CSS code: Difference between revisions

Content deleted Content added

Inline

Revision as of 09:43, 7 October 2020

In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of Stabilizer codes constructed from classical codes with some special properties. An example of a CSS code is the Steane code.

Construction

Let $C_{1}$ and $C_{2}$ be two (classical) $[n,k_{1}]$ , $[n,k_{2}]$ codes such, that $C_{2}^{\perp }\subset C_{1}$ 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 ${\text{CSS}}(C_{1},C_{2})$ , the CSS code of $C_{1}$ over $C_{2}$ as an $[n,k_{1}-k_{2},d]$ code, with $d\geq 2t+1$ as follows:

Define for $x\in C_{1}:{|}x+C_{2}\rangle :=$ $1/{\sqrt {{|}C_{2}{|}}}$ $\sum _{y\in C_{2}}{|}x+y\rangle$ , where $+$ is bitwise addition modulo 2. Then ${\text{CSS}}(C_{1},C_{2})$ is defined as $\{{|}x+C_{2}\rangle \mid x\in C_{1}\}$ .

References

Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 844974180.

External links

This quantum mechanics-related article is a stub. You can help Wikipedia by expanding it.

@@ Line 4: / Line 4: @@
 == Construction ==
-Let <math>C_1</math> and <math>C_2</math> be two (classical) <math> [n,k_1]</math>, <math> [n,k_2]</math> codes such, that <math> C_2 \subset C_1 </math> and <math> C_1 , C_2^\perp</math> both have [[Block code|minimal distance]] <math> \geq 2t+1</math>, where <math> C_2^\perp</math> is the code [[Dual code|dual]] to <math> C_2</math>. Then define <math> \text{CSS}(C_1,C_2)</math>, the CSS code of <math> C_1</math> over <math> C_2</math> as an <math> [n,k_1 - k_2, d]</math> code, with <math> d \geq 2t+1 </math> as follows:
+Let <math>C_1</math> and <math>C_2</math> be two (classical) <math> [n,k_1]</math>, <math> [n,k_2]</math> codes such, that <math> C_2^\perp \subset C_1 </math> and <math> C_1 , C_2^\perp</math> both have [[Block code|minimal distance]] <math> \geq 2t+1</math>, where <math> C_2^\perp</math> is the code [[Dual code|dual]] to <math> C_2</math>. Then define <math> \text{CSS}(C_1,C_2)</math>, the CSS code of <math> C_1</math> over <math> C_2</math> as an <math> [n,k_1 - k_2, d]</math> code, with <math> d \geq 2t+1 </math> as follows:
 Define for <math> x \in C_1 : {{|}} x + C_2 \rangle  := </math> <math> 1 / \sqrt{ {{|}} C_2 {{|}} } </math> <math> \sum_{y \in C_2} {{|}} x + y \rangle</math>, where <math> + </math> is bitwise addition modulo 2. Then <math> \text{CSS}(C_1,C_2) </math> is defined as <math> \{ {{|}} x + C_2 \rangle \mid x \in C_1 \} </math>.