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Conditions for the existence of quantum error correction codes

Daniel Mark Lucas

Published Jan 1, 2020• 0 citations
Publisher

Abstract

In this dissertation, we consider quantum errors, represented by quantum channels defined on the space of n x n complex matrices, M(n). We begin with a quantum channel T defined on M(n). The Kraus operators of this channel can be used to generate an operator system S, which is guaranteed to have a basis of Hermitian matrices A_1,...,A_m in M(n). Given natural numbers k and m, we find a lower bound on n above which there is always an n x k matrix U with orthonormal columns such that U*A_1U,...,U*A_mU are diagonal k x k matrices. We then extend this to a lower bound on n above which there is always a k-dimensional quantum error correction code for any quantum channel T on M(n) with an associated operator system S with dim(S) < m+2. Such a bound is equivalent to a lower bound on n above which the joint rank-k numerical range of any m Hermitian matrices in M(n) is non-empty. So, we further extend our result to a lower bound on n above which the rank-k numerical range of any m Hermitian matrices in M(n) is star-shaped.

DOI
10.31274/etd-20200902-93

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In this dissertation, we consider quantum errors, represented by quantum channels defined on the space of n x n complex matrices, M(n). We begin with a quantum channel T defined on M(n). The Kraus operators of this channel can be used to generate an operator system S, which is guaranteed to have a basis of Hermitian matrices A_1,...,A_m in M(n). Given natural numbers k and m, we find a lower bound on n above which there is always an n x k matrix U with orthonormal columns such that UA_1U,...,UA_mU are diagonal k x k matrices. We then extend this to a lower bound on n above which there is always a k-dimensional quantum error correction code for any quantum channel T on M(n) with an associated operator system S with dim(S) < m+2. Such a bound is equivalent to a lower bound on n above which the joint rank-k numerical range of any m Hermitian matrices in M(n) is non-empty. So, we further extend our result to a lower bound on n above which the rank-k numerical range of any m Hermitian matrices in M(n) is star-shaped.
For a noisy quantum channel, a quantum error correcting code exists if and only if the joint higher rank numerical ranges associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint higher rank numerical ranges are obtained and their implications to quantum computing are discussed. It is shown that if the dimension of the underlying Hilbert space of the quantum states is sufficiently large, the joint higher rank numerical range of operators is always star-shaped and contains a non-empty convex subset. In case the operators are infinite dimensional, the joint infinite rank numerical range of the operators is a convex set closely related to the joint essential numerical ranges of the operators.
We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.
We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty, and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.
The rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. For a noisy quantum channel, a quantum error correcting code of dimension k exists, if and only if the associated joint rank-k numerical range is non-empty. In this paper, the no- tion of joint rank-k numerical range is generalized, and some state- ments of (2011, Generalized numerical ranges and quantum error correction, J. Operator Theory, 66: 2, 335-351.) are extended.
Approximate quantum error correction (AQEC) extends the framework of discrete- and continuous-variable quantum error correction beyond the Knill-Laflamme (KL) conditions, where the recovery performance is quantified by entanglement fidelity. Recent studies have enabled efficient evaluation of near-optimal entanglement fidelity using transpose-channel recovery. Yet, determining the global optimal recovery map and its entanglement fidelity for general codes beyond the KL conditions remains a major computational challenge. Direct optimization becomes prohibitive as the number of noise Kraus operators grows rapidly with system size, and existing approaches lack rigorous guarantees for reducing this optimization to a tractable dimension. Here, we derive an explicit characterization of the optimal environmental state of complement channel, which transforms the optimization over recovery channels into an equivalent optimization over quotient unitaries. For a broader class of codes that satisfy only the orthogonality part of the KL conditions, we show that the optimal recovery map admits an explicit analytical form. Building on this form, we derive novel rigorous lower bounds of entanglement fidelity that strictly improve upon the transpose-recovery bound. We further develop a novel recovery strategy based on principle components, and derive a rigorous bound on the error introduced by noise truncation. Our approach enables efficient searches for approximate recovery maps for AQEC codes, avoiding the need to optimize over the full Kraus-operator space.