Encoding One Logical Qubit Into Six Physical Qubits

          @misc{ pirsa_08070001,
            doi = {10.48660/08070001},
            url = {https://pirsa.org/08070001},
            author = {Shaw, Bilal},
            keywords = {Quantum Information},
            language = {en},
            title = {Encoding One Logical Qubit Into Six Physical Qubits},
            publisher = {Perimeter Institute},
            year = {2008},
            month = {jul},
            note = {PIRSA:08070001 see, \url{https://pirsa.org}}
          }
          

Abstract

We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuits, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver\'s half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender\'s side. We discuss the advantages and disadvantages for each of the two codes.