Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space

We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space. This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time. We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a counting argument. This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model.Comment: Presented at QIP 201

Efficient discrete-time simulations of continuous-time quantum query algorithms

The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. Interesting algorithms have been discovered in this model, such as an algorithm for evaluating nand trees more efficiently than any classical algorithm. Subsequent work has shown that there also exists an efficient algorithm for nand trees in the discrete query model; however, there is no efficient conversion known for continuous-time query algorithms for arbitrary problems. We show that any quantum algorithm in the continuous-time query model whose total query time is T can be simulated by a quantum algorithm in the discrete query model that makes O[T log(T) / log(log(T))] queries. This is the first upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corollary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of \Omega[T log(log(T))/log (T)] in the continuous-time query model.Comment: 12 pages, 6 fig

Cuban Treefrogs, \u3ci\u3eOsteopilus septentrionalis\u3c/i\u3e (Duméril & Bibron 1841) (Anura: Hylidae), and other nonindigenous herpetofauna interdicted in Grenada, Lesser Antilles.

The number of introduced nonindigenous species of amphibians and reptiles within the greater Caribbean, including Grenada, is escalating and has become an ever-increasing critical conservation concern (Daudin and de Silva 2011; Powell et al. 2011; Powell and Henderson 2012). The amount of development, tourism, and consequent import commerce is increasing, requiring careful regulation of the pet trade and fauna introduced for biological control as well as diligence in cargo inspection. Herein we document the first records of nonindigenous Cuban Treefrogs, Osteopilus septentrionalis (Duméril and Bibron 1841), interdicted from cargo, along with recent interceptions of two species of nonindigenous lizards already established on Grenada in the Lesser Antilles. Osteopilus septentrionalis is indigenous to Cuba and portions of The Bahamas, with nonindigenous populations established in Florida, USA, Costa Rica, and a number of islands throughout the Caribbean (Meshaka 2001, 2011; Kraus 2009; Rödder and Weinsheimer 2009; Krysko et al. 2011a, 2011b; Powell et al. 2011; Powell and Henderson 2012; Somma 2012; Rivalta González 2014). This highly invasive and potentially ecologically injurious hylid is established on several islands in the Lesser Antilles. Within the Grenada Bank, it is known only on Mustique (Kraus 2009; Powell et al. 2011; Henderson and Breuil 2012; Somma 2012; Yokoyama 2012). On 9 November 2013, an adult O. septentrionalis (UF-Herpetology 174214) was collected from ornamental horticultural cargo by PRG at the main shipping port in Saint George’s Harbour, Grenada Island, Grenada (12.047808°N, 61.748347°W, datum WGS84)

Mpemba effect and phase transitions in the adiabatic cooling of water before freezing

An accurate experimental investigation on the Mpemba effect (that is, the freezing of initially hot water before cold one) is carried out, showing that in the adiabatic cooling of water a relevant role is played by supercooling as well as by phase transitions taking place at 6 +/- 1 oC, 3.5 +/- 0.5 oC and 1.3 +/- 0.6 oC, respectively. The last transition, occurring with a non negligible probability of 0.21, has not been detected earlier. Supported by the experimental results achieved, a thorough theoretical analysis of supercooling and such phase transitions, which are interpreted in terms of different ordering of clusters of molecules in water, is given.Comment: revtex, 4 pages, 2 figure

Spectral Gap Amplification

A large number of problems in science can be solved by preparing a specific eigenstate of some Hamiltonian H. The generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of H for that eigenstate and the cost of evolving with H for some fixed time. The goal of spectral gap amplification is to construct a Hamiltonian H' with the same eigenstate as H but a bigger spectral gap, requiring that constant-time evolutions with H' and H are implemented with nearly the same cost. We show that a quadratic spectral gap amplification is possible when H satisfies a frustration-free property and give H' for these cases. This results in quantum speedups for optimization problems. It also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states (PEPS), which play an important role in quantum many-body physics. Defining a suitable black-box model, we establish that the quadratic amplification is optimal for frustration-free Hamiltonians and that no spectral gap amplification is possible, in general, if the frustration-free property is removed. A corollary is that finding a similarity transformation between a stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the black-box model, setting limits to the power of some classical methods that simulate quantum adiabatic evolutions.Comment: 14 pages. New version has an improved section on adiabatic simulations of quantum circuit