Soham Bhandary | Searcharxiv

arXiv2025

Tensor Factorized Hamiltonian Downfolding To Optimize The Scaling Complexity Of The Electronic Correlations Problem on Classical and Quantum Computers

Achieving chemical accuracy for strongly correlated molecules is a defining milestone for first-generation, fault-tolerant quantum computers, yet the factorial growth of three, four, and six-index tensor contractions in coupled-cluster CCSD(T), full configuration interaction (FCI), and multireference CI (MRCI) makes current classical and quantum approaches prohibitive. We introduce tensor-factorized Hamiltonian downfolding (TFHD) and its quantum analogue, qubitized downfolding (QD)- a hybrid classical-quantum framework that collapses every high-rank object to rank-2 networks executed in depth-optimal, block-encoded circuits. The complexity of these operations scales exponentially with the system size. We aim to find properties of chemical systems by optimizing this scaling through mathematical transformations on the Hamiltonian and the state space. By defining a bi-partition of the many-body Hilbert space into electronoccupied and electron-unoccupied blocks for a given orbital, we perform a downfolding transformation that decouples the electron-occupied block from its complement. We factorize high-rank electronic integrals and cluster amplitude tensors into low-rank tensor factors of a downfolding transformation, mapping the full many-body Hamiltonian into a smaller dimensional block-Hamiltonians. This reduces the computational complexity of solving the residual equations for Hamiltonian downfolding from O(N7) for CCSD(T) and O(N9) - O(N10) for CI and MRCI to O(N3). This operations can be implemented as a family of tensor networks solely made from two-rank tensors. Additionally, we create block-encoding quantum circuits of the tensor networks, generating circuits of O(N2) depth with O(logN) qubits. We demonstrate super-quadratic speedups of expensive quantum chemistry algorithms on both classical and quantum computers.

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arXiv2022

Random Quantum Neural Networks (RQNN) for Noisy Image Recognition

Classical Random Neural Networks (RNNs) have demonstrated effective applications in decision making, signal processing, and image recognition tasks. However, their implementation has been limited to deterministic digital systems that output probability distributions in lieu of stochastic behaviors of random spiking signals. We introduce the novel class of supervised Random Quantum Neural Networks (RQNNs) with a robust training strategy to better exploit the random nature of the spiking RNN. The proposed RQNN employs hybrid classical-quantum algorithms with superposition state and amplitude encoding features, inspired by quantum information theory and the brain's spatial-temporal stochastic spiking property of neuron information encoding. We have extensively validated our proposed RQNN model, relying on hybrid classical-quantum algorithms via the PennyLane Quantum simulator with a limited number of \emph{qubits}. Experiments on the MNIST, FashionMNIST, and KMNIST datasets demonstrate that the proposed RQNN model achieves an average classification accuracy of $94.9\%$. Additionally, the experimental findings illustrate the proposed RQNN's effectiveness and resilience in noisy settings, with enhanced image classification accuracy when compared to the classical counterparts (RNNs), classical Spiking Neural Networks (SNNs), and the classical convolutional neural network (AlexNet). Furthermore, the RQNN can deal with noise, which is useful for various applications, including computer vision in NISQ devices. The PyTorch code (https://github.com/darthsimpus/RQN) is made available on GitHub to reproduce the results reported in this manuscript.

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