Hamiltonian Simulation by Qubitization

We present the problem of approximating the time-evolution operator $e^{-i\hat{H}t}$ to error $\epsilon$, where the Hamiltonian $\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\mathcal{O}\big(t+\log({1/\epsilon})\big)$ to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio

Comment on "Kinetic decoupling of WIMPs: Analytic expressions"

Visinelli and Gondolo (2015, hereafter VG15) derived analytic expressions for the evolution of the dark matter temperature in a generic cosmological model. They then calculated the dark matter kinetic decoupling temperature $T_{\mathrm{kd}}$ and compared their results to the Gelmini and Gondolo (2008, hereafter GG08) calculation of $T_{\mathrm{kd}}$ in an early matter-dominated era (EMDE), which occurs when the Universe is dominated by either a decaying oscillating scalar field or a semistable massive particle before Big Bang nucleosynthesis. VG15 found that dark matter decouples at a lower temperature in an EMDE than it would in a radiation-dominated era, while GG08 found that dark matter decouples at a higher temperature in an EMDE than it would in a radiation-dominated era. VG15 attributed this discrepancy to the presence of a matching constant that ensures that the dark matter temperature is continuous during the transition from the EMDE to the subsequent radiation-dominated era and concluded that the GG08 result is incorrect. We show that the disparity is due to the fact that VG15 compared $T_\mathrm{kd}$ in an EMDE to the decoupling temperature in a radiation-dominated universe that would result in the same dark matter temperature at late times. Since decoupling during an EMDE leaves the dark matter colder than it would be if it decoupled during radiation domination, this temperature is much higher than $T_\mathrm{kd}$ in a standard thermal history, which is indeed lower than $T_{\mathrm{kd}}$ in an EMDE, as stated by GG08.Comment: 4 pages, 1 figure; comment on arXiv: 1501.0223

Condensation of degrees emerging through a first-order phase transition in classical random graphs

Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a first-order phase transition between a Poisson-like distribution and a condensed phase, the latter characterized by a large fraction of nodes having degrees in a limited sector of their configuration space. The mechanism underlying the first-order transition is discussed in light of standard concepts in statistical physics. We uncover the phase diagram characterizing the ensemble space of the model and we evaluate the rate function governing the probability to observe a condensed state, which shows that condensation of degrees is a rare statistical event akin to similar condensation phenomena recently observed in several other systems. Monte Carlo simulations confirm the exactness of our theoretical results.Comment: 8 pages, 6 figure

Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

We calculate the level compressibility $\chi(W,L)$ of the energy levels inside $[-L/2,L/2]$ for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in $[-W/2,W/2]$. We show that $\chi(W,L)$ approaches the limit $\lim_{L \rightarrow 0^+} \chi(W,L) = 0$ for a broad interval of the disorder strength $W$ within the extended phase, including the region of $W$ close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erd\"os-R\'enyi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.Comment: 7 pages, 3 figure

The Benefits of Achieving the Chesapeake Bay TMDLs (Total Maximum Daily Loads): A Scoping Study

Concerns about nutrient pollution in the Chesapeake Bay have led to the establishment of pollution limits—total maximum daily loads (TMDLs)—which, by 2025, are expected to reduce nitrogen loadings to the Bay by 25 percent and phosphorous loadings by 24 percent from current levels. This paper outlines how the benefits associated with achieving the Chesapeake Bay TMDLs could be measured and monetized. We summarize studies that measure the benefits of improved water quality in the Bay and evaluate whether these studies could be used to value the water quality benefits associated with the TMDLs.In cases where studies conducted in the Bay watershed either do not exist or are out of date, we discuss whether results from studies conducted elsewhere could be transferred to the Chesapeake Bay. We also discuss original studies that would be useful to conduct in the future.Chesapeake Bay restoration, total maximum daily loads, benefits of water quality improvements