Fast binomial-code holonomic quantum computation with ultrastrong light-matter coupling

We propose a protocol for bosonic binomial-code nonadiabatic holonomic quantum computation in system composed of an artificial atom ultrastrongly coupled to cavity resonator. In our protocol, the binomial codes, formed by superpositions Fock states, can greatly save physical resources correct errors computation. apply strong driving fields designed shortcuts-to-adiabatic methods. This reduces gate time tens nanoseconds. Decoherence accumulated over be effectively reduced such fast evolution. Noise induced control imperfections suppressed systematic-error-sensitivity nullification method, thus is mostly insensitive noises. As result, this rapidly ($\sim 35$ ns) generate fault-tolerant and high-fidelity ($\gtrsim 98\%$ with experimentally realistic parameters) gates.