Physical optimization of quantum error correction circuits

Quantum error-correcting codes have been developed to protect a quantum computer from decoherence due to a noisy environment. In this paper, we present two methods for optimizing the physical implementation of such error correction schemes. First, we discuss an optimal quantum circuit implementation of the smallest error-correcting code (the three bit code). Quantum circuits are physically implemented by serial pulses, i.e., by switching on and off external parameters in the Hamiltonian one after another. In contrast to this, we introduce a parallel switching method which allows faster gate operation by switching all external parameters simultaneously, and which has potential applications for arbitrary quantum computer architectures. We apply both serial and parallel switching to electron spins in coupled quantum dots subject to a Heisenberg coupling $H=J\left(t\right)\mathrm{}{\mathbf{S}}_1\cdot {\mathbf{S}}_2.$ We provide a list of steps that can be implemented experimentally and used as a test for the functionality of quantum error correction.