Fault-Tolerant Encoding of Logical Qudits in Spin Systems

Abstract: The universal quantum computer will enable the simulation of arbitrary quantum states governed by arbitrary Hamiltonians. In this context, it is essential to equip future quantum processors with fault-tolerant logical qudits, since qudits naturally align with the simulation of multi-level physical systems. In this study, we present a general framework and working examples of fault-tolerant logical qudit encoding using spin systems, which are among the most coherent and robust finite multi-level physical platforms. The d-dimensional logical qudit encoding with distance-3 (or 5) codewords can be designed within a 12d (or 40d)-dimensional Hilbert space, and the design can be further generalized to 2t+1-distance codes and to encodings exploiting multiple physical qudits. A quantitative comparison shows that the logical qudit encoding proposed here offers an exponential resource advantage over multi-level mappings from logical qubits, and therefore we believe this strategy can pave the way for realizing logical qudit encodings in finite multi-level physical systems.