<p><em>What is it that quantumness provides that sieve techniques cannot provide?</em></p>

<p>It is more a question why quantum computers are believed to be more powerful than classical  computation models. Short: Quantum computers provide superpositions of quantum states.</p>

<p>Randomized algorithms are restricted to be in a certain state with some (constant) error probablility. Quantumness provides the posibility of evaluating function on every possible state and afterwards measuring a certain state with some error probability.</p>

<p><em>What exact barrier does Shor's algorithm break that randomness cannot?</em></p>

<p>The quantum states represent the accepted states as well as the rejected states. Shor's algorithm evaluates the accepting states in such an extent that the accepting states are more likely to be measured. Also the quantum algorithm can manipulate all states at the same time. A randomized algorithm is bound to increase the probability of being in such a accepting state, looping often enough to get a certain probability (or extending some other variable).</p>

CC BY-SA 3.0
2015-07-14T09:36:57.233Z
I think the answer is too vague to be of any help, although to be fair, the question is too broad as well. If I would replace factoring with almost any other problem, the same answer would hold. I believe also that definitively answering the questions posed is extremely difficult, since it would require essentially proving that $\mathbb{BPP} \neq \mathbb{BQP}$, since we cannot even exclude the possibility of an efficient deterministic algorithm for factoring, let alone a randomized one. The best I would hope as an answer is some kind of intuition.
chazisop
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2015-07-14T10:21:01.087Z
chazisop is right, the answer does not really solve the question. Although i believe for the problem of factorization the question can be answered comparing shor to the sieve algorithms. Taken quantumness provides a solution for the abelian hidden subgroup problem, which cannot in such a way be computed with classical computation models.
Fleeep
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2015-07-14T07:55:53.787Z
2015-07-14T07:55:53.787Z
Fleeep
<p>The best deterministic factorization algorithm that is currently known runs in $O(N^{\frac{1}4+\epsilon})$ arithmetic steps.</p>

<p>Randomness and quantumness improves upon this.</p>

<p>I believe Quadratic/Number field sieve run in randomized subexponential time. What is it that quantumness provides that sieve techniques cannot provide? What exact barrier does Shor's algorithm break that randomness cannot?</p>

CC BY-SA 3.0
2015-07-12T13:31:27.950Z
Those three questions are not very related and probably should be separate questions....
usul
<factoring>
Randomness versus Quantumness in factorization
Turbo
# Randomness versus Quantumness in factorization

(Asked by: username_0 on 2015-07-12)

The best deterministic factorization algorithm that is currently known runs in $O(N^{\frac{1}4+\epsilon})$ arithmetic steps.

Randomness and quantumness improves upon this.

I believe Quadratic/Number field sieve run in randomized subexponential time. What is it that quantumness provides that sieve techniques cannot provide? What exact barrier does Shor's algorithm break that randomness cannot?

**Tags:** factoring

> **Comments:**
> **username_1**: Those three questions are not very related and probably should be separate questions....  

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## 1 answers

### Answer by username_2 on 2015-07-14. Score: -1

*What is it that quantumness provides that sieve techniques cannot provide?*

It is more a question why quantum computers are believed to be more powerful than classical computation models. Short: Quantum computers provide superpositions of quantum states.

Randomized algorithms are restricted to be in a certain state with some (constant) error probablility. Quantumness provides the posibility of evaluating function on every possible state and afterwards measuring a certain state with some error probability.

*What exact barrier does Shor's algorithm break that randomness cannot?*

The quantum states represent the accepted states as well as the rejected states. Shor's algorithm evaluates the accepting states in such an extent that the accepting states are more likely to be measured. Also the quantum algorithm can manipulate all states at the same time. A randomized algorithm is bound to increase the probability of being in such a accepting state, looping often enough to get a certain probability (or extending some other variable).

> **Comments:**
> **username_3**: I think the answer is too vague to be of any help, although to be fair, the question is too broad as well. If I would replace factoring with almost any other problem, the same answer would hold. I believe also that definitively answering the questions posed is extremely difficult, since it would require essentially proving that $\mathbb{BPP} \neq \mathbb{BQP}$, since we cannot even exclude the possibility of an efficient deterministic algorithm for factoring, let alone a randomized one. The best I would hope as an answer is some kind of intuition.  
> **username_2**: username_3 is right, the answer does not really solve the question. Although i believe for the problem of factorization the question can be answered comparing shor to the sieve algorithms. Taken quantumness provides a solution for the abelian hidden subgroup problem, which cannot in such a way be computed with classical computation models.  

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