scholarly journals Transforming graph states using single-qubit operations

2018 ◽  
Vol 376 (2123) ◽  
pp. 20170325 ◽  
Author(s):  
Axel Dahlberg ◽  
Stephanie Wehner
Keyword(s):  
Quantum Information ◽  
Error Correcting Codes ◽  
Graph States ◽  
Target State ◽  
Single Qubit ◽  
Minor Problem ◽  
Fixed Parameter ◽  
Source State ◽  
Quantum Error ◽  
Stabilizer States

Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G ′ is a vertex-minor of another graph G . The vertex-minor problem is, in general, -Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O (| G | 3 ), where | G | is the size of the graph G , whenever the rank-width of G and the size of G ′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

Constructing new q-ary quantum MDS codes with distances bigger than q/2 from generator matrices

2018 ◽  
Vol 18 (3&4) ◽  
pp. 223-230
Author(s):  
Xianmang He
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Minimum Distance ◽  
Quantum Information Theory ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Error ◽  
Generator Matrices ◽  
Quantum Mds Codes

The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.

Measures of 7-qubit Stabilizer Codes for Graph States

2014 ◽  
Vol 1049-1050 ◽  
pp. 1844-1847
Author(s):  
Yun Yun Wang ◽  
Li Zhen Jiang
Keyword(s):  
Iterative Algorithm ◽  
Error Correcting Codes ◽  
Graph States ◽  
Stabilizer Codes ◽  
Entanglement Measures ◽  
Quantum Error

Most quantum error-correcting codes constructed are stabilizer codes which are potentially more efficient and significant.Under the concept of graph states theory, we focused on 7-qubit graph states characteristics of stabilizer codes, by calculating and analyzing the entanglement properties of graphical codes. For the instance of code ((7,1,3)) with 16 inequitable graphs, the figure of entanglement properties can be measured.And we analyzed 7-qubit graph characteristics of stabilizers codes and calculated the entanglement measures by iterative algorithm.

scholarly journals Non-Pauli topological stabilizer codes from twisted quantum doubles

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 398
Author(s):  
Julio Carlos Magdalena de la Fuente ◽  
Nicolas Tarantino ◽  
Jens Eisert
Keyword(s):  
Quantum Information ◽  
Error Correction ◽  
Lattice Models ◽  
Full Rank ◽  
Error Correcting Codes ◽  
Quantum Error Correction ◽  
Topological Order ◽  
Topological Quantum ◽  
Surface Code ◽  
Quantum Error

It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.

An information theoretical model for quantum secret sharing schemes

2005 ◽  
Vol 5 (1) ◽  
pp. 68-79 ◽  
Author(s):  
H. Imai ◽  
J. Mueller-Quade ◽  
A.C.A. Nascimento ◽  
P. Tuyls ◽  
A. Winter
Keyword(s):  
Quantum Information ◽  
Theoretical Model ◽  
Secret Sharing ◽  
Quantum Secret Sharing ◽  
Error Correcting Codes ◽  
Secret Sharing Schemes ◽  
Secret Sharing Scheme ◽  
Sharing Scheme ◽  
Quantum Error ◽  
Quantum Secret Sharing Scheme

Similarly to earlier models for quantum error correcting codes, we introduce a quantum information theoretical model for quantum secret sharing schemes. This model provides new insights into the theory of quantum secret sharing. By using our model, among other results, we give a shorter proof of Gottesman's theorem that the size of the shares in a quantum secret sharing scheme must be as large as the secret itself. Also, we introduced approximate quantum secret sharing schemes and showed robustness of quantum secret sharing schemes by extending Gottesman's theorem to the approximate case.

scholarly journals Quantum-error-correcting codes using qudit graph states

2008 ◽  
Vol 78 (4) ◽  
Author(s):  
Shiang Yong Looi ◽  
Li Yu ◽  
Vlad Gheorghiu ◽  
Robert B. Griffiths
Keyword(s):  
Error Correcting Codes ◽  
Graph States ◽  
Quantum Error

scholarly journals Experimental exploration of five-qubit quantum error correcting code with superconducting qubits

