Holonomic Quantum Computation in Surface Codes

It is well known that surface codes are favorable candidates toward realizing large-scale fault-tolerant quantum computation. While methods to reduce errors by surface codes have been well explored, it is still challenging to achieve small-enough errors as stated by the threshold theorem in practical experimental settings. This prompts the integration of surface codes and protected quantum operations to further mitigate the negative effect of errors, creating the challenge of executing protected surface codes. In this work, we propose one scheme to perform holonomic quantum operations in a physical system by use of auxiliary qubits. The system Hamiltonian describes a quantum spin model with adjustable coupling strength. Various noncommutable single-qubit and nontrivial two-qubit holonomic gates are applicable on the basis of the interaction between the auxiliary qubits and physical qubits. By considering a two-dimensional lattice of qubits, we explore the execution of surface codes. The utilization rate of physical qubits as data qubits is about $24.7\mathrm{%}$ in executing the protected surface codes. Our scheme is based on an experimentally achievable Hamiltonian and thus it brings us closer to realizing protected quantum operations on quantum-error-correction codes with a modest resource.