[Paper Review] Actual computational time-cost of the Quantum Fourier Transform in a quantum computer using nuclear spins

Akihiro Saito, K. Kioi+3 more|ArXiv.org|Jan 31, 2000

Quantum Computing Algorithms and Architecture2 references7 citations

TL;DR

This paper analyzes the actual computational time-cost of the Quantum Fourier Transform (QFT) in nuclear spin-based quantum computers, specifically Bulk Spin Resonance (BSR) and Spin Resonance Transistor (SRT) implementations. It shows that due to gate-dependent operation times—especially for phase rotations and non-adjacent controlled-phase gates—the actual time-cost scales as $O(n2^n)$, drastically exceeding the ideal $O(n^2)$ cost, raising feasibility concerns for large-scale factoring.

ABSTRACT

We found that the actual computational time-cost of the QFT is O(n 2^n) for large n in a quantum computer using nuclear spins. The computational cost of a quantum algorithm has usually been estimated as the sum of the universal gates required in such ideal mathematical models as the Quantum Turing Machine(QTM) and the quantum circuit. This cost is proportional to an actual time-cost in the physical implementation where all quantum operations can be achieved in the same time. However, if the implementation takes a different time for each quantum gate, there is a possibility that the actual time-cost will have a different behavior from the ideal cost. So we estimated the actual time-cost of the QFT in these implementations by considering the gating time. The actual time-cost is drastically different from O(n^2) estimated by complexity analysis.

Motivation & Objective

• To assess the actual time-cost of the Quantum Fourier Transform (QFT) in physical quantum computers using nuclear spins, moving beyond idealized complexity models.
• To investigate how gate operation times vary in BSR and SRT implementations, particularly for single-qubit and two-qubit gates.
• To evaluate the impact of non-adjacent gate construction via adjacent swaps on total execution time.
• To compare idealized $O(n^2)$ complexity with actual hardware-dependent time-costs in realistic physical implementations.
• To assess the feasibility of large-scale quantum factoring under realistic physical constraints of field control and gate timing.

Proposed method

• Models the QFT using standard quantum gates: Hadamard ($H_j$), phase rotations ($R_{yj}( heta)$, $R_{zj}( heta)$), controlled-phase gates ($C_{j,k}( heta)$), and controlled-phase rotations ($D_{j,k}( heta)$).
• Analyzes gate operations in BSR and SRT, where gate times depend on pulse duration or field intensity, with phase rotations requiring time proportional to the rotation angle.
• Models non-adjacent two-qubit gates using adjacent swap operations ($S_{j,k}$), with sequences like $S_{k,k-1} \cdots S_{j+1,j} U_{j,j+1} S_{j,j+1} \cdots S_{k,k-1}$ to transfer control qubits.
• Derives the total time-cost by summing the time for all phase rotations and swap operations, accounting for the exponential growth in required rotation angles with $n$.
• Uses symmetry in the QFT circuit to reduce redundant swaps, showing that only $O(n^2)$ swaps remain after optimization.
• Compares duration control (time proportional to angle) and intensity control (fixed time per gate), showing the former leads to exponential time scaling.

Experimental results

Research questions

• RQ1 How does the actual time-cost of the QFT in nuclear spin quantum computers differ from the idealized $O(n^2)$ complexity?
• RQ2 What is the impact of gate-dependent operation times—especially for phase rotations—on the total execution time of the QFT?
• RQ3 How does the construction of non-adjacent controlled-phase gates via adjacent swaps affect the total time-cost?
• RQ4 Can the symmetry of the QFT circuit reduce the number of required swap operations, and if so, to what extent?
• RQ5 What are the practical limitations on implementing large-scale QFT due to the exponential scaling of required magnetic field intensities?

Key findings

• The actual time-cost of the QFT in BSR and SRT implementations scales as $O(n2^n)$ for large $n$, significantly exceeding the ideal $O(n^2)$ cost.
• In duration control mode, the time for each phase rotation is proportional to the rotation angle, leading to exponential time scaling with $n$.
• In intensity control mode, while gate times are constant, the required magnetic field intensity increases exponentially with $n$, reaching $B \sim 10^{27}$ T for a 100-qubit QFT if minimum rotation is at $10^{-3}$ T.
• Non-adjacent controlled-phase gates require $O(n^3)$ adjacent swaps in naive construction, but symmetry reduces this to $O(n^2)$ swaps after optimization.
• The actual time-cost is dominated by phase rotations, not swap operations, making gate timing a critical bottleneck in physical implementations.
• The study concludes that idealized complexity analysis is insufficient for practical feasibility; actual time-costs must be considered for large-scale quantum algorithms.