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Bulletin of the American Physical Society

Continuous Quantum Error Correction with two-qubit Annealing Hamiltonian

Arman Babakhani, C. H. Herman Chan, Jeffrey M. Epstein, Song Zhang, Juan Atalaya, K. Birgitta Whaley

Published Mar 6, 2019• 0 citations
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We develop a protocol for continuous operation of a quantum error correcting code for protection of coherent evolution due to an encoded Hamiltonian against environmental errors, using the three-qubit bit-flip code and bit-flip errors as a canonical example. To detect errors in real time, we filter the output signals from continuous measurement of the error syndrome operators and use a double thresholding protocol for error diagnosis, while correction of errors is done as in the conventional operation. We optimize our continuous operation protocol for evolution under quantum memory and under quantum annealing, by maximizing the fidelity between the target and actual logical states at a specified final time. In the case of quantum memory, we show that our continuous operation protocol yields a logical error rate that is slightly larger than the one obtained from using the optimal Wonham filter for error diagnosis while being simpler to implement. For quantum annealing, we show that our continuous quantum error correction protocol can significantly reduce the final logical state infidelity when the continuous measurements are sufficiently strong relative to the strength of the time-dependent Hamiltonian. We also show that this continuous quantum error correction protocol can reduce the time-to-solution relative to the value obtained from a classical parallelization scheme. These results suggest that a continuous implementation is suitable for quantum error correction in the presence of encoded time-dependent Hamiltonians, opening the possibility of many applications in quantum simulation and quantum annealing.
Abstract We present a general error-correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. Given any Ising model optimization problem, the encoding replaces each logical qubit by a complete graph of degree C, representing the distance of the error-correcting code. A subsequent minor-embedding step then implements the encoding on the underlying hardware graph of the quantum annealer. We demonstrate experimentally that the performance of a D-Wave Two quantum annealing device improves as C grows. We show that the performance improvement can be interpreted as arising from an effective increase in the energy scale of the problem Hamiltonian or, equivalently, an effective reduction in the temperature at which the device operates. The number C thus allows us to control the amount of protection against thermal and control errors, and, in particular, to trade qubits for a lower effective temperature that scales as C−η, with η⩽2. This effective temperature reduction is an important step towards scalable quantum annealing.
We study the effect of continuous quantum error correction in the case where each qubit in a codeword is subject to a general Hamiltonian interaction with an independent bath. We first consider the scheme in the case of a trivial single-qubit code, which provides useful insights into the workings of continuous error correction and the difference between Markovian and non-Markovian decoherence. We then study the model of a bit-flip code with each qubit coupled to an independent bath qubit and subject to continuous correction, and find its solution. We show that for sufficiently large error-correction rates, the encoded state approximately follows an evolution of the type of a single decohering qubit, but with an effectively decreased coupling constant. The factor by which the coupling constant is decreased scales quadratically with the error-correction rate. This is compared to the case of Markovian noise, where the decoherence rate is effectively decreased by a factor which scales only linearly with the rate of error correction. The quadratic enhancement depends on the existence of a Zeno regime in the Hamiltonian evolution which is absent in purely Markovian dynamics. We analyze the range of validity of this result and identify two relevant time scales. Finally, we extend the result to more general codes and argue that the performance of continuous error correction will exhibit the same qualitative characteristics.