Scalable nonadiabatic holonomic quantum computation on a superconducting qubit lattice

Geometric phase is an indispensable element for achieving robust and high-fidelity quantum gates due to its built-in noise-resilience feature. However, due to the complexity of manipulation and the intrinsic leakage of the encoded quantum information to a non-logical-qubit basis, the experimental realization of universal nonadiabatic holonomic quantum computation is very difficult. Here we propose to implement scalable nonadiabatic holonomic quantum computation with a decoherence-free subspace encoding on a two-dimensional square superconducting transmon-qubit lattice, where only the two-body interaction of neighboring qubits, from the simplest capacitive coupling, is needed. Meanwhile, we introduce qubit-frequency driving to achieve tunable resonant coupling for the neighboring transmon qubits, and thus avoid the leakage problem. In addition, our presented numerical simulation shows that high-fidelity quantum gates can be obtained, verifying the advantages of the robustness and scalability of our scheme. Therefore, our scheme provides a promising way towards the physical implementation of robust and scalable quantum computation.

Figure 1

Our proposed setup. (a) A scalable 2D square lattice, where each ellipse denotes a DFS encoded logical qubit, which consists of a pair of transmons with different frequencies connected by an auxiliary transmon, the corresponding circuits are shown in (b). (c) Energy spectrum for two parametrically tunable coupled transmons, where single- and two-excitation subspaces can be used to achieve single- and two-logical-qubit holonomic gates, respectively.

Figure 2

The gate fidelities as functions of the qubit frequency differences ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, the results of the not and Hadamard gates are shown in (a) and (b), respectively, with the same color bar. State populations and the state fidelity dynamics of the not (c) and Hadamard gate (d) operations with the initial state being ${|0\rangle }_{\text{a}}{|0\rangle }_L$.

Figure 3

(a) The gate fidelities as functions of the qubit frequency differences ${\mathrm{\Delta }}_3$ and ${\mathrm{\Delta }}_4$, for the controlled-not holonomic gate. State populations and the state fidelity dynamics of the controlled-not (b) and controlled-phase (c) gates with the initial state being $\frac{1}{\sqrt{2}}{\left(\right|00\rangle }_L+{|11\rangle }_L)$.

Figure 4

Gate fidelities with respect to the different uniform decoherence rate $\kappa$.