Higherorder composite pulses Kenneth Brown Isaac Chuang Aram
Higher-order composite pulses Kenneth Brown, Isaac Chuang, Aram Harrow Background Contribution General problem: An algorithm to construct a pulse sequence to correct to order n for any positive integer n. This is often hard. There are two methods to achieve this: Method 1: This means that when we think we are applying a rotation by q, we actually apply a rotation by q(1+e) where e is unknown, but constant. Physically, this might correspond to a macroscopic NMR sample that experiences an inhomogenous RF field. Since e varies through the sample, it cannot be eliminated by calibration. This means that our solution must work for all sufficiently small values of e. Write the desired pulse as e. X+Y for some feasible X and Y. Expand e. X+Y=ea. Xea. Yeb. Xeb. Yec. Xec. Y… This sequence (a, b, c, …) is a universal method of creating composite pulses (i. e. works for any X and Y). Unfortunately, its length is exponential in n, and the only known algorithm for finding the sequence is also exponential in n. Solution idea: Construct composite pulses in which several faulty pulses simulate one good pulse. Notation used in graphs: Our goal is to reduce the error from e to All pulses in NMR are rotations by angles in the plane, so they can be represented by vectors. O(en) for some n>1. Previous work: Previously, only constructions based on Tycko’s average Hamiltonian theory that map e to O(e 2) or O(e 3) were known. [Wimperis, J. Magn. Reson. B 109, 221 (1994)] first order composite pulse sequence desired correction Method 2: Use the Trotter-Suzuki expansion to approximate the desired correction as a sequence of pulses. Consider only systematic errors in pulse power. Second order: 6 pulses Use an adapted version of the Solovay-Kitaev theorem from quantum computing. The elements of the algorithm are: • Order-by order cancellation: First cancel the e 1 term, which introduces e 2 and higher-order errors. Then cancel the e 2 term, again introducing higher-order errors, etc… • Recursive construction of the en step: Use the fact that exp(ea. A)exp(eb. B)exp(-ea. A)exp(-eb. B)¼exp(ea+b(ABBA)) Third order: 18 pulses The running time is poly(n), and we conjecture that the number of pulses is also poly(n) (roughly n 3), but haven’t proven this. Effective error vs. base error with method 2 Fourth order: 44 pulses If we perform a rotation by q, then our correction should be qe. These corrections can be built out of rotations by 2 pe. To first-order, these rotations add (so the vectors will always sum to zero), but the higher-order terms are more complicated. second order composite pulse sequence desired correction First order: 2 pulses Our goal: Coherent control of quantum systems, as in nuclear magnetic resonance (NMR), requires the ability to precisely control the timing, power, frequency and phase of radiofrequency (RF) pulses. Simplified problem: Results Fifteenth order: 1, 884 pulses
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