Phase-Programmable Gaussian Boson Sampling Using Stimulated Squeezed Light

We report phase-programmable Gaussian boson sampling (GBS) which produces up to 113 photon detection events out of a 144-mode photonic circuit. A new high-brightness and scalable quantum light source is developed, exploring the idea of stimulated emission of squeezed photons, which has simultaneously near-unity purity and efficiency. This GBS is programmable by tuning the phase of the input squeezed states. The obtained samples are efficiently validated by inferring from computationally friendly subsystems, which rules out hypotheses including distinguishable photons and thermal states. We show that our GBS experiment passes a nonclassicality test based on inequality constraints, and we reveal nontrivial genuine high-order correlations in the GBS samples, which are evidence of robustness against possible classical simulation schemes. This photonic quantum computer, Jiuzhang 2.0, yields a Hilbert space dimension up to $\sim {10}^{43}$, and a sampling rate $\sim {10}^{24}$ faster than using brute-force simulation on classical supercomputers.

Figure 1

The key experimental parameters. (a) Squeezed light amplification by stimulated parametric down-conversion. Both the pump and parametric photons are refocused by the concave mirror back to the PPKTP for stimulated emission. The down-converted photons are polarization exchanged by the QWP and its relative phase with pump light are tuned by the wedge phase plate. The squeezed light passed a KTP crystal for birefringence compensation and an antireflection coated silicon plate for filtering out residual pumping light, before being collected into the single-mode fiber. (b) The purities of the 25 squeezed light sources with an average of 0.961. The purity is defined as photon indistinguishability, that is, the overlap of the wave function of two independent photons. (c) The collection efficiencies—defined as the ratio of generated photons collected in the single mode fiber—of the 25 squeezed light sources with an average of 0.918. (d) An overview of the GBS setup. 25 pairs of two-mode squeezed photons are sent into a 144-mode interferometer, and the output distribution is readout by 144 single-photon detectors. (e) Phase stability of the whole setup. The histogram shows that the phase fluctuations are within 0.06 rad in an hour.

Figure 2

Validation of GBS. (a) Photon clicks distribution at different pump power and focal waists. (b) Output photon-click distribution of the experiment (red), theoretical prediction (orange), thermal state (blue), and distinguishable photons (purple). The experimental GBS data perfectly overlap with the theoretical curve, while strongly deviates from the other three mockups. (c) Bayesian test against the hypothesis of distinguishable photons with subsystem mode number ranging from 100 to 144. For clarity, we added an artificial offset to the Bayesian counter in order to make the different dataset clearly visible. (d) The calculated Bayesian confidence $\mathrm{\Delta }H$ as a function of the subsystem mode number. For all the hypotheses, it is clear that the validation confidence is stronger for a larger mode number. (e)–(h) The experimental results of Bayesian confidence $\mathrm{\Delta }H$ at different laser pump power and focal waists for the validations against the uniform samplers (e), coherent state (f), distinguishable photons (g), and thermal state (h). The color coding is the same as panel (a). Error bars represent 1 standard deviation of uncertainty. From (e) to (h), the $Y$ axis scale is different for different mockups. The validation appears to be easier for uniform sampler (e) and harder for thermal state sampler (h).

Figure 3

Truncated correlation functions of the GBS machine. (a) Nonclassicality test of experiment as proposed in Ref. [45]. The colored areas correspond to the classically simulable regimes, while our experiment results (red dots) are well above color area. Here, the classically simulable means that the experimentally generated state and the measurement operator can be written as $t$-ordered phase-space quasi-probability distributions without negative probabilities and therefore the experiment can be simulate efficiently [45]. (b)–(e) are the truncated first- to fourth-order correlation functions of calculated theoretical values ($y$ axis) and extracted experimental results ($x$ axis), respectively. The results clearly indicate that our data have genuine correlations. (f) Extracted $p$ values of Spearman’s rank-order tests. The vertical axis is $-\ln (p)/L$, where $L$ is the number of $k$-order correlations. The red dots are from our experimental dataset within 200 s, which drops exponentially when the correlation order linearly increases. By fitting, we extrapolate that there is up to correlations of $19\pm 1$ order in our experiment for $p<0.05$. The blue dots are from a hypothetic dataset within 100 h. Using the same method, we estimate that correlations of $43\pm 7$ order can be determined.

Figure 4

The programmability of the GBS machine. (a) 30 different groups of phase settings of all 25 TMSSs. All phase curves are stacked with a $2\pi$ offset in the vertical axis direction. (b) Total variance distances between experimental two-point correlation functions with different groups of TMSS phases. The average of diagonal elements is 0.068(1), which represents the statistical fluctuating noise due to finite samples. The average of off-diagonal elements is 0.32(1), represent successfully distinguishing samples with 30 groups of TMSS phases. (c) Normalized mean and coefficient of variation of the distributions of two-point correlation functions, including experimental samples (red square), theoretical predictions (blue dot), thermal state (black triangle) mockup and distinguishable photons (purple triangle) mockup. Each point is from one setting of the TMSS phases.

Figure 5

Dimension of Hilbert space and quantum advantage ratio. (a) Summary of the output Hilbert spaces of boson sampling and random circuit sampling. This work (red star) has a Hilbert space dimension of $\sim {10}^{43}$. (b) The quantum advantage ratio (compared to direct simulation with classical computers) as a function of the maximally detected photon clicks. Based on brute-force simulation, a tentative quantum advantage ratio of ${10}^{24}$ is observed in this work.