2021 ◽  
Author(s):  
Ming Gong ◽  
Xiao Yuan ◽  
Shiyu Wang ◽  
Yulin Wu ◽  
Youwei Zhao ◽  
...  
Keyword(s):  
Error Correction ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Perfect Code ◽  
Experimental Realization ◽  
Single Qubit ◽  
Universal Quantum ◽  
Quantum Error

Abstract Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this challenge, we experimentally realise the [[5, 1, 3]] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [[5, 1, 3]] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\%$ while with a high fidelity of $98.6(1)\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of $97.2(2)\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of $74.5(6)\%$, in total with 92 gates. Our work demonstrates each key aspect of the [[5, 1, 3]] code and verifies the viability of experimental realization of quantum error correcting codes with superconducting qubits.

scholarly journals Limits on the storage of quantum information in a volume of space

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 4 ◽  
Author(s):  
Steven T. Flammia ◽  
Jeongwan Haah ◽  
Michael J. Kastoryano ◽  
Isaac H. Kim
Keyword(s):  
Quantum Information ◽  
Constant Width ◽  
Error Correcting Codes ◽  
Fundamental Limits ◽  
Code Distance ◽  
Logical Operators ◽  
Euclidean Metric ◽  
Accuracy Parameter ◽  
Quantum Error ◽  
Minimal Region

We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubitsn, the number of encoded qubitsk, the code distanced, the accuracy parameterδthat quantifies how well the erasure channel can be reversed, and the locality parameterℓthat specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracyϵthat is exponentially small inℓ, which is the case for perturbations of local commuting projector codes, our bound readskd2D−1≤O(n(log⁡n)2DD−1)for codes onD-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceedO(ℓn(D−1)/D)ifδ≤O(ℓn−1/D). As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distancedand the sized~of a minimal region that can support all approximate logical operators satisfiesd~d1D−1≤O(nℓDD−1), where the logical operators are accurate up toO((nδ/d)1/2)in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.

scholarly journals Hypermap-homology quantum codes

2014 ◽  
Vol 12 (01) ◽  
pp. 1430001 ◽  
Author(s):  
Martin Leslie
Keyword(s):  
Quantum Information ◽  
Error Correcting Code ◽  
Chain Complex ◽  
Error Correcting Codes ◽  
Quantum Computers ◽  
Quantum Codes ◽  
Straightforward Generalization ◽  
New Type ◽  
Quantum Error ◽  
Toric Code

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 × m toric code to hypermap-homology codes gives us a [(3/2)m2, 2, m] code as compared to the toric code which is a [2m2, 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

scholarly journals NOTES ON THE FIDELITY OF SYMPLECTIC QUANTUM ERROR-CORRECTING CODES

2003 ◽  
Vol 01 (04) ◽  
pp. 443-463 ◽  
Author(s):  
MITSURU HAMADA
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Quantum Information Processing ◽  
Exponential Convergence ◽  
Error Correcting Codes ◽  
Quantum Channels ◽  
General Quantum ◽  
Quantum Capacity ◽  
Stabilizer Code ◽  
Quantum Error

Two observations are given on the fidelity of schemes for quantum information processing. In the first one, we show that the fidelity of a symplectic (stabilizer) code, if properly defined, exactly equals the "probability" of the correctable errors for general quantum channels. The second observation states that for any coding rate below the quantum capacity, exponential convergence of the fidelity of some codes to unity is possible.

[[2n,n]] Additive quantum error correcting codes

2008 ◽  
Vol 25 (4) ◽  
pp. 519-522
Author(s):  
Yongjun Du ◽  
Yuefei Ma
Keyword(s):  
Error Correcting Codes ◽  
Quantum Error
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