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Shor's algorithm, first introduced by mathematician Peter Shor, is a quantum algorithm for integer factorization. On a quantum computer, to factor an integer $N$, Shor's algorithm takes polynomial time in $log\{N\}$, specifically $O((log\{N\})^3)$, demonstrating that integer factorization is in the complexity class BQP. This is exponentially faster than the best-known classical factoring algorithm, the general number field sieve, which works in sub-exponential time - about $O(2^\{\{(log\; N)\}^\{1/3\}\})$. Peter Shor discovered the eponymous algorithm in 1994.

Shor's algorithm is important because it can, in theory, be used to "break" the widely used public-key cryptography scheme known as RSA. RSA is based on the assumption that factoring large numbers is computationally infeasible. So far as is known, this assumption is valid for classical computers. No classical algorithm is known that can factor in time polynomial in log N. However, Shor's algorithm shows that factoring is efficient on a quantum computer, so an appropriately large quantum computer can "break" RSA. It was also a powerful motivator for the design and construction of quantum computers and for the study of new quantum computer algorithms.

In 2001, Shor's algorithm was demonstrated by a group at IBM, who factored 15 into 3 x 5, using an NMR implementation of a quantum computer with 7 qubits. However, some doubts have been raised as to whether their IBM's experiment was a true demonstration of quantum computation, since no entanglement was observed. Since IBM's implementation, several other groups have implemented Shor's algorithm using photonic qubits, emphasizing that entanglement was observed.

The problem we are trying to solve is: given a composite number N, find an integer p, strictly between 1 and N, that divides N.

Shor's algorithm consists of two parts:

- A reduction of the factoring problem to the problem of order-finding, which can be done on a classical computer.
- A quantum algorithm to solve the order-finding problem.

- Pick a random number a < N
- Compute gcd(a, N). This may be done using the Euclidean algorithm.
- If gcd(a, N) ≠ 1, then there is a nontrivial factor of N, so we are done.
- Otherwise, use the period-finding subroutine (below) to find r, the period of the following function:
- $f(x)\; =\; a^x\; mbox\{mod\}\; N$,

- If r is odd, go back to step 1.
- If a
^{r /2}≡ -1 (mod N), go back to step 1. - gcd(a
^{r/2}± 1, N) is a nontrivial factor of N. We are done.

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f(x) = a^{x} mod N. Given N, find Q = 2^{q} such that $N^2\; le\; Q\; <\; 2N^2$, which implies $Q/r\; >\; N$. The input and output qubit registers need to hold superpositions of values from 0 to Q − 1, and so have q qubits each. Using what might appear to be twice as many qubits as necessary guarantees that there are at least N different x which produce the same f(x), even as the period r approaches N/2.

Proceed as follows:

- Initialize the registers to
- $Q^\{-1/2\}\; sum\_\{x=0\}^\{Q-1\}\; left|xrightrangle\; left|0rightrangle$

where x runs from 0 to Q − 1. This initial state is a superposition of Q states.

- Construct f(x) as a quantum function and apply it to the above state, to obtain
- $Q^\{-1/2\}\; sum\_x\; left|xrightrangle\; left|f(x)rightrangle$.

This is still a superposition of Q states.

- Apply the quantum Fourier transform to the input register. This transform (operating on a superposition of power-of-two Q = 2
^{q}states) uses a Q^{th}root of unity such as $omega\; =\; e^\{2\; pi\; i\; /Q\}$ to distribute the amplitude of any given $left|xrightrangle$ state equally among all Q of the $left|yrightrangle$ states, and to do so in a different way for each different x:- $U\_\{QFT\}\; left|xrightrangle$

This leads to the final state

- $Q^\{-1\}\; sum\_x\; sum\_y\; omega^\{x\; y\}\; left|yrightrangle\; left|f(x)rightrangle$.

This is a superposition of many more than Q states, but many fewer than Q

^{2}states. Although there are Q^{2}terms in the sum, the state $left|yrightrangle\; left|f(x\_0)rightrangle$ can be factored out whenever x_{0}and x produce the same value. Let- $omega\; =\; e^\{2\; pi\; i\; /Q\}$ be a Q
^{th}root of unity,

- r be the period of f,

- x
_{0}be the smallest of a set of x which yield the same given f(x) (we have x_{0}< r), and

- b run from 0 to $lfloor(Q-x\_0-1)/rrfloor$ so that $x\_0\; +\; rb\; <\; Q$.

- $sum\_\{x:,\; f(x)=f(x\_0)\}\; omega^\{x\; y\}\; =\; sum\_\{b\}\; omega^\{(x\_0\; +\; r\; b)\; y\}\; =\; omega^\{x\_0y\}\; sum\_\{b\}\; omega^\{r\; b\; y\}$.

