In quantum error correction, CSS codes, named after their inventors, Robert Calderbank,

Peter Shor Peter Williston Shor (born August 14, 1959) is an American professor of applied mathematics at MIT. He is known for his work on quantum computation, in particular for devising Shor's algorithm, a quantum algorithm for factoring exponentially fa ...

and

Andrew Steane Andrew Martin Steane is Professor of physics at the University of Oxford. He is also a fellow of Exeter College, Oxford. He was a student at St Edmund Hall, Oxford where he obtained his MA and DPhil. His major works to date are on error correc ...

, are a special type of

stabilizer code The theory of quantum error correction plays a prominent role in the practical realization and engineering of quantum computing and quantum communication devices. The first quantum error-correcting codes are strikingly similar to classical block ...

constructed from classical codes with some special properties. An example of a CSS code is the

Steane code The Steane code is a tool in quantum error correction introduced by Andrew Steane in 1996. It is a CSS code (Calderbank-Shor-Steane), using the classical binary ,4,3Hamming code to correct for qubit flip errors (X errors) and the dual (mathema ...

Construction

Let

C_1

and

C_2

be two (classical)

,k_1 /math>,,k_2 /math> codes such, that C_2 \subset C_1 and C_1 , C_2^\perp both have minimal distance \geq 2t+1, where C_2^\perp is the code dual to C_2 . Then define \text(C_1,C_2), the CSS code of C_1 over C_2 as an,k_1 - k_2, d /math> code, with d \geq 2t+1 as follows: 

Define for x \in C_1 :  x + C_2 \rangle  := 1 / \sqrt \sum_  x + y \rangle, where + is bitwise addition modulo 2. Then \text(C_1,C_2) is defined as \ .

References

External links

Linear algebra Quantum information science {{Quantum-stub