Random Projection using Random Quantum Circuits

Abstract

The random sampling task performed by Google's Sycamore processor gave us a glimpse of the "Quantum Supremacy era". This has definitely shed some spotlight on the power of random quantum circuits in this abstract task of sampling outputs from the (pseudo-) random circuits. In this manuscript, we explore a practical near-term use of local random quantum circuits in dimensional reduction of large low-rank data sets. We make use of the well-studied dimensionality reduction technique called the random projection method. This method has been extensively used in various applications such as image processing, logistic regression, entropy computation of low-rank matrices, etc. We prove that the matrix representations of local random quantum circuits with sufficiently shorter depths ($\sim O(n)$) serve as good candidates for random projection. We demonstrate numerically that their projection abilities are not far off from the computationally expensive classical principal components analysis on MNIST and CIFAR-100 image data sets. We also benchmark the performance of quantum random projection against the commonly used classical random projection in the tasks of dimensionality reduction of image datasets and computing Von Neumann entropies of large low-rank density matrices. And finally using variational quantum singular value decomposition, we demonstrate a near-term implementation of extracting the singular vectors with dominant singular values after quantum random projecting a large low-rank matrix to lower dimensions. All such numerical experiments unequivocally demonstrate the ability of local random circuits to randomize a large Hilbert space at sufficiently shorter depths with robust retention of properties of large datasets in reduced dimensions.

arXiv commentMinor typos fixed in this version

References in corpus (39)

• Quantum Computing in the NISQ era and beyond
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Barren plateaus in quantum neural network training landscapes
• Quantum principal component analysis
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Sketching as a Tool for Numerical Linear Algebra
• Quantum autoencoders for efficient compression of quantum data
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• An initialization strategy for addressing barren plateaus in parametrized quantum circuits
• Random Quantum Circuits are Approximate 2-designs
• Local random quantum circuits are approximate polynomial-designs
• A quantum linear system algorithm for dense matrices
• Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment
• Efficient estimation of Pauli observables by derandomization
• Quantum Machine Learning for Chemistry and Physics
• Variational Quantum State Eigensolver
• Quantum singular value decomposition of non-sparse low-rank matrices
• Expressibility of the alternating layered ansatz for quantum computation
• Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicator
• Random quantum circuits are approximate unitary $t$-designs in depth $O\left(n{t}^{5+o(1)}\right)$
• Variational Quantum Singular Value Decomposition
• The geometry of quantum spin networks
• Anticoncentration theorems for schemes showing a quantum speedup
• Quantum Machine-Learning for Eigenstate Filtration in Two-Dimensional Materials
• Robust and Efficient Hamiltonian Learning
• Imaginary components of out-of-time correlators and information scrambling for navigating the learning landscape of a quantum machine learning model
• Maximal entropy approach for quantum state tomography
• Applying a random projection algorithm to optimize machine learning model for breast lesion classification
• A Variational Ansatz for the Ground State of the Quantum Sherrington-Kirkpatrick Model
• Variational Approach to Quantum State Tomography based on Maximal Entropy Formalism
• A comparison between D-wave and a classical approximation algorithm and a heuristic for computing the ground state of an Ising spin glass
• Magnetic Phases of Spatially-Modulated Spin-1 Chains in Rydberg Excitons: Classical and Quantum Simulations
• Rényi free energy and variational approximations to thermal states
• Hamiltonian learning from time dynamics using variational algorithms
• Dimension reduction and redundancy removal through successive Schmidt decompositions
• Statistical Analysis on Random Quantum Sampling by Sycamore and Zuchongzhi Quantum Processors

Cited by in corpus (1)

• Secure authentication via Quantum Physical Unclonable Functions: a review