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## Code reduction shoro

bequiv ar/2bmod N is a square root of 1displaystyle 1 modulo Ndisplaystyle N that is different from 1displaystyle. ArXiv : 1411.6758 quant-ph. 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. Discrete logarithms edit Given prime pdisplaystyle p with generator gdisplaystyle g where 1 g p1displaystyle 1 g p-1, suppose we know that xgrmodpdisplaystyle xgrbmod p, nyx makeup coupons for some rdisplaystyle r, and we wish to compute rdisplaystyle r, which is the discrete logarithm. There are several approaches to constructing and optimizing circuits for modular exponentiation. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer. Since Ndisplaystyle N is not a power of a prime, it is the product of two coprime integers greater than 1displaystyle. The input and output qubit registers need to hold superpositions of values from 0displaystyle 0 to Q1displaystyle Q-1, and so have qdisplaystyle q qubits each. On a quantum computer, to factor an integer Ndisplaystyle N, Shor's algorithm runs in polynomial time (the time taken is polynomial in logNdisplaystyle log N, the size of the integer given as input). "Constant-Optimized Quantum Circuits for Modular Multiplication and Exponentiation". Otherwise, try again starting from step 1 of this subroutine.

Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm that runs on a quantum computer) for integer factorization, formulated in 1994.

Informally, it solves the following problem: Given an integer N displaystyle N, find its prime factors.

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9 In April 2012, the factorization of 1431113displaystyle 14311times 13 was achieved, although this used adiabatic quantum computation rather than Shor's algorithm. Scott Aaronson suggests the following 12 references as further reading (out of "the quantum algorithm tutorials that are already on the web. 2, specifically, it takes quantum gates of order O!left(log N)2(log log N log log log N)right) using fast multiplication, 3 thus demonstrating that the integer-factorization problem can be efficiently solved on a quantum computer and is consequently in the complexity class, bQP. There are several approaches to constructing and optimizing circuits for modular exponentiation. "Shor's Algorithm for Factoring Large Integers". Displaystyle arequiv 1bmod. This is a superposition of many more than Qdisplaystyle Q states, but many fewer than Q2displaystyle Q2 states, as there are fewer than Qdisplaystyle Q distinct values of zf(x)displaystyle zf(x). Since yrQdisplaystyle frac yrQ is close to some integer cdisplaystyle c, the known value yQdisplaystyle dfrac yQ is close to the unknown value crdisplaystyle dfrac. Quantum Computation and Shor's Factoring Algorithm, Ronald de Wolf, CWI and University of Amsterdam, January 12, 1999, 9 page postscript document. Otherwise, at least one of gcd(ar/21,N)displaystyle gcd(ar/21,N) and gcd(ar/21,N)displaystyle gcd(ar/2-1,N) is a nontrivial factor of Ndisplaystyle N, so we are done. Proceedings 35th Annual Symposium on Foundations of Computer Science.

"Quantum factorization of 56153displaystyle 56153 with only 4displaystyle 4 qubits". To see this, we notice that then e2ibyrQ1displaystyle e-frac 2pi ibyrQ1 for all integers bdisplaystyle. Moreover, for the algorithm to work, we need Ndisplaystyle N not to be any power of a prime. 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. A Step Toward Quantum Computing: Entangling 10 Billion Particles, from "Discover Magazine Dated January 19, 2011. The second part finds the period using the quantum Fourier transform, and is responsible for the quantum speedup. To achieve this, Shor used repeated squaring for his modular exponentiation transformation.

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