Speculative Growth: Hints from the US Economy

We propose a framework for understanding recurrent historical episodes of vigorous economic expansion accompanied by extreme asset valuations, as exhibited by the U.S. in the 1990s. We interpret this phenomenon as a high-valuation equilibrium with a low effective cost of capital based on optimism about the future availability of funds for investment. The key to the sustainability of such an equilibrium is feedback from increased growth to an increase in the supply of effective funding. We show that such feedback arises naturally when an expansion comes with technological progress in the capital producing sector, when fiscal rules generate sustained fiscal surpluses, when the rest of the world has lower expansion potential, and when financial constraints are relaxed by the expansion itself. Arguably, these ingredients were all simultaneously present in the U.S. during the 1990s. We also show that such expansions can be welfare improving but they can crash. The latter is more likely if bubbles develop along the expansionary path. These (rational) bubbles can emerge even when the interest rate exceeds the rate of growth of the economy.

Quantum Energies of Interfaces

We present a method for computing the one-loop, renormalized quantum energies of symmetrical interfaces of arbitrary dimension and codimension using elementary scattering data. Internal consistency requires finite-energy sum rules relating phase shifts to bound state energies.Comment: 8 pages, 1 figure, minor changes, Phys. Rev. Lett., in prin

For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances

The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth quantum circuit to produce a parameter dependent state. For a given combinatorial optimization problem instance, the quantum expectation of the associated cost function is the parameter dependent objective function of the QAOA. We demonstrate that if the parameters are fixed and the instance comes from a reasonable distribution then the objective function value is concentrated in the sense that typical instances have (nearly) the same value of the objective function. This applies not just for optimal parameters as the whole landscape is instance independent. We can prove this is true for low depth quantum circuits for instances of MaxCut on large 3-regular graphs. Our results generalize beyond this example. We support the arguments with numerical examples that show remarkable concentration. For higher depth circuits the numerics also show concentration and we argue for this using the Law of Large Numbers. We also observe by simulation that if we find parameters which result in good performance at say 10 bits these same parameters result in good performance at say 24 bits. These findings suggest ways to run the QAOA that reduce or eliminate the use of the outer loop optimization and may allow us to find good solutions with fewer calls to the quantum computer.Comment: 16 pages, 1 figur

The least common multiple of a sequence of products of linear polynomials

Let $f(x)$ be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate: $\log {\rm lcm}(f(1), ..., f(n))\sim An$ as $n\rightarrow\infty$, where $A$ is a constant depending on $f$.Comment: To appear in Acta Mathematica Hungaric

Gravitational Force and the Cardiovascular System

Cardiovascular responses to changes in gravitational force are considered. Man is ideally suited to his 1-g environment. Although cardiovascular adjustments are required to accommodate to postural changes and exercise, these are fully accomplished for short periods (min). More challenging stresses are those of short-term microgravity (h) and long-term microgravity (days) and of gravitational forces greater than that of Earth. The latter can be simulated in the laboratory and quantitative studies can be conducted

Hamiltonian and measuring time for analog quantum search

We derive in this study a Hamiltonian to solve with certainty the analog quantum search problem analogue to the Grover algorithm. The general form of the initial state is considered. Since the evaluation of the measuring time for finding the marked state by probability of unity is crucially important in the problem, especially when the Bohr frequency is high, we then give the exact formula as a function of all given parameters for the measuring time.Comment: 5 page

Adiabatic quantum computation and quantum phase transitions

We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the mass gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR

Speculative growth : hints from the US economy

Original date of pub.: November 27, 200

Sufficiency Criterion for the Validity of the Adiabatic Approximation

We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it along with additional conditions is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to all $N-$dimensional quantum systems. Moreover, we illustrate the use of the general quantitative criterion in some physical models.Comment: 9 pages, no figure,appearing in PRL98(2007)15040