- Perform a measurement.
We obtain some outcome y in the input register and $f(x\_0)$ in the output register.
Since f is periodic, the probability of measuring some pair y and $f(x\_0)$ is given by
- $left|\; Q^\{-1\}\; sum\_\{x:,\; f(x)=f(x\_0)\}\; omega^\{x\; y\}\; right|^2$

Analysis now shows that this probability is higher, the closer unit vector $omega^\{ry\}$ is to the positive real axis, or the closer yr/Q is to an integer.

- Turn y/Q into an irreducible fraction, and extract the denominator r′, which is a candidate for r.
- Check if f(x) = f(x + r′) $Leftrightarrow$ $a^r\; equiv\; 1\; pmod\{N\}$. If so, we are done.
- Otherwise, obtain more candidates for r by using values near y, or multiples of r′. If any candidate works, we are done.
- Otherwise, go back to step 1 of the subroutine.

The algorithm is composed of two parts. 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. The second part finds the period using the quantum Fourier transform, and is responsible for the quantum speedup.

The integers less than N and coprime with N form a finite group under multiplication modulo N. By the end of step 3, we have an integer a in this group. Since the group is finite, a must have a finite order r, the smallest positive integer such that

- $a^r\; equiv\; 1\; mbox\{mod\}\; N.,$

Therefore, N Divides (a ^{r} − 1 ). Suppose we are able to obtain r, and it is even. Then

- $a^r\; -\; 1\; =\; (a^\{r/2\}\; -\; 1)\; (a^\{r/2\}\; +\; 1)\; equiv\; 0\; mbox\{mod\}\; N$

- $Rightarrow\; N\; |\; (a^\{r/2\}\; -\; 1)\; (a^\{r/2\}\; +\; 1).,$

r is the smallest positive integer such that a ^{r} ≡ 1, so N cannot divide (a ^{r / 2} − 1). If N also does not divide (a ^{r / 2} + 1), then N must have a nontrivial common factor with each of (a ^{r / 2} − 1) and (a ^{r / 2} + 1).

Proof: For simplicity, denote (a ^{r / 2} − 1) and (a ^{r / 2} + 1) by u and v respectively. N | uv, so kN = uv for some integer k. Suppose gcd(u, N) = 1; then mu + nN = 1 for some integers m and n (this is a property of the greatest common divisor.) Multiplying both sides by v, we find that mkN + nvN = v, so N | v. By contradiction, gcd(u, N) ≠ 1. By a similar argument, gcd(v, N) ≠ 1.

This supplies us with a factorization of N. If N is the product of two primes, this is the only possible factorization.

Shor's period-finding algorithm relies heavily on the ability of a quantum computer to be in many states simultaneously. Physicists call this behavior a "superposition" of states. To compute the period of a function f, we evaluate the function at all points simultaneously.

Quantum physics does not allow us to access all this information directly, though. A measurement will yield only one of all possible values, destroying all others. But for the no cloning theorem, we could first measure f(x) without measuring x, and then make a few copies of the resulting state (which is a superposition of states all having the same f(x)). Measuring x on these states would provide different x values which give the same f(x), leading to the period. Because we cannot make exact copies of a quantum state, this method does not work. Therefore we have to carefully transform the superposition to another state that will return the correct answer with high probability. This is achieved by the quantum Fourier transform.

Shor thus had to solve three "implementation" problems. All of them had to be implemented "fast", which means that they can be implemented with a number of quantum gates that is polynomial in $log\; N$.

- Create a superposition of states.
This can be done by applying Hadamard gates to all qubits in the input register. Another approach would be to use the quantum Fourier transform (see below).

- Implement the function f as a quantum transform.
To achieve this, Shor used repeated squaring for his modular exponentiation transformation. It is important to note that this step is more difficult to implement than the quantum Fourier transform, in that it requires ancillary qubits and substantially more gates to accomplish.

- Perform a quantum Fourier transform.
By using controlled rotation gates and Hadamard gates Shor designed a circuit for the quantum Fourier transform (with Q = 2

^{q}) that uses just $q(q-1)/2\; =\; O((log\; Q)^2)$ gates. arXiv:quant-ph/9508027v2 p. 14

After all these transformations a measurement will yield an approximation to the period r. For simplicity assume that there is a y such that yr/Q is an integer. Then the probability to measure y is 1. To see that we notice that then

- $e^\{-2\; pi\; i\; b\; yr/Q\}\; =\; 1,$

Note: another way to explain Shor's algorithm is by noting that it is just the quantum phase estimation algorithm in disguise.

