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Generation and sampling of quantum states of light in a silicon chip
Stefano Paesani, Yunhong Ding, Raffaele Santagati, Levon Chakhmakhchyan, Caterina Vigliar, Karsten Rottwitt, Leif K. Oxenløwe, Jianwei Wang, Mark G. Thompson, Anthony Laing
TL;DR
Integrated photonics needs scalable ways to generate and process many photons for quantum algorithms. This paper uses silicon SFWM sources and configurable pumping to implement boson-sampling protocols and benchmark vibronic-spectrum calculations, while identifying noise and scalability boundaries.
Problem
Generating and processing many photons remains a bottleneck for integrated photonics, motivating validation and modeling of scalable boson-sampling implementations.
Source: Test models for validating Gaussian… · p. 16, Noises in Gaussian boson sampling with… · p. 20Method
The study validates boson-sampling data against ideal and alternative models and uses photonic squeezed-state sampling to simulate and benchmark Franck–Condon profiles.
Source: Additional validation tests for… · p. 14, Test models for validating Gaussian… · p. 16, Franck-Condon profiles · p. 23Results
The validation procedure distinguishes boson-sampler outcomes from distinguishable-photon sampling using a likelihood-ratio counter with 4-fold and 6-fold experimental data.
Source: Additional validation tests for… · p. 14Takeaways & Limitations
Photonic sampling can provide a benchmarking tool for Franck–Condon profiles and related molecular vibronic-spectrum simulations.
Source: Franck-Condon profiles · p. 23Losses transform single-mode squeezed states toward thermal states, while higher-order terms make the Franck–Condon simulation approach non-scalable.
Source: Noises in Gaussian boson sampling with… · p. 20, Franck-Condon profiles · p. 25
Abstract
from arXiv · showhide
Implementing large instances of quantum algorithms requires the processing of many quantum information carriers in a hardware platform that supports the integration of different components. While established semiconductor fabrication processes can integrate many photonic components, the generation and algorithmic processing of many photons has been a bottleneck in integrated photonics. Here we report the on-chip generation and processing of quantum states of light with up to eight photons in quantum sampling algorithms. Switching between different optical pumping regimes, we implement the Scattershot, Gaussian and standard boson sampling protocols in the same silicon chip, which integrates linear and nonlinear photonic circuitry. We use these results to benchmark a quantum algorithm for calculating molecular vibronic spectra. Our techniques can be readily scaled for the on-chip implementation of specialised quantum algorithms with tens of photons, pointing the way to efficiency advantages over conventional computers.
Supplementary Materials
The experiment generates and detects photons on silicon using SFWM sources, wavelength-selective filtering, fiber coupling, and SNSPD detection.
- Single photons are generated in 450 nm × 250 nm silicon waveguides using SFWM in four 1.4 cm spiral sources.Source: Experimental setup details · p. 7
- A 500 MHz tunable laser provides near-telecom pump pulses that are amplified with an EDFA before chip injection.Source: Experimental setup details · p. 7
- A 16-channel fiber array and off-chip filters collect and spectrally clean photons before SNSPD detection.Source: Experimental setup details · p. 7
Chip design, fabrication, and components characterisation
The chip combines silicon photonic sources, tunable AMZI filters, grating couplers, and a 12-waveguide random walk fabricated on an SOI platform.
- The platform uses a 250 nm silicon layer on 3 µm buried oxide, with Au/Ti electrode pads formed by UV lithography and lift-off.Source: Chip design, fabrication, and… · pp. 7–8
- The device integrates four spiral sources, two AMZI filter layers, and a 12-waveguide continuous random walk connected to 16 grating couplers.Source: Figure S1 · p. 8
- Al-mirror-assisted grating couplers achieve −1.1 dB coupling efficiency with a 40 nm 1 dB coupling bandwidth.Source: Chip design, fabrication, and… · p. 8
- The two pumping schemes use either a single wavelength for SBS or two spectrally selected wavelengths recombined for GBS.Source: Figure S2 · p. 9
- The 12-waveguide random walk uses 450 nm-wide waveguides, 180 nm coupling gaps, and 110 µm coupling length.Source: Chip design, fabrication, and… · p. 9
System efficiency
System characterization quantifies photon transmission, detector performance, component loss, and the random walk’s transmission efficiency.
