Raman scattering plays a key role in unraveling the quantum dynamics of graphene, perhaps the most promising material of recent times. It is crucial to correctly interpret the meaning of the spectra. It is therefore very surprising that the widely accepted understanding of Raman scattering, i.e., Kramers-Heisenberg-Dirac theory, has never been applied to graphene. Doing so here, a remarkable mechanism we term"transition sliding" is uncovered, explaining the uncommon brightness of overtones in graphene. Graphene's dispersive and fixed Raman bands, missing bands, defect density and laser frequency dependence of band intensities, widths of overtone bands, Stokes, anti-Stokes anomalies, and other known properties emerge simply and directly.

We give a path integral formulation of the time evolution of qudits of odd dimension. This allows us to consider semiclassical evolution of discrete systems in terms of an expansion of the propagator in powers of . The largest power of required to describe the evolution is a traditional measure of classicality. We show that the action of the Clifford operators on stabilizer states can be fully described by a single contribution of a path integral truncated at order 0 and so are "classical," just like propagation of Gaussians under harmonic Hamiltonians in the continuous case. Such operations have no dependence on phase or quantum interference. Conversely, we show that supplementing the Clifford group with gates necessary for universal quantum computation results in a propagator consisting of a finite number of semiclassical path integral contributions truncated at order 1 , a number that nevertheless scales exponentially with the number of qudits. The same sum in continuous systems has an infinite number of terms at order 1 .

We show that qubit stabilizer states can be represented by non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive stateindependent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two-producing an exterior, or Grassmann, algebra-which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each stabilizer state's three-generator Wigner function, and these distributions evolve deterministically to one another under Clifford gates. This corresponds to a hidden variable theory that is non-contextual and local for qubit Clifford gates while Clifford (Pauli) measurements have a context-dependent representation. Equivalently, we show that qubit Clifford gates can be expressed as propagators within the three-generator Wigner-Weyl-Moyal formalism whose semiclassical expansion is truncated at order 0 with a finite number of terms. The T -gate, which extends the Clifford gate set to one capable of universal quantum computation, require a semiclassical expansion of the propagator to order 1 . We compare this approach to previous quasi-probability descriptions of qubits that relied on the two-generator Wigner-Weyl-Moyal formalism and find that the twogenerator Weyl symbols of stabilizer states result in a description of evolution under Clifford gates that is state-dependent, in contrast to the three-generator formalism. We have thus extended Wigner non-negative quasi-probability distributions from the odd d-dimensional case to d = 2 qubits, which describe the non-contextuality of Clifford gates and contextuality of Pauli measurements on qubit stabilizer states.

Polyacetylene has been a paradigm conjugated organic conductor since well before other conjugated carbon systems such as nanotubes and graphene became front and center. It is widely acknowledged that Raman spectroscopy of these systems is extremely important to characterize them and understand their internal quantum behavior. Here we show, for the first time, what information the Raman spectrum of polyacetylene contains, by solving the 35-year-old mystery of its spectrum. Our methods have immediate and clear implications for other conjugated carbon systems. By relaxing the nearly universal approximation of ignoring the nuclear coordinate dependence of the transition moment (Condon approximation), we find the reasons for its unusual spectroscopic features. When the Kramers–Heisenberg–Dirac Raman scattering theory is fully applied, incorporating this nuclear coordinate dependence, and also the energy and momentum dependence of the electronic and phonon band structure, then unusual line shapes, growth, and dispersion of the bands are explained and very well matched by theory.

Abstract:The Gottesman-Knill theorem established that stabilizer states and Clifford operations can be efficiently simulated classically. For qudits with odd dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-d qudits that has the same time and space complexity as the Aaronson-Gottesman algorithm for qubits. We show that the efficiency of both algorithms is due to harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm for odd-d and the Aaronson-Gottesman algorithm for qubits are likely due only to the fact that the Weyl-Heisenberg group is not in SU(d) for d = 2 and that qubits exhibit state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits.