Revision #1 Authors: Dieter van Melkebeek, Thomas Watson

Accepted on: 13th January 2012 11:40

Downloads: 2811

Keywords:

We give two time- and space-efficient simulations of quantum computations with

intermediate measurements, one by classical randomized computations with

unbounded error and the other by quantum computations that use an arbitrary

fixed universal set of gates. Specifically, our simulations show that every

language solvable by a bounded-error quantum algorithm running in time $t$ and

space $s$ is also solvable by an unbounded-error randomized algorithm running

in time $O(t\cdot\log{t})$ and space $O(s+\log{t})$, as well as by a

bounded-error quantum algorithm restricted to use an arbitrary universal set

and running in time $O(t\cdot\polylog{t})$ and space $O(s+\log{t})$, provided

the universal set is closed under adjoint. We also develop a quantum model

that is particularly suitable for the study of general computations with

simultaneous time and space bounds.

As an application of our randomized simulation, we obtain the first nontrivial

lower bound for general quantum algorithms solving problems related to

satisfiability. Our bound applies to MajSAT and MajMajSAT, which are

the problems of determining the truth value of a given Boolean formula whose

variables are fully quantified by one or two majority quantifiers,

respectively. We prove that for every real $d$ and every positive real

$\delta$ there exists a real $c>1$ such that either:

(i) MajMajSAT does not have a bounded-error quantum algorithm running in

time $O(n^c)$, or

(ii) MajSAT does not have a bounded-error quantum algorithm running in

time $O(n^d)$ and space $O(n^{1-\delta})$.

In particular, MajMajSAT does not have a bounded-error quantum algorithm

running in time $O(n^{1+o(1)})$ and space $O(n^{1-\delta})$ for any

$\delta>0$. Our lower bounds hold for any reasonable uniform model of quantum

computation, in particular for the model we develop.

TR10-147 Authors: Dieter van Melkebeek, Thomas Watson

Publication: 22nd September 2010 04:13

Downloads: 3470

Keywords:

We give two time- and space-efficient simulations of quantum computations with

intermediate measurements, one by classical randomized computations with

unbounded error and the other by quantum computations that use an arbitrary

fixed universal set of gates. Specifically, our simulations show that every

language solvable by a bounded-error quantum algorithm running in time $t$ and

space $s$ is also solvable by an unbounded-error randomized algorithm running

in time $O(t\cdot\log{t})$ and space $O(s+\log{t})$, as well as by a

bounded-error quantum algorithm restricted to use an arbitrary universal set

and running in time $O(t\cdot\polylog{t})$ and space $O(s+\log{t})$, provided

the universal set is closed under adjoint. We also develop a quantum model

that is particularly suitable for the study of general computations with

simultaneous time and space bounds.

As an application of our randomized simulation, we obtain the first nontrivial

lower bound for general quantum algorithms solving problems related to

satisfiability. Our bound applies to MajSAT and MajMajSAT, which are

the problems of determining the truth value of a given Boolean formula whose

variables are fully quantified by one or two majority quantifiers,

respectively. We prove that for every real $d$ and every positive real

$\delta$ there exists a real $c>1$ such that either:

(i) MajMajSAT does not have a bounded-error quantum algorithm running in

time $O(n^c)$, or

(ii) MajSAT does not have a bounded-error quantum algorithm running in

time $O(n^d)$ and space $O(n^{1-\delta})$.

In particular, MajMajSAT does not have a bounded-error quantum algorithm

running in time $O(n^{1+o(1)})$ and space $O(n^{1-\delta})$ for any

$\delta>0$. Our lower bounds hold for any reasonable uniform model of quantum

computation, in particular for the model we develop.