Discovering approximate quantum error correcting codes via similarity renormalization group flow

A quantum error-correcting code embeds lower-dimensional logical qubits into a higher-dimensional physical space via an encoding map, and the physically encoded logical states satisfy the Knill-Laflamme error correction criteria. Given a physical noise channel, its error effects can be used to construct an Hermitian block matrix. This matrix can serve as the input Hamiltonian for a similarity renormalization group (SRG) flow. SRG flows unitarily transform Hamiltonians, often making them band-diagonal. In this work, we use SRG flows to block-diagonalize the effects block matrix, while simultaneously and identically conjugating each effects block. The result of the flow is a quantum error-correcting code when the conjugated effects blocks satisfy the approximate Knill-Laflamme error correction criteria.