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arXiv:2404.13773

A route to quantum computing through the theory of quantum graphs

Фаррох Разавиниа

Published Apr 21, 2024• 0 citations
arXivHTMLPDF

Abstract

Based on our previous works, and in order to relate them with the theory of quantum graphs and the quantum computing principles, we once again try to introduce some newly developed technical structures just by relying on our toy example, the coordinate ring of $n\times n$ quantum matrix algebra $M_q(n)$, and the associated directed locally finite graphs $\mathcal{G}(Π_n)$, and the Cuntz-Krieger $C^*$-graph algebras. Meaningly, we introduce a $(4i-6)$-qubit quantum system by using the Cuntz-Krieger $\mathcal{G}(Π_i)$-families associated to the $4i-6$ distinct Hamiltonian paths of $\mathcal{G}(Π_i)$, for $i\in\{2,\cdots,n\}$. We also will present a proof of a claim raised in our previous paper concerning the graph $C^*$-algebra structure and the associated Cuntz-Krieger $\mathcal{G}(Π_n)$-families.

DOI
10.48550/arxiv.2404.13773

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Quantum computation is one of the very promising new fields of the new millennium for basic research in theoretical computer science and computational mathematics. Quantum computers have stimulated great interests due to their promise of being able to solve problems considered infeasible on current computers. From a computer scientist's and mathematician's point of view, the growth in the capacities of the underlying hardware is vital since an inefficient algorithm can solve the problem with the faster new hardware. For decades, this attitude was irrelevant but the deadline to change it is near. That is why quantum computation has led to rapid developments in physics, mathematics and computer science.This tutorial paper summarizes our year long series of multidisciplinary seminars on the topics of using computational algebra in quantum computation for finding new theoretical models for the upcoming generation of computers together with finding the models' computational abilities.
The appearance of the Cuntz algebras Od is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. In view of this apparent discrepancy, we develop a comprehensive theory of quasi-free endomorphisms of the CAR and CCR algebras which give rise to representations of the Cuntz algebras Od on Fock space. The number d is the statistics dimension of the endomorphism. It can be any power of 2 (including 1 and infinite) in the CAR case, but takes only the values 1 and infinite in the CCR case. We obtain necessary and sufficient conditions for implementability of quasi- free endomorphisms. By studying extensions of partial Fock states, we find that the semigroup of implementable quasi-free endomorphisms has a simple structure: It can be written as a product of a group of automorphisms which are close to the identity, and the semigroup of endomorphisms which leave the given Fock state invariant. We describe the construction of the implementers of a quasi-free endomorphism in terms of annihilation and creation operators in is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. We compare the structure of the semigroup of (gauge invariant) quasi-free endomorphisms with the generic superselection structure of quantum field theory.
Summary form only given, as follows. Gives a system theoretic approach to the concept of quantum computing. A quantum computer is envisaged to be a system of quantum circuits, acting on a state space, which is a finite-say 2/sup n/-dimensional-complex Hilbert space. The circuitry is a sequence of unitary transforms U/sub t/ /spl epsiv/ SU(2/sup n/) followed by a measurement. These transforms, so-called quantum gates, are controlled by a classical computer, usually especially electromagnetic fields. The state space of a quantum computer has the structure of a Hermitian vector space. Thus it allows "simultaneous" superposition of orthogonal basis states (corresponding to classical states) with the possibility of constructive and destructive interference between different paths of computation. This principle allows the usage of so-called entangled states by preparing the superposition of special "bent" configurations of basis vectors in the product Hilbert space, e.g., as they are known from error control and cryptography. This latter "entanglement" not only promises to make a quantum computer much more powerful than a probabilistic one, but it also allows a method of parallelism. The reason for this lies not only in the fact that the 2/sup n/ dimensional Hilbert space is the n-fold tensor product of 2-dimensional spaces as it is given by spin-1/2-particles, like photons, representing a quantum bit. First of all, we shall address the fact that the power of quantum computing lies in the properties of entangled states as opposed to those of separable states which essentially would resemble classical circuits without giving the feature of exponential speed-up. We shall briefly describe possible physical realisations of quantum (optic, electronic, magnetic,...) devices allowing the engineering of Hamiltonians needed for quantum systems. Basic quantum gates, especially the quantum Horner-Toffoli gate will be discussed to display the principles and methodology for designing quantum circuits. An outlook towards the possible availability of nano- and mesoscopic technologies supporting this new architecture in future generations concludes based on a speculative worst/best case forecast of possible applications, e.g., in public key and conventional cryptography.
Recent work has demonstrated the existence of universal Hamiltonians-simple spin-lattice models that can simulate any other quantum many-body system to any desired level of accuracy.Until now, proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians.In this work, we go beyond this approach and completely classify the simulation ability of quantum Hamiltonians by their complexity classes.We do this by deriving necessary and sufficient complexitytheoretic conditions characterizing universal quantum Hamiltonians.Although the result concerns the theory of analog Hamiltonian simulation-a promising application of near-term quantum technology-the proof relies on abstract complexity-theoretic concepts and the theory of quantum computation.As well as providing simplified proofs of previous Hamiltonian universality results and offering a route to new universal constructions, the results in this paper give insight into the origins of universality; for example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory.