Boson-sampling with photons of arbitrary spectral structure

Peter P. Rohde

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Boson-sampling with photons of arbitrary spectral structure

Peter P. Rohde

arXiv:1410.3979v1Oct 15, 2014quant-ph

TL;DR

Boson-sampling relies on indistinguishable photons, but real photons have spectral structure. This paper generalizes the model to arbitrary spectra and relates sampling statistics to matrix permanents. It finds that indistinguishable photons recover the usual permanent result, whereas distinguishable photons yield a computationally easy limiting case.

Problem
Boson-sampling requires indistinguishable photons, although real photons exhibit spectral structure that can distinguish them.
Source: §I · p. 1
Method
The paper models photons with arbitrary spectral structure and relates their sampling statistics to functions of matrix permanents for both spectrally resolving and non-spectrally-resolving detectors.
Source: §I · p. 1
Results
Indistinguishable photons recover the usual boson-sampling permanent result, while distinguishable photons yield Per(|U′|2), a computationally easy case.
Source: D. Limiting cases · pp. 3–4, §V · p. 5
Takeaways & Limitations
The sampling amplitudes for arbitrary spectra are linear combinations of products of functions of permanents, connecting photon distinguishability to the sampling statistics.
Source: §V · p. 5

Abstract

from arXiv · show

Boson-sampling has attracted much interest as a simplified approach to implementing a subset of optical quantum computing. Boson-sampling requires indistinguishable photons, but far fewer of them than universal optical quantum computing architectures. In reality, photons are never indistinguishable, and exhibit a rich spectral structure. Here we consider the operation of boson-sampling with photons of arbitrary spectral structure and relate the sampling statistics of the device to matrix permanents. This sheds light on the computational complexity of different regimes of the photons' spectral characteristics, and provides very general results for the operation of linear optics interferometers in the presence of partially distinguishable photons. Our results apply to both the cases of spectrally resolving and non-spectrally resolving detectors.

I. INTRODUCTION

Boson-sampling uses passive linear optics to evolve single photons and samples the outputs by coincidence photodetection. This paper studies how spectral structure changes the permanent-based sampling model; its standard setup assumes a binary output regime for large systems.

I. INTRODUCTION

Boson-sampling evolves single photons through passive linear optics and samples the output using coincidence photodetection.The model is presented as a basis for a recent field whose sampling problem has been shown to be classically intractable.Source: §I · p. 1
Standard boson-sampling amplitudes are related to matrix permanents, while networks with time delays have been related to matrix immanents.The paper situates its generalization against these prior results.Source: §I · p. 1
The paper generalizes the model to arbitrary photon spectra, relating sampling probabilities to functions of matrix permanents and considering both spectrally resolving and non-spectrally-resolving detectors.It also reports the expected limiting behavior: indistinguishable photons recover ideal boson-sampling, while distinguishable photons are computationally trivial.Source: §I · p. 1

II. BOSON-SAMPLING

For large systems with m = O(n^2) modes, the paper assumes the binary regime, with zero or one photon in each output mode.This assumption is adopted to simplify notation.Source: §II · p. 2

III. SPECTRAL STRUCTURE OF PHOTONS

Because a single creation operator does not capture photons’ spectral structure, the paper represents photons with mode operators and decomposes their spectra in an orthonormal basis.

III. SPECTRAL STRUCTURE OF PHOTONS

The paper uses mode operators to represent photons with spectral structure that a single photonic creation operator does not characterize.The mode operator creates a photon with spectral distribution ψ(ω) in a specified spatial mode.Source: §III · p. 2
Each photon’s spectral distribution can be expanded in a discrete orthonormal basis, with coefficients describing its decomposition.The coefficients are calculated using the basis functions.Source: §III · p. 2
The mode operators can then be expressed as a decomposition into orthonormal-basis mode operators.Source: §III · p. 2

A. Spectrally pure photons

For spectrally pure inputs, the paper decomposes the photons into configurations of spectral basis modes and applies the usual permanent rule to indistinguishable photons within each spectral mode.

A. Spectrally pure photons

The input state is specified as a tensor product of photons with distinct spectral distributions and is re-expressed in the spectral-mode basis.The mode groups are identified by T(v, i), the spatial modes assigned basis index i in configuration v.Source: A. Spectrally pure photons · p. 2
The number of spectral configurations is exponential in photon number, except when all photons are indistinguishable, for which there is one configuration and the input reduces to the ideal case.A configuration records which spectral basis function is assigned to each input photon.Source: A. Spectrally pure photons · p. 2
For each spectral configuration, the photons assigned to basis mode ξ_i form a set of indistinguishable photons that evolves by the standard permanent rule.The relevant submatrix is formed from the input and output spatial modes for that group.Source: A. Spectrally pure photons · pp. 2–3

B. Spectrally-resolved detectors

With spectrally resolving detectors, output configurations are associated with individual spectral modes; indistinguishable photons recover ordinary permanent sampling, while otherwise amplitudes involve linear combinations of permanents.

