Quantum Computing in Fock Space Systems

M. L. Dalla Chiara, Roberto Giuntini, Stanley Gudder +1 more

Paul J. Werbos

Ordinary Fock space, defined as concatenated combination of three-dimensional spaces is the main workhorse of the canonical version of quantum field theory. It is also fundamental to the modern version of quantum field theory without observers, as in the theory of Everett, Wheeler and Deutsch, which led to the theory of the universal quantum computer. It also has many practical uses, from quantum optics to the closure of turbulence and the derivation of the emergent behavior of space-time dynamical systems. But many of us believe that time and space are interchangeable to some degree, and that methods based on 4D Fock space are needed. This letter describes two simple mathematical tools to help make that possible. It discusses how computing the emergent statistics of such systems is a proper generalization of value function estimation in machine learning, which may therefore carry the same complexities with it.

Leszek Z. Stolarczyk, Hendrik J. Monkhorst

The quasiparticle Fock-space coupled-cluster (QFSCC) theory, introduced by us in 1985, is described. This is a theory of many-electron systems which uses the second-quantisation formalism based on the algebraic approximation: one chooses a finite spin-orbital basis, and builds a fermionic Fock space to represent all possible antisymmetric electronic states of a given system. The algebraic machinery is provided by the algebra of linear operators acting in the Fock space, generated by the fermion (creation and annihilation) operators. The Fock-space Hamiltonian operator then determines the system's stationary states and their energies. Within the QFSCC theory, the Fock space and its operator algebra are subject to a unitary transformation which effectively changes electrons into some fermionic quasiparticles. A generalisation of the coupled-cluster method is achieved by enforcing the principle of quasiparticle-number conservation. The emerging quasiparticle model of many-electron systems offers useful physical insights and computational effectiveness. The QFSCC theory requires a substantial reformulation of the traditional second-quantisation language, by making full use of the algebraic properties of the Fock space and its operator algebra. In particular, the role of operators not conserving the number of electrons (or quasiparticles) is identified.

Shoupu Wan

María Luisa Dalla Chiara, Roberto Giuntini, Roberto Leporini

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence α is identified with a quantum information quantity, represented by a density operator of a Hilbert space, whose dimension depends on the logical complexity of α. At the same time, the logical connectives of the language are interpreted as operations defined in terms of quantum logical gates. Standard quantum computational models can be described as special cases of Fock space models, where the meaning of any sentence is localized in a precise sector of a Fock space ℱ. From an intuitive point of view, the increasing number of particles described in the different sectors of ℱ can be interpreted as increasing information.