[Paper Review] A quantum Fourier transform algorithm

Chris Lomont|ArXiv.org|Apr 9, 2004

Quantum Computing Algorithms and Architecture15 references3 citations

TL;DR

This paper presents a corrected and rigorously analyzed quantum Fourier transform (QFT) algorithm for odd-order cyclic groups, addressing flaws in prior proofs. It provides exact bounds on qubit requirements and approximation error, enabling precise simulation and showing that practical implementations often require significantly fewer qubits than theoretical bounds suggest.

ABSTRACT

Algorithms to compute the quantum Fourier transform over a cyclic group are fundamental to many quantum algorithms. This paper describes such an algorithm and gives a proof of its correctness, tightening some claimed performance bounds given earlier. Exact bounds are given for the number of qubits needed to achieve a desired tolerance, allowing simulation of the algorithm.

Motivation & Objective

• To correct flawed proofs in earlier works on the quantum Fourier transform over odd-order cyclic groups.
• To provide exact, non-asymptotic bounds on the number of qubits and approximation error for the QFT algorithm.
• To enable accurate simulation of the QFT algorithm by specifying concrete resource requirements for a given tolerance.
• To investigate the gap between theoretical bounds and practical performance, revealing potential for significant qubit savings.
• To tighten performance bounds for the QFT algorithm in the context of the Hidden Subgroup Problem and related quantum algorithms.

Proposed method

• Proposes a quantum algorithm for computing the Fourier transform over a cyclic group of odd order N, using a modular arithmetic mapping via a larger modulus M.
• Defines key mappings: i' as the nearest integer to (M/N)i, and a division-like map Δ that splits k ∈ {0,...,M-1} into (s,t) with s ∈ {0,...,N-1} and t ∈ {-α,...,α}.
• Uses the standard QFT over Z_M (via Coppersmith's method) as the core quantum operation, approximated within error ε using O(log M (log log M + log 1/ε)) operations.
• Applies a sawtooth function |x|_M to measure distance modulo M, crucial for bounding error in the transform approximation.
• Derives error bounds using trigonometric inequalities and properties of roots of unity, particularly |1 - ω_M^α| ≤ |2πα/M|.
• Establishes that M ≤ 2 × 735 × N^{3/2}/ε^3 ensures the desired error tolerance, and that ⌈log M⌉ + 2 qubits suffice for the algorithm.

Experimental results

Research questions

• RQ1 What are the exact bounds on the number of qubits required to implement the quantum Fourier transform over an odd-order cyclic group with a given error tolerance?
• RQ2 How can the flawed correctness proofs in prior works on the cyclic QFT be rigorously corrected and tightened?
• RQ3 To what extent do theoretical resource bounds overestimate actual qubit requirements in practice?
• RQ4 Can tighter bounds on the approximation error be derived using exact analysis rather than asymptotic estimates?
• RQ5 What is the minimal number of qubits needed for a given N and ε, and how does this compare to the theoretical upper bound?

Key findings

• The paper establishes a precise upper bound on the number of qubits: ⌈log M⌉ + 2, where M ≤ 2 × 735 × N^{3/2}/ε^3, ensuring the desired approximation error.
• Simulation results show that the observed error is consistently much lower than the theoretical bound—e.g., for N=25, ε=0.3, the max observed error was only 0.0182.
• For practical implementations, the required number of qubits can be nearly halved: e.g., for N=501 and ε=0.2, the theoretical bound requires 30 qubits, but 15 suffice in simulation with max error 0.18.
• The algorithm's performance is robust: even with conservative bounds, the actual error is significantly smaller than the target ε, indicating room for tightening theoretical bounds.
• The minimal M and L values satisfying the error condition are often much smaller than those implied by the general bound, suggesting that optimized choices of M and L can drastically reduce qubit cost.
• The paper proves the correctness of the QFT algorithm over odd-order cyclic groups with concrete, simulation-ready bounds, resolving prior proof errors in [7] and [8].