# Encoding One Logical Qubit Into Six Physical Qubits

### APA

Shaw, B. (2008). Encoding One Logical Qubit Into Six Physical Qubits. Perimeter Institute. https://pirsa.org/08070001

### MLA

Shaw, Bilal. Encoding One Logical Qubit Into Six Physical Qubits. Perimeter Institute, Jul. 09, 2008, https://pirsa.org/08070001

### BibTex

@misc{ pirsa_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}} }

University of Southern California

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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.