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.

Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the quantum approximate optimization algorithm (QAOA). These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters. In this work, we explore connections between these analog algorithms, as well as limits in which they become approximations of the optimal procedure. Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state that effect the coherent error cancellation of diabatic transitions. Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern. Additionally, the ratios of the QAOA bangs are determined by the smooth, non-oscillatory part of the optimal procedure. We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure. With these arguments, we conclude that different analog algorithms can emulate the optimal protocol under different limits and approximations. Finally, we present a new algorithm for better approximating the optimal protocol using the analytic and numeric insights from the rest of the paper. In practice, numerically, we find that this algorithm outperforms standard QAOA and naive quantum annealing procedures.

We present a new paradigm for understanding optical absorption and hot electron dynamics experiments in graphene. Our analysis pivots on assigning proper importance to phonon-assisted indirect processes and bleaching of direct processes. We show indirect processes figure in the excess absorption in the UV region. Experiments which were thought to indicate ultrafast relaxation of electrons and holes, reaching a thermal distribution from an extremely nonthermal one in under 5-10 fs, instead are explained by the nascent electron and hole distributions produced by indirect transitions. These need no relaxation or ad-hoc energy removal to agree with the observed emission spectra and fast pulsed absorption spectra. The fast emission following pulsed absorption is dominated by phonon-assisted processes, which vastly outnumber direct ones and are always available, connecting any electron with any hole any time. Calculations are given, including explicitly calculating the magnitude of indirect processes, supporting these views.

Contextuality is a necessary resource for universal quantum computation and non-contextual quantum mechanics can be simulated efficiently by classical computers in many cases. Orders of Planck's constant, ÿ, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of ÿ-all orders higher than ÿ 0 can be interpreted as quantum corrections to the order ÿ 0 term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner-Weyl-Moyal (WWM) formalism that require higher than order ÿ 0 terms to be included in order to violate the classical bounds on their expectation values. As a result, we show that contextuality as a resource is closely related to orders of ÿ as a resource within the WWM formalism. This offers an explanation for why qubits can only exhibit state-independent contextuality under Pauli observables as in the Peres-Mermin square while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, qubit states exhibit contextuality when measured by qubit Pauli observables regardless of the state being measured and so the Weyl symbol of these observables lack an order ÿ 0 contribution altogether. On the other hand, odddimensional qudit states exhibit contextuality when measured by qudit observables depending on the state measured and so odd-dimensional qudit observables generally possess non-zero order ÿ 0 terms, and higher, in their WWM formulation: odd-dimensional qudit states that exhibit measurement contextuality have an order ÿ 1 contribution in their expectation values with the observable that allows for the violation of classical bounds while states that have insufficiently large order ÿ 1 contributions do not exhibit measurement contextuality.