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We present an efficient method for estimating the eigenvalues of a Hamiltonian H from the expectation values of the evolution operator for various times. For a given quantum state ρ, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of ρ in those eigenstates of H associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter ϵ, which is the gap between eigenvalue estimates. Unlike the well-known quantum phase estimation algorithm that uses the quantum Fourier transform, our method does not require large ancillary systems, large sequences of controlled operations, or preserving coherence between experiments, and is therefore more attractive for near-term applications. The output of our method can be used to estimate spectral properties of H and other expectation values efficiently, within additive error proportional to ϵ. [ABSTRACT FROM AUTHOR]

We study the incommensurate charge ordered states in the model using the Gutzwiller mean field theory on large systems. In particular, we explore the properties of incommensurate charge modulated states referred to as nodal pair density waves (nPDW) in the literature. nPDW states intertwine site and bond charge order with modulated d-wave pair order, and are characterized by a nonzero amplitude of uniform pairing; they also manifest a dominant intra-unit cell d-density wave form factor. To compare with a recent scanning tunneling microscopy (STM) study (Hamidian et al 2015 Nat. Phys.12 150) of the cuprate superconductor BSCCO-2212, we compute the continuum local density of states (LDOS) at a typical STM tip height using the Wannier function based approach. By Fourier transforming Cu and O sub-lattice LDOS we also obtain bias-dependent intra-unit cell form factors and spatial phase difference. We find that in the nPDW state the behavior of form factors and spatial phase difference as a function of energy agrees remarkably well with the experiment.This is in contrast to commensurate charge modulated states, which we show do not agree with experiment. We propose that the nPDW states are good candidates for the charge density wave phase observed in the superconducting state of underdoped cuprates. [ABSTRACT FROM AUTHOR]

A general prescription is given for finding uncertainty relations that dictate the lower bounds on the measures of spread corresponding to two different representations of a partially coherent wave field or mixed quantum state, for a given measure of overall coherence or purity. In particular it is shown that the coherent modes of the fields/states that achieve the lower bounds are independent of the measure of purity being used, and that this measure determines only the amount in which these modes contribute. Our results are important in the design of optical systems with partially coherent light and in quantum mixed states, for which maximal joint localization is desired. These ideas are illustrated for the case of optical beams, pulses propagating in dispersive media and quantum phase. [ABSTRACT FROM AUTHOR]