Operator quantum error-correcting subsystems for self-correcting quantum memories

The most general method for encoding quantum information is not to encode the information into a subspace of a Hilbert space, but to encode information into a subsystem of a Hilbert space. Recently this notion has led to a more general notion of quantum error correction known as operator quantum error correction. In standard quantum error-correcting codes, one requires the ability to apply a procedure which exactly reverses on the error-correcting subspace any correctable error. In contrast, for operator error-correcting subsystems, the correction procedure need not undo the error which has occurred, but instead one must perform corrections only modulo the subsystem structure. This does not lead to codes which differ from subspace codes, but does lead to recovery routines which explicitly make use of the subsystem structure. Here we present two examples of such operator error-correcting subsystems. These examples are motivated by simple spatially local Hamiltonians on square and cubic lattices. In three dimensions we provide evidence, in the form a simple mean field theory, that our Hamiltonian gives rise to a system which is self-correcting. Such a system will be a natural high-temperature quantum memory, robust to noise without external intervening quantum error-correction procedures.

Figure 1

Above we have represented elements of the $\mathcal{T}$. Each set of operators enclosed in a rectangle represents a Pauli operator acting on the particular qubits tensored with the identity on all other qubits. Each of the operators enclosed in the dotted rectangles are elements of $\mathcal{T}$.

Figure 2

A nontrivial element of the group $\mathcal{S}$. This element has an even number of columns which are entirely $X$ operators multiplied by an even number of rows which are entirely $Y$ operators. Notice how the $Y$ elements appear where both of these conditions are met.

Figure 3

A nontrivial element of the set $\mathcal{L}$. This element has an odd number of columns which are entirely $X$ operators multiplied by an odd number of rows which are entirely $Y$ operators. It represents an encoded $Y$ operator on the encoded qubit as described in Sec. 4D.