Implementing universal nonadiabatic holonomic quantum gates with transmons

Geometric phases are well known to be noise resilient in quantum evolutions and operations. Holonomic quantum gates provide us with a robust way towards universal quantum computation, as these quantum gates are actually induced by non-Abelian geometric phases. Here we propose and elaborate how to efficiently implement universal nonadiabatic holonomic quantum gates on simpler superconducting circuits, with a single transmon serving as a qubit. In our proposal, an arbitrary single-qubit holonomic gate can be realized in a single-loop scenario by varying the amplitudes and phase difference of two microwave fields resonantly coupled to a transmon, while nontrivial two-qubit holonomic gates may be generated with a transmission-line resonator being simultaneously coupled to the two target transmons in an effective resonant way. Moreover, our scenario may readily be scaled up to a two-dimensional lattice configuration, which is able to support large scalable quantum computation, paving the way for practically implementing universal nonadiabatic holonomic quantum computation with superconducting circuits.

Figure 1

Illustration of the proposed scheme. (a) The coupling configuration for the single-qubit gates with two microwave fields resonantly coupled to the three levels of a transmon qubit. (b) Geometric illustration of the proposed single-qubit gate. (c) For nontrivial two-qubit gates, both qubits are coupled a TLR and driven by microwave fields, which induced effective resonant qubit-TLR coupling. (d) In this single-excitation subspace, the effective coupling configuration for the two qubits and the TLR for nontrivial two-qubit gates. (e) Scale-up of our scheme, where the transmon qubits and TLRs are denoted by filled circles and bonds, respectively.

Figure 2

State population and fidelity dynamics of different $(\gamma ,\theta )$ gates, (a) $(\pi ,\pi /2)$ and (b) $(\pi /2,\pi /2)$, as a function of ${\mathrm{\Omega }}_{N/E}t/\pi$ with the initial state being $|g\rangle$. (c) Dynamics of the gate fidelities. (d) The stability of the not gate to certain random fluctuations $\epsilon \mathrm{\Omega }$.

Figure 3

State population and fidelity dynamics for gates as a function of time with the initial state being $|f0g\rangle$.

Figure 4

Illustration of the ac Stark shift to be compensated for fixed $g$ with respect to $\mathrm{\Omega }$.

Figure 5

Illustration of the effective transmon-TLR coupling strength with respect to $\mathrm{\Omega }$, with fixed $g$.