Timeline for In quantum dynamical simulations, what is the symmetric (Riemannian) analog of a Poisson bracket?
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6 events
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| Jun 2, 2011 at 14:06 | comment | added | John Sidles | Similar operator product expansions, with implicit rather than explicit evaluation, are discussed in (e.g.) Hogben, Hore, and Kuprov "Strategies for state space restriction in densely coupled spin systems with applications to spin chemistry" (2010). But these restriction methods don't work so well in integrating long-time trajectories of very large, grossly inhomogeneous dynamical systems, in essence because these methods restrict the density matrix rather than the trajectories themselves. The motivation of the question is to leave Hilbert space (and thus density matrices) wholly behind... | |
| Jun 1, 2011 at 22:42 | comment | added | Alex 'qubeat' | Yes, the Clifford algebra are isomorphic with algebra of $2^n \times 2^n$ complex matixes, but it is not always necessary to write down this matrices. Expressions with products of few generators correspond to some subspaces of the exponential space, e.g. products of two generators -- is simply Lie algebra so(2n). | |
| Jun 1, 2011 at 21:43 | comment | added | John Sidles | The practical difficulty associated to canonical anticommutation relations is that (as Nielsen's manuscript discusses) these operators have exponentially large dimension ... this is bad for code efficiency! Canonical commutators are even worse ... they formally require infinite-dimension Hilbert spaces. For the case of commutators, a well-known technique is to modify the commutation relation by putting $I_z$ on the right, thus converting it to an SU(N) algebra (where N can be small). I know of no similar code-speeding trick for anti-commutators... that's one practical point of the question. | |
| Jun 1, 2011 at 19:12 | comment | added | Alex 'qubeat' | Canonical anticommutation relations with $n$ generators has a standard description using complex Clifford algebras with $2n$ generators, i.e. $a_n = (e_{2n}+ i e_{2n+1})/2$, $a^*_n = (e_{2n}- i e_{2n+1})/2$. Could it help? | |
| Jun 1, 2011 at 14:42 | answer | added | Theo Johnson-Freyd | timeline score: 2 | |
| May 31, 2011 at 0:13 | history | asked | John Sidles | CC BY-SA 3.0 |