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SUMMARY:Some algebraic structures in computational quantum mechanics - Pra
 nav Singh (University of Cambridge)
DTSTART:20151126T141500Z
DTEND:20151126T151500Z
UID:TALK62618@talks.cam.ac.uk
CONTACT:William Matthews
DESCRIPTION:*Background and Motivation*\nWhen Feynman first proposed the n
 otion of quantum computers in 1982\, his primary motivation was efficient 
 computation of quantum physics and quantum chemistry. On a classical compu
 ter we encounter exponential slowdown when attempting to simulate multi-pa
 rticle systems due to the large Hilbert space such quantum systems reside 
 in. A universal quantum computer should\, in principle\, be able to overco
 me this curse of dimensionality by utilising entanglement. \n\nHowever\, t
 here are two major reasons why advances in classical algorithms for solvin
 g equations of quantum mechanics are desperately needed.\n\nFirstly and ir
 onically\, the process of creating quantum computers will itself require r
 outinely solving equations of quantum mechanics such as the time-dependent
  Schrodinger equation for predicting the dynamics of qubits and designing 
 controls and gates using lasers\, magnetic fields etc. These will need to 
 be solved on the only devices available to us -- the classical computers. 
 \n\nSecondly\, a major stumbling block will still remain even once quantum
  computers become available: Trotter splitting based quantum computing alg
 orithms have already been devised and\, unsurprisingly\, they need a very 
 small time step for a reasonably accurate simulation. The Trotter splittin
 g is the lowest order method among exponential splittings which allow us t
 o propagate the actions of different components of a Hamiltonian separatel
 y. We need higher-order exponential splittings for overcoming the small ti
 me step barrier.\n\n*Higher-Order Methods and Algebraic Structures*\n\nThe
  main focus of my talk will be on some higher-order exponential splitting 
 algorithms for efficiently solving the time-dependent Schrödinger equatio
 n. These algorithms are entirely in the classical language but it should b
 e possible to utilise them to devise corresponding quantum algorithms.\n\n
 In particular\, I will talk about the Zassenhaus splittings whose costs gr
 ow quadratically in contrast to the exponential growth in cost of competin
 g Yosida splittings. This exponential speedup\, along with unitarity\, sta
 bility\, convergence and error bounds of these splittings is traced to the
  structural properties of a Z2 graded Lie algebra. \n\nWe will see how the
 se Lie algebras arise in the context of any associative algebra with a com
 mutative subalgebra and its non-trivial Lie idealiser. The generality of t
 his framework suggests that the results should extend to many other equati
 ons of quantum mechanics. 
LOCATION:MR4\, Centre for Mathematical Sciences\, Wilberforce Road\, Cambr
 idge
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