For How Long Is It Possible To Quantum Compute? Mucciolo, Eduardo
One of the key problems in quantum information processing is to understand the physical limits to quantum computation. Several strategies have been proposed to attenuate errors caused by the interaction of the computer with its surrounding environment and quantum error correction (QEC) is likely the most versatile. A large effort has been devoted to proving that resilience can be achieved by concatenating QEC codes in logical structures. In our work we look into this question at a different angle: we provide an upper bound on the time available to computation given a certain computer, a QEC code, and a decohering environment. We consider a broad class of environments, including those where correlation effects can be induced by gapless modes. Our approach is based on a Hamiltonian formulation where we use coarse graining in time to derive an explicit quantum evolution operator for the logical qubits, taking into account the QEC code. We show that this evolution operator has the same form as that for the original physical qubits, except for a reduced coupling to the environment which can be evaluated systematically for a given geometry and QEC code structure. To quantify the effectiveness of QEC, we compute the trace distance between the real and ideal states of a logical qubit after an arbitrary number of QEC cycles. We derive expressions for the long-time trace distance for several for super-ohmic-, ohmic-, and sub-ohmic-like baths. Given a confidence threshold for the trace distance, we establish the maximum time available for computation in those three cases. This maximum time is controlled by an exponent related to the spatial dimensions and other characteristics of the computer and the environment. For the super-ohmic regime, we find that computation can continue indefinitely, while in the other regimes the maximum time depends strongly on the QEC code, on the number of logical qubits, and on the original environment-computer strength interaction.
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