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Quantum Information Processing

An optimal quantum error-correcting procedure using quantifier elimination

Ying-Ji Sun, Ming Xu, Yuxin Deng

Published May 1, 2021• 1 citation
Publisher
DOI
10.1007/s11128-021-03109-w

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In this short notes, we present the formulation of a kind of quantum algorithm, applicable to the polynomial optimization. The present algorithm introduces the quantum effect by means of the coordinate variables, the momentum variables, and the commutators, and the Plank constant, which shall establish the correspondence between the quantum and the classical cases. The formalism is an extension to the polynomial optimization in the ``classical'' sense; the relations among quantum mechanical values are represented by a semi-algebraic set ( by the set of polynomials and sign conditions), and the optimum is searched in this semi-algebraic set. In the context of this formalism, the optimization of the Ising model [QUBO model] will be a special case. However, the computational procedure proposed here is different from the quantum annealing which is now in vogue. We demonstrate the exemplary calculation through quantifier elimination (which is a kind of symbolic computation applicable to optimization and other problems).
We develop a theory for finding quantum error correction (QEC) procedures which are optimized for given noise channels. Our theory accounts for uncertainties in the noise channel, against which our QEC procedures are robust. We demonstrate, via numerical examples, that our optimized QEC procedures always achieve a higher channel fidelity than the standard error correction method, which is agnostic about the specifics of the channel. This demonstrates the importance of channel characterization before QEC procedures are applied. Our main novel finding is that in the setting of a known noise channel the recovery ancillas are redundant for optimized quantum error correction. We show this using a general rank minimization heuristic and supporting numerical calculations. Therefore, one can further improve the fidelity by utilizing all the available ancillas in the encoding block.