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Boson Sampling on a Photonic Chip
Justin B. Spring, Benjamin J. Metcalf, Peter C. Humphreys, W. Steven Kolthammer, Xian-Min Jin, Marco Barbieri, Animesh Datta, Nicholas Thomas-Peter, Nathan K. Langford, Dmytro Kundys, James C. Gates, Brian J. Smith, Peter G. R. Smith, Ian A. Walmsley
TL;DR
The paper addresses whether a simpler photonic computation can experimentally sample a distribution believed to be classically hard, without requiring universal quantum-computing capabilities. It builds and characterizes an integrated-chip QBSM, models photon loss and source imperfections, and finds that the modeled distribution agrees with measured data within finite-sample variation. The study establishes a benchmark for improving larger boson-sampling machines.
Problem
The paper investigates how to realize and validate boson sampling, a photonic computation thought to be exponentially hard to sample classically but requiring only linear evolution, photons, and measurement.
Method
The authors construct an integrated photonic QBSM, characterize its lossy transformation, collect post-selected coincidence data, and model higher-order source terms and photon distinguishability.
Results
The modeled distribution Pmod agrees with the experimental distribution Pexp within the output variance expected from an ideal finite-sample machine.
Takeaways & Limitations
The validated error model can be used to predict realistic QBSM performance and guide designs of future machines with larger N.
Takeaways & Limitations
The characterization determines only the E00 Kraus operator, so transformations involving photon loss processes such as E20 are not characterized.
Abstract
from arXiv · showhide
While universal quantum computers ideally solve problems such as factoring integers exponentially more efficiently than classical machines, the formidable challenges in building such devices motivate the demonstration of simpler, problem-specific algorithms that still promise a quantum speedup. We construct a quantum boson sampling machine (QBSM) to sample the output distribution resulting from the nonclassical interference of photons in an integrated photonic circuit, a problem thought to be exponentially hard to solve classically. Unlike universal quantum computation, boson sampling merely requires indistinguishable photons, linear state evolution, and detectors. We benchmark our QBSM with three and four photons and analyze sources of sampling inaccuracy. Our studies pave the way to larger devices that could offer the first definitive quantum-enhanced computation.
MATERIALS AND METHODS
The experiment characterizes and implements a six-mode integrated photonic circuit for boson sampling, using photon transmission and interference measurements to reconstruct its transformation. Photon-counting data are collected with background subtraction and calibrated to estimate the circuit’s output distributions.
- Photon and circuit implementation: The photon sources use filtered type-II parametric downconversion, while UV-written silica-on-silicon fabrication provides the integrated circuit.The sources generate photons at 830 nm after frequency conversion and filtering to improve indistinguishability.
- Data collection: The experiment collects coincidence events with avalanche photodiodes, subtracting background contributions that comprise approximately 5% of total counts.Three-photon measurements are conditioned on a herald detection, while four-photon measurements count combinations across six output detectors.
- Circuit characterization: The chip implements a non-unitary complex-valued mapping Λij=τijeiφij, reconstructed from one- and two-photon measurements.Single-photon data determine transmission magnitudes, while two-photon interference constrains phases.
- Circuit characterization: Transmission magnitudes are obtained from single-photon output probabilities and relative-port ratios, with repeated measurements providing variances for error bars.The ratio characterization is sufficient because the fixed input state makes the associated normalization factor cancel.
- Circuit characterization: Two-photon interference measurements are fit by least squares to determine the circuit phases and account for an overconstrained set of measured visibilities.The same procedure uses Monte Carlo sampling of characterized matrix elements to estimate uncertainty.
- Circuit characterization: The characterization requires O(M^2) measurements for a general M-mode linear transformation and supports prediction of boson distributions through matrix permanents.The predicted distribution is constructed from the experimentally determined Λ for selected input and output states.
SUPPLEMENTARY TEXT
The supplementary derivation expresses boson-sampling probabilities through matrix permanents. It develops the transformation from input creation operators to projected output states and identifies the assumptions distinguishing bosonic from fermionic statistics.
- Boson-distribution derivation: The derivation begins with N bosons entering an M-mode unitary network, assuming at most one boson per input and output mode for the experimental case.Input photons occupy modes 1 through N, while the remaining input modes contain vacuum.
- Boson-distribution derivation: The unitary evolves input creation operators into output operators, after which projection onto an output number state yields its measurement probability.The output state is represented by an N-element vector whose entries specify the detected modes.
- Boson-distribution derivation: Expanding the projected state produces sums over assignments of N photons to M modes, allowing repeated occupation of a mode.These assignments are organized as permutations with repetitions permitted.
- Boson-distribution derivation: Only permutations compatible with the selected output state contribute, because incompatible terms leave an annihilation operator acting on vacuum.This reduction gives the permanent structure of the boson-sampling amplitude.
- Boson-distribution derivation: The boson distribution is computed from permanents of submatrices formed by retaining input-photon rows and output-state columns.The general case allows more than one photon per input or output mode.
