594 research outputs found

    The Quantum Monadology

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    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

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    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

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    This is a brief introduction to link homology theories that categorify Reshetikhin--Turaev SL(N)\mathsf{SL}(N)-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

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    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 wEnd(Cn)wEnd(C_n) and End(Cn)End(C_n) of all weak endomorphisms and all endomorphisms of an undirected cycle graph CnC_n with nn 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

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    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 (Z/mZ)n(\Z/m\Z)^n. 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

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    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

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    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

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    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 Fib\mathbf{Fib}, the index is a complete invariant for the group of QCA modulo finite depth circuits.Comment: 21 page
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