- The average channel efficiency from generation to detection was −11.5 dB across the 16 channels.Source: System efficiency · p. 9
- The integrated components include grating couplers and thermally tuned AMZI filters characterized through optical and scanning-electron-microscope measurements.Source: Figure S3 · p. 10, Figure S4 · p. 10
- The 12-mode random walk achieved a transmission efficiency of 0.995, while the 1.4 cm spiral waveguides had average losses of −7.1 dB.Source: System efficiency · p. 10
Degenerate and non-degenerate SFWM experimental details
Single- and dual-wavelength pumping select the SFWM regimes used for SBS and GBS, with source parameters and photon-event rates measured experimentally.
- Non-degenerate SFWM sources operated at SBS conditions with two-mode squeezing parameters {0.25, 0.21, 0.18, 0.17}.Source: Degenerate and non-degenerate SFWM… · p. 11
- The dual-wavelength scheme uses spectral broadening and WDM selection to avoid synchronization and phase-locking requirements between separate lasers.Source: Degenerate and non-degenerate SFWM… · p. 11
- Four-photon GBS events were observed at approximately 5 Hz with 1 dBm input power.Source: Degenerate and non-degenerate SFWM… · p. 12
- The dual-wavelength scheme reduces count rates because of its low injected pump power, while separate pulsed lasers could increase power at the cost of synchronization.Source: Degenerate and non-degenerate SFWM… · p. 12
Photon purity characterisation
In SBS, high spectral purity of heralded signal photons is important for achieving good quantum interference. The experiment estimates photon purity quantitatively using unheralded second-order correlation measurements in the non-degenerate SFWM case.
- High spectral purity of each heralded signal photon is important for achieving good-quality quantum interference between signal photons.Source: Photon purity characterisation · p. 12
Bayesian model comparison for boson sampling validation.
Bayesian model comparison evaluates whether measured samples are more likely under an ideal boson-sampling model or a classically tractable alternative. The framework updates model confidence from observed samples while accounting for collision-free detection and model priors.
- Bayesian model comparison tests whether measured samples are more likely under an ideal SBS or GBS model than a classical adversary model.Source: Bayesian model comparison for boson… · p. 12
- The confidence in the ideal model is computed from the likelihoods of the observed data and the prior probabilities assigned to competing models.Source: Bayesian model comparison for boson… · p. 13
- Assuming independent events, the data likelihood is the product of the ideal model’s probabilities for each measured outcome.Source: Bayesian model comparison for boson… · p. 13
- For collision-free detection, model probabilities are normalized over collision-free outcomes because models generally assign different probabilities to such events.Source: Bayesian model comparison for boson… · p. 13
- Uniform priors of 0.5 are used, and model confidences are dynamically updated as experimental samples accumulate.Source: Bayesian model comparison for boson… · p. 13
- With multiple test models, simultaneous confidence in the ideal model converges to one exactly when all corresponding pairwise validation confidences converge to one.Source: Bayesian model comparison for boson… · p. 13
Scattershot boson sampling
Standard boson sampling suffers an exponentially decreasing probability that all nondeterministic sources emit photons. Scattershot boson sampling addresses this by pumping more sources and heralding photons from randomly selected modes, yielding a combinatorial generation-rate advantage in principle.
- ϵ^n is the probability that all n nondeterministic single-photon sources fire in standard boson sampling, decreasing exponentially with n.Source: Scattershot boson sampling · p. 13
- For k ≫ n, SBS has an exponential combinatorial speed-up in photon-pair generation rate relative to the standard approach.Source: Scattershot boson sampling · p. 13
- The supplementary validation compares experimental SBS data with uniform and distinguishable samplers using row-norm and likelihood-ratio tests.Source: Figure S7 · p. 14
Additional validation tests for scattershot boson sampling
Additional SBS validation tests compare experimental outcomes with uniform and distinguishable-photon models. The tests use efficiently computable estimators or likelihood ratios and classify the data from the final value of an accumulated counter.
- The row-norm estimator test distinguishes boson-sampler outputs from a trivial uniform distribution without calculating ideal output probabilities.Source: Additional validation tests for… · p. 14
- The row-norm protocol updates a counter using efficiently computable row-norm estimators and accepts boson-sampler data when the final counter satisfies C > 0.Source: Additional validation tests for… · p. 14
- The likelihood-ratio test distinguishes ideal boson sampling from sampling with distinguishable photons using their respective output probabilities.Source: Additional validation tests for… · p. 14
- The likelihood-ratio estimator is L_j,k = p_ind(k|j)/p_dist(k|j), and the counter is updated according to thresholded likelihood-ratio ranges.Source: Additional validation tests for… · p. 14
- Using a1 = 0.75 and a2 = 2, the test classifies outcomes as boson-sampler data when the final counter is positive; results are reported for 4-fold and 6-fold data.Source: Additional validation tests for… · p. 14
Gaussian boson sampling
Gaussian boson sampling generates photon-counting samples from vacuum squeezed states evolved through a linear-optical circuit. Its output probabilities involve Hafnians, making sampling classically intractable under stated complexity assumptions, while scattershot sampling appears as a special case.
- GBS samples photon-counting patterns from m vacuum single-mode squeezed states after linear-optical evolution.Source: Gaussian boson sampling · p. 15
- The output probability for a detection pattern is expressed through the Hafnian of a matrix derived from the circuit transformation and squeezing parameters.The matrix is formed by repeating rows and columns according to detected photon occupations.Source: Gaussian boson sampling · p. 15
- Hafnians generalize permanents, and computing both matrix functions is hard for classical computers.Source: Gaussian boson sampling · p. 15
- Under stated complexity-theoretic assumptions, GBS sampling is classically intractable because its photon-counting probabilities are given by Hafnians.Source: Gaussian boson sampling · p. 15
- Scattershot boson sampling can be realized as a special GBS case by interfering adjacent squeezed states, heralding one arm, and sending the other through the circuit.The resulting scattershot probabilities are expressed using permanents of the circuit matrix.Source: Gaussian boson sampling · pp. 15–16
- The experiment uses four input modes and twelve output modes, equivalently represented by a 12 × 12 circuit with eight vacuum inputs.Source: Gaussian boson sampling · p. 16
Test models for validating Gaussian boson sampling
The validation framework compares experimental GBS data with classically tractable alternative models. Coherent and thermal states, distinguishable squeezed states, TMS inputs, and uniform sampling provide distinct test distributions and validation benchmarks.
- Bayesian model comparison validates experimental GBS data by comparing an ideal model with test models whose output probabilities are computable.Source: Test models for validating Gaussian… · p. 16
Coherent states
- Coherent-state evolution remains separable, producing a product of Poisson distributions that can be sampled and computed efficiently classically.The coherent-state model is used as a physically relevant validation test despite its computational triviality.Source: Test models for validating Gaussian… · p. 16
Thermal states
- Thermal-state photon-counting probabilities involve permanents, but sampling from the resulting distribution is classically tractable.Source: Test models for validating Gaussian… · pp. 16–17
Distinguishable squeezed states
- Distinguishable squeezed states do not interfere and are equivalent to accumulating detection events from separate single-squeezer experiments.The treatment specializes the probability expression to four distinguishable squeezed states in a 12-mode circuit.Source: Test models for validating Gaussian… · p. 17
Two-mode squeezed states
- Two-mode squeezed-state inputs can be mapped to an analogous single-mode GBS circuit using pairs of single-mode squeezers, phase shifts, and beam splitters.Source: Test models for validating Gaussian… · p. 18
- TMS-input sampling remains classically hard but is interpreted as evidence for correct experimental implementation rather than validation of sampler complexity.Source: Test models for validating Gaussian… · p. 18
Uniform sampling
- Uniform sampling assigns equal probability to all allowed output patterns and is included as a validation model.Source: Test models for validating Gaussian… · p. 18, Figure S9 · p. 19
Gaussian boson sampling with pseudo photon number resolving detection
The experiment extends GBS validation beyond collision-free detection by probabilistically resolving up to two photons per output mode. Bayesian tests identify the ideal model with high confidence after 125 events.
- Pseudo photon-number resolution uses 50:50 fiber beam splitters and SNSPDs to probabilistically resolve up to two photons per output mode.Source: Gaussian boson sampling with pseudo… · p. 19
- After 125 events, Bayesian validation gives high confidence in the ideal GBS model against thermal, coherent, distinguishable SMS, and distinguishable TMS alternatives.Source: Gaussian boson sampling with pseudo… · p. 19
Noises in Gaussian boson sampling with integrated SFWM sources
Integrated SFWM implementations of GBS face noise from losses and spurious non-degenerate emissions. The analysis quantifies how these effects depend on squeezing and system size and discusses mitigation through idler detection.
