scholarly journals On the relation between a graph code and a graph state

2016 ◽  
Vol 16 (3&4) ◽  
pp. 237-250
Author(s):  
Yongsoo Hwang ◽  
Jun Heo
Keyword(s):  
Simple Graph ◽  
Graph State ◽  
Stabilizer Codes ◽  
Logical Qubits ◽  
Related Graph ◽  
Local Complementation ◽  
Stabilizer Group

A graph state and a graph code respectively are defined based on a mathematical simple graph. In this work, we examine a relation between a graph state and a graph code both obtained from the same graph, and show that a graph state is a superposition of logical qubits of the related graph code. By using the relation, we first discuss that a local complementation which has been used for a graph state can be useful for searching locally equivalent stabilizer codes, and second provide a method to find a stabilizer group of a graph code.

Locality bounds on Hamiltonians for stabilizer codes

2009 ◽  
Vol 9 (5&6) ◽  
pp. 487-499
Author(s):  
S.S. Bullock ◽  
D.P. O'Leary
Keyword(s):  
Quantum Computing ◽  
Error Correction ◽  
Energy Gap ◽  
Stabilizer Codes ◽  
Adiabatic Quantum Computing ◽  
Group A ◽  
Stabilizer Code ◽  
Stabilizer Group ◽  
The Cost ◽  
Insight Into

In this paper, we study the complexity of Hamiltonians whose groundstate is a stabilizer code. We introduce various notions of $k$-locality of a stabilizer code, inherited from the associated stabilizer group. A choice of generators leads to a Hamiltonian with the code in its groundspace. We establish bounds on the locality of any other Hamiltonian whose groundspace contains such a code, whether or not its Pauli tensor summands commute. Our results provide insight into the cost of creating an energy gap for passive error correction and for adiabatic quantum computing. The results simplify in the cases of XZ-split codes such as Calderbank-Shor-Steane stabilizer codes and topologically-ordered stabilizer codes arising from surface cellulations.

scholarly journals Small Majorana fermion codes

2017 ◽  
Vol 17 (13&14) ◽  
pp. 1191-1205
Author(s):  
Mathew B. Hastings
Keyword(s):  
Upper Bound ◽  
Majorana Fermion ◽  
Hamming Code ◽  
Stabilizer Codes ◽  
Numerical Studies ◽  
Logical Qubits ◽  
Stabilizer Code ◽  
Majorana Modes ◽  
The Given

We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance 4 codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance 4 and 6 codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.

scholarly journals Weight reduction for quantum codes

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1307-1334
Author(s):  
Mathew B. Hastings
Keyword(s):  
Weight Reduction ◽  
High Dimensional ◽  
Quantum Codes ◽  
Stabilizer Codes ◽  
Linear Distance ◽  
Logical Qubits ◽  
Stabilizer Code ◽  
Logarithmic Weight

We present an algorithm that takes a CSS stabilizer code as input, and outputs another CSS stabilizer code such that the stabilizer generators all have weights O(1) and such that O(1) generators act on any given qubit. The number of logical qubits is unchanged by the procedure, while we give bounds on the increase in number of physical qubits and in the effect on distance and other code parameters, such as soundness (as a locally testable code) and “cosoundness” (defined later). Applications are discussed, including to codes from high-dimensional manifolds which have logarithmic weight stabilizers. Assuming a conjecture in geometry[11], this allows the construction of CSS stabilizer codes with generator weight O(1) and almost linear distance. Another application of the construction is to increasing the distance to X or Z errors, whichever is smaller, so that the two distances are equal.

Stabilizer codes and equientangled bases from phase states

2017 ◽  
Vol 31 (20) ◽  
pp. 1750132 ◽  
Author(s):  
Mostafa Mansour ◽  
Mohammed Daoud
Keyword(s):  
Comprehensive Approach ◽  
Stabilizer Codes ◽  
Finite Dimensional ◽  
The Common ◽  
Stabilizer Group ◽  
Maximally Entangled Basis

We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.

scholarly journals Mapping graph state orbits under local complementation

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 305
Author(s):  
Jeremy C. Adcock ◽  
Sam Morley-Short ◽  
Axel Dahlberg ◽  
Joshua W. Silverstone
Keyword(s):  
Quantum Computing ◽  
Graph Theory ◽  
Quantum Entanglement ◽  
Graph State ◽  
Graph States ◽  
Potential Applications ◽  
Graph Operation ◽  
Local Complementation ◽  
Equivalent Graph ◽  
Component Graph

Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation – the graph operation that links all local-Clifford equivalent graph states – allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the 587 orbits up to 9 qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay – our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.

Graph Indices

2020 ◽  
pp. 66-91
Author(s):  
V. R. Kulli
Keyword(s):  
Molecular Structure ◽  
Carbon Atom ◽  
Organic Molecule ◽  
Molecular Graph ◽  
Simple Graph ◽  
Chemical Graph ◽  
Zagreb Indices ◽  
Mathematical Properties ◽  
Chemical Graph Theory ◽  
Related Graph

A molecular graph is a finite simple graph representing the carbon-atom skeleton of an organic molecule of a hydrocarbon. Studying molecular graphs is a constant focus in chemical graph theory: an effort to better understand molecular structure. Many types of graph indices such as degree-based graph indices, distance-based graph indices, and counting-related graph indices have been explored recently. Among degree-based graph indices, Zagreb indices are the oldest and studied well. In the last few years, many new graph indices were proposed. The present survey of these graph indices outlines their mathematical properties and also provides an exhaustive bibliography.

Fuzzy Labeling on Cycle Related Graph

2019 ◽  
Vol 7 (6) ◽  
pp. 1192-1194
Author(s):  
M.K. Pandurangan ◽  
T. Bharathi ◽  
S.Antony Vinoth
Keyword(s):  
Related Graph

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

scholarly journals Entangling logical qubits with lattice surgery

Nature ◽  
2021 ◽  
Vol 589 (7841) ◽  
pp. 220-224
Author(s):  
Alexander Erhard ◽  
Hendrik Poulsen Nautrup ◽  
Michael Meth ◽  
Lukas Postler ◽  
Roman Stricker ◽  
...  
Keyword(s):  
Logical Qubits
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