Abhishek Setty

Abhishek Setty

Block encoding of sparse matrices underpins powerful quantum algorithms such as quantum singular value transformation, Hamiltonian simulation, and quantum linear solvers, yet its efficient gate-level realization for general sparse matrices remains a major challenge. We introduce a unified framework that addresses key obstacles including the overhead of multi-controlled X (MCX) gates, amplitude reordering, and hardware connectivity, enabling simplified block encoding constructions with explicit gate-level implementations. Central to our approach is a connection to combinatorial optimization, which enables systematic assignment of control qubits to satisfy nearest-neighbor connectivity constraints, along with coherent permutation operators that preserve superposition while enabling structured amplitude reordering. We demonstrate our methods on structured sparse matrices, achieving systematic reductions in control overhead and circuit depth. Our framework bridges the gap between theoretical formulations and hardware-efficient quantum circuit implementations.

Abhishek Setty, Rasul Abdusalamov, Felix Motzoi

Chebyshev polynomials have shown significant promise as an efficient tool for both classical and quantum neural networks to solve linear and nonlinear differential equations. In this work, we adapt and generalize this framework in a quantum machine learning setting for a variety of problems, including the 2D Poisson's equation, second-order linear differential equation, system of differential equations, nonlinear Duffing and Riccati equation. In particular, we propose in the quantum setting a modified Self-Adaptive Physics-Informed Neural Network (SAPINN) approach, where self-adaptive weights are applied to problems with multi-objective loss functions. We further explore capturing correlations in our loss function using a quantum-correlated measurement, resulting in improved accuracy for initial value problems. We analyse also the use of entangling layers and their impact on the solution accuracy for second-order differential equations. The results indicate a promising approach to the near-term evaluation of differential equations on quantum devices.