Fault-tolerant quantum computation in spin systems using cat codes

In this work, we construct a class of quantum error-correcting codes for spin systems using the cat codes which is similar in spirit to the cat codes for the bosonic mode. We studied how concatenation of codes gives us the possibility of fault-tolerant quantum computation, for doing fault-tolerant quantum computation we developed a full set of universal gates which respects the error set, one key ingredient was the development of bias preserving CNOT gate which respects the error set. We categorized the errors as phase errors and amplitude errors, the phase errors are very similar to phase flip errors for qubits and can be corrected in a similar way. To correct amplitude errors, we could use the higher dimensional nature of our qubit and use a SWAP gate approach followed by destructive measurement of the data qubits. We also studied the threshold needed to correct for errors including the measurement errors and found that we could get a reasonable threshold including errors for the CNOT gate and errors in the measurement of X.