1 / 17

Can Quantum Search take place in Nature ? Lov K. Grover

Can Quantum Search take place in Nature ? Lov K. Grover . Searching – – some background and the algorithm Applications. Problem Structure. Most of theoretical computer science is about making use of the structure of a problem to devise efficient ways of solving the problem.

tallis
Télécharger la présentation

Can Quantum Search take place in Nature ? Lov K. Grover

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Can Quantum Search take place in Nature ?Lov K. Grover • Searching – – some background and the algorithm • Applications NKS 2008

  2. Problem Structure • Most of theoretical computer science is about making use of the structure of a problem to devise efficient ways of solving the problem. • Slab design problem is reduced to bin-packing NKS 2008

  3. Problems without structure • Searching an unsorted dictionary. • Classically, not much scope for improvement over exhaustive searching. Find fastest pizza delivery time? Is there a pizza delivery time of 14.9 minutes? NKS 2008

  4. Answer - piranha Quantum Search You are given a dictionary with 1,000,000 words. Crossword Puzzle Challenge: _ _ r _ n h _ • Classical system needs to examine ~ 500,000 words. • Quantum mechanical superposition is in multiple states simultaneously. • Quantum search algorithm does exhaustive search • over N items in O(N) steps (about 1,000 steps). NKS 2008

  5. Walsh-Hadamard Transformation (W) • N = 2n state quantum system built out of n 2-state quantum systems called qubits. • In W, each of the n qubits is independently transformed by H. • W-1=WWt0=1/√N for any t NKS 2008

  6. f(x) x (n qubits) x (n qubits) Input Output B XOR f(x) B (1 qubit) CNOT Selective Inversion • Diagonal matrix with all 1’s on diagonal except certain selected (target) elements where it is -1 (denoted by It ). • Circuit to selectively invert amplitudes in states where f(x) is 1 (amplitude unchanged in states where f(x)=0) NKS 2008

  7. Quantum Search Algorithm • Encode N states with log2N qubits. • Start with all qubits in 0 state. • Apply the following operations: • It and Is are selective inversions of source & target states. • Observe the state. NKS 2008

  8. f(x) Given the following block - 0/1 Optimality of quantum search algorithm We are allowed to hook up O(log N) hardware. Problem - find the single point at which f(x) ≠ 0. • Classically we need N steps. • Quantum mechanically, we need only √N steps. Quantum search algorithm is best possible algorithm for exhaustive searching. - Chris Zalka, Phys. Rev. A, 1999 NKS 2008

  9. Amplitude Amplification • - Vts can be amplified by a factor of 2h by iterating the • operations IsV-1ItV, h times. • - True for any unitary operation V. • Choosing V as the Walsh-Hadamard transformation gives • the quantum search algorithm. NKS 2008

  10. Quantum Searching is tolerant of certain kinds of errors • V can be arbitrary. • Is & It can be replaced by arbitrary phase shifts. • This flexibility indicates that the algorithm will be more widely applicable. NKS 2008

  11. Biological computation • DNA & protein synthesis in cells requires the same type of computation as in computers. • In computers, binary representation is most efficient (information theory). • Mystery – DNA computation in cells uses an alphabet of size 4 and protein synthesis uses an alphabet of size 19 (19 amino acids). NKS 2008

  12. Does biological computation use quantum searching? • Mystery – DNA computation in cells uses an alphabet of size 4 and protein synthesis uses an alphabet of size 20 (20 amino acids). • Observation – (Apoorva Patel, 2000) – Quantum searching is most efficient when carried out for databases of size: 4 - (requires 1 query) 10 - (requires 2 queries) 20 – ( requires 3 queries) NKS 2008

  13. Spatial Quantum Searching • Quantum search type algorithms can be used to search a distributed database using only neighbor to neighbor communications.with a square-root speedupover a classical database. (Aaronson 2001) NKS 2008

  14. Spatial Quantum Searching • A robot needs to go to one marked cell in a multidimensional array of N items. • A classical robot will need O(N) steps to find and reach the marked cell. • A quantum robot will need only O(√N) steps O(N) steps O(√N) steps NKS 2008 NKS 2008

  15. Photosynthesis • Photosynthesis in plants is extremely efficient – unlike other chemical & biological reactions, it is able to transport almost 100% of the photons to the desired locations. • This used to be a mystery – one possible explanation was given by Fleming et al (Nature – April 14, 2007) where they analyzed the way the energy flows in the cells as a spatial quantum search problem. NKS 2008

  16. Quantum Computing? • Can FMO act as a dedicated quantum computational device? • No strict one-to-one mapping onto quantum search algorithms such as Grover’s algorithm But • Coherent superposition states allow the complex to sample many states very rapidly. • A trap ‘ decoheres’ the system • Is the system ‘designed’ to preserve coherence? “Quantum Mechanics Helps in Searching for a Needle in a Haystack.” Grover, PRL, 79:325 1997

  17. Does Nature use Quantum Searching? Original algorithm - Database search & function inversion Applications: • Collision problem & Element Distinctness • Communication algorithms • Precision Measurements • Pendulum Modes Natural applications • Genetic Pattern Matching (??) • Photosynthesis (??) “A good idea finds application in contexts beyond where it was originally conceived.” NKS 2008

More Related