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Quantum error-correcting codes and devices

Daniel Gottesman

Published Jan 23, 2023• 0 citations
OpenAlex

Abstract

A method of forming quantum error-correcting codes by first forming a stabilizer for a Hilbert space. A quantum information processing device can be formed to implement such quantum codes.

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The quantum computing must be the initial mark of the second technological revolution caused by the deepening knowledge of the humankind about the peculiar features of the quantum mechanics. If we want to unlock all the potential of this new technology, a development of a fault-tolerant quantum computing will be needed. In order to reach this stage, it is necessary an implementation of quantum error correcting codes that guarantee an exponential suppression of the errors at each step of the execution of an algorithm. Besides that, we need to learn how to isolate our qubits aiming a suppression of the decoherence effects. Quantum mechanics imposes various constraints to the process of construction of the error correction codes, one of them is the impossibility of using copies as redundancy mechanism, once no-cloning theorem does not allow the creation of an universal copy machine of quantum states. Another issue to be noted within this context, is that the measurement process causes a collapse in the quantum state, thus leading to a corruption of the information encoded in the state. Fortunately, there are some ways to bypass the problems pointed out through the entanglement and the stabilizer formalism, the first one allows us to recover the redundancy through the delocalization of the information in the Hilbert spaces of the parts that share quantum correlations, while the second one provides a process of gathering information about the errors that can occur in the quantum state without measuring it. The surface codes are one of the quantum codes that use the features highlighted to correct the errors, this type of code is one of the main candidates to be experimentally implemented on a large scale and it presents promising results in the preliminary studies.
We discuss the connection between quantum error-correcting codes (QECCS) and algebraic coding theory. We start with an introduction to the relevant concepts of quantum mechanics, including the general error model. A quantum error-correcting code is a subspace of a complex Hilbert space, and its error-correcting properties are characterized by the Knill-Laflamme conditions. Using the stabilizer formalism, we illustrate how QECCs for can be constructed using techniques from algebraic coding theory. We also sketch how the information obtained via a quantum measurement can be interpreted as syndrome of the related classical code. Additionally, we present secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs. This includes the puncture code by Rains and construction X for quantum codes.
A major difficulty in quantum computation is the ability to implement fault tolerant computations, protecting information against undesired interactions with the environment. Stabiliser codes were introduced as a means to protect information when storing or applying computations in Hilbert spaces where the local dimension is fixed, i.e. in Hilbert spaces of the form (CD)⊗n({\mathbb C}^D)^{\otimes n}(CD)⊗n. If DDD is a prime power then one can consider stabiliser codes over finite fields \cite{KKKS2006}, which allows a deeper mathematical structure to be used to develop stabiliser codes. However, there is no practical reason that the subsystems should have the same local dimension and in this article we introduce a stabiliser formalism for mixed dimensional Hilbert spaces, i.e. of the form CD1⊗⋯⊗CDn{\mathbb C}^{D_1} \otimes \cdots \otimes {\mathbb C}^{D_n}CD1​⊗⋯⊗CDn​. More generally, we define and prove a Singleton bound for quantum error-correcting codes of mixed dimensional Hilbert spaces. We redefine entanglement measures for these Hilbert spaces and follow \cite{HESG2018} and define absolutely maximally entangled states as states which maximise this entanglement measure. We provide examples of absolutely maximally entangled states in spaces of dimensions not previously known to have absolutely maximally entangled states.