Decoherence and interferometric sensitivity of boson sampling in superconducting resonator networks

Multiple bosons undergoing coherent evolution in a coupled network of sites constitute a so-called quantum walk system. The simplest example of such a two-particle interference is the celebrated Hong-Ou-Mandel interference. When scaling to larger boson numbers, simulating the exact distribution of bosons has been shown, under reasonable assumptions, to be exponentially hard. We analyze the feasibility and expected performance of a globally connected superconducting resonator based quantum walk system, using the known characteristics of state-of-the-art components. We simulate the sensitivity of such a system to decay processes and to perturbations and compare with coherent input states.

Figure 1

A scheme for implementing BosonSampling. The resonators are loaded from qubits (here represented as the green schematic xmons [21]), and their final state is dispersively measured from readout resonators (magenta) at the desired time when qubits are brought back into resonance with their respective resonators (blue). The resonators are capacitively coupled to each other (black waveguides) with capacitance $C_{ij}$, and this graph is made possible by air bridges [22] (arcs on control lines and graph coupling lines) that add a negligible and known capacitive coupling between lines.

Figure 2

Normalized variance of the occupation probabilities for a single boson in the network $Var\left(〈P_i\left(t\right)〉\right)M$ for $M=10$ (red line for $M=250$) randomly and globally connected resonators. The horizontal line ($\frac{1}{M}$ for $M=10$) intersects the blue curve at $T_{\mathrm{rich}}$. Note that the normalized variance can rise again over $1/M$ as there is no decay in this simulation. For 250 resonators the richness time is reduced to $\sim 5$ $\mathrm{ns}$.

Figure 3

Distribution distance, $\mathrm{\Delta }(\rho ,\sigma )$ for a decohering (with relaxation $T_1=50\mu s$ and dephasing $T_{\varphi }=50\mu s$ processes) quantum walk of $N=3$ bosons in a 10 site resonator network, simulated up to $T_f=25\mathrm{ns}$. In each panel plot we simulate the outcome of varying one of the above default values and compare to the phenomenological Eq. (3). Top left: varying $T_1$, top right: varying $T_{\varphi }$, bottom left: varying $T_f$, bottom right: varying $N$.

Figure 4

Histogram of the distribution distances, $\mathrm{\Delta }$, for initial single photons in resonators and coherent states. The simulation was carried out in a 10 site network, with an ensemble of perturbations, and $N=3$ bosons. The same perturbations were applied for both kinds of initial states. For the coherent state a classical “amplitude simulation” was performed, with $\alpha =1$ in three resonators in the initial state.

Figure 5

The distribution overlap, $\mathcal{S}$ as defined in Eq. (4) vs $N$, the number of bosons, decreases, as $N$ grows. Inserts visualize $\mathcal{S}$ and this effect as the shaded region for $N=1$ and $N=5$.