Optimal control of circuit quantum electrodynamics in one and two dimensions

Optimal control can be used to significantly improve multi-qubit gates in quantum information processing hardware architectures based on superconducting circuit quantum electrodynamics. We apply this approach not only to dispersive gates of two qubits inside a cavity, but, more generally, to architectures based on two-dimensional (2D) arrays of cavities and qubits. For high-fidelity gate operations, simultaneous evolutions of controls and couplings in the two coupling dimensions of cavity grids are shown to be significantly faster than conventional sequential implementations. Even under experimentally realistic conditions speedups by a factor of three can be gained. The methods immediately scale to large grids and indirect gates between arbitrary pairs of qubits on the grid. They are anticipated to be paradigmatic for 2D arrays and lattices of controllable qubits.

Figure 1

(Color online) The superconducting cavity grid, (Ref. 20) with two layers of vertical (bottom) and horizontal (top layer) transmission line resonators, coupled to qubits (small red squares). Two-qubit gates between qubits 1 and 3 are mediated indirectly via qubit 2, employing the dispersive interaction inside the two highlighted resonators.

Figure 2

Analytical pulse sequences for time-optimal implementations of two-qubit gates: (a) the CNOT gate, where 1 is the control qubit and 2 is the target qubit and (b) the SWAP gate.

Figure 3

The standard decomposition of an indirect two-qubit gate into direct two-qubit gates via the SWAP operation.

Figure 4

(a) Maximum achievable fidelity as a function of pulse duration in the three-qubit idealized model for an ${\text{i}\mathrm{SWAP}}_{13}$ gate. (b) On a logarithmic scale we observe a sharp convergence to the threshold fidelity of $1–{10}^{-5}$, where the algorithm terminates.

Figure 5

(Color online) Logarithmic negativity between qubit pairs 1–2 (blue line) and 2–3 (red, dashed line) for (a) sequential and (b) optimized implementations of an ${\text{i}\mathrm{SWAP}}_{13}$.

Figure 6

(Color online) Maximum fidelity as a function of pulse duration in the two-qubit realistic model for (a) a CNOT gate, and (b) a SWAP gate. Maxima obtained with no restrictions on the controls are shown in blue (●), while those obtained under the restrictions in Eq. (4) are shown in red (○). The arrows indicate the minimal times at which the threshold fidelity is achieved.

Figure 7

(Color online) Sample controls obtained by the GRAPE algorithm for the minimal-time implementation of a CNOT in the two-qubit realistic model: ${\Delta }_1$ (red, dotted line), ${\Delta }_2$ (green, dashed line), and $\Omega$ (blue, solid line) with (a) unrestricted controls, and (b) restricted controls. Note that the time course of the controls in (a) is perfectly palindromic as the controls may equally well be read forward and backward in time.

Figure 8

(Color online) Maximum fidelity as a function of pulse duration in the three-qubit realistic model for (a) a ${\mathrm{CNOT}}_{13}$ gate, and (b) an ${\text{i}\mathrm{SWAP}}_{13}$ gate. Maxima obtained with no restrictions on the controls are shown in blue (●), while those obtained under the restrictions in Eq. (4) are shown in red (○). The arrows indicate the minimal times at which the threshold fidelity is achieved.

Figure 9

(Color online) Sample controls to implement an ${\text{i}\mathrm{SWAP}}_{13}$ gate in the three-qubit realistic model with restrictions (ii) in place: (a) ${\Omega }_{12}^x$ (green, dashed line), ${\Omega }_{23}^x$ (blue, solid line). (b) ${\Omega }_1^z$ (blue, solid line), ${\Omega }_2^z$ (green, dashed line), and ${\Omega }_3^z$ (red, dotted line).