- Revised version of the original paper by Peter Shor ("28 pages, LaTeX. This is an expanded version of a paper that appeared in the Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 1996"). This preprint was eventually published as SIAM J.Sci.Statist.Comput. 26 (1997) 1484.

- Quantum Computation and Quantum Information, Michael A. Nielsen, Isaac L. Chuang, Cambridge University Press, 2000.

- A general textbook on quantum computing.

- This book was recommended (includes "a complete review of Shor’s algorithm") in the discussion of Aaronson's blog article (see below).

- Efficient Networks for Quantum Factoring, David Beckman, Amalavoyal N. Chari, Srikrishna Devabhaktuni, and John Preskill, Phys. Rev. A 54, 1034–1063 (1996).

- The authors investigate and optimize the resource requirements of Shor's algorithm. They determine the time complexity of factoring N to be about $72\; (log\; N)^3$, using a quantum computer with about $5\; log\; N$ qubits.

- Lieven M. K. Vandersypen, Matthias Steffen, Gregory Breyta, Costantino S. Yannoni, Mark H. Sherwood & Isaac L. Chuang, Nature 414, 883–887 (20 Dec 2001). abstract

- An implementation of Shor's Algorithm that factorizes the number 15.

- Juha J. Vartiainen, Antti O. Niskanen, Mikio Nakahara, Martti M. Salomaa

- Austin G. Fowler, Simon J. Devitt, Lloyd C. L. Hollenberg

- Quant. Info. Comput. 4, 237-251 (2004)

- David McAnally. 45 pages. A refinement of Shor's Algorithm for determining order is introduced, which determines a divisor of the order after any one run of a quantum computer with almost absolute certainty. The information garnered from each run is accumulated to determine the order, and for any k greater than 1, there is a guaranteed minimum positive probability that the order will be determined after at most k runs. The probability of determination of the order after at most k runs exponentially approaches a value negligibly less than one, so that the accumulated information determines the order with almost absolute certainty. The probability of determining the order after at most two runs is more than 60%, and the probability of determining the order after at most four runs is more than 90%.

- "Explanation for the man in the street" by Scott Aaronson, " approved" by Peter Shor. (Shor wrote "Great article, Scott! That’s the best job of explaining quantum computing to the man on the street that I’ve seen."). Scott Aaronson suggests the following 12 sites as further reading (out of "the 10
^{10}5000 quantum algorithm tutorials that are already on the web."): - arXiv quant-ph/9508027 Shor's revised paper See above for details.
- Quantum Computing and Shor's Algorithm, Matthew Hayward, 2005-02-17, imsa.edu, LaTeX2HTML version of the original 2750 line LaTeX document, also available as a 61 page PDF or postscript document.
- Quantum Computation and Shor's Factoring Algorithm, Ronald de Wolf, CWI and University of Amsterdam, January 12, 1999, 9 page postscript document.
- Shor's Factoring Algorithm, Notes from Lecture 9 of Berkeley CS 294-2, dated 4 Oct 2004, 7 page postscript document.
- Chapter 6 Quantum Computation, 91 page postscript document, Caltech, Preskill, PH229.
- Quantum computation: a tutorial by Samuel L. Braunstein
- The Quantum States of Shor's Algorithm, by Neal Young, Last modified: Tue May 21 11:47:38 1996.
- A now-circular reference via the Avoid self-references copy of this article; clearly Aaronson's link originally reached the 20 Feb 2007 version
- III. Breaking RSA Encryption with a Quantum Computer: Shor's Factoring Algorithm, LECTURE NOTES ON QUANTUM COMPUTATION, Cornell University, Physics 481-681, CS 483; Spring, 2006 by N. David Mermin. Last revised 2006-03-28, 30 page PDF document.
- arXiv quant-ph/0303175 Shor's Algorithm for Factoring Large Integers. C. Lavor, L.R.U. Manssur, R. Portugal Submitted on 29 Mar 2003. This work is a tutorial on Shor's factoring algorithm by means of a worked out example. Some basic concepts of Quantum Mechanics and quantum circuits are reviewed. It is intended for non-specialists which have basic knowledge on undergraduate Linear Algebra. 25 pages, 14 figures, introductory review.
- arXiv quant-ph/0010034 Shor's Quantum Factoring Algorithm, Samuel J. Lomonaco, Jr, Submitted on 9 Oct 2000, This paper is a written version of a one hour lecture given on Peter Shor's quantum factoring algorithm. 22 pages.
- Chapter 20 Quantum Computation, from Computational Complexity: A Modern Approach, Draft of a book: Dated January 2007, Comments welcome!, Sanjeev Arora and Boaz Barak, Princeton University.
- A demonstration of Shor's algorithm in PHP

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Last updated on Thursday September 25, 2008 at 08:32:01 PDT (GMT -0700)

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