Hamiltonian simulation is a fundamental application of quantum computing, particularly in quantum chemistry, where solving the electronic structure problem requires efficiently representing electron repulsion integrals (ERI). This thesis analyzes the computational complexity of Hamiltonian simulation using the Linear Combination of Unitaries (LCU) algorithm and Tensor Hypercontraction (THC), a factorization method that reduces the cost of ERI representation.

By decomposing the Coulomb tensor via Interpolative Separable Density Fitting (ISDF) and mapping the THC representation to an LCU form using the Jordan-Wigner transformation, we achieve a more efficient block encoding of the Hamiltonian in a basis of N molecular orbitals. This enables quantum simulation with an asymptotic Toffoli complexity of Õ(Nλ/ε) and a space complexity of Õ(N), where ε = 0.0016 Hartree is the target accuracy and λ denotes the 1-norm of the coefficients of the electronic Hamiltonian in the THC representation.  

Beyond Hamiltonian simulation, we explore how Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) leverage block encoding for eigenvalue transformations and time evolution, extending its utility to a broader class of quantum algorithms.  

Despite these advancements, large-scale quantum chemistry simulations are far from experimental feasibility—for instance, simulating the FeMoCo molecule in an active space model under surface code constraints would require billions of Toffoli gates, millions of physical qubits and several days of runtime using error-corrected qubits; and that is assuming it ever becomes feasible.

This work contributes to the effort of making quantum chemistry simulations more computationally viable by investigating how LCU and THC can be integrated to reduce costs and maintain chemical accuracy, while exploring different methods and their combinations to identify key challenges in quantum chemistry simulations.