Error suppression in Hamiltonian based quantum computation using energy penalties

We consider the use of quantum error detecting codes, together with energy penalties against leaving the codespace, as a method for suppressing environmentally induced errors in Hamiltonian based quantum computation. This method was introduced in [1] in the context of quantum adiabatic computation, but we consider it more generally. Specifically, we consider a computational Hamiltonian, which has been encoded using the logical qubits of a single-qubit error detecting code, coupled to an environment of qubits by interaction terms that act one-locally on the system. Energy penalty terms are added that penalize states outside of the codespace. We prove that in the limit of infinitely large penalties, one-local errors are completely suppressed, and we derive some bounds for the finite penalty case. Our proof technique involves exact integration of the Schrodinger equation, making no use of master equations or their assumptions. We perform long time numerical simulations on a small (one logical qubit) computational system coupled to an environment and the results suggest that the energy penalty method achieves even greater protection than our bounds indicate.