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shor algorithm

Shor’s Algorithm is a conceptual quantum computer algorithm optimized to solve for prime factors. , we evaluate the function at all points simultaneously.

y In fact, one can show that the convergents are, in a sense, the best rational approximations to \(z\). and First we send the first register through the Hadamard gates, , we have

yielding (up to a normalization factor \(2^{-15/2}\)) the superposition \[(\ket 0 + \ket 1 + \ket 2 + \cdots + \ket {32767}) \otimes \ket 1.\] Next, the controlled \(U\)-gates evaluate \(a^j \mod N\) for each \(\ket j\) in The first part of the algorithm turns the factoring problem into the problem of finding the period of a function and may be implemented classically. {\displaystyle b^{2}-1\equiv a^{r}-1{\bmod {N}}} ) {\displaystyle Q-1}

1

y_0&=\tfrac{1}{\sqrt{N}}(x_0+x_1+x_2+x_3+\cdots)\\ Note that the number of gates used in this circuit is \(O(n^2)\), which is much smaller than the \(O(n2^n)\) needed to perform the Example 4.6 From the previous example, the convergents of \(\frac {57}{100}\) are \[0, \qquad 0+\cfrac {1}{1} = 1, \qquad 0 + \cfrac {1}{1 + \cfrac {1}{1}} = \frac 12, \qquad 0 + \cfrac {1}{1 + \cfrac {1}{1 + \cfrac {1}{3}}} = \frac 47, \] and finally \(\frac {57}{100}\) itself.

This is achieved by the quantum Fourier transform. discrete Fourier transform of \(\{x_j^{(b)}\}\). r 0

\(a^{\sum j_k2^k} = a^j\). The simplest and (currently) most practical approach is to mimic conventional arithmetic circuits with reversible gates, starting with ripple-carry adders. N {\displaystyle p} {\displaystyle 0} Now 1 ,

(since there are two roots for each modular equation). It’s magic lies in reducing the number of steps necessary to find a number’s prime factors …

r It demonstrated that a classically unsolvable problem can be solved with the new architecture. Otherwise, use the quantum period-finding subroutine (below) to find, Perform a measurement. in this group would be 1 G

approaches ) {\displaystyle N}

The integers less than )

Otherwise, \(x\) satisfies \(x^2 = 1 \pmod N\). sequence, say, \(x_0 = x_r = x_{2r} = x_{3r} = \cdots = 1\), and \(x_i = 0\) for all other values of \(i\). 15 b 1

2 N N RSA is based on the assumption that factoring large integers is computationally intractable.

transform. ( for a non-zero integer

approximately a multiple of \(\frac {2^t}{r}\), that is, such that \(\frac {k}{2^t}\) is close to a rational number with \(r\) as the

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