Optimal control of two qubits via a single cavity drive in circuit quantum electrodynamics

Optimization of the fidelity of control operations is of critical importance in the pursuit of fault-tolerant quantum computation. We apply optimal control techniques to demonstrate that a single drive via the cavity in circuit quantum electrodynamics can implement a high-fidelity two-qubit all-microwave gate that directly entangles the qubits via the mutual qubit-cavity couplings. This is performed by driving at one of the qubits' frequencies which generates a conditional two-qubit gate, but will also generate other spurious interactions. These optimal control techniques are used to find pulse shapes that can perform this two-qubit gate with high fidelity, robust against errors in the system parameters. The simulations were all performed using experimentally relevant parameters and constraints.

Figure 1

Example of a pulse shape where the pulse has been broken up into 16 piecewise constant parts over an interval of 12.5 ns each. $c\left(t\right)=\varepsilon \left(t\right)/2\pi$ gives the pulse amplitude in $\mathrm{GHz}$ and is over an interval of $200\mathrm{ns}$ total time. Each separate part can be varied to optimize over some problem. This one takes the form of a Gaussian turn on, with a flat top and a Gaussian turn-off.

Figure 2

The initial pulse sequence for the SCP algorithm (colored area), with $\mathcal{F}=0.1461$, and the optimal pulse sequence (outlined area) showing the variation from initial to final, when the pulse is broken into 16 piecewise constant parts. As is shown, the solution tends to stay close to the initial solution if a good initial guess is chosen. Here, $c\left(t\right)=\varepsilon \left(t\right)/2\pi$ is given in $\mathrm{GHz}$ while $t$ is given in $\mathrm{ns}$. This optimal pulse sequence generates the desired unitary with $\mathcal{F}=0.9945$.

Figure 3

16 pulses are applied over an interval of $\tau =12.5\mathrm{ns}$ each. The entanglement of a pure state is given by $2|ad-bc|$ for an arbitrary two-qubit state $|\psi \rangle =a|00\rangle +b|01\rangle +c|10\rangle +d|11\rangle$. The fidelity curve is given by $F=|\langle {\mathrm{\Phi }}^{-}|{\tilde{U}}^m{|+y\rangle |+y\rangle |}^2$ where ${\tilde{U}}^m|+y\rangle |+y\rangle ={\mathrm{\Pi }}_{k=1}^m\text{exp}[-iH\left(t_k\right)\tau ]|+y\rangle |+y\rangle$.

Figure 4

The initial (colored area) and optimal (outlined area) pulses when the control is split up into 100 piecewise constant parts, each over an interval of $2\mathrm{ns}$ which is close to the limit of microwave generators. In this case, the final pulse, while showing a similar overall structure to the initial pulse, has large amplitude changes between each piece. This could off resonantly drive unwanted higher-frequency terms and cause problems in actual implementation. $c\left(t\right)=\varepsilon \left(t\right)/2\pi$ is in $\mathrm{GHz}$ and $t$ is in $\mathrm{ns}$. The fidelity that this pulse generates is $\mathcal{F}=0.9940$.

Figure 5

The optimal pulse when there is no filtering (colored area) gives a fidelity of $\mathcal{F}=0.8303$ when the simulation takes into account filtering. Upon reoptimizing, a new pulse is generated (shaded area) which generates the desired unitary with $\mathcal{F}=0.9947$.

Figure 6

Fidelity against ${\omega }_a^{\left(2\right)}/2\pi$ when there is no filter in the simulation. It can be seen that the fidelity for each point $i{\mathcal{F}}_i>0.986$, and that in the range $4.83\mathrm{GHz}<{\omega }_a^{\left(2\right)}/2\pi <4.88\mathrm{GHz}$ the fidelities are all ${\mathcal{F}}_i>0.995$. Therefore, a robust pulse has been generated for the range of qubit-2 values: ${\omega }_a^{\left(2\right)}/2\pi =4.85\pm 0.5\mathrm{GHz}$.

Figure 7

Fidelity against ${\omega }_a^{\left(2\right)}/2\pi$ when the filter effect has been taken into account. In this case, a robust pulse has been generated that gives ${\mathcal{F}}_i>0.992$ for all points in the range ${\omega }_a^{\left(2\right)}/2\pi =4.85\pm 0.5\mathrm{GHz}$.

Figure 8

Fidelity against ${\omega }_a^{\left(1\right)}/2\pi$ and ${\omega }_a^{\left(2\right)}/2\pi$ when filtering is turned on in the simulation. Here, we have ${\mathcal{F}}_i>0.9875$ for all points $i$ in the ranges of ${\omega }_a^{\left(2\right)}/2\pi =4.85\pm 0.05\mathrm{GHz}, {\omega }_a^{\left(1\right)}/2\pi =4.50\pm 0.005\mathrm{GHz}$.

Figure 9

Fidelity of optimized pulses using a multilevel system Hamiltonian comprised of two Duffing oscillators each coupled to a common cavity mode with a single cavity drive. The initial pulse had the form of a flat-top Gaussian with each piecewise constant part being 2 ns long, for different total times ranging from 2 to 200 ns. For $T<100\mathrm{ns}$, the optimized pulses perform poorly, however, for all times $T>100\mathrm{ns}$ the fidelity converges to $\mathcal{F}>0.999$ and even to $\mathcal{F}>0.9999$ for certain times in this range.

Figure 10

Variation in the fidelity $\mathcal{F}$, with changing ${\omega }_a^{\left(2\right)}$ for a pulse given without taking into account an error range in this parameter during the optimization. The area of interest is highlighted by the red rectangle: at the ideal parameter, with ${\omega }_a^{\left(2\right)}/2\pi =4.85\mathrm{GHz}, \mathcal{F}>0.9999$, but the fidelity rapidly decreases as the value moves away from the optimal.

Figure 11

Fidelity against ${\omega }_a^{\left(2\right)}/2\pi$ for an optimized pulse using the robust methods on a multilevel system comprised of two Duffing oscillators each coupled to a common cavity mode with a single cavity drive. Here, ${\mathcal{F}}_i=0.9937$ for all points $i$ in the range ${\omega }_a^{\left(2\right)}/2\pi =4.85\pm 0.05\mathrm{GHz}$, which is the range that has been optimized for. It can be seen that outside of this range, the fidelity falls off rapidly.

Figure 12

Fidelity against ${\omega }_a^{\left(2\right)}/2\pi , {\omega }_a^{\left(1\right)}/2\pi$ for an optimized pulse using the robust methods on a multilevel system comprised of two Duffing oscillators each coupled to a common cavity mode with a single cavity drive. Here, ${\mathcal{F}}_i=0.9639$ for all points $i$ in the range ${\omega }_a^{\left(2\right)}/2\pi =4.85\pm 0.05\mathrm{GHz}, {\omega }_a^{\left(1\right)}/2\pi =4.50\pm 0.005\mathrm{GHz}$, which the pulse has been optimized for. Outside of this range, the fidelity falls off rapidly.