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What is the code distance in quantum information theory? Code distance seems to be a very important concept in fault tolerant quantum computation and topological quantum computation.

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Xiao-Gang Wen
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  • Since this is way out of my field of expertise I will not presume to answer, but the link provided in another question http://www.qip2010.ethz.ch/tutorialprogram/JiannisPachosLecture defines also the code distance as "The distance k of the code C is the minimal length among the elements of Z(S)\S up to a sign. Such elements serve as encoded logical operations. For an efficient encoding we thus assume that the errors are less than [k/2]-local." – anna v Jun 03 '12 at 11:36
  • Thanks for the link. The code distance suppose to be a distance between two objects. But what are these two objects? – Xiao-Gang Wen Jun 03 '12 at 11:53
  • still wading: at 2.1.1 "A general k-local operator O is an operator that acts non-trivially to at most k subsystems of H (also known as operator of length k)." seems to define length for a local operator. So one has to count the subsystems of H on which the operator acts non trivially to get its length, I guess. – anna v Jun 03 '12 at 13:28

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Rather than give a more mathematical answer, to which I'll refer you to another answer, let me give you slightly less precise explanation. Basically, the distance is the shortest path in a certain "space of errors" which maps between two orthogonal quantum states that are in the code. The natural space of errors is that of single qubit errors of the form $\sigma_X$, $\sigma_Y$ or $\sigma_z$, in the case where the Hilbert space is that of $n$ qubits. So you can think of distance as the shortest path to get from one state to another by operations on single qubits, applied one at a time sequentially.

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Steve Flammia
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    If two states can be connected by a sequence of $\sigma_X$, $\sigma_Y$, and $\sigma_Z$ operations, then the code distance is the minimal number of the single qubit operations that transform one state into the other. But if two states can NOT be connected by a sequence of $\sigma_X$, $\sigma_Y$, and $\sigma_Z$ operations, can we still define the code distance? – Xiao-Gang Wen Jun 06 '12 at 02:29
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    The code distance can be defined in terms of the weight of the correctible Kraus operators using the Knill-Laflamme quantum error correction criterion. For example, a rigorous definition in this framework can be found in Page 2 of the reference: http://www.sciencedirect.com/science/article/pii/S0024379517303956?via%3Dihub or https://arxiv.org/abs/1604.07925 – Yingkai Ouyang Oct 30 '17 at 04:30