Fault-Tolerant Encoding of Logical Qudits in Spin Systems

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.