[Paper Review] Targeted Fibonacci Exponentiation

Burton S. Kaliski|arXiv (Cornell University)|Nov 5, 2017

Quantum Computing Algorithms and Architecture6 references3 citations

TL;DR

This paper presents three targeted Fibonacci exponentiation algorithms that compute $ a^k $ using reversible quantum circuits with minimal ancilla qubits, ensuring only the result $ a^k $ and fixed garbage values remain. By leveraging Fibonacci addition chains and novel variants of the Hofstadter G pair algorithm, the methods reduce register usage and iteration count, offering tradeoffs between space and time efficiency for quantum cryptanalytic applications.

ABSTRACT

A targeted exponentiation algorithm computes a group exponentiation operation $a^k$ with a reversible circuit in such a way that the initial state of the circuit consists of only the base $a$ and fixed values, and the final state consists of only the exponential $a^k$ and fixed values. Three targeted exponentiation algorithms based on Fibonacci addition chains are considered, offering tradeoffs in terms of the number of working registers and the number of iterations. The approaches also motivate related results on the Fibonacci Zeckendorf array, including a new \emph{modular Hofstadter G problem} and an improvement to Anderson's recent algorithm for locating pairs of adjacent integers in the extended Fibonacci Zeckendorf array. The algorithms have applications in quantum computing.

Motivation & Objective

• To design reversible quantum circuits that compute $ a^k $ with only $ a^k $ and fixed ancilla values in the final state, avoiding entanglement with side values.
• To reduce the number of working qubits and iterations in quantum exponentiation by exploiting Fibonacci addition chains instead of binary representations.
• To enable efficient exponentiation in superpositions where the base $ a $ is variable or generated dynamically during computation.
• To improve upon existing algorithms for locating adjacent pairs in the extended Fibonacci Zeckendorf array and solve a new modular Hofstadter G problem.
• To provide practical, optimized tools for quantum cryptanalysis, particularly in Shor’s algorithm variants involving group exponentiation.

Proposed method

• Uses Fibonacci addition chains to construct reversible circuits that evolve $ (a, b) \mapsto (b, ab) $, preserving reversibility and minimizing ancilla.
• Employs a low-to-high Fibonacci exponentiation algorithm as the base approach, with state transitions tracking powers $ a^{F_i} $.
• Introduces a 'dual' approach that processes two chains in parallel, improving register efficiency at the cost of increased iterations.
• Develops a hybrid method combining high-to-low Hofstadter G pair exponentiation with low-to-high Fibonacci exponentiation to balance qubit and iteration costs.
• Applies a new modular Hofstadter G problem solution to enable efficient computation of adjacent Fibonacci pairs in the Zeckendorf array.
• Improves Anderson’s algorithm for locating adjacent integers in the extended Fibonacci Zeckendorf array, enhancing search efficiency.

Experimental results

Research questions

• RQ1 Can Fibonacci-based exponentiation be made targeted, so that only $ a^k $ and fixed ancilla remain, with no entanglement to intermediate values?
• RQ2 What tradeoffs exist between the number of working registers and the number of iterations in targeted Fibonacci exponentiation?
• RQ3 How can the Hofstadter G pair exponentiation algorithm be adapted for use in targeted quantum exponentiation circuits?
• RQ4 Can the Zeckendorf representation and Fibonacci array properties be leveraged to improve the efficiency of quantum exponentiation?
• RQ5 What are the minimal qubit and iteration requirements for targeted exponentiation, and can lower bounds be established?

Key findings

• The basic Fibonacci exponentiation algorithm uses 3 working registers and requires $ 3h $ iterations, where $ h = \lceil \log_\phi r \rceil $.
• The dual approach uses 4 registers and $ 4h $ iterations, achieving better register efficiency at the cost of more iterations.
• The hybrid method uses 3 registers and $ 3h $ iterations, offering a balanced tradeoff between space and time.
• The paper introduces a new modular Hofstadter G problem and provides a solution that enables efficient computation in the Fibonacci Zeckendorf array.
• An improved algorithm is presented for locating adjacent integer pairs in the extended Fibonacci Zeckendorf array, reducing search complexity.
• The targeted exponentiation framework enables clean superposition-based exponentiation in quantum circuits, crucial for applications like transforming eigenstates in Shor’s algorithm.