Fig. 1. Time-bin encoding machines and Borealis structure. (a) Example of universal time-bin interferometers. In such an encoding, the optical modes are discrete time bins. The unitary operation over the modes is performed by two concatenated fiber loops. The shortest loop covers the time separation between two consecutive time bins, whereas the longest one covers the whole duration of the number of modes. The same squeezing source is excited at each pulse, and photon-counting measurements are performed at the end of the evolution. (b) Structure of Borealis.^5,58 The interferometer comprises three consecutive loops of increasing length. (c) Example of a unitary matrix Uij performed by Borealis that shows the limited connectivity among the modes due to the loop structure.

Fig. 2. Detected photons and orbit probability distributions. The top row: the distributions of the number of detected photons for the three levels of squeezing: (a) “low,” (b) “medium,” and (c) “high.” The bottom row: the distributions of the orbit configurations for the number of detected photons associated with the highest probability, namely, n=26 for low squeezing, n=62 for medium, and n=97 for high squeezing. The x axis represents the various orbits’ configuration ordered as follows, first [1,1,1,…], then [2,1,1,…], [2,2,1,…], etc. The y axis represents the number of samples for that orbit configuration. The size of each sample was 250,000 shots.

Fig. 3. Stability check of Borealis total efficiency over days. Orbit distribution for the 2 weeks of runs with the same circuit settings. The variations in the different days are due to changes in the total efficiency of the apparatus, which is mostly dominated by the variations in common efficiency. The points correspond to the three orbits’ probability for detected photons in the range between 18 and 32 photons. More precisely, each point on the plot is associated with a specific detected photon number. The points at the bottom left corner correspond to 32 photons, while the points on the right are the orbits for 18 postselected photons.

Fig. 4. Role of losses in the orbit estimation. (a), (b) Orbits for the thermal sampler simulations. All points, except the green and dark gold ones, represent orbits where only the common efficiency has been changed. The lower the efficiency is, the larger the radius of the orbits is. The green points represent the case where one of the detectors was turned off, thus producing an unbalanced loss. The dark gold points are from the lossless indistinguishable SMSV states simulation. The details on the parameters used for the simulations can be found in the Supplementary Material. The lines are obtained by making a linear fit on all points with the same number of detected photons, excluding the points of the orbit with unbalanced losses and the SMSV simulation. Each line represents a different number of photons. We highlighted the number of corresponding photons for some of the lines in this figure. The effect of balanced losses is to move points along the lines. The unbalanced ones move the points away from the lines, but keep them on the same hyperplane. However, such an effect is evident only in the case of a strong amount of imbalance. The lossless SMSV states points are not on those lines but are still on the same hyperplane. We show the projection of the orbits in the plane in (b). (c), (d) Experimental orbits from the runs on Borealis. The lines are the same as the panel above calculated from the simulation with thermal states.

Fig. 5. Validation of Borealis samples. (a), (b) Comparison of the orbits of Borealis (red), thermal states simulation (blue), squashed states simulation (orange), a second-order greedy sampler with the ideal U transformation and no loss (purple) and with the effective transfer matrix and loss (cyan), and coherent states simulation (green). The coherent, squashed, lossy greedy, and thermal states simulations used the same parameters and device certificate of the Borealis runs. All the points were calculated on a sample size of 250,000 shots with squeezing set to low. (a) Borealis data collected on different days with different U transformations. Such runs display a similar average number of detected photons and device certificates. (b) Borealis data collected on the same day with different unitary transformations. On that day, the fluctuation in the average number of detected photons between runs on Borealis was compatible with statistical fluctuations, and the deviations of the average number of detected photons predicted by the simulations according to the certificate with those measured on the device were small. (c)–(h) Comparison of the covariances of lossy thermal states, lossy squashed states, and Borealis for a given transformation U of the device. The same circuit parameters were used for the simulations and the measurements with Borealis. The same device certificate was used for all the simulations. (c) Covariance of the simulated thermal sampler with losses. (d) Covariance of the thermal sampler with losses calculated from 250,000 samples, to reproduce the additional noise due to a limited number of samples. (e) Simulated covariance of a squashed light sampler with the losses of the Borealis circuit. (f) Covariance of squashed light sampler from a finite sample of 250,000 shots. (g) Simulated covariance of an SMSV GBS with losses. (h) Covariance of Borealis samples.

Fig. 6. Two-point correlators. Scatter plot of two-point correlators Cij of samples collected from Borealis (experiment) as a function of the corresponding values calculated with different possible models for the device (simulation). Comparison with a (a) thermal sampler, (b) squashed state sampler, and (c) lossy SMSV state GBS. Black dashed line: the expected trend for the ground truth, corresponding to a linear function with a slope equal to 1. Colored dashed lines: trends obtained by a linear fit of the data shown in each figure.

Fig. 7. Orbits of the data from the Borealis experiment of Ref. 5. Red, the Borealis data; blue and orange, the simulated data from the thermal and squashed samplers, respectively. The size of the sample was ∼106. The points correspond to the number of detected photons from 18 (from the right) to 32 (to the left). The Borealis data are those of Fig. 3(a) of Ref. 5. The lines are those we obtained from the thermal simulations in Fig. 4.