Lecture 021 - Quantum for Classical

Strong Exponential Time Hypothesis: no less than $O(2^n)$ algorithm for SAT (proved only with blackbox model)

Quantum Strong Exponential Time Hypothesis: no less than $O(2^{n/2})$ quantum algorithm for SAT (proved only with blackbox model)

Grover: solve SAT with high probability using $2^{n/2}c$ instruction given AND/OR/NOT circuit $C$ with length $c$.

Bernstein-Vazirani (XOR): $O(c)$ with $100\%$. (while classical need $O(nc)$), verifiable output, unverifiable input

Simon's Algorithm: not a natural problem, unverifiable input

Quantum Factoring: polynomial (while classical need $O(\exp(n^{1/3}))$), verifiable

• Decision Factoring: whether there exist a factor of $F$ in range $[2, k]$

• when decision factoring return "Yes": verifiable

• when decision factoring return "No": still verifiable (unlike SAT)

If we assume $NP \neq coNP$, then $\text{NP-Complete}$ problems should not have such "no-witness". And therefore, factoring is not $\text{NP-Complete}$.

Bias-Busting: if code $C$ biased then $Pr\{0...0\} > 0$, else $Pr\{0...0\} = 0$.

• $SAT \leq_p \text{Bias-Busting}$ (so it is NP-Hard)

• If satisfiable, then $C$ must be biased. If unsatisfiable, then $C$ must be unbiased.

• But $Pr\{\} > 0$ is too weak to solve $NP-Hard$ problems.

• $\text{Bias-Busting}$ probably not in $NP$, since otherwise, a witness will be similar to Bias-Busting itself.

Toda-Ogiwara Theorem: Bias-Busting not in NP assuming $P^{\Sigma_2} \neq NP^{\Sigma_2}$