Implementing BosonSampling with time-bin encoding: Analysis of loss, mode mismatch, and time jitter

It was recently shown by Motes, Gilchrist, Dowling, and Rohde [Phys. Rev. Lett. 113, 120501 (2014)] that a time-bin encoded fiber-loop architecture can implement an arbitrary passive linear optics transformation. This was shown in the case of an ideal scheme whereby the architecture has no sources of error. In any realistic implementation, however, physical errors are present, which corrupt the output of the transformation. We investigate the dominant sources of error in this architecture—loss and mode mismatch—and consider how it affects the BosonSampling protocol, a key application for passive linear optics. For our loss analysis we consider two major components that contribute to loss—fiber and switches—and calculate how this affects the success probability and fidelity of the device. Interestingly, we find that errors due to loss are not uniform (unique to time-bin encoding), which asymmetrically biases the implemented unitary. Thus loss necessarily limits the class of unitaries that may be implemented, and therefore future implementations must prioritize minimizing loss rates if arbitrary unitaries are to be implemented. Our formalism for mode mismatch is generalized to account for various phenomenon that may cause mode mismatch, but we focus on two—errors in fiber-loop lengths and time jitter of the photon source. These results provide a guideline for how well future experimental implementations might perform in light of these error mechanisms.

Figure 1

Complete fiber-loop architecture fed by a pulse train of photonic modes, each separated in time by $\tau$. The squares represent optical switches. A single length $\tau$ inner fiber loop is embedded inside a length $>m\tau$ outer fiber loop. The outer loop allows an arbitrary number of inner loops to be applied consecutively. When $m-1$ inner loops are implemented this architecture realizes an arbitrary unitary transformation on $m$ modes given that no loss is present.

Figure 2

Lossy inner fiber loop fed by a pulse train of photonic modes, each separated in time by $\tau$. We model the loss of the loop with a beamsplitter of reflectivity ${\eta }_f$ and the loss of the switch with an efficiency ${\eta }_s$. Each mode experiences different amounts of loss, i.e., the first mode traverses the loop up to $m$ times, the second up to $m-1$ times, …, and the $m\mathrm{th}$ mode at most once.

Figure 3

Full architecture which implements the lossy transformation ${\widehat{U}}^{\prime }$. Each mode experiences ${{\eta }_f}^m$ loss per outer loop since they each take time $m\tau$ to traverse the outer loop. For an arbitrary unitary to be implemented in the ideal case the photons will traverse the outer loop $L-1$ times. This yields a net fiber loss from the outer loop of ${{\eta }_f}^{m(L-1)}$ that can be factored out of ${\widehat{U}}^{\prime }$, since it affects all paths equally. The net switch loss from the outer two switches is ${{\eta }_s}^{2(L-1)}$ and can also be factored out of ${\widehat{U}}^{\prime }$. The losses within the inner loop, on the other hand, affect different paths differently, and in general cannot be factored out.

Figure 4

(a) Similarity $\mathcal{S}$ vs mode or photon number $m=n$ and loop efficiency ${\eta }_f$ for $m-1$ loops. The map remains similar to the uniform unitary for low loss rates, implying that nontrivial unitary transformations may be implemented. $m-1$ loops are considered because this is the number of loops required to implement an arbitrary unitary transformation in the lossless case. (b) Postselection probability ${\mathcal{P}}_{\mathrm{S}}$ vs mode or photon number $m=n$ and loop efficiency ${\eta }_f$ for $m-1$ loops. These two plots are related in that each point in ${\mathcal{P}}_{\mathrm{S}}$ was calculated from the switching sequence ${\widehat{U}}^{\prime }$ corresponding to that which maximizes $\mathcal{S}$. In both (a) and (b) the data was averaged over 1750 Monte Carlo iterations and we let ${\eta }_s=1$, i.e., the switches are ideal but the fibers are not.

Figure 5

(a) Similarity $\mathcal{S}$, and (b) postselection probability ${\mathcal{P}}_{\mathrm{S}}$ vs loop efficiency ${\eta }_f$ and switch efficiency ${\eta }_s$ with $m=3$ modes, one photon per input mode, and $m-1$ loops. These two plots are again related in that ${\mathcal{P}}_{\mathrm{S}}$ is calculated from the switching sequence that maximizes $\mathcal{S}$. This data was averaged over 1750 iterations.

Figure 6

Three regions we consider in the mode-mismatch formalism. Region $\mathcal{A}$ corresponds to the modes coming from the source, region $\mathcal{B}$ to the modes inside the inner loop, and region $\mathcal{C}$ to the modes exiting the loop.

Figure 7

Inner fiber loop with an error $\delta$ in its intended length. Every time a pulse traverses the inner loop it is shifted from its expected temporal position by $\delta$, thereby reducing the Hong-Ou-Mandel visibility at $U_{\mathrm{BS}}$.

Figure 8

(a) Average fidelity $\mathcal{F}$ between the ideal state $|{\mathrm{\Psi }}_{\mathrm{i}}\rangle$ and the actual experimental state $|{\mathrm{\Psi }}_{\mathrm{a}}\rangle$ vs the error in the intended length of the inner loop $\delta$. (b) The worst (bottom) and best (top) case fidelity $\mathcal{F}$ between the ideal state $|{\mathrm{\Psi }}_{\mathrm{i}}\rangle$ and the actual experimental state $|{\mathrm{\Psi }}_{\mathrm{a}}\rangle$ vs the error in the intended length of the inner loop $\delta$ and number of modes $m$. In (a) and (b) there are $m$ modes with one photon per mode and the data was obtained over 250 implementations each with a unique randomly generated unitary.

Figure 9

(a) Fidelity $\mathcal{F}$ between the ideal state $|{\mathrm{\Psi }}_i\rangle$ and the actual experimental state $|{\mathrm{\Psi }}_a\rangle$ with random time jitter in the input source vs modes $m$ and standard deviation $\sigma$ with no fiber length error $\delta =0$. (b) The worst (bottom) and best (top) case fidelity $\mathcal{F}$ between the ideal state $|{\mathrm{\Psi }}_{\mathrm{i}}\rangle$ and the actual experimental state $|{\mathrm{\Psi }}_{\mathrm{a}}\rangle$ with time jitter. In (a) and (b) there is one photon per mode, the data was averaged over 250 implementations each with a unique randomly generated unitary, and the time jitter was drawn from the normal distribution.