Discrete Quantum Mechanics by H Thomas Williams

Author:H Thomas Williams
Language: eng
Format: epub
ISBN: 9781681741253
Publisher: IOP Publishing
Published: 2015-12-11T16:00:00+00:00

Another is contributed to the phase, therefore, whenever corresponding digits in x and y are both equal to 1. We will use the notation

to capture this notion, where we mean by the sum, modulo 2, of the products of corresponding digits of x and y.8 We thus rewrite equation (3.12) as

and the input register, as a result of the applied Hadamard gates, becomes

(3.13)

The final form of this expression shows that the sum is limited to those values of y for which The final step in the algorithm, measuring the input register in its natural basis, will give as a result only values of y that satisfy this condition, thus revealing information about the parameter s.

Each time the algorithm is repeated, a random value of y satisfying is revealed from among the possibilities. We need to find linearly independent values of y (necessarily nonzero) to completely determine s. The first nonzero y to be picked, y1, has a pattern of 1s, implying that if one sums the bits of s found in that same pattern of locations, the sum is even. Half of the possible s values have that property. The probability of obtaining information about s from a y pick is thus

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