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.

In response to stimulation, B lymphocytes pursue a large number of distinct fates important for immune regulation. Whether each cell's fate is determined by external direction, internal stochastic processes, or directed asymmetric division is unknown. Measurement of times to isotype switch, to develop into a plasmablast, and to divide or to die for thousands of cells indicated that each fate is pursued autonomously and stochastically. As a consequence of competition between these processes, censorship of alternative outcomes predicts intricate correlations that are observed in the data. Stochastic competition can explain how the allocation of a proportion of B cells to each cell fate is achieved. The B cell may exemplify how other complex cell differentiation systems are controlled.

Deletion of Bak and Bax, the effectors of mitochondrial apoptosis, does not affect platelet production, however, loss of prosurvival Bcl-xL results in megakaryocyte apoptosis and failure of platelet shedding.

Whether the class Quantum Merlin Arthur is equal to QMA_1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a quantum oracle relative to which QMA\neqQMA_1. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such trivial containments as BQP\subseteq{ZQEXP}.

We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, U, to the optimal control cost associated to the synthesis of U. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of ͓Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 ͑2006͔͒, which showed that the gate complexity can be related to distances on a Riemannian manifold.

Stochastic variation in cell cycle time is a consistent feature of otherwise similar cells within a growing population. Classic studies concluded that the bulk of the variation occurs in the G 1 phase, and many mathematical models assume a constant time for traversing the S/G 2 /M phases. By direct observation of transgenic fluorescent fusion proteins that report the onset of S phase, we establish that dividing B and T lymphocytes spend a near-fixed proportion of total division time in S/G 2 /M phases, and this proportion is correlated between sibling cells. This result is inconsistent with models that assume independent times for consecutive phases. Instead, we propose a stretching model for dividing lymphocytes where all parts of the cell cycle are proportional to total division time. Data fitting based on a stretched cell cycle model can significantly improve estimates of cell cycle parameters drawn from DNA labeling data used to monitor immune cell dynamics.he kinetic relationship between phases of the cell cycle first came to attention with the advent of autoradiographic techniques for detecting DNA synthesis in the 1950s (1, 2). It was realized that such data could be used to resolve the dynamics of the proliferating population if combined with an appropriate cell cycle model. However, direct filming of times to divide revealed remarkable variation, even among cloned, presumed identical, cells (3-6), eliminating simple deterministic models as the basis for cell cycle control. Working toward developing a general model, Smith and Martin made the striking observation that plotting the proportion of undivided cells versus time (so-called "alpha plots"), gave curves suggestive of two distinct phases, one relatively constant and another stochastic (7). They proposed that the two phases mapped to discrete states of the cell cycle. A resting "A state," they suggested, was contained within the G 1 phase from which cells could exit with constant probability per unit time (analogous to radioactive decay). The cells then entered the "B phase," which includes that part of G 1 not included in A state, as well as the entirety of S/G 2 /M. In B phase, cells' activities were first described to be "deterministic, and directed towards replication," implying a constant B phase. However, in the same paper, this assumption was relaxed and the duration of B phase was described with a relatively constant random variable (7).Although details of the quantitative relationship and biological interpretation have been debated (7-12), the rule that the bulk of kinetic variation is in G 1 phase, and that time in S/G 2 /M is relatively fixed, is widely accepted. Furthermore, mathematical models adopting this mechanical description (so-called "transition probability" or "compartment" models) remain popular and form the basis of many studies of lymphocyte and cancer kinetics in vitro and in vivo today (13-21).More recently, a molecular description of cell cycle regulation, including the discovery of key regulatory proteins such as cyclins a...