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Towards Lagrangian approach to quantum computations

Alexander Yu. Vlasov

Published Aug 4, 2003• 1 citation
OpenAlex

Abstract

In this work is discussed possibility and actuality of Lagrangian approach to quantum computations. Finite-dimensional Hilbert spaces used in this area provide some challenge for such consideration. The model discussed here can be considered as an analogue of Weyl quantization of field theory via path integral in L. D. Faddeev's approach. Weyl quantization is possible to use also in finite-dimensional case, and some formulas may be simply rewritten with change of integrals to finite sums. On the other hand, there are specific difficulties relevant to finite case. This work has some allusions with phase space models of quantum computations developed last time by different authors.

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In this work is discussed possibility and actuality of Lagrangian approach to quantum computations. Finite-dimensional Hilbert spaces used in this area provide some challenge for such consideration. The model discussed here can be considered as an analogue of Weyl quantization of field theory via path integral in L. D. Faddeev's approach. Weyl quantization is possible to use also in finite-dimensional case, and some formulas may be simply rewritten with change of integrals to finite sums. On the other hand, there are specific difficulties relevant to finite case. This work has some allusions with phase space models of quantum computations developed last time by different authors.
In this work is discussed possibility and actuality of Lagrangian approach to quantum computations. Finite-dimensional Hilbert spaces used in this area provide some challenge for such consideration. The model discussed here can be considered as an analogue of Weyl quantization of field theory via path integral in L. D. Faddeev's approach. Weyl quantization is possible to use also in finite-dimensional case, and some formulas may be simply rewritten with change of integrals to finite sums. On the other hand, there are specific difficulties relevant to finite case.
Abstract A Lagrangian description of the qubit based on Schwinger’s picture of Quantum Mechanics that allows for a Feynman-like computation of its probability amplitudes is presented. The Lagrangian is a function on the groupoid that describes the qubit and at the same time determines a self-adjoint element on its associated algebra. Feynman’s paths are replaced by histories on the groupoid which form a groupoid again, and a simple method to compute the sum over all histories is discussed. The unitarity of the theory described in this way imposes quantization conditions on the parameters determining the Lagrangian, and some particular instances are solved completely.
The Lagrangian formulation of quantum dynamics in terms of path-integrals due to Feynman describes systems for which the Hamiltonian is classical in form and quantization is carried out in terms of commutators rather than anticommutators. The difficulty with the Feynman method is the actual evaluation of the pathintegral itself. An explicit evaluation is given for classical wave motion in one dimension. This requires an extension of the Feynman method which was introduced by Tobocman and studied in detail by Davies. The work of Corson on the question of a unified formulation of dynamics is discussed. (Author)
The Hilbert space of quantum mechanics has a dual representation in lattice theory, called the Hilbert lattice. In addition to offering the potential for new insights, the lattice-theoretical approach may be computationally efficient for certain kinds of quantum mechanics problems, particularly if, in the future, we are able to exploit what may be a “natural” fit with quantum computation. The equations that hold in the Hilbert space lattice representation are not completely known and are poorly understood, although much progress has been made in the last several years. This work contributes to the development of these equations, with special attention to the so-called generalized orthoarguesian equations. Many new results that do not appear in the literature are given, along with their detailed proofs. In addition, possible approaches for work towards answering some remaining open questions are discussed.
It is discussed an opportunity to introduce new class of quantum algorithms based on possibility to express amplitude of transition between two states of quantum system as sum of some function along all possible classical paths. Continuous analogue of the property with integral on all possible paths is well known due to Feynman approach and ensures the correspondence with classical minimal action principle. It is less technical and rather popular introduction to earlier work about application of Lagrangian methods in quantum computing [quant-ph/0308017].
I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the algebra of perturbation Lagrangians. Moreover, historical considerations on the go-and-stop path performed by the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices till up to group theory and C*-algebras. But there was no progress in approaching closer the foundations of physics; therefore the problem of discovering an algebraic formulation of QM organized as a problem-based theory and making use of no more than constructive mathematics is open.