Quantum computational advantage with a programmable photonic processor

Abstract

A quantum computer attains computational advantage when outperforming the best classical computers running the best-known algorithms on well-defined tasks. No photonic machine offering programmability over all its quantum gates has demonstrated quantum computational advantage: previous machines^1,2 were largely restricted to static gate sequences. Earlier photonic demonstrations were also vulnerable to spoofing³, in which classical heuristics produce samples, without direct simulation, lying closer to the ideal distribution than do samples from the quantum hardware. Here we report quantum computational advantage using Borealis, a photonic processor offering dynamic programmability on all gates implemented. We carry out Gaussian boson sampling⁴ (GBS) on 216 squeezed modes entangled with three-dimensional connectivity⁵, using a time-multiplexed and photon-number-resolving architecture. On average, it would take more than 9,000 years for the best available algorithms and supercomputers to produce, using exact methods, a single sample from the programmed distribution, whereas Borealis requires only 36 μs. This runtime advantage is over 50 million times as extreme as that reported from earlier photonic machines. Ours constitutes a very large GBS experiment, registering events with up to 219 photons and a mean photon number of 125. This work is a critical milestone on the path to a practical quantum computer, validating key technological features of photonics as a platform for this goal.

Fig. 1. High-dimensional GBS from a fully programmable photonic processor.

A periodic pulse train of single-mode squeezed states from a pulsed OPO enters a sequence of three dynamically programmable loop-based interferometers. Each loop contains a VBS, including a programmable phase shifter, and an optical fibre delay line. At the output of the interferometer, the Gaussian state is sent to a 1-to-16 binary switch tree (demux), which partially demultiplexes the output before readout by PNRs. The resulting detected sequence of 216 photon numbers, in approximately 36 μs, comprises one sample. The fibre delays and accompanying beamsplitters and phase shifters implement gates between both temporally adjacent and distant modes, enabling high-dimensional connectivity in the quantum circuit. Above each loop stage is depicted a lattice representation of the multipartite entangled Gaussian state being progressively synthesized. The first stage (τ) effects two-mode programmable gates (green edges) between nearest-neighbour modes in one dimension, whereas the second (6 τ) and third (36 τ) mediate couplings between modes separated by six and 36 time bins in the second and third dimensions (red and blue edges, respectively). Each run of the device involves the specification of 1,296 real parameters, corresponding to the sequence of settings for all VBS units.

Fig. 2. Experimental validation of the GBS device.

Each panel compares experimentally obtained sample probabilities, against those calculated from the ground truth (r, T), for up to six-photon events in a 16-mode state. A total of 84.1 × 10⁶ samples were collected and divided according to their total photon number N and further split according to the collision pattern, from no collision (no more than one photon detected per PNR) to collisions of different densities (more than one photon per PNR). The overall fidelity (F) and TVD to simulations for each photon-number event is shown below. Further analysis of TVD for classical adversaries in the 16-mode GBS instance can be found in the Supplementary Information.

Fig. 3. Benchmarks against the ground truth.

a, Cross-entropy benchmark against the ground truth. Experimental samples from a high-dimensional GBS instance of 216 modes, averaging $\overline{N}=21.120\pm 0.006$ photons per sample, are bundled according to their total photon number N, from 10 to 26. Each point (score) corresponds to an average (equation (1)) over 10,000 samples per N. Genuine samples from the quantum hardware score higher than all classical spoofers, validating the high device fidelity with the ground truth. Error bars are standard errors of the mean. b, Bayesian log average score against the ground truth. Experimental samples from a 72-mode GBS instance and $\overline{N}=22.416\pm 0.006$ photon number per sample. Each score is averaged over 2,000 samples with N from 10 to 26. Error bars are standard errors of the mean. All scores are above zero, including error bar, indicating that the samples generated by Borealis are closer to the ground truth than from the adversarial distribution corresponding to squashed, thermal, coherent and distinguishable squeezed spoofers.

Fig. 4. Quantum computational advantage.

a, Measured photon statistics of 10⁶ samples of a high-dimensional Gaussian state compared with those generated numerically from different hypotheses. The inset shows the same distribution in a log scale having significant support past 160 photons, up to 219. b, Scatter plot of two-mode cumulants C_ij for all the pairs of modes comparing experimentally obtained ones versus the ones predicted by four different hypotheses. A perfect hypothesis fit (shown in plot) would correspond to the experimentally obtained cumulants lying on a straight line at 45° (shown in plot). Note that the ground truth is the only one that explains the cumulants well. Moreover, to make a fair comparison all the hypothesis have exactly the same first-order cumulants (mean photon in each mode). c, Distribution of classical simulation times for each sample from this experiment, shown as Borealis in red and for Jiuzhang 2.0 in blue. For each sample of both experiments, we calculate the pair (N_c, G) and then construct a frequency histogram populating this two-dimensional space. Note that because the samples from Jiuzhang 2.0 are all threshold samples they have G = 2, whereas samples from Borealis, having collisions and being photon-number resolved, have G ≥ 2. Having plotted the density of samples for each experiment in (N_c, G) space, we indicate with a star the sample with the highest complexity in each experiment. For each experiment, the starred sample is at the very end of the distribution and occurs very rarely; for Jiuzhang 2.0 this falls within the line G = 2. Finally, we overlay lines of equal simulation time as given by equation (4) as a function of N_c and G. To guide the eye we also show boundaries delineating two standard deviations in plotted distributions (dashed lines).