Quantum computers hold great promise, but it remains a challenge to find efficient quantum circuits that solve interesting computational problems. We show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms, or to prove limitations on the power of quantum computers.Quantum computers have the potential to efficiently solve problems considered intractable on conventional classical computers, the most famous example being Shor's algorithm (1) for finding the prime factors of an integer. Despite this great promise, as yet there is no general method for constructing good quantum algorithms, and very little is known about the potential power (or limitations) of quantum computers.A quantum computation is usually described as a sequence of logical gates, each coupling only a small number of qubits. The sequence of gates determines a unitary evolution U per-
The magnitude of an adaptive immune response is controlled by the interplay of lymphocyte quiescence, proliferation, and apoptosis. How lymphocytes integrate receptor-mediated signals influencing these cell fates is a fundamental question for understanding this complex system. We examined how lymphocytes interleave times to divide and die to develop a mathematical model of lymphocyte growth regulation. This model provides a powerful method for fitting and analyzing fluorescent division tracking data and reveals how summing receptor-mediated kinetic changes can modify the immune response progressively from rapid tolerance induction to strong immunity. An important consequence of our results is that intrinsic variability in otherwise identical cells, usually dismissed as noise, may have evolved to be an essential feature of immune regulation.
T cell responses are initiated by antigen and promoted by a range of costimulatory signals. Understanding how T cells integrate alternative signal combinations and make decisions affecting immune response strength or tolerance poses a considerable theoretical challenge. Here, we report that T cell receptor (TCR) and costimulatory signals imprint an early, cell-intrinsic, division fate, whereby cells effectively count through generations before returning automatically to a quiescent state. This autonomous program can be extended by cytokines. Signals from the TCR, costimulatory receptors, and cytokines add together using a linear division calculus, allowing the strength of a T cell response to be predicted from the sum of the underlying signal components. These data resolve a long-standing costimulation paradox and provide a quantitative paradigm for therapeutically manipulating immune response strength.
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