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 $m \times m$ toric code to hypermap-homology codes gives us a $[(3/2)m^2,2,m]$ code as compared to the toric code which is a $[2m^2,2,m]$ code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.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 $m \times m$ toric code to hypermap-homology codes gives us a $[(3/2)m^2,2,m]$ code as compared to the toric code which is a $[2m^2,2,m]$ code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

arXiv:1310.5376Oct 20, 20130 citations

Relation between surface codes and hypermap-homology quantum codes

Pradeep Kiran Sarvepalli

Recently, a new class of quantum codes based on hypermaps has been proposed. These codes are obtained from embeddings of hypergraphs as opposed to surface codes which are obtained from the embeddings of graphs. It is natural to compare these two classes of codes and their relation to each other. In this context two related questions are addressed in this paper: Can the parameters of hypermap-homology codes be superior to those of surface codes, and what is precisely the relation between these two classes of quantum codes? We show that a canonical hypermap code is identical to a surface code, whereas a noncanonical hypermap code can be transformed to a surface code by cnot gates alone. Our approach is constructive; we construct the related surface code and the transformation involving cnot gates.Recently, a new class of quantum codes based on hypermaps has been proposed. These codes are obtained from embeddings of hypergraphs as opposed to surface codes which are obtained from the embeddings of graphs. It is natural to compare these two classes of codes and their relation to each other. In this context two related questions are addressed in this paper: Can the parameters of hypermap-homology codes be superior to those of surface codes, and what is precisely the relation between these two classes of quantum codes? We show that a canonical hypermap code is identical to a surface code, whereas a noncanonical hypermap code can be transformed to a surface code by cnot gates alone. Our approach is constructive; we construct the related surface code and the transformation involving cnot gates.

arXiv:1312.6344Physical Review AMay 13, 20144 citations

Homological error correction: Classical and quantum codes

Héctor Bombín, M. A. Martin-Delgado

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension D>2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension D. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor’s 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension D>2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension D. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor’s 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.

arXiv:quant-ph/0605094Journal of Mathematical PhysicsMay 1, 200792 citations

Homological Error Correction: Classical and Quantum Codes

Héctor Bombín, Martín-Delgado, M A

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$ , while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$ . We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.

May 10, 200674 citations

Homological quantum codes beyond the toric code

Nikolas P. Breuckmann

Computer architectures which exploit quantum mechanical effects can solve computing tasks that are otherwise impossible to perform. A quantum computer operates on a number of small quantum mechanical systems, known as quantum bits, or qubits. Since these systems are realized on the scale of atoms, they are very prone to errors. Errors occur when the environment interacts with the qubits, a process called decoherence. It is widely accepted that it will not be possible to shield qubits completely from the outside world. If one were to perform a quantum computation on the qubits directly, then after a short period of time the information present in the qubits would be lost. To counter decoherence the state of a qubit can be encoded into multiple physical ones. This is called a quantum error correcting code. Performing quantum error correction allows one to extend the life time of the encoded qubit arbitrarily, assuming that the rate of errors remains below a certain threshold value. The use of quantum codes creates an overhead in resources, as for every logical qubit many more physical qubits are needed. The resource overhead for fault-tolerance is problematic, since realizing qubits will be costly, and in the early stages of building quantum computers the number of physical qubits will be limited. The currently favored coding architecture is the toric code and its variant the surface code in which the physical qubits are put on a square grid in which interactions are only between nearest neighbors. In this thesis we will explore quantum codes in which qubits interact as if they were nearest neighbors in more exotic spaces. In the first part we will consider closed surfaces with constant negative curvature. We show how such surfaces can be constructed and enumerate all quantum codes derived from them which have less than 10.000 physical qubits. For codes that are extremal in a certain sense we perform numerical simulations to determine the value of their threshold. Furthermore, we give evidence that these codes can be used for more overhead efficient storage as compared to the surface code by orders of magnitude. We also show how to read and write the encoded qubits while keeping their connectivity low. In the second part we consider codes in which qubits are layed-out according to a four- dimensional geometry. Such codes allow for much simpler decoding schemes compared to codes which are two-dimensional. In particular, measurements do not necessarily have to be repeated to obtain reliable information about the error and the classical hardware performing the error correction is greatly simplified. We perform numerical simulations to analyze the performance of these codes using decoders based on local updates. We also introduce a novel decoder based on techniques from machine learning and image recognition to decode four-dimensional codes.Computer architectures which exploit quantum mechanical effects can solve computing tasks that are otherwise impossible to perform. A quantum computer operates on a number of small quantum mechanical systems, known as quantum bits, or qubits. Since these systems are realized on the scale of atoms, they are very prone to errors. Errors occur when the environment interacts with the qubits, a process called decoherence. It is widely accepted that it will not be possible to shield qubits completely from the outside world. If one were to perform a quantum computation on the qubits directly, then after a short period of time the information present in the qubits would be lost. To counter decoherence the state of a qubit can be encoded into multiple physical ones. This is called a quantum error correcting code. Performing quantum error correction allows one to extend the life time of the encoded qubit arbitrarily, assuming that the rate of errors remains below a certain threshold value. The use of quantum codes creates an overhead in resources, as for every logical qubit many more physical qubits are needed. The resource overhead for fault-tolerance is problematic, since realizing qubits will be costly, and in the early stages of building quantum computers the number of physical qubits will be limited. The currently favored coding architecture is the toric code and its variant the surface code in which the physical qubits are put on a square grid in which interactions are only between nearest neighbors. In this thesis we will explore quantum codes in which qubits interact as if they were nearest neighbors in more exotic spaces. In the first part we will consider closed surfaces with constant negative curvature. We show how such surfaces can be constructed and enumerate all quantum codes derived from them which have less than 10.000 physical qubits. For codes that are extremal in a certain sense we perform numerical simulations to determine the value of their threshold. Furthermore, we give evidence that these codes can be used for more overhead efficient storage as compared to the surface code by orders of magnitude. We also show how to read and write the encoded qubits while keeping their connectivity low. In the second part we consider codes in which qubits are layed-out according to a four- dimensional geometry. Such codes allow for much simpler decoding schemes compared to codes which are two-dimensional. In particular, measurements do not necessarily have to be repeated to obtain reliable information about the error and the classical hardware performing the error correction is greatly simplified. We perform numerical simulations to analyze the performance of these codes using decoders based on local updates. We also introduce a novel decoder based on techniques from machine learning and image recognition to decode four-dimensional codes.

Jan 1, 201715 citations

Homological Quantum Error Correction with Torsion

S. Novák

Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.

arXiv:2405.03559May 6, 20240 citations

m \times m

toric code to hypermap-homology codes gives us a

[(3/2)m^2,2,m]

code as compared to the toric code which is a

[2m^2,2,m]

code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

D>2

, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension

D

. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.