Daniel Cano

The present study analyzes the efficiency of employing quantum error correction codes (QECC) to encode quantum information states in the context of Quantum Key Distribution (QKD). Specifically, the possibility of enhancing the security and reliability of QKD systems by adding a secondary layer of quantum coding to the states emitted by Alice in the \textit{Prepare-and-Measure} protocols is exhaustively quantified. Such an encoding scheme would be expected to be achievable by means of quantum hardware potentially available in the mid-term. This last statement refers to the assumed reasonable interconnectivity and scalability requirements that may be imposed on the physical encoding capabilities of a quantum processor for the case here considered, since only 1-qubit states are used in QKD. The model for quantum states transmission here considered does not impose any restrictions on the quantum channel, but does assume that the noise and errors to which qubits may be subject in QKD links can be characterized by discrete transformations. That is, depending on the physical encoding scheme chosen for photon's polarization, errors such as bit-flip or phase-shift errors (among others) can be corrected through logical gates derived from Pauli operators, which, along with identity, form the basis $\{I,X,Y,Z\}$ for 1-qubit discrete error operators of the form: \begin{equation} E = \left(\begin{array}{cc} \alpha_0 & \alpha_1\\ \alpha_2 & \alpha_3 \end{array}\right) \end{equation} \medskip Such a consideration imposes the need to be able to identify and correct up to a total of $k = 1 + 3n$ different types of errors (including no error at all, bit-flip, phase-shift, and combinations of the previous) that may affect a QKD state (encoded in an $n-$qubit physical state). In line with previous scalability arguments, that requires for the number of physical qubits needed to achieve such encoding to be lower bounded by the product of the previous magnitude and the dimension of the quantum code $C$ used (which, in the context of QKD, shall be $\mathrm{dim}(C) = 2$). Therefore, if $m=1$ is the number of qubits to be encoded for each transmitted state in QKD, the condition: \begin{equation} 2^n \geq dim(C)(1+3n) \end{equation} imposes a a minimum of $n=5$ physical qubits in a quantum algorithm to carry out encoding and correction of a 1-qubit quantum state. However, it should be noted that, beyond the anticipated error types, the efficiency of identifying errors in a key distilled through QKD (i.e., for all purposes, the correctable QBER associated with each transmission) will be all the more efficient the greater the number n of physical qubits available in a processor for such encoding (of the order of $2^{n-1}$). Thus, the minimum requirements of the quantum hardware topology for the feasibility of this type of encoding are specified, as well as the optimal trade-off in terms of the assumable QBER against different types of attacks, supported by future advances in quantum processor scalability. In this sense, beyond the security considerations associated with QKD implementations of this nature, the goal of this analysis is to parallelly discern the potential speed-up of employing quantum algorithms to carry out error correction of QKD keys and their potential superiority over classical error correction processes in the future. In this regard, two types of QECC are tested in this work. On the one hand, the widespread use of low-density parity check (LDPC)-type linear codes (being linearity a requirement that quantum error correction codes must necessarily satisfy) naturally leads to considering their use in Quantum CSS (Calderbank, Shor \& Steane) codes. The efficiency and performance benefits of LDPC codes applied to QKD are therefore as well transferable to a quantum processor in this context. The performance of these codes applied in QKD is contrasted, secondly, with stabilizer codes. It can be anticipated that the latter may present challenges in the initial algorithm for the encoding of the states emitted by Alice, however the decoding circuit algorithms can be implemented with relative simplicity -albeit scalability limitations- through 1-qubit logical gates (such is also the case with CSS codes once the parity matrix of the LDPC code is known, whose speed advantages over the classical use of belief-propagation algorithms are showed here). On the previous precepts, this study focuses on carrying out a comparative analysis of the convenience of potentially benefitting from the performance of either type of code, while analyzing technical considerations derived from the experimental implementation of QECC protocols in this QKD hybrid approach. The most important considerations are the following: -Complexity. From the point of view of reliability of these types of implementations, potential disadvantages are analyzed in terms of complexity added to real physical systems. Not only is the experimental complexity increase of combining quantum hardware with QKD optical transmissions estimated, but also the anticipation of additional error sources, considering the acceptable threshold values of decoding techniques and calibration errors for real applications and security proofs. - Efficiency. In terms of efficiency and overall code performance, estimated times (for different prospective states of quantum processor advancement) for quantum key generation through these techniques are simulated, and the circumstances under which each may be most convenient are identified. - Components demands. Increased demand for quality of the optics involved in the QKD protocol is expected. Protocols of this nature further increase the demand for high-quality transmissions, especially regarding photon sources, which may have a significant impact on both implementability and its associated costs. - Overhead. Additional overhead needs are projected in terms of code design, number of qubits required depending on the use case, as well as measurement operators necessary for error detection and correction. Consequently, partial limits have been found on the amount of data that can be transmitted in a QKD system that integrates this methodology, which is projected to be overcome when widely available quantum hardware reaches sufficient maturity. - Side channel attacks. Possible vulnerabilities to quantum hacking are preliminarily identified, and a testing method is suggested for this type of QECC-based QKD systems. In addition to the previous analyses, the authors note that one of the most significant features of -both of- the codes here analyzed is that they carry out the identification of errors that affect quantum states at the time of reception, while preserving the encoded quantum information in photons. In this sense, a protocol of these characteristics allows to anticipate, in some applications, the error correction process to the security analysis (although the syndromes of each of the states can be stored classically and the correction processed once the QBER estimation is finished). This can constitute a significant disadvantage in unnecessary computational energy costs when the transmission is not considered secure, but may also be exploited for beneficial applications on certain use cases. With all of the above, the work here presented collects the results on the aforementioned considerations, quantitative cost analysis and future feasibility prospects of this QECC-QKD proposal, as well as details on design and integration considerations, and requirements of both the QKD and quantum hardware components that support this type of implementation.