Direct Interpolative Construction of the Discrete Fourier Transform as an Efficient Matrix Product Operator

Core Concepts

The quantum Fourier transform (QFT) can be efficiently represented as a low-rank matrix product operator (MPO) using an interpolative decomposition, providing a simple closed-form construction with guaranteed near-optimal compression error.

Abstract

The content discusses an efficient construction of the discrete Fourier transform (DFT) and the quantum Fourier transform (QFT) as a matrix product operator (MPO).
Key highlights:

The QFT can be viewed as a reindexing of the DFT and has been shown to be compressible as a low-rank MPO. However, the original proof did not provide a construction with guaranteed error bounds.
The authors present a simple closed-form construction of the QFT MPO using the interpolative decomposition, which provides a near-optimal compression error for a given rank.
This construction can speed up the application of the QFT and DFT in quantum circuit simulations and quantized tensor train (QTT) applications.
The authors also connect their interpolative construction to the approximate quantum Fourier transform (AQFT), showing that the AQFT can be viewed as an MPO constructed using a different interpolation scheme.
The interpolative construction yields a MPO representation of the QFT with lower ranks compared to the AQFT MPO for the same error tolerance, indicating that the long-range gates in the QFT circuit play a role in reducing entanglement.

Stats

The quantum Fourier transform (QFT) has a gate complexity of O(n^2) acting on an n-qubit quantum state, while the fast Fourier transform (FFT) has a complexity of O(N log N) = O(2^n n), where N = 2^n.

Quotes

"The quantum Fourier transform (QFT) [2] is one of the most important algorithmic primitives in quantum computing."
"If one makes use of low-rank structure of the input state, the QFT can often be simulated classically using efficient operations with matrix product states (MPS) and matrix product operators (MPO) [3, 4, 5, 6, 7]."
"Equivalently, the DFT of functions compressible as quantized tensor trains (QTT) [8, 9] can be computed with exponential speedup."

Key Insights Distilled From

Direct interpolative construction of the discrete Fourier transform as a matrix product operator

by Jielun Chen,... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03182.pdf

Deeper Inquiries

How can the interpolative construction of the QFT MPO be extended to other quantum algorithms or operators beyond the Fourier transform

The interpolative construction of the QFT MPO can be extended to other quantum algorithms or operators by adapting the same principles to the specific structure and requirements of those algorithms. The key idea is to identify the underlying low-rank structure or patterns in the quantum circuit representation and use interpolative decomposition to construct an efficient MPO representation. This approach can be applied to algorithms that exhibit similar characteristics to the QFT, such as those with translational invariance or specific symmetries that can be exploited through tensor decompositions. By analyzing the circuit structure and identifying the relevant patterns, one can apply the interpolative decomposition method to construct MPO representations that offer significant computational advantages.

What are the potential limitations or challenges in applying the interpolative decomposition approach to construct efficient MPO representations of quantum circuits

While the interpolative decomposition approach offers a powerful method for constructing efficient MPO representations of quantum circuits, there are potential limitations and challenges to consider. One limitation is the scalability of the method to larger quantum circuits with increasing qubit counts. As the size of the circuit grows, the computational complexity of the interpolative decomposition process may become prohibitive, leading to challenges in efficiently constructing MPO representations. Additionally, the accuracy of the interpolative construction may be affected by the complexity of the circuit and the presence of high entanglement, requiring careful optimization and parameter tuning to achieve desired error bounds. Furthermore, the interpolative decomposition approach may struggle with circuits that lack clear low-rank structures or exhibit complex interactions that are challenging to capture effectively through tensor decompositions.

Could the insights from the connection between the interpolative construction and the AQFT lead to the development of new quantum circuit design techniques that balance the trade-off between circuit depth and entanglement

The insights from the connection between the interpolative construction and the AQFT could inspire the development of new quantum circuit design techniques that optimize the trade-off between circuit depth and entanglement. By leveraging the understanding of how the AQFT can be viewed as an MPO constructed using a specific interpolation scheme, designers can explore novel approaches to designing quantum circuits that balance entanglement requirements with circuit complexity. This could involve developing hybrid circuit designs that incorporate elements of the interpolative construction method to reduce entanglement while maintaining circuit efficiency. Additionally, the insights could lead to the exploration of new interpolation schemes tailored to specific quantum algorithms, enabling the creation of optimized circuit designs that maximize computational performance while minimizing resource requirements.