[Paper Review] Quantum vs. Classical Read-once Branching Programs

Martin Sauerhoff|ArXiv.org|Apr 26, 2005

Quantum Computing Algorithms and Architecture27 references3 citations

TL;DR

This paper establishes the first nontrivial upper and lower bounds for non-oblivious quantum read-once branching programs, demonstrating that quantum and classical models have incomparable computational power: quantum programs can solve a specific total function exponentially faster than classical randomized ones, while requiring exponential size for the set-disjointness function, despite classical deterministic OBDDs solving it in linear size.

ABSTRACT

The paper presents the first nontrivial upper and lower bounds for (non-oblivious) quantum read-once branching programs. It is shown that the computational power of quantum and classical read-once branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by error-free quantum read-once branching programs of size O(n^3), while each classical randomized read-once branching program and each quantum OBDD for this function with bounded two-sided error requires size 2^{Ω(n)}. (ii) Quantum branching programs reading each input variable exactly once are shown to require size 2^{Ω(n)} for computing the set-disjointness function DISJ_n from communication complexity theory with two-sided error bounded by a constant smaller than 1/2-2\sqrt{3}/7. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multi-partition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented.

Motivation & Objective

• To establish the first nontrivial upper and lower bounds for non-oblivious quantum read-once branching programs.
• To demonstrate the incomparable computational power of quantum and classical read-once branching programs by constructing explicit functions where each model outperforms the other.
• To extend information cost techniques to a new model of quantum multi-partition communication protocols to prove a lower bound for the set-disjointness function.
• To show that quantum read-once branching programs can achieve exponential size advantages over classical randomized counterparts for a total function.
• To prove that quantum read-once branching programs require exponential size for computing the set-disjointness function, despite its tractability for classical deterministic OBDDs.

Proposed method

• Introduces a new model of quantum multi-partition communication protocols to analyze quantum read-once branching programs.
• Applies an extension of the Jain-Radhakrishnan-Sen information cost technique to the new quantum multi-partition model.
• Uses the weighted sum function and its extension, the mixed weighted sum function (MWSₙ), as a building block for constructing an exponential quantum advantage.
• Employs mutual information and conditional entropy techniques to bound the information flow in quantum branching programs.
• Reduces the analysis of branching program size to the information cost of quantum communication protocols via a reduction from input conditioning and averaging arguments.
• Leverages the independence of input bits under conditional distributions to simulate quantum 2-partition protocols from branching program behavior.

Experimental results

Research questions

• RQ1 Can quantum read-once branching programs achieve exponential size advantages over classical randomized read-once branching programs for total functions?
• RQ2 Is there a total function for which quantum read-once branching programs are exponentially smaller than classical randomized ones?
• RQ3 What is the space complexity of quantum read-once branching programs for the set-disjointness function, and how does it compare to classical models?
• RQ4 Can information cost techniques from communication complexity be extended to quantum multi-partition protocols to prove lower bounds?
• RQ5 How does the non-obliviousness of quantum read-once branching programs affect their computational power compared to oblivious models?

Key findings

• An error-free quantum read-once branching program of size $O(n^3)$ computes the mixed weighted sum function MWSₙ, a total function on $2n$ input bits.
• Any classical randomized read-once branching program or quantum OBDD with two-sided error less than $1/2$ requires size $2^{ ilde{ ext{O}}(n)}$ to compute MWSₙ.
• Quantum read-once branching programs require size $2^{ ilde{ ext{O}}(n)}$ to compute the set-disjointness function DISJₙ with two-sided error bounded by a constant less than $1/2 - 2 ilde{3}/7$.
• The lower bound for DISJₙ is proven via a novel quantum multi-partition communication protocol model and an extension of the information cost technique to this model.
• The information cost of an $ ilde{ ext{O}}$-error quantum 2-partition protocol for the AND function is bounded below by a positive constant, enabling the size lower bound.
• The results show that quantum and classical read-once branching programs have incomparable computational power: each can solve problems that the other cannot efficiently solve.