[Paper Review] Algorithms for Quantum Branching Programs Based on Fingerprinting

Farid Ablayev, Alexander Vasiliev|arXiv (Cornell University)|Nov 12, 2009

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper presents a fingerprinting-based method for constructing optimal quantum ordered branching programs (QOBDDs) for Boolean functions using characteristic polynomials. By leveraging quantum fingerprinting and unitary transformations conditioned on classical inputs, the approach achieves width-efficient QOBDDs for functions like $MOD_m$, $EQ_n$, $Palindrome_n$, and $PERM_n$, with a generalization to the Boolean Hidden Subgroup Problem using $O(n)$-width QOBDDs.

ABSTRACT

In the paper we develop a method for constructing quantum algorithms for computing Boolean functions by quantum ordered read-once branching programs (quantum OBDDs). Our method is based on fingerprinting technique and representation of Boolean functions by their characteristic polynomials. We use circuit notation for branching programs for desired algorithms presentation. For several known functions our approach provides optimal QOBDDs. Namely we consider such functions as Equality, Palindrome, and Permutation Matrix Test. We also propose a generalization of our method and apply it to the Boolean variant of the Hidden Subgroup Problem.

Motivation & Objective

• To develop a systematic method for constructing quantum ordered read-once branching programs (QOBDDs) that are more efficient than classical counterparts.
• To address the challenge of designing width-optimal QOBDDs for fundamental Boolean functions such as $MOD_m$, $EQ_n$, $Palindrome_n$, and $PERM_n$.
• To generalize the fingerprinting technique to the quantum branching program model for solving the Boolean variant of the Hidden Subgroup Problem.
• To establish a connection between characteristic polynomials of Boolean functions and efficient quantum computation via quantum fingerprinting.

Proposed method

• Represents Boolean functions via their characteristic polynomials over $\mathbb{Z}_{2^n}$, enabling algebraic manipulation for quantum algorithm design.
• Applies quantum fingerprinting by encoding function values into quantum states using controlled unitary operations conditioned on classical input variables.
• Uses circuit notation to model QOBDDs as quantum circuits with classical control, where each instruction applies a unitary $U_j(\sigma_{i_j})$ based on input bit $\sigma_{i_j}$.
• Employs projective measurement via $M_{\text{accept}}$ to compute acceptance probability $||M_{\text{accept}}|\psi_\sigma\rangle||^2_2$ for final output.
• Constructs two key polynomials: $g_1(x)$ to verify constant values within cosets and $g_2(x)$ to verify distinct values across cosets in the Hidden Subgroup Problem.
• Leverages the generalized fingerprinting technique to implement $O(n)$-width QOBDDs for the Boolean Hidden Subgroup Problem with one-sided error.

Experimental results

Research questions

• RQ1 Can fingerprinting techniques be adapted to construct optimal QOBDDs for fundamental Boolean functions?
• RQ2 How can characteristic polynomials of Boolean functions be used to design efficient quantum branching programs?
• RQ3 What is the quantum complexity of computing $MOD_m$, $EQ_n$, $Palindrome_n$, and $PERM_n$ using ordered read-once branching programs?
• RQ4 Can the fingerprinting method be generalized to solve the Boolean Hidden Subgroup Problem with bounded error in a width-efficient manner?

Key findings

• The method constructs optimal QOBDDs for $MOD_m$, $EQ_n$, $Palindrome_n$, and $PERM_n$ using fingerprinting and characteristic polynomials.
• For $MOD_m$, the QOBDD width is $O(m)$, matching known lower bounds and proving optimality.
• For $EQ_n$, the QOBDD width is $O(n)$, achieving optimal efficiency for this function.
• For $Palindrome_n$, the QOBDD width is $O(n)$, demonstrating exponential advantage over classical OBDDs.
• For $PERM_n$, the QOBDD width is $O(n)$, showing efficient quantum computation of permutation matrix testing.
• The generalized fingerprinting approach yields an $O(n)$-width QOBDD for the Boolean Hidden Subgroup Problem with one-sided error, where $n = |G|\lceil\log(G:K)\rceil$.