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arXiv:1307.4677 · Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions

An application of Khovanov homology to quantum codes

Benjamin Audoux

Published Jul 3, 2014• 7 citations
arXivHTMLPDF

Abstract

We use Khovanov homology to define families of LDPC quantum error-correcting codes: unknot codes with asymptotical parameters [[ \frac{3^{2 \ell+1}}{\sqrt{8\pi\ell}};1;2^\ell]] ; unlink codes with asymptotical parameters [[\sqrt{\frac{3}{2\pi \ell}}6^\ell;2^\ell;2^\ell ]] and (2,\ell) -torus link codes with asymptotical parameters [[n;1;d_n]] where d_n>\frac{\sqrt{n}}{1.62} .

DOI
10.4171/aihpd/6

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We use Khovanov homology to define families of LDPC quantum error-correcting codes: unknot codes with asymptotical parameters [[ \frac{3^{2 \ell+1}}{\sqrt{8\pi\ell}};1;2^\ell]] ; unlink codes with asymptotical parameters [[\sqrt{\frac{3}{2\pi \ell}}6^\ell;2^\ell;2^\ell ]] and (2,\ell) -torus link codes with asymptotical parameters [[n;1;d_n]] where d_n>\frac{\sqrt{n}}{1.62} .
Error-correcting codes for quantum computing are crucial to address the fundamental problem of communication in the presence of noise and imperfections. Audoux used Khovanov homology to define families of quantum error-correcting codes with desirable properties. We explore Khovanov homology and some of its many extensions, namely reduced, annular, and sl3 homology, to generate new families of quantum codes and to establish several properties about codes that arise in this way, such as behavior of distance under Reidemeister moves or connected sums.
Error correction is the set of techniques used in order to store, process and transmit information reliably in a noisy context. The classical theory of error correction is based on encoding classical information redundantly. A major endeavor of the theory is to find optimal trade-offs between redundancy, which we try to minimize, and noise tolerance, which we try to maximize. The quantum theory of error correction cannot directly imitate the redundant schemes of the classical theory because it has to cope with the no-cloning theorem: quantum information cannot be copied. Quantum error correction is nonetheless possible by spreading the information on more quantum memory elements than would be necessary. In quantum information theory, dilution of the information replaces redundancy since copying is forbidden by the laws of quantum mechanics. Besides this conceptual difference, quantum error correction inherits a lot from its classical counterpart. In this PhD thesis, we are concerned with a class of quantum error correcting codes whose classical counterpart was defined in 1961 by Gallager [Gal62]. At that time, quantum information was not even a research domain yet. This class is the family of low density parity check (LDPC) codes. Informally, a code is said to be LDPC if the constraints imposed to ensure redundancy in the classical setting or dilution in the quantum setting are local. More precisely, this PhD thesis focuses on a subset of the LDPC quantum error correcting codes: the homological quantum error correcting codes. These codes take their name from the mathematical field of homology, whose objects of study are sequences of linear maps such that the kernel of a map contains the image of its left neighbour. Originally introduced to study the topology of geometric shapes, homology theory now encompasses more algebraic branches as well, where the focus is more abstract and combinatorial. The same is true of homological codes: they were introduced in 1997 by Kitaev [Kit03] with a quantum code that has the shape of a torus. They now form a vast family of quantum LDPC codes, some more inspired from geometry than others. Homological quantum codes were designed from spherical, Euclidean and hyperbolic geometries, from 2-dimensional, 3-dimensional and 4- dimensional objects, from objects with increasing and unbounded dimension and from hypergraph or homological products. After introducing some general quantum information concepts in the first chapter of this manuscript, we focus in the two following ones on families of quantum codes based on 4-dimensional hyperbolic objects. We highlight the interplay between their geometric side, given by manifolds, and their combinatorial side, given by abstract polytopes. We use both sides to analyze the corresponding quantum codes. In the fourth and last chapter we analyze a family of quantum codes based on spherical objects of arbitrary dimension. To have more flexibility in the design of quantum codes, we use combinatorial objects that realize this spherical geometry: hypercube complexes. This setting allows us to introduce a new link between classical and quantum error correction where classical codes are used to introduce homology in hypercube complexes.
We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m×mm \times mm×m toric code to hypermap-homology codes gives us a [(3/2)m2,2,m][(3/2)m^2,2,m][(3/2)m2,2,m] code as compared to the toric code which is a [2m2,2,m][2m^2,2,m][2m2,2,m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.
Khovanov homology is a topological knot invariant that categorifies the Jones polynomial, recognizes the unknot, and is conjectured to appear as an observable in 4D4D4D supersymmetric Yang--Mills theory. Despite its rich mathematical and physical significance, the computational complexity of Khovanov homology remains largely unknown. To address this challenge, this work initiates the study of efficient quantum algorithms for Khovanov homology. We provide simple proofs that increasingly accurate additive approximations to the ranks of Khovanov homology are DQC1-hard, BQP-hard, and #P-hard, respectively. For the first two approximation regimes, we propose a novel quantum algorithm. Our algorithm is efficient provided the corresponding Hodge Laplacian thermalizes in polynomial time and has a sufficiently large spectral gap, for which we give numerical and analytical evidence. Our approach introduces a pre-thermalization procedure that allows our quantum algorithm to succeed even if the Betti numbers of Khovanov homology are much smaller than the dimensions of the corresponding chain spaces, overcoming a limitation of prior quantum homology algorithms. We introduce novel connections between Khovanov homology and graph theory to derive analytic lower bounds on the spectral gap.