Gate Efficient Composition of Hamiltonian Simulation and Block-Encoding with its Application on HUBO, Chemistry and Finite Difference Method

TL;DR

This paper introduces a gate-efficient formalism for Hamiltonian simulation that simplifies circuit design, reduces gate count, and applies effectively to HUBO, fermionic, and finite difference problems, outperforming traditional methods.

Contribution

The paper presents a novel, unified formalism for Hamiltonian simulation that improves gate efficiency and simplifies circuit construction across multiple quantum computing applications.

Findings

Reduces quantum circuit depth and gate count significantly.

Achieves exponential gate reduction for high-order HUBO cost functions.

Enables error-free implementation of electronic transitions in fermionic Hamiltonians.

Abstract

This article proposes a formalism which unifies Hamiltonian simulation techniques from different fields. This formalism leads to a competitive method to construct the Hamiltonian simulation with a comprehensible, simple-to-implement circuit generation technique. It leads to a gate decomposition and a scaling different from the usual strategy based on a Linear Combination of Unitaries (LCU) reformulation of the problem. It can significantly reduce the quantum circuit number of rotational gates, multi-qubit gates, and the circuit depth. This method leads to one exact Hamiltonian simulation for each summed term and Trotter step. Each of these Hamiltonian simulation unitary matrices also allows the construction of the non-exponential terms with a maximum of six unitary matrices to be Block-encoding (BE). The formalism is easy to apply to the widely studied Highorder Unconstrained Binary Optimization (HUBO), fermionic transition Hamiltonian, and basic finite difference method instances. For the HUBO, our implementation exponentially reduces the number of gates for high-order cost functions with respect to the HUBO order. The individual electronic transitions are implemented without error for the second-quantization Fermionic Hamiltonian. Finite difference proposed matrix decompositions are straightforward, very versatile, and scale as the state-of-the-art proposals.