Zeeshan Ahmed supervised by Dr. Shantanav Chakraborty received his Master of Science – Dual Degree in Computer Science and Engineering (CSD). Here’s a summary of his research work on Early fault-tolerant quantum algorithms for Hamiltonian simulation: benchmarking and applications

Classical computers face fundamental barriers when simulating quantum phenomena due to the exponential growth of quantum state spaces. This limitation motivates quantum simulation, particularly Hamiltonian simulation, as a central application for quantum computers. Simulating quantum dynamics is essential for studying materials and molecules, and Hamiltonian simulation is an integral subroutine of important quantum algorithms including ground state preparation, quantum phase estimation, quantum linear systems, and quantum singular value transformation. While various Hamiltonian simulation methods have been developed, those offering optimal complexity such as qubitization and LCU-based methods require oracular assumptions, complicated multi-qubit control, and extensive ancilla qubits, making them unsuitable for near and intermediate-term quantum computers. In contrast, hardware-friendly methods such as Trotterization, qDRIFT, and Single-Ancilla LCU appear more practical, but their performance for concrete physical systems and applications of interest remains unclear. This thesis provides comprehensive numerical benchmarks comparing Hamiltonian simulation algorithms for early fault-tolerant quantum computers. We analyze when and why different methods achieve optimal performance based on system structure, precision requirements, and hardware constraints. We benchmark Single-Ancilla LCU (SA-LCU), first and second-order Trotterization, and qDRIFT across two distinct classes of quantum systems. For sparse Ising chains with nearest-neighbor interactions, first-order Trotterization achieves better performance at moderate precision due to tight commutator bounds, while SA-LCU dominates at high precision owing to its polylogarithmic scaling. For dense quantum chemistry Hamiltonians, SA-LCU consistently achieves the lowest gate counts across nearly all regimes. These results show that no single algorithm dominates universally, optimal method selection depends on Hamiltonian structure and target precision. We extend this comparative framework to two applications that use Hamiltonian simulation as a subroutine. First, we consider the problem of estimating the magnetization of the ground state of the transverse-field Ising Hamiltonian. We vary the transverse-field coupling strength across a quantum phase transition and benchmark the performance of various Hamiltonian simulation procedures. Our numerical simulations show that SA-LCU outperforms other techniques, requiring shorter circuit depths while using minimal ancilla overhead. Secondly, we consider the problem of simulating open systems dynamics, an avenue where simulating Hamiltonian dynamics finds widespread applications. However, most quantum algorithms for this problem require a lot of resources, rendering them infeasible for early fault-tolerant quantum computers. We develop a collision model framework, where a single environment qubit repeatedly interacts with the underlying system. This sequence of simple interactions allows for the simulation of both Markovian and non-Markovian open systems dynamics, while naturally incorporating near-term Hamiltonian simulation techniques. Our numerical validation shows that second-order Trotter methods achieve the best asymptotic scaling for long-time evolution, while SA-LCU remains competitive for high-precision, short-time regimes. Our findings offer concrete guidance for the choice of Hamiltonian simulation method driven by the structure of the system, target precision, and available hardware resources. In the early fault-tolerant era, no single algorithm dominates universally, and understanding these trade-offs is essential. 

July 2026