Pasting quantum codes

I describe a method for pasting together certain quantum error-correcting codes that correct one error to make a single larger one-error quantum code. I show how to construct codes encoding 7 qubits in 13 qubits using the method, as well as 15 qubits in 21 qubits and all the other “perfect” codes. 03.65.Bz,89.80.+h Typeset using REVTEX [email protected] 1 Quantum computers have a great deal of promise, but they are likely to be inherently much noisier than classical computers. One approach to dealing with noise and decoherence in quantum computers and quantum communications is to encode the data using a quantum error-correcting code. A number of such codes and classes of codes are known [1–8]. However, the only known method of automatically generating such codes is to find a suitable classical error-correcting code and convert it into a quantum code [2,3]. This method is limited to producing less efficient codes (i.e., with smaller ratio of encoded qubits to total qubits) than dedicated quantum codes, so a method of automatically producing highly efficient quantum codes is desirable. I will present here a method to create one-error quantum codes from smaller ones with almost no effort. The conditions for a set of n-qubit states |ψ1〉, . . . , |ψ2k〉 to form an error-correcting code for the errors Ea is 〈ψi|E † aEb|ψj〉 = Cabδij , (1) where Cab is independent of i and j [5,9]. A code with 2 k states encodes k qubits. Typically, a code will be designed to correct all possible errors affecting less than or equal to t qubits. The basis errors Ea are usually tensor products of I = 