By Goong Chen, Louis Kauffman, Samuel J. Lomonaco
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As observed by Shor, a way to work around this difficulty is to push the HSP. In particular, as illustrated by the following commutative diagram ϕ Z μ −→ τ Z× N ϕ = Push (ϕ ) = ϕ ◦ τ , ZQ a push ϕ = Push (ϕ ) is constructed by selecting the epimorphism μ : Z −→ ZQ of Z onto the finite cyclic group ZQ of order Q, where the integer Q is the unique power of 2 such that N 2 ≤ Q < 2N 2 , and then choosing the transversal9 τ : ZQ −→ Z m mod Q −→ m , where 0 ≤ m < Q. This push ϕ = Push (ϕ ) is called Shor’s oracle.
Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium,” Proceedings of the Symposia of Applied Mathematics, vol. 58, American Mathematical Society, Providence, Rhode Island, (2002). , and Howard E. ), “Quantum Computation and Information,” AMS Contemporary Mathematics, vol. 305, American Mathematical Society, Providence, RI, (2002). , Grover’s quantum search algorithm, AMS PSAPM/58, (2002), 181–192. , and Louis H. Kauffman, Quantum Hidden Subgroup Algorithms: A Mathematical Perspective, AMS CONM/ 305 (2002), 139–202.
In light of this and of the results outlined in the previous section, it is a natural objective to generalize Shor’s algorithm to free groups of finite rank. 12. , concatenation followed by full reduction. 10 A set W of reduced words in a free group F = F x1 , x2 , . . , xn is said to be a 2-sided Schreier system provided • The empty word 1 lies in W . • w = a1 a2 · · · a −1 a ∈ W ⇒ wLe f t = a1 a2 · · · a • w = a1 a2 · · · a −1 a ∈ W ⇒ wRight = a2 · · · a −1 −1 a ∈ W , and ∈W. Given an epimorphism ν : F −→ G of the free group F onto a group G, a 2-sided Schreier transversal τ : G −→ F for ν is a transversal of ν for which there exists a 2-sided Schreier system such that τ (G) = W .