Simulating and assessing boson sampling experiments with phase-space representations

The search for new, application-specific quantum computers designed to outperform any classical computer is driven by the ending of Moore's law and the quantum advantages potentially obtainable. Photonic networks are promising examples, with experimental demonstrations and potential for obtaining a quantum computer to solve problems believed classically impossible. This introduces a challenge: how does one design or understand such photonic networks? One must be able to calculate observables using general methods capable of treating arbitrary inputs, dissipation, and noise. We develop complex phase-space software for simulating these photonic networks, and apply this to boson sampling experiments. Our techniques give sampling errors orders of magnitude lower than experimental correlation measurements for the same number of samples. We show that these techniques remove systematic errors in previous algorithms for estimating correlations, with large improvements in errors in some cases. In addition, we obtain a scalable channel-combination strategy for assessment of boson sampling devices.

Figure 1

Channel diagram of a boson sampling photonic network gedanken experiment.

Figure 2

Contour integral for $z$, where $\alpha =rz$. Here $z$ is chosen randomly with unit modulus for sampling purposes, to give a Monte Carlo sampled contour integral.

Figure 3

Actual error of the Gurvits method (solid blue lines) and the QCP representation with $d=2$ (dashed orange lines) and $d\to \infty$ (dotted green lines) for the modulus squared of the permanent, relative to the value of the permanent squared, as a function of the total number of samples $L=L_1{L}_2$. For each point we have used $N_{\mathrm{m}}=100$ random unitary matrices and $L_2=1000$ subensembles of size (a) $L_1={10}^4\cdots {10}^6$ and (b) $L_1={10}^6\cdots {10}^8$. Here we consider $k=4$, (a) $N=10$ and (b) $N=20$. Dash-dotted gray lines denote the estimated experimental error $\sqrt{{\langle P\rangle }_{\mathrm{e},\mathrm{m}}/L}$.

Figure 4

Actual error of the Gurvits method (solid blue lines) and the QCP representation with $d\to \infty$ (dashed orange lines) for the modulus squared of the permanent, expressed in the units of the sampling error, as a function of the total number of samples $L$. Graphs have $N=10$ and $N=20$, respectively, with $k=4$. The dotted line indicates a $3\sigma$ confidence interval. Other parameters are the same as in Fig. 3.

Figure 5

Estimation of the error for unitary-averaged coincidence rate ${\overline{P}}_{N|kN}$ as a function of $N$ using the QCP method with a random discrete phase $d\to \infty$. For each point we have used $N_{\mathrm{m}}=100$ random unitary matrices and $L_2=100$ ensembles of $L_1={10}^5$ samples each. Dashed orange line corresponds to the average for the exact value of $P_{N|kN}$. Solid blue line is the average error $\epsilon$, compared to the exact value. Dash-dotted green line denotes the estimated experimental error $\sqrt{{\langle P\rangle }_{\mathrm{e},\mathrm{m}}/L}$. This is always greater than the simulation error, for the same number of samples. Here we consider (a) $k=1$, (b) $k=2$, (c) $k=6$, and (d) $k=10$.

Figure 6

A schematic of the combined correlations strategy. Photodetectors on output channels are indicated by triangles. Counter outputs must be binary or passed through a step function filter. Thus, channel counts are always 0 or 1, even for multiphoton events, which give total counts less than $N$. As a result, the final output counts all events with exactly $N$ photons in $N$ distinct channels. This sums over all possible channel combinations.

Figure 7

A schematic of the channel deleted combined correlations strategy. One or more channels are deleted randomly by switching off their counters. Therefore, only the events with $N$ counts in $N$ distinct channels, excluding deleted channels, are counted. This gives a high total count rate. At the same time, it provides a successively more detailed fingerprint of the unitary. This becomes effectively unique as the number of deleted channels is increased.

Figure 8

Combined correlations $C_{N|M}^{\left(M\right)}$ given in Eq. (5.3) evaluated using the QCP method. Here we have used $d\to \infty$, $k=6$, $N_{\mathrm{m}}=1$, and $L_2={10}^4$ ensembles of $L_1={10}^6$ samples. The dotted gray line is the analytical result given in Eq. (5.4), where $N=M/k$. The dashed blue line corresponds to our numerical results. The solid blue line is the average estimate of the error in the mean $E_S={\langle \sqrt{\langle C^2\rangle -{\langle C\rangle }^2}\rangle }_{\mathrm{m}}/\sqrt{L_2}$. The dash-dotted orange line is the estimate of the error for the case of $d=2$.

Figure 9

Modulus of the difference between the numerically obtained combined correlation, and the conjecture $C_{N|M}^{\left(M\right),\mathrm{conj}}$ (dashed orange line) plotted against the error estimate from Fig. 8, normalized on $C_{N|M}^{\left(M\right),\mathrm{conj}}$ (solid blue line). All other parameters as in Fig. 8.

Figure 10

Standard deviation of the analytic conjecture $C_{N|M}^{\left(M\right),\mathrm{conj}}$ for ${10}^4$ random matrices (dashed orange line) plotted against its mean value (solid blue line), showing that there is a substantial variation between unitaries even for relatively large $N$. All other parameters as in Fig. 8.