Arbitrary multimode Gaussian operations on mechanical cluster states

We consider opto- and electromechanical quantum systems composed of a driven cavity mode interacting with a set of mechanical resonators. It has been proposed that the latter can be initialized in arbitrary cluster states, including universal resource states for measurement-based quantum computation (MBQC). We show that, despite the unavailability in this setup of direct measurements over the mechanical resonators, computation can still be performed to a high degree of accuracy. In particular, it is possible to indirectly implement the measurements necessary for arbitrary Gaussian MBQC by properly coupling the mechanical resonators to the cavity field and continuously monitoring the leakage of the latter. We provide a thorough theoretical analysis of the performances obtained via indirect measurements, comparing them with what is achievable when direct measurements are instead available. We show that high levels of fidelity are attainable in parameter regimes within reach of present experimental capabilities.

Figure 1

The circuit representing gate teleportation. Upper circuit: An input state $|\mathrm{\Psi }\rangle$ is linked via a control-phase gate to an ancilla state ${|0\rangle }_p$. A measurement of $p$ with outcome $m$ teleports the input state to the ancilla along with some by-product known operations [$F$ being the Fourier transform and $X\left(m\right)=e^{-\mathrm{imp}}$]. Lower circuit: If a unitary operation $U$ is carried out on the input before the entangling gate then it also will be teleported, as $U|\mathrm{\Psi }\rangle$ is simply another legitimate input state. However, if $U$ is diagonal in the computational basis then the two circuits are equivalent, since $U$ commutes with the control-phase operation and can be incorporated into the measurement by a rotation of the measurement basis. Thus the measurement itself can be used to induce the operation $U$ on an input state.

Figure 2

A linear cluster of five nodes. Before measurement commences (a) all nodes (blue with solid border) are equally linked by balanced CZ gates and each node has an equal degree of squeezing. After a sequence of measurements (b) the (red with dashed border) measured nodes are disconnected from the cluster and are discarded, leaving behind a single node (green with no border) modified by the required operation. A two-mode operation requiring a dual rail can be minimally simulated by (c) taking the four-node ancilla and manipulating it topologically.

Figure 3

A driven single-mode cavity hosts a collection of noninteracting mechanical resonators. The latter are supposed to be already prepared in a cluster state and to have nonoverlapping frequencies (this can be achieved following the proposal of Ref. [53]). They are coupled to a thermal environment with dissipation rate $\gamma$. The cavity decays at a rate $\kappa$ into a monitored mode which is eventually measured via a homodyne detector. Unmonitored losses at rate $\tau$ are also taken into account.

Figure 4

From top to bottom, effect of the variation of the detector efficiency $\eta$, unmonitored losses $\tau$, and squeezing of the postmeasurement state of the homodyned mode $r_{\text{postmeas}}$ on the highest achievable fidelity on application of the gate $S\left(1\right)$ to a 5-node linear cluster against the monitoring time per step. Except for the noise parameter under consideration all the parameters are set as per Table 1 with solid and dashed lines denoting close-to-ideal (set 2) and realistic (set 1) parameters, respectively. In each plot the curves corresponding to each set of parameters bunch together (particularly for long monitoring times). From top to bottom of each bunch the parameter under consideration is varied: (a) the detector efficiency $\eta =1,0.9,0.8$, (b) the unmonitored losses $\tau =0.01\kappa ,0.05\kappa ,0.1\kappa$, and (c) the squeezing of postmeasurement state of the homodyned mode $r_{\text{postmeas}}=20,10,5$ dB. These plots show that these three noise mechanisms are only minimally detrimental for successfully implementing Gaussian transformations on a mechanical cluster state, and that their effect is more and more negligible for long monitoring times.

Figure 5

Variation of (a) the interaction strength $\alpha g$ and (b) the cavity decay rate $\kappa$. These plots show the maximum fidelity achieved over long monitoring times using realistic parameters as per set 1 of Table 1 (except for the mechanical losses $\gamma$ whose value is indicated in the legend).

Figure 6

Applying the operations of identity $\mathbb{I}$, Fourier transform $F$, and shearing $S\left(\lambda \right)$ with $\lambda =1,3,5$ under the realistic parameters of Table 1. The final fidelity of the output of these operations is plotted against the monitoring time for each step.

Figure 7

The fidelity of the identity operation with equally spaced step intervals (left) and with optimized steps (right). Without optimization there are long periods where the fidelity is not increased by the monitoring scheme, and in fact is actively decreased by exposure to the mechanical damping. With optimization, the ceiling for the fidelity is raised and the time required to reach high fidelities is significantly lessened. These simulations also consider realistic parameters (set 1), with the exception of $T=10$ K, emphasizing the effect of the optimized monitoring steps. The optimized steps for this scenario are $21.4,21.2,21.1$, and $21.1\mu \mathrm{s}$.

Figure 8

The maximum achievable fidelity with optimized monitoring intervals for the continuous monitoring implementation of the shearing $S\left(1\right)$ and $CZ$ operations. Increased squeezing increases the susceptibility of the output to damage from the thermal environment.