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arXiv:1912.11725

Quantum Error Correction in Loop Quantum Gravity

Deepak Vaid

Published Dec 25, 2019• 1 citation
arXivHTMLPDF

Abstract

Previous works (by Almiehri, Dong, Harlow, Pastakawski, Preskill, Yoshida and others) have established that quantum error correction plays an important role in understanding how the bulk degrees of freedom of an Anti-deSitter spacetime are encoded in the degrees of freedom of the boundary Conformal Field Theory. In previous work \cite{Vaid2013Elementary} I have argued that the Bilson-Thompson model \cite{Bilson-Thompson2006Quantum,Vaid2010Embedding} of elementary particles allows us to view elementary particles as gates for universal quantum computation. In the present work I show that the Bilson-Thompson model, where elementary particles are represented by elements of the framed braid group on three strands, provides an explicit model for the generation of qutrit (three-qubit) states which are the ingredients of Shor's quantum error correcting code. This allows, for the first time, to connect in a concrete manner the proposals of Almheiri, Pastawski, Preskill and others regarding the role of quantum error correction in quantum gravity, to a viable model of elementary particles. Loop Quantum Gravity (LQG), the theory of quantum gravity in which such topological excitations exist, can thus serve as the glue which can connect AdS/CFT based approaches to quantum gravity to the well understood physics of the Standard Model.

DOI
10.48550/arxiv.1912.11725

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Previous works (by Almiehri, Dong, Harlow, Pastakawski, Preskill, Yoshida and others) have established that quantum error correction plays an important role in understanding how the bulk degrees of freedom of an Anti-deSitter spacetime are encoded in the degrees of freedom of the boundary Conformal Field Theory. In previous work \cite{Vaid2013Elementary} I have argued that the Bilson-Thompson model \cite{Bilson-Thompson2006Quantum,Vaid2010Embedding} of elementary particles allows us to view elementary particles as gates for universal quantum computation. In the present work I show that the Bilson-Thompson model, where elementary particles are represented by elements of the framed braid group on three strands, provides an explicit model for the generation of qutrit (three-qubit) states which are the ingredients of Shor's quantum error correcting code. This allows, for the first time, to connect in a concrete manner the proposals of Almheiri, Pastawski, Preskill and others regarding the role of quantum error correction in quantum gravity, to a viable model of elementary particles. Loop Quantum Gravity (LQG), the theory of quantum gravity in which such topological excitations exist, can thus serve as the glue which can connect AdS/CFT based approaches to quantum gravity to the well understood physics of the Standard Model.
In this work, we investigate some applications of quantum information theory motivated by high-energy physics. There is strong evidence suggesting that entanglement is deeply connected with the geometry of spacetime, which leads to surprising applications of quantum information theory in the AdS/CFT correspondence and holography. We start by reviewing the fundamental concepts of the AdS/CFT correspondence which play a key role in bulk-boundary reconstruction, in particular, we explore some features which suggest that one must interpret the encoding of information in the correspondence as a quantum errorcorrecting code. We discuss the fundamentals of error correction, exploring the formalisms of operator algebra and stabilizer codes. Then, we establish the concrete connection between the two main concepts by showing examples of quantum error-correcting codes that serve as a toy model for AdS/CFT. We illustrate how the 3-qutrit code and the HaPPY code can be powerful tools to explore the correspondence analytically and to solve apparent paradoxes. Following recent results, using quantum error correction, that suggest an intrinsic incompatibility of quantum gravity with global symmetries, we explore approximate error-correcting codes and asymmetric codes as a way to better understand the consequences in a quantum resource-theoretic way. Finally, we discuss our original contribution: geometric bounds for approximate quantum error correction. We calculate our bounds for three typical quantum channels that model the lack of exactness in error correction, namely, dephasing, depolarizing, and amplitude damping channels. The implications of our bounds for AdS/CFT are somewhat elusive; nonetheless, we provide a new approach to benchmark approximations in error correction performance, which may be of high interest for AdS/CFT and its corresponding absence of global symmetries.
Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code (HyPeC). Relating its logical code basis to Majorana dimers, we efficiently compute boundary-state properties even for the non-Gaussian case of generic logical input. The dimers characterizing these boundary states coincide with discrete bulk geodesics, leading to a geometric picture from which properties of entanglement, quantum error correction, and bulk/boundary operator mapping immediately follow. We also elaborate upon the emergence of the Ryu-Takayanagi formula from our model, which realizes many of the properties of the recent bit thread proposal. Our work thus elucidates the connection among bulk geometry, entanglement, and quantum error correction in AdS/CFT and lays the foundation for new models of holography.
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The study of critical quantum many-body systems through conformal field theory (CFT) is one of the pillars of modern quantum physics. Certain CFTs are also understood to be dual to higher-dimensional theories of gravity via the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. To reproduce various features of AdS/CFT, a large number of discrete models based on tensor networks have been proposed. Some recent models, most notably including toy models of holographic quantum error correction, are constructed on regular time-slice discretizations of AdS. In this work, we show that the symmetries of these models are well suited for approximating CFT states, as their geometry enforces a discrete subgroup of conformal symmetries. Based on these symmetries, we introduce the notion of a quasiperiodic conformal field theory (qCFT), a critical theory less restrictive than a full CFT and with characteristic multi-scale quasiperiodicity. We discuss holographic code states and their renormalization group flow as specific implementations of a qCFT with fractional central charges and argue that their behavior generalizes to a large class of existing and future models. Beyond approximating CFT properties, we show that these can be best understood as belonging to a paradigm of discrete holography.
Abstract: It was shown by [2] how bulk operators in the AdS/CFT correspondence can be represented on the boundary analogously to the way logical qubits are represented in an encoded subspace in quantum error correction. Then in [1] holographic tensor networks that serve as toy models of the bulk boundary. This paper reviews some of the developments of [1] and [2]. Then it is demonstrated explicitly how to construct perfect tensors, which are essential to the tensor networks mentioned in [2]. Lastly a new example of a holographic quantum error-correcting code based on an eight index perfect tensor is presented.