Quantum error correcting codes from the compression formalism

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We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. By solving an algebraic compression problem, we construct qubit codes for such channels on arbitrary dimension Hilbert space, and identify correctable codes for Pauli-error models not obtained by the stabilizer formalism. This is accomplished through an application of a new tool for error correction in quantum computing called the “higher-rank numerical range”. We describe its basic properties and discuss possible further applications.

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      Citation Excerpt :

      The main effort was to eliminate the error factors created during transmission of quantum information and to describe possible corruption induced in the quantum system. Motivated by a physical problem, Choi et al. in their pioneering articles [7–9], reduced this problem to a purely mathematical introducing the notion of higher rank numerical ranges, and triggering the interest of many authors leading to an extensive literature [1,2,16,17,21]. Due to the unitary invariance property (P2) of the rank-k numerical range, we may assume that A already is in the form (2.1).

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