Loss noise
- Losses before the interferometer transform single-mode squeezed inputs toward thermal states, inserting noise into the output distribution.The losses are modeled by coupling each source to an ancillary vacuum mode and tracing out the ancillary modes.Source: Noises in Gaussian boson sampling with… · p. 20
- For a 12-mode interferometer, the protocol shows good resilience to pre-interferometer losses, while noise increases with photon number, source number, and squeezing.Source: Noises in Gaussian boson sampling with… · p. 20
Spurious-emission noise
- Spurious non-degenerate SFWM can produce erroneous samples when undetected photons from two-mode squeezing imitate photons expected from an ideal single-mode squeezed state.Source: Noises in Gaussian boson sampling with… · p. 20
Signal-to-noise ratio
- At low squeezing, the SNR decreases rapidly with photon number; at higher squeezing, the cosh4k(ξ) factor becomes dominant and restores good SNR.Source: Noises in Gaussian boson sampling with… · p. 22
Mitigation
- Detecting idler photons and discarding events with additional idler detections can in principle make the SNR arbitrarily high.Practical limits would still constrain the achievable SNR.Source: Noises in Gaussian boson sampling with… · p. 22
Franck-Condon profiles
The paper maps molecular vibrational transitions to photonic detection patterns, allowing Franck–Condon profiles to be simulated with displaced squeezed states and linear-optical networks. It also reverses this mapping to benchmark experimentally reconstructed profiles, while identifying fidelity limits from finite squeezing, losses, and photon-number truncation.
Photonic representation
- Franck–Condon factors are probabilities of detecting molecular excitation patterns after linearly evolving a squeezed coherent state.This photon–phonon analogy supports photonic simulation of molecular vibrational transition profiles.Source: Franck-Condon profiles · pp. 22–23
- Photonic simulation samples excitation patterns from displaced squeezed states evolved through a linear-optical network, with photon numbers not fixed as in standard GBS.The resulting samples approximate the Franck–Condon profile across vibrational transitions.Source: Franck-Condon profiles · pp. 23–24
Benchmarking
- The experiment’s device can be mapped to a synthetic molecule, using its transfer matrix as UL and characterized squeezing values as ξi.This reverse protocol constructs a corresponding molecular benchmark, although the mapping yields a set rather than a unique synthetic molecule.Source: Franck-Condon profiles · pp. 23–24
- With low source squeezing, the reconstructed profile had FQ > 99% fidelity but only C = 0.4% quantum enhancement because classical vacuum contributions dominated.Higher quantum enhancements require molecules involving higher squeezing values.Source: Franck-Condon profiles · p. 24
Limitations
- The post-processing reconstruction is faithful only when ideal contributions above the measured n-photon cutoff are negligible, generally requiring η and γ close to unity at large n.The approach is therefore not scalable, though it can benefit near-term quantum devices.Source: Franck-Condon profiles · pp. 24–25
Estimation of photon number scaling with current silicon photonics technology
The analysis estimates photon-event rates for silicon photonic boson-sampling circuits using experimentally characterised component parameters and scaling scenarios. Losses eventually offset the combinatorial benefits of SBS and GBS, but the estimates indicate feasible multi-photon experiments before rates become impractical.
Parameters and scaling assumptions
- The estimates use measured silicon-device parameters, including ηdet = 80%, ηch = 64%, ηu = 99.95%, and R0 = 500 MHz.They consider low-efficiency spiral sources with ξ = 0.17 and more efficient integrated ring-resonator sources.Source: Estimation of photon number scaling… · p. 25
- Scaling estimates extend to interferometers with up to 1000 modes and 1000 signal detectors, assuming sources, signal modes, and detectors scale together.The analysis uses ring sources and integrated detectors for this larger-scale scenario.Source: Estimation of photon number scaling… · p. 25
- A 100-mode interferometer requires 100 signal detectors for GBS and 200 total detectors for SBS, including 100 idler detectors.The configuration assumes m = k = 100 and is described as realistic for near-term silicon photonics.Source: Estimation of photon number scaling… · p. 25, Table S1 · p. 26
Event-rate scaling
- Increasing the number of sources initially improves event rates but becomes detrimental in oversized circuits because interferometer losses dominate.These losses suppress the combinatorial enhancement of scattershot and Gaussian boson sampling.Source: Estimation of photon number scaling… · p. 26
- GBS reaches the 1 event/week threshold at approximately 70 signal photons, whereas SBS reaches it at approximately 48 signal photons.The event rate decreases exponentially with photon number, yet these photon numbers are estimated to approach the tractability limit of classical supercomputers.Source: Estimation of photon number scaling… · p. 26
Protocol comparison
- Compared with SBS and GBS, standard boson sampling using a time-demultiplexed quantum-dot source is estimated to have significantly lower event rates.The comparison includes photon-demultiplexing losses modeled with ηdemux = ⌈ηswitch log2 n⌉n and ηswitch = 99.5%.Source: Estimation of photon number scaling… · pp. 26–27