32 research outputs found
Performance Analysis of Quantum Error-Correcting Codes via MacWilliams Identities
One of the main challenges for an efficient implementation of quantum
information technologies is how to counteract quantum noise. Quantum error
correcting codes are therefore of primary interest for the evolution towards
quantum computing and quantum Internet. We analyze the performance of
stabilizer codes, one of the most important classes for practical
implementations, on both symmetric and asymmetric quantum channels. To this
aim, we first derive the weight enumerator (WE) for the undetectable errors of
stabilizer codes based on the quantum MacWilliams identities. The WE is then
used to evaluate the error rate of quantum codes under maximum likelihood
decoding or, in the case of surface codes, under minimum weight perfect
matching (MWPM) decoding. Our findings lead to analytical formulas for the
performance of generic stabilizer codes, including the Shor code, the Steane
code, as well as surface codes. For example, on a depolarizing channel with
physical error rate it is found that the logical error rate
is asymptotically for the
Shor code, for the
Steane code, for the surface
code, and for the surface
code.Comment: 25 pages, 5 figures, submitted to an IEEE journal. arXiv admin note:
substantial text overlap with arXiv:2302.1301
Quantum weight enumerators and tensor networks
We examine the use of weight enumerators for analyzing tensor network
constructions, and specifically the quantum lego framework recently introduced.
We extend the notion of quantum weight enumerators to so-called tensor
enumerators, and prove that the trace operation on tensor networks is
compatible with a trace operation on tensor enumerators. This allows us to
compute quantum weight enumerators of larger codes such as the ones constructed
through tensor network methods more efficiently. We also provide an analogue of
the MacWilliams identity for tensor enumerators.Comment: 21 pages, 3 figures. Sets up the tensor enumerator formalis
Equivalence of Classical and Quantum Codes
In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture
Shor–Laflamme distributions of graph states and noise robustness of entanglement
The Shor–Laflamme distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify k-body correlations. We show that the SLD of graph states can be derived by solving a graph-theoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLD-based entanglement criterion from the purity criterion and apply it to derive meaningful noise thresholds for entanglement. Our new entanglement criterion is easy to use and also applies to the case of higher-dimensional qudits. In the bigger picture, our results foster the understanding both of quantum error-correcting codes, where a closely related notion of SLDs plays an important role, and of the geometry of quantum states, where SLDs are known as sector length distributions
Sector length distributions of graph states
The sector length distribution (SLD) of a quantum state is a collection of
local unitary invariants that quantify -body correlations. We show that the
SLD of graph states can be derived by solving a graph-theoretical problem. In
this way, the mean and variance of the SLD are obtained as simple functions of
efficiently computable graph properties. Furthermore, this formulation enables
us to derive closed expressions of SLDs for some graph state families. For
cluster states, we observe that the SLD is very similar to a binomial
distribution, and we argue that this property is typical for graph states in
general. Finally, we derive an SLD-based entanglement criterion from the
majorization criterion and apply it to derive meaningful noise thresholds for
entanglement.Comment: 20+21 pages, 8+8 figure
Towards Fault-Tolerant Quantum Computation with Higher-Dimensional Systems
The main focus of this thesis is to explore the advantages of using higher-dimensional quantum systems (qudits) as building blocks for fault-tolerant quantum computation. In particular, we investigate the two main essential ingredients of many state-of-the-art fault-tolerant schemes [133], which are magic state distillation and topological error correction. The theory for both of these components is well established for the qubit case, but little has been known for the generalised qudit case. For magic state distillation, we first present a general numerical approach that can be used to investigate the distillation properties of any stabilizer code. We use this approach to study small threedimensional (qutrit) codes and classify, for the first time, new types of qutrit magic states. We then provide an analytic study of a family of distillation protocols based on the quantum Reed-Muller codes. We discover a particular five-dimensional code that, by many measures, outperforms all known qubit codes. For the topological error correction, we study the qudit toric code serving as a quantum memory. For this purpose we examine an efficient renormalization group decoder to estimate the error correction threshold. We find that when the qudit toric code is subject to a generalised bit-flip noise, and for a sufficiently high dimension, a threshold of 30% can be obtained under perfect decoding
Trellis Decoding And Applications For Quantum Error Correction
Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost.
Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products.
Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.Ph.D
Applications of Derandomization Theory in Coding
Randomized techniques play a fundamental role in theoretical computer science
and discrete mathematics, in particular for the design of efficient algorithms
and construction of combinatorial objects. The basic goal in derandomization
theory is to eliminate or reduce the need for randomness in such randomized
constructions. In this thesis, we explore some applications of the fundamental
notions in derandomization theory to problems outside the core of theoretical
computer science, and in particular, certain problems related to coding theory.
First, we consider the wiretap channel problem which involves a communication
system in which an intruder can eavesdrop a limited portion of the
transmissions, and construct efficient and information-theoretically optimal
communication protocols for this model. Then we consider the combinatorial
group testing problem. In this classical problem, one aims to determine a set
of defective items within a large population by asking a number of queries,
where each query reveals whether a defective item is present within a specified
group of items. We use randomness condensers to explicitly construct optimal,
or nearly optimal, group testing schemes for a setting where the query outcomes
can be highly unreliable, as well as the threshold model where a query returns
positive if the number of defectives pass a certain threshold. Finally, we
design ensembles of error-correcting codes that achieve the
information-theoretic capacity of a large class of communication channels, and
then use the obtained ensembles for construction of explicit capacity achieving
codes.
[This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi