594 research outputs found
The Quantum Monadology
The modern theory of functional programming languages uses monads for
encoding computational side-effects and side-contexts, beyond bare-bone program
logic. Even though quantum computing is intrinsically side-effectful (as in
quantum measurement) and context-dependent (as on mixed ancillary states),
little of this monadic paradigm has previously been brought to bear on quantum
programming languages.
Here we systematically analyze the (co)monads on categories of parameterized
module spectra which are induced by Grothendieck's "motivic yoga of operations"
-- for the present purpose specialized to HC-modules and further to set-indexed
complex vector spaces. Interpreting an indexed vector space as a collection of
alternative possible quantum state spaces parameterized by quantum measurement
results, as familiar from Proto-Quipper-semantics, we find that these
(co)monads provide a comprehensive natural language for functional quantum
programming with classical control and with "dynamic lifting" of quantum
measurement results back into classical contexts.
We close by indicating a domain-specific quantum programming language (QS)
expressing these monadic quantum effects in transparent do-notation, embeddable
into the recently constructed Linear Homotopy Type Theory (LHoTT) which
interprets into parameterized module spectra. Once embedded into LHoTT, this
should make for formally verifiable universal quantum programming with linear
quantum types, classical control, dynamic lifting, and notably also with
topological effects.Comment: 120 pages, various figure
Light-matter interaction in the ZXW calculus
In this paper, we develop a graphical calculus to rewrite photonic circuits
involving light-matter interactions and non-linear optical effects. We
introduce the infinite ZW calculus, a graphical language for linear operators
on the bosonic Fock space which captures both linear and non-linear photonic
circuits. This calculus is obtained by combining the QPath calculus, a
diagrammatic language for linear optics, and the recently developed qudit ZXW
calculus, a complete axiomatisation of linear maps between qudits. It comes
with a 'lifting' theorem allowing to prove equalities between infinite
operators by rewriting in the ZXW calculus. We give a method for representing
bosonic and fermionic Hamiltonians in the infinite ZW calculus. This allows us
to derive their exponentials by diagrammatic reasoning. Examples include phase
shifts and beam splitters, as well as non-linear Kerr media and Jaynes-Cummings
light-matter interaction.Comment: 27 pages, lots of figures, a previous version accepted to QPL 202
From finite state automata to tangle cobordisms: a TQFT journey from one to four dimensions
This is a brief introduction to link homology theories that categorify
Reshetikhin--Turaev -quantum link invariants. A recently
discovered surprising connection between finite state automata and Boolean
TQFTs in dimension one is explained as a warm-up.Comment: 40 pages, many figure
On monoids of endomorphisms of a cycle graph
In this paper we consider endomorphisms of an undirected cycle graph from
Semigroup Theory perspective. Our main aim is to present a process to determine
sets of generators with minimal cardinality for the monoids and
of all weak endomorphisms and all endomorphisms of an undirected
cycle graph with vertices. We also describe Green's relations and
regularity of these monoids and calculate their cardinalities
An Easily Checkable Algebraic Characterization of Positive Expansivity for Additive Cellular Automata over a Finite Abelian Group
We provide an easily checkable algebraic characterization of positive
expansivity for Additive Cellular Automata over a finite abelian group. First
of all, an easily checkable characterization of positive expansivity is
provided for the non trivial subclass of Linear Cellular Automata over the
alphabet . Then, we show how it can be exploited to decide positive
expansivity for the whole class of Additive Cellular Automata over a finite
abelian group.Comment: 12 page
Realizing Finitely Presented Groups as Projective Fundamental Groups of SFTs
Subshifts are sets of colourings - or tilings - of the plane, defined by local constraints. Historically introduced as discretizations of continuous dynamical systems, they are also heavily related to computability theory. In this article, we study a conjugacy invariant for subshifts, known as the projective fundamental group. It is defined via paths inside and between configurations. We show that any finitely presented group can be realized as a projective fundamental group of some SFT
Efficient and Side-Channel Resistant Implementations of Next-Generation Cryptography
The rapid development of emerging information technologies, such as quantum computing and the Internet of Things (IoT), will have or have already had a huge impact on the world. These technologies can not only improve industrial productivity but they could also bring more convenience to people’s daily lives. However, these techniques have “side effects” in the world of cryptography – they pose new difficulties and challenges from theory to practice. Specifically, when quantum computing capability (i.e., logical qubits) reaches a certain level, Shor’s algorithm will be able to break almost all public-key cryptosystems currently in use. On the other hand, a great number of devices deployed in IoT environments have very constrained computing and storage resources, so the current widely-used cryptographic algorithms may not run efficiently on those devices. A new generation of cryptography has thus emerged, including Post-Quantum Cryptography (PQC), which remains secure under both classical and quantum attacks, and LightWeight Cryptography (LWC), which is tailored for resource-constrained devices. Research on next-generation cryptography is of importance and utmost urgency, and the US National Institute of Standards and Technology in particular has initiated the standardization process for PQC and LWC in 2016 and in 2018 respectively.
Since next-generation cryptography is in a premature state and has developed rapidly in recent years, its theoretical security and practical deployment are not very well explored and are in significant need of evaluation. This thesis aims to look into the engineering aspects of next-generation cryptography, i.e., the problems concerning implementation efficiency (e.g., execution time and memory consumption) and security (e.g., countermeasures against timing attacks and power side-channel attacks). In more detail, we first explore efficient software implementation approaches for lattice-based PQC on constrained devices. Then, we study how to speed up isogeny-based PQC on modern high-performance processors especially by using their powerful vector units. Moreover, we research how to design sophisticated yet low-area instruction set extensions to further accelerate software implementations of LWC and long-integer-arithmetic-based PQC. Finally, to address the threats from potential power side-channel attacks, we present a concept of using special leakage-aware instructions to eliminate overwriting leakage for masked software implementations (of next-generation cryptography)
An index for quantum cellular automata on fusion spin chains
Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms
of the Jones index for subfactors leads to a generalization of the index
defined for QCA on more general abstract spin chains. These include fusion spin
chains, which arise as the local operators invariant under a global
(categorical/MPO) symmetry, as the boundary operators on 2D topologically
ordered spin systems. We show that for the fusion spin chains built from the
fusion category , the index is a complete invariant for the group
of QCA modulo finite depth circuits.Comment: 21 page
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