC offer new potential for optical interconnects; it also brings the ubiquitous technology of high-speed fiber optics to bear on photonic quantum information, making wavelength-configurable and robust optical quantum systems within reach.

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Section: Protocol Componentsmentioning

confidence: 99%

Frequency-encoded photonic qubits for scalable quantum information processing

Lukens

Lougovski

2016

Optica

269

232

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“…Since an arbitrary unitary matrix of size (2 N sb + 1) × (2 N sb + 1) has real degrees of freedom whereas N f ≤2 N sb , we conclude that a single modulated ring is insufficient to approximate an arbitrary unitary matrix to a high degree of accuracy, even if all modulation tones up to 2 N sb Ω R are used. To overcome this problem, notice that products of unitary transformations are also unitary 37 . Therefore, as shown in Fig.…”

Section: Resultsmentioning

confidence: 99%

Arbitrary linear transformations for photons in the frequency synthetic dimension

et al. 2021

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Arbitrary linear transformations are of crucial importance in a plethora of photonic applications spanning classical signal processing, communication systems, quantum information processing and machine learning. Here, we present a photonic architecture to achieve arbitrary linear transformations by harnessing the synthetic frequency dimension of photons. Our structure consists of dynamically modulated micro-ring resonators that implement tunable couplings between multiple frequency modes carried by a single waveguide. By inverse design of these short- and long-range couplings using automatic differentiation, we realize arbitrary scattering matrices in synthetic space between the input and output frequency modes with near-unity fidelity and favorable scaling. We show that the same physical structure can be reconfigured to implement a wide variety of manipulations including single-frequency conversion, nonreciprocal frequency translations, and unitary as well as non-unitary transformations. Our approach enables compact, scalable and reconfigurable integrated photonic architectures to achieve arbitrary linear transformations in both the classical and quantum domains using current state-of-the-art technology.

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“…While difficult to extrapolate with such a limited number of data points, we observe that all matrices fall under the scaling cap Q = 2N + 1; in other words, for a given N , no operation requires more than 2N + 1 elements to be realized with F, P ≥ 0.99. Should this cap hold for larger dimensions as well, it would show that-at least for these particular matrices-the linear scaling Q ∝ N expected for arbitrary temporal/spectral AFP patterns [12,20] can hold even for the significantly restricted class of sinewave-only temporal modulation. In order to examine scaling in our AFP approach in greater detail, though, it is important not only consider the circuit depth Q, but also the effective bandwidth.…”

Section: B Sinewave Modulationmentioning

confidence: 99%

All-Optical Frequency Processor for Networking Applications

Lukens

et al. 2020

J. Lightwave Technol.

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We propose an electro-optic approach for transparent optical networking, in which frequency channels are actively transformed into any desired mapping in a wavelength-multiplexed environment. Based on electro-optic phase modulators and Fourier-transform pulse shapers, our all-optical frequency processor (AFP) is examined numerically for the specific operations of frequency channel hopping and broadcasting, and found capable of implementing these transformations with favorable component requirements. Extending our analysis via a mutual-information-based metric for system optimization, we show how to optimize transformation performance under limited resources in a classical context, contrasting the results with those found using metrics motivated by quantum information, such as fidelity and success probability. Given its compatibility with on-chip implementation, as well as elimination of optical-to-electrical conversion in frequency channel switching, the AFP looks to offer valuable potential in silicon photonic network design.

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Factoring Matrices into the Product of Circulant and Diagonal Matrices

Cited by 17 publications

References 12 publications

Frequency-encoded photonic qubits for scalable quantum information processing

Frequency-encoded photonic qubits for scalable quantum information processing

Arbitrary linear transformations for photons in the frequency synthetic dimension

All-Optical Frequency Processor for Networking Applications

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