B. Spectrally-resolved detectors

For distinguishable spectral structures, the resolved-detector amplitudes are linear combinations of permanents rather than a single permanent.Each output configuration is associated with a spectral mode, and the measurement probability is the squared magnitude of its amplitude.Source: B. Spectrally-resolved detectors · p. 3
When all photons are indistinguishable, the spectral basis can be chosen so that there is one configuration and the amplitude reduces to ordinary permanent sampling.Source: B. Spectrally-resolved detectors · p. 3

C. Non-spectrally-resolved detectors

Non-spectrally-resolving detectors report photon counts without identifying spectral modes, so the probability of an outcome sums over all compatible spectral partitions. The resulting expression combines a classical sum over unresolved spectral components with quantum permanents for their individual sampling outcomes.

C. Non-spectrally-resolved detectors

A measurement signature records the output configuration irrespective of the spectral modes in which photons were measured.Its total photon count is the sum across spectral modes.Source: C. Non-spectrally-resolved detectors · p. 3
The probability of a measurement signature sums over all partitions of that outcome into signatures for individual spectral modes.The detector reports photon counts but not their spectral structure.Source: C. Non-spectrally-resolved detectors · p. 3
The probability expression has a classical positive-real-valued sum over unresolved spectral components and quantum permanents for sampling within each component.The classical term sums spectrally distinct elements that the detector cannot resolve.Source: C. Non-spectrally-resolved detectors · p. 3

D. Limiting cases

The limiting cases recover ordinary permanent-based boson-sampling for indistinguishable photons and a classically easy probability calculation for distinguishable photons. Intermediate distinguishability requires a combinatoric expression involving permanents of varying sizes.

D. Limiting cases

For indistinguishable photons, the limiting expression reduces to the expected boson-sampling result with a permanent of an n × n submatrix of U.All photons occupy the same spectral mode.Source: D. Limiting cases · p. 3
For distinguishable photons, the sampling probabilities are a permanent of positive real-valued classical probabilities, which is computationally easy to approximate.Each photon’s evolution can be evaluated independently.Source: D. Limiting cases · pp. 3–4
The two endpoints differ in where the absolute square is taken: after the permanent of quantum amplitudes for indistinguishable photons, but before the permanent of classical probabilities for distinguishable photons.Source: D. Limiting cases · p. 4
The intermediate regime requires a combinatoric expression over permanents of matrices ranging in size from 1 to n.Source: D. Limiting cases · p. 4

E. Hong-Ou-Mandel interference

The two-photon Hong-Ou-Mandel example uses a 50/50 beamsplitter and a parameter α for distinguishability. Its outcome depends on whether detection resolves spectral modes: filtering can restore no anti-bunching, while unresolved detection gives coincidence probability 1/2 for distinguishable photons and zero for indistinguishable photons.

E. Hong-Ou-Mandel interference

For two photons at a 50/50 beamsplitter, α ranges from 1 for indistinguishable photons to 0 for distinguishable photons.The example uses a superposition across two spectral modes.Source: E. Hong-Ou-Mandel interference · p. 4
With a spectrally resolving detector projecting onto ξ1, anti-bunching is never observed for indistinguishable photons or partially distinguishable photons with spectral filtering.Narrowband filtering is given as a possible implementation when the spectral basis states are frequency eigenstates.Source: E. Hong-Ou-Mandel interference · p. 4
With non-resolving detection, the coincidence probability is 0 for α = 1 and 1/2 for α = 0.The α = 0 photons behave as classical particles.Source: E. Hong-Ou-Mandel interference · p. 4

F. Spectrally mixed photons

Spectrally mixed inputs can be modeled as mixtures of spectral states, and the output probability sums over combinations of their components. That sum requires q^n amplitudes and grows exponentially with photon number, except when photons are spectrally pure.

F. Spectrally mixed photons

Heralded spontaneous parametric down-conversion can produce spectral mixing through signal-idler correlations, depending on the heralding photodetector’s spectral characteristics.Source: F. Spectrally mixed photons · pp. 4–5
Each input photon can be modeled as a mixture of q spectral states, with ψi,j denoting the spectral distribution of component j for photon i.Source: F. Spectrally mixed photons · p. 5
Summing the output probability over all combinations of components in the input mixture requires q^n amplitudes, growing exponentially in n except for spectrally pure photons.Source: F. Spectrally mixed photons · p. 5

V. CONCLUSION

For arbitrary photon spectra, boson-sampling amplitudes become linear combinations of products of permanent functions, recovering the usual indistinguishable and distinguishable limits. The results apply to spectrally resolving and spectrally blind detectors.

V. CONCLUSION

Indistinguishable photons yield measurement probabilities |Per(U ′)|2, while distinguishable photons yield Per(|U ′|2), corresponding to computationally hard and easy-to-approximate regimes, respectively.Source: §V · p. 5
With arbitrary spectral structure, amplitudes are generally linear combinations of products of functions of permanents.Source: §V · p. 5
The model recovers the usual permanent result for indistinguishable photons and verifies computational ease for distinguishable photons.Source: §V · p. 5
The results cover both spectrally resolving and spectrally blind detectors and reproduce expected two-photon Hong-Ou-Mandel interference in both regimes.Source: §V · p. 5