[Paper Review] Communication complexity of promise problems and their applications to finite automata

Jozef Gruska, Daowen Qiu+1 more|arXiv (Cornell University)|Sep 30, 2013

Complexity and Algorithms in Graphs21 references3 citations

TL;DR

This paper establishes an exponential gap between classical deterministic and quantum communication complexity for a specific promise version of the disjointness problem, demonstrating that quantum finite automata can solve certain promise problems with exponentially fewer states than classical finite automata. Using communication complexity techniques, the authors prove that while deterministic one-way finite automata require exponentially many states, one-way quantum finite automata can solve the same problems with only polynomial state complexity.

ABSTRACT

Equality and disjointness are two of the most studied problems in communication complexity. They have been studied for both classical and also quantum communication and for various models and modes of communication. Buhrman et al. [Buh98] proved that the exact quantum communication complexity for a promise version of the equality problem is ${\bf O}(\log {n})$ while the classical deterministic communication complexity is $n+1$ for two-way communication, which was the first impressively large (exponential) gap between quantum and classical (deterministic and probabilistic) communication complexity. If an error is tolerated, both quantum and probabilistic communication complexities for equality are ${\bf O}(\log {n})$. However, even if an error is tolerated, the gaps between quantum (probabilistic) and deterministic complexity are not larger than quadratic for the disjointness problem. It is therefore interesting to ask whether there are some promise versions of the disjointness problem for which bigger gaps can be shown. We give a positive answer to such a question. Namely, we prove that there exists an exponential gap between quantum (even probabilistic) communication complexity and classical deterministic communication complexity of some specific versions of the disjointness problem. Klauck [Kla00] proved, for any language, that the state complexity of exact quantum/classical finite automata, which is a general model of one-way quantum finite automata, is not less than the state complexity of an equivalent one-way deterministic finite automata (1DFA). In this paper we show, using a communication complexity result, that situation may be different for some promise problems. Namely, we show for certain promise problem that the gap between the state complexity of exact one-way quantum finite automata and 1DFA can be exponential.

Motivation & Objective

• To investigate whether promise versions of the disjointness problem can exhibit larger communication complexity gaps between quantum and classical models than previously known.
• To explore the implications of such gaps for the state complexity of finite automata, particularly comparing classical and quantum models.
• To demonstrate that quantum finite automata can solve certain promise problems with significantly fewer states than classical finite automata.
• To establish a connection between communication complexity results and the state complexity of one-way quantum finite automata (1QCFA).

Proposed method

• Define a promise problem $\text{DISJ}_{\lambda}^\prime$ where inputs are either disjoint ($\sum x_i \wedge y_i = 0$) or have a large intersection ($\lambda n \leq \sum x_i \wedge y_i \leq (1-\lambda)n$), with $\lambda = \frac{1}{4}$.
• Prove that the quantum communication complexity of $\text{DISJ}_{\lambda}^\prime$ is $O(\log n)$, using quantum protocols that leverage amplitude amplification and phase estimation techniques.
• Establish a lower bound of $\Omega(n)$ on the deterministic communication complexity of $\text{DISJ}_{\lambda}^\prime$, using a fooling set argument or similar combinatorial method.
• Construct a one-way quantum finite automaton (MO-1QCFA) with $2n$ quantum basis states and $O(n)$ classical states that solves the promise problem $A_D(n)$ with one-sided error $\frac{1}{4}$.
• Use the automaton's behavior to derive a communication protocol: Alice sends her state after reading $x\#$, Bob continues computation and sends his state after $x\#y\#$, and Alice checks acceptance after reading $x$ again.
• Derive a lower bound on the size of any 1DFA solving the same problem by relating it to the deterministic communication complexity, showing $N \in 2^{\Omega(n)}$.

Experimental results

Research questions

• RQ1 Can promise versions of the disjointness problem exhibit communication complexity gaps larger than quadratic between quantum and classical models?
• RQ2 Is there a promise problem for which quantum communication complexity is logarithmic while classical deterministic communication complexity is linear?
• RQ3 Can quantum finite automata achieve an exponential state complexity advantage over classical finite automata for promise problems?
• RQ4 What is the relationship between communication complexity of promise problems and the state complexity of one-way quantum finite automata?
• RQ5 Does the existence of an exponential gap in communication complexity imply an exponential gap in state complexity for finite automata?

Key findings

• For $\lambda = \frac{1}{4}$, the quantum communication complexity of $\text{DISJ}_{\lambda}^\prime$ is at most $3 + 2\log n$, which is $O(\log n)$.
• The deterministic communication complexity of $\text{DISJ}_{\lambda}^\prime$ is $\Omega(n)$, with a lower bound of $0.0073n - 1$.
• An MO-1QCFA can solve the promise problem $A_D(n)$ with one-sided error $\frac{1}{4}$ using $2n$ quantum basis states and $O(n)$ classical states.
• Any 1DFA solving the same promise problem $A_D(n)$ must have at least $2^{\Omega(n)}$ states, demonstrating an exponential state complexity gap.
• The communication complexity lower bound implies that any deterministic protocol for $\text{DISJ}_{\lambda}^\prime$ requires $\Omega(n)$ bits, while quantum protocols require only $O(\log n)$ bits.
• The construction of the MO-1QCFA and its acceptance behavior under measurement confirms that acceptance probability is 1 for yes-instances and at most $\frac{1}{4}$ for no-instances, ensuring one-sided error.