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arXiv:1507.07185 · Physical Review A

Implementing BosonSampling with time-bin encoding: Analysis of loss, mode mismatch, and time jitter

Keith R. Motes, Jonathan P. Dowling, Alexei Gilchrist, Peter P. Rohde

Published Nov 17, 2015• 18 citations

Abstract

It was recently shown by Motes, Gilchrist, Dowling, and Rohde [Phys. Rev. Lett. 113, 120501 (2014)] that a time-bin encoded fiber-loop architecture can implement an arbitrary passive linear optics transformation. This was shown in the case of an ideal scheme whereby the architecture has no sources of error. In any realistic implementation, however, physical errors are present, which corrupt the output of the transformation. We investigate the dominant sources of error in this architecture---loss and mode mismatch---and consider how it affects the BosonSampling protocol, a key application for passive linear optics. For our loss analysis we consider two major components that contribute to loss---fiber and switches---and calculate how this affects the success probability and fidelity of the device. Interestingly, we find that errors due to loss are not uniform (unique to time-bin encoding), which asymmetrically biases the implemented unitary. Thus loss necessarily limits the class of unitaries that may be implemented, and therefore future implementations must prioritize minimizing loss rates if arbitrary unitaries are to be implemented. Our formalism for mode mismatch is generalized to account for various phenomenon that may cause mode mismatch, but we focus on two---errors in fiber-loop lengths and time jitter of the photon source. These results provide a guideline for how well future experimental implementations might perform in light of these error mechanisms.

DOI: 10.1103/physreva.92.052319

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It was recently shown by Motes, Gilchrist, Dowling, and Rohde [Phys. Rev. Lett. 113, 120501 (2014)] that a time-bin encoded fiber-loop architecture can implement an arbitrary passive linear optics transformation. This was shown in the case of an ideal scheme whereby the architecture has no sources of error. In any realistic implementation, however, physical errors are present, which corrupt the output of the transformation. We investigate the dominant sources of error in this architecture---loss and mode mismatch---and consider how it affects the BosonSampling protocol, a key application for passive linear optics. For our loss analysis we consider two major components that contribute to loss---fiber and switches---and calculate how this affects the success probability and fidelity of the device. Interestingly, we find that errors due to loss are not uniform (unique to time-bin encoding), which asymmetrically biases the implemented unitary. Thus loss necessarily limits the class of unitaries that may be implemented, and therefore future implementations must prioritize minimizing loss rates if arbitrary unitaries are to be implemented. Our formalism for mode mismatch is generalized to account for various phenomenon that may cause mode mismatch, but we focus on two---errors in fiber-loop lengths and time jitter of the photon source. These results provide a guideline for how well future experimental implementations might perform in light of these error mechanisms.

We present an architecture for arbitrarily scalable boson sampling using two nested fiber loops. The architecture has fixed experimental complexity, irrespective of the size of the desired interferometer, whose scale is limited only by fiber and switch loss rates. The architecture employs time-bin encoding, whereby the incident photons form a pulse train, which enters the loops. Dynamically controlled loop coupling ratios allow the construction of the arbitrary linear optics interferometers required for boson sampling. The architecture employs only a single point of interference and may thus be easier to stabilize than other approaches. The scheme has polynomial complexity and could be realized using demonstrated present-day technologies.

Recently, Motes, Gilchrist, Dowling, and Rohde [Phys. Rev. Lett. 113, 120501 (2014).] presented a scheme for photonic boson sampling using a fiber-loop architecture. Here we show that the same architecture can be modified to implement full, universal linear-optics quantum computing, in various incarnations. The scheme employs two embedded fiber loops, a single push-button photon source, three dynamically controlled beamsplitters, and a single time-resolved photodetector. The architecture has only a single point of interference, and thus may be significantly easier to align than other schemes. The experimental complexity of the scheme is constant, irrespective of the size of the computation, limited only by fiber lengths and their respective loss rates.

A major challenge for postclassical boson sampling experiments is the need for a large number of coupled optical modes, detectors, and single-photon sources. Here we show that these requirements can be greatly eased by time-bin encoding and dispersive optics-based unitary transformations. Detecting consecutively heralded photons after time-independent dispersion performs boson sampling from unitaries for which an efficient classical algorithm is lacking. We also show that time-dependent dispersion can implement general single-particle unitary operations. More generally, this scheme promises an efficient architecture for a range of other linear optics experiments.

Understanding how photon loss changes the classical complexity of BosonSampling is central to assessing the robustness of near-term photonic quantum advantage. In this work, we present a polynomial-time classical simulation algorithm for lossy BosonSampling that applies whenever a constant fraction of the input photons survives. We consider the standard fixed-loss model, where

N

input photons propagate through an

M

-mode interferometer and exactly

N-n

photons are lost uniformly at random, leaving

n

surviving photons at the output. For any constant surviving fraction

n/N\in(0,1)

, the algorithm samples from a distribution within total variation distance

\varepsilon

of the fixed-loss BosonSampling distribution with probability at least

1-δ

over Haar-random interferometers, in time

\operatorname{poly}(M,N,\varepsilon^{-1},δ^{-1})

. We further extend the fixed-loss analysis to the uniform-transmission-loss model and obtain analogous polynomial-time simulability for any fixed transmission rate

η\in(0,1)

. These results substantially extend the known provably simulable regime of

n=O(\sqrt{\varepsilon N})

surviving photons or

η=O(\sqrt{\varepsilon/N})

transmission rate, and demonstrate that even the experimentally relevant linear-survival regime can admit efficient classical simulation under the conditions considered here.

We demonstrate the robustness of BosonSampling against imperfections in the linear optical network that cause a small deviation in the matrix it implements. We show that applying a noisy matrix $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$ that is within $\ensuremath{\epsilon}$ of the desired matrix

U

in operator norm leads to an output distribution that is within $\ensuremath{\epsilon}n$ of the desired distribution in variation distance, where

n

is the number of photons. This lets us derive a sufficient tolerance for beam splitters and phase shifters in the network. This result only concerns errors that result from the network encoding a different unitary than desired and not from other sources of noise such as photon loss and partial distinguishability.