Quantum physics and signal processing in the line R are strictly related to Fourier transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io (2). A rigged Hilbert space is found and a new discrete basis in R obtained. All operators defined on R are shown to belong to the universal enveloping algebra of io (2) allowing, in this way, their algebraic treatment. Introducing in the half-line a Fourier-like transform, the procedure is extended to R + and can be easily generalized to R n and to spherical cohordinate systems.
The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs for the partition algebra
Pn(d), Brauer algebra
Bn(d), and walled Brauer algebra
Br,s(d). These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms.
Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter
d is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras
A from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity
poly(n,logd,log(1/ε)) for approximating the Fourier transform to error
(d−1/2+ε)⋅poly(∣A∣). Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.
Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A Rigged Hilbert space is found and a new discrete basis in R obtained. The operators {O[R]} defined on R are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in this way, their algebraic discussion. Introducing in the half-line a Fourier-like Transform, the procedure is extended to R^+ and can be easily generalized to R^n and to spherical reference systems.
The Heisenberg-Weyl group
HW(d) related to a
d-dimensional Hilbert space
H(d), is enlarged into the Heisenberg-Weyl-parity group
HWP(d) that incorporates parity transformations. It consists of
2d3 elements, of which
d3 elements belong to the
HW(d) subgroup, and extra
d3 elements which are related through a Fourier transform with the former ones. It is shown that
HWP(d) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed.
HWP(d) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.
2d2 coherent states related to the
HWP(d) group are introduced, which consist of
d2 coherent states related to the
HW(d) subgroup, and extra
d2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the
2d2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the
d2 coherent states related to
HW(d). One of the consequences of the
HWP(d) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.