A major obstacle towards realizing a practical quantum computer is the `noise’ that arises due to system-environment interactions. While quantum error correction (QEC) provides a way to protect against errors that arise due to the noise affecting the system, a `perfect’ quantum code requires at least five physical qubits to protect against single-qubit noise. In cases where the noise structure in the system is already known, it might be more useful to consider quantum codes that are adapted to specific noise models. Such codes are known to be resource efficient and perform on par with the standard codes. With this motivation, we address the following questions concerning channel-adapted quantum codes.




a) Construction: Given a noise model, we propose a simple and fast numerical optimization algorithm to search for good quantum codes. Specifically, we use the Cartan form of the encoding unitaries to search over non-local parts of the encoding space, thereby reducing the number of search parameters. We also provide simple quantum circuits which can implement the optimal codes obtained via our search.




b) Application: We propose an adaptive QEC protocol that allows transmission of quantum information from one site to another across a 1-d spin chain with high fidelity. We obtain numerical and analytical results, both for ideal and disordered, provided the nearest neighbor interaction is governed bya total spin-preserving Hamiltonian.




c) Fault tolerance: Finally, we address the question of whether such noise-adapted QEC protocols can be made fault-tolerant. We show by explicit construction, that it is possible to obtain a fault-tolerant, universal gate set and a fault-tolerant error correction unit, for a 4-qubit code adapted to protect against amplitude damping noise. We show that our gadgets admit reliable quantum computing so long as the threshold error-rate does not exceed 10^{-4}.