Encoding a Qubit in a Qudit for fault tolerant quantum computation

We construct a class of quantum error-correcting cat codes for atomic spin systems, similar in spirit to the cat codes used for a bosonic mode, by encoding a logical qubit in a spin qudit. We study how a repetition code based on the cat codes in the spin systems can be used for fault-tolerant quantum computation and we develop a universal set of gates that respects the error set. One key ingredient is the development of a bias-preserving CNOT gate protocol suitable for errors in the spins systems. Using the well-known Rydberg blockade for entanglement in neutral atom quantum computing, we construct a bias-preserving CNOT gate. We categorize the errors as phase errors and amplitude errors. The phase errors are analogous to phase-flip errors for qubits and can be corrected in a similar way by constructively measuring the syndrome followed by an X-gate. To correct amplitude errors, we use the higher dimensional nature of the qudit, couple the data qubits with ancilla, and use a swap gate to swap the states. Following this, a destructive measurement of the data qubits determines the syndrome for amplitude errors. We also study the fault-tolerant threshold for error correction by studying the errors on the CNOT gates; the thresholds are close to the error-biased cat qubits [1]. Thus, our results provide a path to fault tolerance with fewer qubits by using the extra available internal degrees of freedom in atoms.

[1] Shruti Puri, Lucas St-Jean, Jonathan A Gross, Alexander Grimm, Nicholas E Frattini, Pavithran S Iyer, Anirudh Krishna, Steven Touzard, Liang Jiang, Alexandre Blais, et al., “Bias-preserving gates with stabilized cat qubits,” Science advances 6, eaay5901 (2020)