- Statistics: The derivation assumes bosonic commutation relations; replacing them with fermionic anticommutation introduces alternating signs and yields a determinant.The determinant is described as classically computable in the supplementary discussion.
Effects of loss
The loss analysis shows that post-selecting trials with no lost photons reduces the experimentally relevant transformation to a non-unitary accessible-mode submatrix. Boson-sampling predictions can therefore still be expressed through matrix permanents of this lossy transformation.
- Loss model: Experimental losses are modeled as couplings between accessible and inaccessible modes, producing a full unitary evolution followed by a mixed accessible output.The inaccessible modes represent environmental loss channels that are traced out.
- Post-selection: The experiment post-selects events in which all N input photons are detected in the accessible modes.This condition isolates the no-loss contribution to the observed boson-sampling data.
- Accessible transformation: Because output projections involve only accessible modes, the calculation can use Λ, the submatrix of the full unitary restricted to accessible input and output modes.The matrix permanents of Λ provide the theoretical predictions for the main experiment.
- Post-selection: Under no-loss post-selection, the relevant evolution is represented by the E00 Kraus operator rather than the full environmental transformation.The experimental results therefore sample a non-unitary transformation equivalent to E00 in this conditioned subspace.
Sources of error
The paper benchmarks sampling accuracy against finite-sample variation and models the principal experimental imperfections. The modeled distribution agrees with the experimental data within the variation expected for an ideal finite-sample machine.
- Benchmarking accuracy: The benchmark compares the inferred experimental distribution Pexp with the theoretical distribution Pth, while recognizing that finite samples necessarily produce nonzero distance.The computational challenge becomes harder for classical machines as the allowed sampling error decreases.
- Benchmarking accuracy: An additional error source beyond finite sampling is indicated by the discrepancy between experimental distances and the Monte Carlo variation expected from ideal sampling.The comparison is made for the experimentally collected three- and four-photon sample sizes.
- Error modeling: The model Pmod incorporates photon distinguishability and higher-order terms from the PDC sources while neglecting photon impurity minimized by the experimental design.The resulting distance d(Pexp,Pmod) falls within the output variance of an ideal machine.
- Error modeling: The agreement between Pexp and Pmod indicates that photon distinguishability and higher-order PDC terms account for the principal modeled sources of experimental error.The validated model is intended to guide designs of larger-N boson-sampling machines.
Effect of using heralded single photon sources
The experiment models how higher-order emission from squeezed PDC sources and photon loss alter the intended boson-sampling distribution. It combines loss modeling with measured source parameters to predict these effects.
- Source imperfections: PDC sources generate undesired multiphoton terms alongside single-photon terms, with higher-order contributions increasing with the squeezing parameter λ.The experiment lowers pump power to reduce |22⟩ and |33⟩ terms while retaining feasible count rates.
- Source imperfections: Losses can make inputs containing more than N photons appear as N-fold detections at the output.This means an observed N-fold event does not necessarily originate from the intended N-photon input.
- Characterization limits: The characterization determines only the E00 Kraus operator, leaving processes involving photons lost to environmental modes uncharacterized.For example, injecting five photons and losing two corresponds to the unknown E20 operator.
- Loss modeling: Losses are modeled with beam splitters linking accessible circuit modes to inaccessible loss modes, with relative losses inferred numerically from the characterized transformation.Source, circuit, and detector losses receive independent scaling factors, including source scaling matched to heralding efficiency.
- Loss modeling: The model uses an extended unitary transformation over accessible and loss modes to predict higher-order PDC effects and sums events with photons distributed into loss modes.First higher-order input distributions are weighted by λ^2 obtained from a conditional second-order correlation measurement.
Photon Distinguishability
The experiment accounts for imperfect photon indistinguishability by parameterizing each photon as a superposition of desired and distinguishable modes. It estimates this imperfection from Hong–Ou–Mandel interference and incorporates the resulting distributions.
- Model: Boson sampling assumes indistinguishable bosons, whereas experimental photons inevitably exhibit some distinguishability.The analysis therefore uses a distinguishability parameter for the input state.
- Model: Each photon is modeled as a superposition of a desired mode ξ0 and another mode ξi, with α representing distinguishability.The creation operator ξ†j,i labels photon i in mode ξj.
- Measurement: α = 0.974 on average, estimated from the reduction in Hong–Ou–Mandel dip visibility at a beamsplitter inside the circuit.This measurement supplies the experiment’s average indistinguishability parameter.
- Probability model: When one photon is distinguishable, output probabilities are formed from incoherent sums of permanents of N−1 matrices.For the example input, the relevant terms are permanents of 2 × 2 matrices.
- Probability model: The model weights the one-distinguishable-photon cases by 1 −α^2 for three photons and |α^3√ 1 −α^2|^2 for four photons.Because α is large, cases with two distinguishable photons are ignored.