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On the experimental verification of quantum complexity in linear optics
Jacques Carolan, Jasmin D. A. Meinecke, Pete Shadbolt, Nicholas J. Russell, Nur Ismail, Kerstin Wörhoff, Terry Rudolph, Mark G. Thompson, Jeremy L. O'Brien, Jonathan C. F. Matthews, Anthony Laing
TL;DR
Boson-sampling distributions are associated with harder #P complexity, and scalable versions lack a known efficient classical method for verifying correct operation. The authors use a row-norm product estimator to test against flat sampling, photon-indistinguishability checks to test against distinguishable sampling, and bosonic clouding as a unitary-specific witness. Confidence that samples were drawn from B rather than F rose to 1−10−35; bosonic clouding persisted for 3, 4, and 5 photons in continuously coupled waveguides but was absent in random-unitary circuits.
Problem
Boson-sampling distributions are associated with harder #P complexity, and scalable versions lack a known efficient classical method for verifying correct operation.
Source: p. 1Method
The authors use a row-norm product estimator to test against flat sampling, photon-indistinguishability checks to test against distinguishable sampling, and bosonic clouding as a unitary-specific witness.
Source: pp. 2–4Results
Confidence that samples were drawn from B rather than F rose to 1−10−35; bosonic clouding persisted for 3, 4, and 5 photons in continuously coupled waveguides but was absent in random-unitary circuits.
Source: pp. 2–3Takeaways & Limitations
Bosonic clouds provide a unitary witness and distinguish quantum from classical regimes without reconstructing parts of the probability distribution.
Source: p. 2The row-norm product estimator distinguishes boson sampling from flat sampling but does not distinguish it from sampling with distinguishable photons.
Source: p. 3
Abstract
from arXiv · showhide
The first quantum technologies to solve computational problems that are beyond the capabilities of classical computers are likely to be devices that exploit characteristics inherent to a particular physical system, to tackle a bespoke problem suited to those characteristics. Evidence implies that the detection of ensembles of photons, which have propagated through a linear optical circuit, is equivalent to sampling from a probability distribution that is intractable to classical simulation. However, it is probable that the complexity of this type of sampling problem means that its solution is classically unverifiable within a feasible number of trials, and the task of establishing correct operation becomes one of gathering sufficiently convincing circumstantial evidence. Here, we develop scalable methods to experimentally establish correct operation for this class of sampling algorithm, which we implement with two different types of optical circuits for 3, 4, and 5 photons, on Hilbert spaces of up to 50,000 dimensions. With only a small number of trials, we establish a confidence >99% that we are not sampling from a uniform distribution or a classical distribution, and we demonstrate a unitary specific witness that functions robustly for small amounts of data. Like the algorithmic operations they endorse, our methods exploit the characteristics native to the quantum system in question. Here we observe and make an application of a "bosonic clouding" phenomenon, interesting in its own right, where photons are found in local groups of modes superposed across two locations. Our broad approach is likely to be practical for all architectures for quantum technologies where formal verification methods for quantum algorithms are either intractable or unknown.
BAYESIAN INFERENCE
Bayesian inference updates confidence that observed events came from boson sampling rather than a flat distribution using the row-norm product estimator R*. The calculation uses probabilities for R* under both hypotheses, with equal prior probabilities; the boson-sampling probability is numerically estimated because the scalable-limit bounds do not apply here.
BAYESIAN INFERENCE
- For p = 3 photons and m = 9 modes, P(R* < 1 | F) = 0.645.The value was determined numerically by averaging over 10^5 Haar-random unitaries.Source: BAYESIAN INFERENCE · p. 8
- R* values update the probability of boson sampling versus flat sampling through Bayes’ theorem, with a prior of 0.5 for each hypothesis.A high R* is more likely under the boson-sampling distribution.Source: BAYESIAN INFERENCE · p. 8
- The boson-sampling probabilities for R* are not scalable to compute, so the authors use numerical values rather than bounds from the large-p, m ≫ p^2 limit.The flat-sampling distribution tends to lognormal in that limit, allowing its probabilities to be computed; the experiment does not operate in that regime.Source: BAYESIAN INFERENCE · p. 8
9 mode unitary
The 9-mode random-unitary chip was fabricated in silicon nitride and uses directional couplers to implement the optical circuit. Its measured fibre-to-fibre coupling efficiency was approximately 5%.
