The Bacon–Shor code is a subsystem

error correcting code In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is ...

. In a subsystem code, information is encoded in a

subsystem A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and expressed ...

of a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

. Subsystem codes lend to simplified error correcting procedures unlike codes which encode information in the subspace of a Hilbert space. This simplicity led to the first claim of fault tolerant circuit demonstration on a quantum computer. It is named after Dave Bacon and

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 ...

. Given the stabilizer generators of Shor's code:

\langle X_X_X_X_X_X_, X_X_X_X_X_X_, Z_Z_, Z_Z_, Z_Z_, Z_Z_, Z_Z_, Z_Z_\rangle

, 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get:

\langle X_X_X_X_X_X_, X_X_X_X_X_X_, Z_Z_Z_Z_Z_Z_, Z_Z_Z_Z_Z_Z_ \rangle

. Error correction is now simplified because 4 stabilizers are needed to measure errors instead of 8. A

gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...

can be created from the stabilizer generators:

\langle Z_Z_, X_X_, Z_Z_, X_X_, Z_Z_, X_X_, Z_Z_, X_X_, X_X_, X_X_, Z_Z_, Z_Z_\rangle

. Given that the Bacon–Shor code is defined on a square lattice where the qubits are placed on the vertices; laying the qubits on a grid in a way that corresponds to the gauge group shows how only 2 qubit nearest-neighbor measurements are needed to infer the error syndromes. The simplicity of deducing the syndromes reduces the overheard for fault tolerant error correction.

    ZZ   ZZ 
  q0---q1--q2
XX,   XX,    , XX
  ,   ZZ,  ZZ, 
  q6--q7--q8
XX,   XX,    , XX
  ,     ,    , 
  q3--q4--q5
   ZZ   ZZ

References

{{DEFAULTSORT:Bacon-Shor code Quantum computing

See also

References