Conservative logic is a comprehensive model of computation which explicitly reflects a number of fundamental principles of physics, such as the reversibility of the dynamical laws and the conservation of certain additit, e quantities (among which energy plays a distinguished role). Because it more closely mirrors physics than traditional models of computation, conservative logic is in a better position to provide indications concerning the realization of high-performance computing systems, i.e., of systems that make very efficient use of the "computing resources" actually offered by nature. In particular, conservative logic shows that it is ideally possible to build sequential circuits with zero internal power dissipation. After establishing a general framework, we discuss two specific models of computation. The first uses binary, variables and is the conservative-logic counterpart of switching theory; this model proves that universal computing capabilities are compatible with the reversibility and conservation constraints. The second model, which is a refinement of the first, constitutes a substantial breakthrough in establishing a correspondence between computation and physics. In fact, this model is based on elastic collisions of identical "balls," and thus is formally identical with the atomic model that underlies the (classical) kinetic theory of perfect gases. Quite literally, the functional behavior of a general-purpose digital computer can be reproduced by a perfect gas placed in a suitably shaped container and given appropriate initial conditions.

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The question of how fast a quantum state can evolve has attracted a considerable attention in connection with quantum measurement, metrology, and information processing. Since only orthogonal states can be unambiguously distinguished, a transition from a state to an orthogonal one can be taken as the elementary step of a computational process.1 Therefore, such a transition can be interpreted as the operation of "flipping a qubit", and the number of orthogonal states visited by the system per unit time can be viewed as the maximum rate of operation.A lower bound on the orthogonalization time, based on the energy spread ∆E, was found by Mandelstam and Tamm. 2 Another bound, based on the average energy E, was established by Margolus and Levitin.3 The bounds coincide, and can be exactly attained by certain initial states if ∆E = E. However, the problem remained open of what the situation is when ∆E = E.Here we consider the unified bound that takes into account both ∆E and E. We prove that there exist no initial states that saturate the bound if ∆E = E. However, the bound remains tight: for any given values of ∆E and E, there exists a oneparameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit value. The relation between the largest energy level, the average energy, and the orthogonalization time is also discussed. These results establish the fundamental quantum limit on the rate of operation of any information-processing system. Starting with the classical result of Mandelstam and Tamm,2 it was later shown by Fleming, 4 Anandan and Aharonov, 5 and Vaidman 6 that the minimum time τ required for arriving to an orthogonal state is bounded bywhere (∆E) 2 = ψ|H 2 |ψ − ( ψ|H|ψ ) 2 , H is the Hamiltonian, and |ψ the wavefunction of the system. A different bound was obtained in,Here, E = ψ|H|ψ is the quantum-mechanical average energy of the system (the energy of the ground state is taken to be zero). Both bounds (1) and (2) are tight, and achieved for a quantum state such that ∆E = E.Since then, a vast literature has been devoted to various aspects of this problem. In particular, inequality (2) has been proved for mixed states and for composite systems both in separable and in entangled states (e.g., Giovannetti et al.,7, 8 Zander et al. 9 ). Bound (2) obtained for an isolated system has been generalized to a system driven by an external Hamiltonian (a "quantum gate") in.10, 11 Various derivations of (1) and (2) However, what remained unnoticed is the paradoxical situation of the existence of two bounds based on two different characteristics of the quantum state, seemingly independent of one another. Since the average energy E and the energy uncertainty ∆E play the most determinative role in quantum evolution, it is important to have a unified bound that would take into account both of these characteristics.In all known cases where bounds (1) and (2) can be exactly attained, the ratio α = ∆E E equals 1. A question arises: what happens if α = 1? Some authors ...

Recently, cellular automata machines with the size, speed, and flexibility for general experimentation at a moderate cost have become available to the scientific community. These machines provide a laboratory in which the ideas presented in this book can be tested and applied to the synthesis of a great variety of systems. Computer scientists and researchers interested in modeling and simulation as well as other scientists who do mathematical modeling will find this introduction to cellular automata and cellular automata machines (CAM) both useful and timely. Cellular automata are the computer scientist's counterpart to the physicist's concept of 'field' They provide natural models for many investigations in physics, combinatorial mathematics, and computer science that deal with systems extended in space and evolving in time according to local laws. A cellular automata machine is a computer optimized for the simulation of cellular automata. Its dedicated architecture allows it to run thousands of times faster than a general-purpose computer of comparable cost programmed to do the same task. In practical terms this permits intensive interactive experimentation and opens up new fields of research in distributed dynamics, including practical applications involving parallel computation and image processing. Contents Introduction • Cellular Automata • The CAM Environment • A Live Demo • The Rules of the Game • Our First rules • Second-order Dynamics • The Laboratory • Neighbors and Neighborhood • Running • Particle Motion • The Margolus Neighborhood • Noisy Neighbors • Display and Analysis • Physical Modeling • Reversibility • Computing Machinery • Hydrodynamics • Statistical Mechanics • Other Applications • Imaging Processing • Rotations • Pattern Recognition • Multiple CAMS • Perspectives and Conclusions Cellular Automata Machines is included in the Scientific Computation Series, edited by Dennis Cannon.