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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.

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

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  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

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