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Studies of Statistical-Mechanical Models Related to Quantum Codes

  • Author(s): Jiang, Yi
  • Advisor(s): Pryadko, Leonid P
  • et al.
Creative Commons 'BY' version 4.0 license
Abstract

As the beginning of the age of quantum supremacy comes closer, researches have been focusing on how to harness the full power of quantum computation. Quantum states that serve as the computational basis, known as qubits, are fragile. The interaction between them and the environment may result in errors, a process named decoherence. In the classical world, redundancy is the easiest way to protect information. But unlike classical information, it's impossible to clone an arbitrary quantum state. Fortunately, it has been shown that a small number of logical qubits can be encoded into a large number of physical ones, a technique known as quantum error correcting codes (QECCs). In the study of stabilizer codes, an important family of QECCs which shares some similarities with classical linear codes, it was discovered that the probability distribution of the decoding result can be mapped to the partition functions of spin models on graphs, a concept in statistical mechanics. Here we will explore the properties of certain families of QECCs and their corresponding statistical-mechanical models.

One of the most famous examples of stabilizer codes, the toric code, has the limitation that it only encodes 2 qubits regardless of how many physical qubits are used. To overcome this limitation, hyperbolic codes were proposed, where the physical qubits are placed on the edges of a quotient graph of a hyperbolic tessellation. Here we study the corresponding Ising models on such graphs with theoretical and numerical methods, and explore their relationship to the quantum codes.

Instead of limiting ourselves to binary codes, we also consider q-ary codes where the computational basis is formed by qudits, a generalization of qubits to q-state quantum systems. We study the properties of qudit stabilizer codes not only in the cases where q is prime, which forms a Galois field, but also where q is composite, which forms a ring of integers modulo q. We find that their corresponding statistical-mechanical models are the Potts models, a q-ary generalization to Ising models. We explore the construction and parameters of such q-ary codes, and extend the known results on qubit stabilizer codes to qudit codes.

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