9 mode unitary
- The 9-mode random-unitary chip was fabricated in silicon nitride with a refractive-index contrast of 27%.The waveguides were 1.5 µm wide and separated by 127 µm outside the interaction region.Source: §9 · p. 8
- Thirty-six directional couplers bring waveguides within 2.5 µm for interaction lengths of approximately 400 µm.Interaction length depends on the desired splitting ratio.Source: §9 · p. 8
- The chip's fibre-to-fibre coupling efficiency was approximately 5%.Source: §9 · p. 8
21 mode quantum walk unitary
The 21-mode waveguide array uses silicon oxynitride and compact, micron-scale single-mode waveguides with nearest-neighbour coupling. Its geometry bends and tapers the guides at the facets to match fibre arrays and improve mode overlap.
21 mode quantum walk unitary
- The 21-mode array was fabricated in silicon oxynitride with 2.4% index contrast and a minimum bend radius of 560 µm.Source: §21 · p. 8
- Waveguides measuring 2.2 µm by 0.85 µm are pitched at 1.3 µm along a 700 µm coupling region for nearest-neighbour coupling.Source: §21 · p. 8
- The guides bend to 127 µm pitch and taper to 0.7 µm at the facets, achieving approximately 30% fibre-to-fibre coupling efficiency.The taper is intended to improve overlap with fibre modes.Source: §21 · p. 8
DETECTION SCHEME
The detection scheme measures off-diagonal correlations directly with APDs, while same-mode photon events require fibre splitters and multiple detectors. The reported three- and four-photon measurements cover subsets of possible events.
DETECTION SCHEME
- Off-diagonal, collision-free correlations can be measured with waveguide outputs connected directly to APDs; diagonal same-mode events require fibre splitters and multiple detectors.The splitters enable nondeterministic number-resolved photon detection.Source: DETECTION SCHEME · p. 8
- For three photons, 524 of 1,771 possible three-fold events were measured across specified waveguide sets.Measurements included |3_i⟩ events and correlations between even- and odd-numbered waveguides.Source: DETECTION SCHEME · p. 8
- For four photons, 1,016 of 10,626 possible events were measured, including collision-free events across 16 waveguides.All possible four-photon coincidences were also measured across waveguides {3, 7, 11, 15}.Source: DETECTION SCHEME · p. 8
FOUR PHOTON INPUT STATE
The four-photon |1111⟩ input is isolated from the full down-conversion state by phase cycling and subtracting the other components. The scheme is independent of phase-fluctuation speed but is not intended to scale to large states.
FOUR PHOTON INPUT STATE
- The four-mode, two-crystal source has unknown mode phases; waveplates apply phase θ to mode 1 to produce the desired mixture.Source: Figure 5 · p. 9, FOUR PHOTON INPUT STATE · p. 9
- Phase cycling yields a four-photon mixture, from which the |2200⟩ and |0022⟩ components are subtracted to obtain |1111⟩ statistics.The challenge is isolating the non-superposed |1111⟩ input from the entire down-conversion state.Source: FOUR PHOTON INPUT STATE · pp. 8–9
- The method yields mixed-state statistics independent of phase-fluctuation speed, assuming the phases φi are on average uncorrelated with θ.Source: FOUR PHOTON INPUT STATE · p. 9
- The authors do not intend this scheme as a scalable way to isolate large states such as |11...1⟩.Source: FOUR PHOTON INPUT STATE · p. 9