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Windows on Theory
Posted by Michael Freedman, managing director of Station Q
Station Q focuses on the physics of those condensed-matter quantum systems that offer the promise of intrinsic or “topological” protection from error and decoherence. Such systems are likely to play an important role in the architecture of quantum computers. We also enjoy trying to understand what quantum computers will be able to do once they are built.The answer is certainly not in asymptotic formulations; the constants matter. During a recent meeting on quantum chemistry, I learned that for problems that will be at the forefront in the next couple of decades, the limiting factor for quantum algorithms is not the number of qubits but the number of gate operations. One easily produces numbers like 10^20 if one does the “obvious”: imitate the unitary evolution you wish to study with fine “Trotter time steps” and build each step—which, because of the fineness, will agree with the identity matrix to a dozen decimal places—by a composition of millions of gates. Here is a hint that we might think of something cleverer.Number theorists know Ramanujan’s constant: e^root(163)*pi = 262,537,412,640,768,743.99999999999925. …. It is interesting—and not a mystery to experts—that the left-hand side, writeable as six standard characters, produces 12 nines on the right-hand side. This example leads me to speculate that the subject “arithmetic dynamical systems” will be of value in finding algorithms for rapid convergence to Trotter steps—which analogously to being nearly integral are nearly identical—and for other problems in quantum computer science.