Introduction to Quantum Algorithms via Linear Algebra by Richard J. Lipton

Author:Richard J. Lipton
Language: eng
Format: epub
Tags: Quantum computing; quantum algorithms; computing; linear algebra
Publisher: MIT Press

The finest point is that, as far as we know, the atoms are identical as they enter the magnetic field. Some go up, and some go down, in ways we cannot tell in advance. We can attenuate the beam so that most of the time only a single atom is traveling through the device. We still cannot tell as it enters which path it will later take. There may be some unknown factors that determine the path, but—and this is also important—our methods of preparing the atoms for their flight are not knowingly biased in regard to them. We say more about possible “hidden variables” in section 14.10 discussing 14.7. The point for now is not what the atoms “are” but what they do:

We have a device that allows us repeatedly to sample results by which we get the outcome ↑ with some probability p and ↓ with probability 1 − p.

We can prepare the atoms to give a known value p, for instance, p = 0.5, from a configurable set of possible values. We can observe that, when we shoot many atoms, almost always the frequency of those giving the outcome ↑ is close to p.

The atoms behave as if controlled by complex numbers a,b such that |a|2 = p and |b|2 = 1 − p. We can also prepare those numbers, but we cannot examine a flying atom and tell what a and b it has unless we already know. We can write the state in Dirac notation as

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