A Faster Quantum Fourier Transform (2501.12414v2)

Abstract: We present an asymptotically improved algorithm for implementing the Quantum Fourier Transform (QFT) in both the exact and approximate settings. Historically, the approximate QFT has been implemented in $\Theta(n \log n)$ gates, and the exact in $\Theta(n^2)$ gates. In this work, we show that these costs can be reduced by leveraging a novel formulation of the QFT that recurses on two partitions of the qubits. Specifically, our approach yields an $\Theta(n(\log \log n)^2)$ algorithm for the approximate QFT using $\Theta(\log n)$ ancillas, and an $\Theta(n(\log n)^2)$ algorithm for the exact QFT requiring $\Theta(n)$ ancillas.

• The paper introduces an asymptotically improved QFT algorithm that cuts exact gate complexity from O(n²) to O(n(log n)²) and approximate complexity from O(n log n) to O(n(log log n)²).
• The methodology leverages a novel recursive partitioning strategy with fast integer multiplication to streamline phase corrections and reduce ancillary qubit usage.
• The improvements significantly lower resource demands for quantum algorithms, enhancing the practicality of implementations like Shor's algorithm and phase estimation techniques.

On the Asymptotic Improvement of Quantum Fourier Transform Algorithms

The paper under discussion presents a compelling advancement in the implementation of the Quantum Fourier Transform (QFT) by introducing an asymptotically improved algorithm in both the exact and approximate settings. This research addresses the core computational challenges associated with QFT gate complexity and ancillary qubit usage, which are pivotal for numerous quantum algorithms like Shor's algorithm and various phase estimation procedures.

Summary of Contributions

The key contribution lies in developing an algorithm that significantly improves upon the traditional O(n$^2$) gate complexity for the exact QFT and O(n log n) for the approximate QFT. Specifically, the paper proposes:

1. An exact QFT implementation with O(n(log n)$^2$) gate complexity using O(n) ancillas.
2. An approximate QFT with O(n(log log n)$^2$) gate complexity employing O(log n) ancillas.

These improvements are achieved through a novel recursive partitioning strategy involving two QFTs over half of the qubits each. This new design leverages fast integer multiplication to streamline correction operations, offering computational efficiency over past formulations.

Technical Framework

The paper presents a detailed technical framework for its proposed algorithms. Using the recursive decomposition strategy, the exact QFT computes the transformation by splitting the initial state into two smaller subsets of qubits, effectively reducing the requirement of phase gates. The correctness of the method is analytically validated, proving that it indeed realizes the QFT transformation with reduced complexity for both classical multiplication and precomputation of the required states.

For the approximate QFT, the paper utilizes a similar recursive approach but with specific modifications that allow ignoring small angle rotations in deeper recursive levels. This adjustment ensures bounded net errors while maintaining effective transformation fidelity. The approximate implementation is validated by measuring the cumulative impact of these simplifications on the overall error, ensuring it remains within acceptable bounds.

Implications and Future Directions

The implications of this research are profound, as reducing the gate complexity and ancilla usage inherently increases the efficiency and feasibility of QFT in practical quantum computing applications. This improvement can potentially spark advancements in existing quantum algorithms by lowering resource demands, leading to more accessible and efficient quantum computations.

Future research could explore the applicability of these recursive techniques to non-power-of-two qubit counts, as adapting these algorithms beyond the assumed input sizes could address broader computational needs in quantum algorithms. Moreover, exploring further reductions in both exact and approximate QFT complexities remains an attractive research avenue. Achieving a complexity closer to O(n log n), akin to classical multiplication, would represent a substantial stride forward for the quantum computing field.

In summary, the paper makes an essential contribution to quantum computing by simplifying QFT implementations, which are a fundamental component in the resource-efficient execution of quantum algorithms. These advancements highlight the continued evolution of quantum algorithmic efficiency and set the stage for future explorations in this vital area of research.