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.

scholarly journals Classification of transversal gates in qubit stabilizer codes

2016 ◽  
Vol 16 (9&10) ◽  
pp. 771-802
Author(s):  
Jonas T. Anderson ◽  
Tomas Jochym-O'Connor
Keyword(s):  
Restricted Class ◽  
Stabilizer Codes ◽  
Single Qubit ◽  
Stabilizer Code ◽  
The Given ◽  
Logical Gate

This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical diagonal gates on the underlying qubits that compose the code are restricted to have entries of the form e iπc/2 k along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. As such, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Moreover, we show that for a given stabilizer code, the two-qubit diagonal transversal gates must belong to the same level of Clifford hierarchy as the single-qubit diagonal transversal gates available for the given code. We use this result to prove a conjecture about arbitrary transversal gates made by Zeng et al. in 2007.

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.

Positioning Effect on Free Convection with an Elliptical Plate Placed in a Circular Enclosure

2020 ◽  
Vol 12 (8) ◽  
pp. 1054-1062
Author(s):  
Parth Patpatiya ◽  
Soumya ◽  
Bhavya Shaan ◽  
Bhavana Yadav
Keyword(s):  
Heat Transfer ◽  
Aspect Ratio ◽  
The Body ◽  
Elliptical Plate ◽  
Numerical Studies ◽  
First Case ◽  
Varied Temperature ◽  
Laminar Natural Convection ◽  
The Given ◽  
Plate System

In this analysis we have examined the process of the steady state laminar natural convection around heated elliptical plate with Rayleigh number 10^6 positioned inside a circular enclosure. The purpose of the numerical analysis is to analyze the behavior of isotherms, streamlines and heat transfer rate in enclosure plate system due to the variation in the position of elliptical plate (r/D =0.00, 0.05, and 0.2) and aspect ratio, where the given diameter of the enclosure is D and r is the distance between the centre of elliptical plate and centre of circle. Elliptical plate is inclined at different angles and results are summed up in relative manner. There are two cases, in first case aspect ratio a/D and b/D is varied and D is kept constant, whereas in second case aspect ratio a/D and b/D is kept constant and D is varied. Temperature difference between the enclosure and the inner body (i.e., temperature of inner body is kept high as compared to the enclosure) is maintained. Two dimensional study is followed by considering air as a fluid in enclosure. The effects of the Heat Transfer and Flow of Fluid are analyzed by the streamlines and isotherms plotted for the body placed inside enclosure. Value of local Nusselt number (Nu) is also plotted along the wall of elliptical plate and along the surface of the circular enclosure. For every aspect ratio isotherms and streamlines had been plotted. This work has been validated with various other numerical studies and was in good conciliation.

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 New Constructions of Quantum Stabilizer Codes Based on Difference Sets

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 655 ◽  
Author(s):  
Duc Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Difference Sets ◽  
Combinatorial Design ◽  
Inner Product ◽  
Code Construction ◽  
Stabilizer Codes ◽  
Stabilizer Code ◽  
The Difference ◽  
Symplectic Inner Product ◽  
Quantum Stabilizer Code ◽  
Quantum Stabilizer Codes

In this paper, new conditions on parameters in difference sets are derived to satisfy symplectic inner product, and new constructions of quantum stabilizer codes are proposed from the conditions. The conversion of the difference sets into parity-check matrices is first explained. Then, the proposed code construction is composed of three steps, which are to choose the generators of quantum stabilizer code, to determine the quantum stabilizer groups, and to determine subspace codewords with large minimum distance. The quantum stabilizer codes with various length are also presented to explain the practicality of the code construction. The proposed design can be applied to quantum stabilizer code construction based on combinatorial design.

ON DAVENPORT'S CONSTANT

2008 ◽  
Vol 04 (01) ◽  
pp. 107-115 ◽  
Author(s):  
P. RATH ◽  
K. SRILAKSHMI ◽  
R. THANGADURAI
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Davenport Constant ◽  
The Given

In this paper, using the idea of Alford, Granville and Pomerance in [1] (or van Emde Boas and Kruyswijk [6]), we obtain an upper bound for the Davenport Constant of an Abelian group G in terms of the number of repetitions of the group elements in any given sequence. In particular, our result implies, [Formula: see text] where n is the exponent of G and k ≥ 0 denotes the number of distinct elements of G that are repeated at least twice in the given sequence.

scholarly journals On subsystem codes beating the quantum Hamming or Singleton bound

2007 ◽  
Vol 463 (2087) ◽  
pp. 2887-2905 ◽  
Author(s):  
Andreas Klappenecker ◽  
Pradeep Kiran Sarvepalli
Keyword(s):  
The Other ◽  
Maximum Distance ◽  
Open Problems ◽  
Prime Field ◽  
Stabilizer Codes ◽  
Singleton Bound ◽  
Other Hand ◽  
Maximum Distance Separable ◽  
Linear Subsystem ◽  
Stabilizer Code

Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

XII.—Further Numerical Studies in Algebraic Equations and Matrices

1932 ◽  
Vol 51 ◽  
pp. 80-90 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Algebraic Equation ◽  
Symmetric Functions ◽  
Algebraic Equations ◽  
Numerical Studies ◽  
The Given

In a former paper on the same subject the writer pointed out that the sequence used by D. Bernoulli for approximating to the greatest root of an algebraic equation could be further utilised in such a way as to give all the roots. It is suggested in the present paper that there is really no need to compute a first Bernoullian sequence at all, but that by the theory of dual symmetric functions the coefficients in the given equation may be used with equal convenience. In a practical respect this simplifies the technique of root-evaluation.

A novel construction for quantum stabilizer codes based on binary formalism

2020 ◽  
Vol 34 (08) ◽  
pp. 2050059 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Minimum Distance ◽  
Binary Vector ◽  
Quantum Codes ◽  
Quantum Code ◽  
Parity Check Matrix ◽  
Stabilizer Codes ◽  
Check Matrix ◽  
Stabilizer Code ◽  
Quantum Stabilizer Code ◽  
Quantum Stabilizer Codes

In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.

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