The quantum speed up as advanced knowledge of the solution

With reference to a search in a database of size N, Grover states: "What is the reason that one would expect that a quantum mechanical scheme could accomplish the search in O(square root of N) steps? It would be insightful to have a simple two line argument for this without having to describe the details of the search algorithm". The answer provided in this work is: "because any quantum algorithm takes the time taken by a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem". This empirical fact, unnoticed so far, holds for both quadratic and exponential speed ups and is theoretically justified in three steps: (i) once the physical representation is extended to the production of the problem on the part of the oracle and to the final measurement of the computer register, quantum computation is reduction on the solution of the problem under a relation representing problem-solution interdependence, (ii) the speed up is explained by a simple consideration of time symmetry, it is the gain of information about the solution due to backdating, to before running the algorithm, a time-symmetric part of the reduction on the solution; this advanced knowledge of the solution reduces the size of the solution space to be explored by the algorithm, (iii) if I is the information acquired by measuring the content of the computer register at the end of the algorithm, the quantum algorithm takes the time taken by a classical algorithm that knows in advance 50% of I, which brings us to the initial statement.Comment: 23 pages, to be published in IJT

The 50% advanced information rule of the quantum algorithms

The oracle chooses a function out of a known set of functions and gives to the player a black box that, given an argument, evaluates the function. The player should find out a certain character of the function through function evaluation. This is the typical problem addressed by the quantum algorithms. In former theoretical work, we showed that a quantum algorithm requires the number of function evaluations of a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem. Here we check that this 50% rule holds for the main quantum algorithms. In the structured problems, a classical algorithm with the advanced information, to identify the missing information should perform one function evaluation. The speed up is exponential since a classical algorithm without advanced information should perform an exponential number of function evaluations. In unstructured database search, a classical algorithm that knows in advance 50% of the n bits of the database location, to identify the n/2 missing bits should perform Order(2 power n/2) function evaluations. The speed up is quadratic since a classical algorithm without advanced information should perform Order(2 power n) function evaluations. The 50% rule identifies the problems solvable with a quantum sped up in an entirely classical way, in fact by comparing two classical algorithms, with and without the advanced information.Comment: 18 pages, submitted with minor changes to the International Journal of Theoretical Physic

Stochastic thermodynamics of holonomic systems

International audienc

Kinetic analysis of channel gating. Application to the cholinergic receptor channel and the chloride channel from Torpedo californica

Identification of the minimum number of ways in which open and closed states communicate is a crucial step in defining the gating kinetics of multistate channels. We used certain correlation functions to extract information about the pathways connecting the open and closed states of the cation channel of the purified nicotinic acetylcholine receptor and of the chloride channel of Torpedo californica electroplax membranes. Single channel currents were recorded from planar lipid bilayers containing the membrane channel proteins under investigation. The correlation functions are conveniently computed from single channel current records and yield information on E, the minimum number of entry/exit states into the open or closed aggregates. E gives a lower limit on the numbers of transition pathways between open and closed states. For the acetylcholine receptor, the autocorrelation analysis shows that there are at least two entry/exit states through which the open and closed aggregates communicate. The chloride channel fluctuates between three conductance substates, here indentified as C, M, and H for closed, intermediate, and high conductance, respectively. Correlation analysis shows that E is greater than or equal to 2 for the M aggregate, indicating that there are at least two distinct entry/exit states in the M aggregate. In contrast, there is no evidence for the existence of more than one entry/exit state in the C or H aggregates. Thus, these correlation functions provide a simple and general strategy to extract information on channel gating kinetics

Electrostatic calculations of amino acid titration and electron transfer, Q-AQB-->QAQ-B, in the reaction center

The titration of amino acids and the energetics of electron transfer from the primary electron acceptor (QA) to the secondary electron acceptor (QB) in the photosynthetic reaction center of Rhodobacter sphaeroides are calculated using a continuum electrostatic model. Strong electrostatic interactions between titrating sites give rise to complex titration curves. Glu L212 is calculated to have an anomalously broad titration curve, which explains the seemingly contradictory experimental results concerning its pKa. The electrostatic field following electron transfer shifts the average protonation of amino acids near the quinones. The pH dependence of the free energy between Q-AQB and QAQ-B calculated from these shifts is in good agreement with experiment. However, the calculated absolute free energy difference is in severe disagreement (by approximately 230 meV) with the observed experimental value, i.e., electron transfer from Q-A to QB is calculated to be unfavorable. The large stabilization energy of the Q-A state arises from the predominantly positively charged residues in the vicinity of QA in contrast to the predominantly negatively charged residues near QB. The discrepancy between calculated and experimental values for delta G(Q-AQB-->QAQ-B) points to limitations of the continuum electrostatic model. Inclusion of other contributions to the energetics (e.g., protein motion following quinone reduction) that may improve the agreement between theory and experiment are discussed