Efficient quantum circuit for singular value thresholding

Abstract: Singular value thresholding (SVT) operation is a fundamental core module in many mathematical models in computer vision and machine learning, particularly for many nuclear norm minimizing-based problems. We presented a quantum SVT (QSVT) algorithm which was used as a subroutine to address an image classification problem. This algorithm runs in $O\left[\log\left(pq\right)\right]$, an exponential speed improvement over the classical algorithm which runs in $O\left[poly\left(pq\right)\right]$. In this study, we investigate this algorithm and design a scalable quantum circuit for QSVT. In the circuit design, we introduce an adjustable parameter $\alpha$ to ensure the high probability of obtaining the final result and the high fidelity of actual and ideal final states. We also show that the value of $\alpha$ can be computed ahead of implementing the quantum circuit when the inputs of the QSVT algorithm, i.e. matrix ${\bf{A}}$ and constant $\tau$, are given. In addition, we propose a small-scale quantum circuit for QSVT. We numerically simulate and demonstrate the performance of this circuit, verifying its capability to solve the intended SVT. The quantum circuit for QSVT implies a tempting possibility for experimental realization on a quantum computer.