Universal control of an oscillator with dispersive coupling to a qubit

We investigate quantum control of an oscillator mode off-resonantly coupled to an ancillary qubit. In the strong dispersive regime, we may drive the qubit conditioned on the number states of the oscillator, which, together with displacement operations, can achieve universal control of the oscillator. Based on our proof of universal control, we provide a straightforward recipe to perform arbitrary unitary operations on the oscillator. With the capability of universal control, we can significantly reduce the number of operations to prepare the number state $\left|n\right.〉$ from $O\left(n\right)$ to $O\left(\sqrt{n}\right)$. This universal control scheme of the oscillator enables us to efficiently manipulate the quantum information stored in the oscillator, which can be implemented using superconducting circuits.

Figure 1

Energy-level diagram of the qubit-oscillator system. In the rotating frame of the oscillator, the states ${\left\{\left|g,n\right.〉\right\}}_n$ have the same energy. After each operation, the population (orange circles) remains in the subspace associated with $\left|g\right.〉$. A weak displacement operation (red dashed arrows) couples the states $\left|g,n-1\right.〉$ and $\left|g,n\right.〉$ with strength $\sqrt{n}\varepsilon$ for all $n$. The snap gate (blue solid arrows) can simultaneously accumulate different Berry phases $\left\{{\theta }_n\right\}$ to states $\left\{\left|g,n\right.〉\right\}$. The Berry phase ${\theta }_n$ is proportional to the enclosed shaded area in the corresponding Bloch sphere, achieved by resonant microwave pulses with frequency ${\omega }_q-n\chi$ (blue oscillatory fields).

Figure 2

Plot of infidelity of the unitary ${\widehat{V}}_n$ [Eq. (10)] constructed by our protocol with respect to the desired target unitary ${\widehat{V}}_{n,\mathrm{target}}$ [Eq. (9)] vs $n$. The target unitaries are $\frac{\pi }{2}$ rotations on the $\{\left|n\right.〉,\left|n+1\right.〉\}$ subspace. In the limit of $\left|{\mathrm{\Omega }}_n\right|/\chi \to 0$, the simple implementation of Eq. (10) can already achieve fidelities $F=\frac{1}{2}\left|\text{Tr}\left({\widehat{V}}_n^{†}{\widehat{V}}_{n,\mathrm{target}}\right)\right|$ of better than $0.998$. In the calculation of the fidelity we use the $2\times 2$ submatrices of interest (acting on the $\{\left|n\right.〉,\left|n+1\right.〉\}$ subspace).

Figure 3

(a) An example target unitary operation, $U_{\mathrm{target}}$ (a permutation). (b) The product is close to identity, ${\widehat{U}}_{\mathrm{construct}}{\widehat{U}}_{\mathrm{target}}^{-1}\approx \widehat{I}$, with small nonzero off-diagonal elements. (c), (d) Comparison of fidelities $F=\frac{1}{N_c}\left|\text{Tr}\left({\widehat{U}}_{\text{construct}}^{†}{\widehat{U}}_{\text{target}}\right)\right|$ after columnwise optimization (black) and fidelities with additional optimization over all displacement parameters (red) for (c) Fourier (triangles) and permutation (squares) operations and (d) randomly generated target unitary operations (hexagons).

Figure 4

The number of snap gates for the preparation of $\left|n\right.〉$ from $\left|0\right.〉$ (with a fixed lower bound of fidelity) for a generic linear scheme (black diamond line) and specialized sublinear schemes with different target fidelities (colored lines). The specialized schemes operate on $\widehat{D}\left(\sqrt{n}\right)\left|0\right.〉$ by folding all population from the subspace $\left\{\left|n-\mathrm{\Delta }n\right.〉,\cdots ,\left|n+\mathrm{\Delta }n\right.〉\right\}$ to the number state $\left|n\right.〉$.

Figure 5

Reprise of Fig. 2 with finite values for $\mathrm{\Omega }$ (represented in units of $\chi )$. The lower panel zooms in on smaller values for $\mathrm{\Omega }$. It is possible to achieve fidelities better than $0.99$ for practical values of $\mathrm{\Omega }=0.05\chi$.