Block encoding lies at the core of many existing quantum algorithms. Meanwhile, efficient and explicit block encodings of dense operators are commonly acknowledged as a challenging problem. This paper presents a comprehensive study of the block encoding of a rich family of dense operators: the pseudo-differential operators (PDOs). First, a block encoding scheme for generic PDOs is developed. Then we propose a more efficient scheme for PDOs with a separable structure. Finally, we demonstrate an explicit and efficient block encoding algorithm for PDOs with a dimension-wise fully separable structure. Complexity analysis is provided for all block encoding algorithms presented. The application of theoretical results is illustrated with worked examples, including the representation of variable coefficient elliptic operators and the computation of the inverse of elliptic operators without invoking quantum linear system algorithms (QLSAs).Block encoding lies at the core of many existing quantum algorithms. Meanwhile, efficient and explicit block encodings of dense operators are commonly acknowledged as a challenging problem. This paper presents a comprehensive study of the block encoding of a rich family of dense operators: the pseudo-differential operators (PDOs). First, a block encoding scheme for generic PDOs is developed. Then we propose a more efficient scheme for PDOs with a separable structure. Finally, we demonstrate an explicit and efficient block encoding algorithm for PDOs with a dimension-wise fully separable structure. Complexity analysis is provided for all block encoding algorithms presented. The application of theoretical results is illustrated with worked examples, including the representation of variable coefficient elliptic operators and the computation of the inverse of elliptic operators without invoking quantum linear system algorithms (QLSAs).

arXiv:2301.08908QuantumJun 2, 202316 citations

Operators for observables in quantum optics

Thomas F. Jordan

Physics LettersAug 1, 196418 citations

Deformation quantization of linear dissipative systems

V. G. Kupriyanov, S. L. Lyakhovich, A. A. Sharapov

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\\partial_t+...$ , differentiating the $\\ast_t$ -product. As in the usual case, the $\\ast_t$ -algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\\mathrm{Tr}_t$ . Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\\partial_t+...$, differentiating the $\\ast_t$-product. As in the usual case, the $\\ast_t$-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\\mathrm{Tr}_t$. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

Jan 1, 20057 citations

Interaction between classical and quantum systems and the measurement of quantum observables

E. C. G. Sudarshan

PramanaMar 1, 197677 citations

Deformation quantization of linear dissipative systems

V G Kupriyanov, S L Lyakhovich, A A Sharapov

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation, we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗ t and an 'extended' operator of time derivative D t = ∂ t + ⋅⋅⋅, differentiating the ∗ t -product. As in the usual case, the ∗ t -algebra of physical observables is shown to admit an essentially unique (time-dependent) trace functional Tr t . Using these ingredients, we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation, we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗ t and an 'extended' operator of time derivative D t = ∂ t + ⋅⋅⋅, differentiating the ∗ t -product. As in the usual case, the ∗ t -algebra of physical observables is shown to admit an essentially unique (time-dependent) trace functional Tr t . Using these ingredients, we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

arXiv:quant-ph/0505023Journal of Physics A Mathematical and GeneralAug 31, 200519 citations

Sets of simple observables in the operational approach to quantum theory

C. Martin Edwards

Annales De L Institut Henri Poincare-physique TheoriqueJan 1, 19713 citations

*_t

and an ``extended'' operator of time derivative

D_t=\\partial_t+...

, differentiating the

\\ast_t

-product. As in the usual case, the

\\ast_t

-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional

\\mathrm{Tr}_t

. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.