CSS code

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. 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. Let C 1 {displaystyle C_{1}} and C 2 {displaystyle C_{2}} be two (classical) [ n , k 1 ] {displaystyle } , [ n , k 2 ] {displaystyle } codes such, that C 2 ⊂ C 1 {displaystyle C_{2}subset C_{1}} and C 1 , C 2 ⊥ {displaystyle C_{1},C_{2}^{perp }} both have minimal distance ≥ 2 t + 1 {displaystyle geq 2t+1} , where C 2 ⊥ {displaystyle C_{2}^{perp }} is the code dual to C 2 {displaystyle C_{2}} . Then define CSS ( C 1 , C 2 ) {displaystyle { ext{CSS}}(C_{1},C_{2})} , the CSS code of C 1 {displaystyle C_{1}} over C 2 {displaystyle C_{2}} as an [ n , k 1 − k 2 , d ] {displaystyle } code, with d ≥ 2 t + 1 {displaystyle dgeq 2t+1} as follows: Define for x ∈ C 1 : | x + C 2 ⟩ := {displaystyle xin C_{1}:{|}x+C_{2} angle :=} 1 / | C 2 | {displaystyle 1/{sqrt {{|}C_{2}{|}}}} ∑ y ∈ C 2 | x + y ⟩ {displaystyle sum _{yin C_{2}}{|}x+y angle } , where + {displaystyle +} is bitwise addition modulo 2. Then CSS ( C 1 , C 2 ) {displaystyle { ext{CSS}}(C_{1},C_{2})} is defined as { | x + C 2 ⟩ ∣ x ∈ C 1 } {displaystyle {{|}x+C_{2} angle mid xin C_{1}}} . 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..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:''''''''''''}.mw-parser-output .citation .cs1-lock-free a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}