Efficient Quantum Circuit Implementations of The qDRIFT Algorithm via Linear Combinations of Unitaries and Quantum Forking

Abstract Quantum computers can be used to simulate the evolution of quantum systems via the discretization of this evolution into unitary gate sequences. While Trotter-Suzuki (TS) methods stand as the most prevalent approach for quantum simulation, their effectiveness is contingent upon the sparsity of the Hamiltonian system. This is due to the precision and gate complexity of the TS method being dependent on the number of terms within the Hamiltonian. Despite attempts to mitigate this dependence through the random permutation of terms within TS methods, scalability issues persist. To address this, Campbell introduced the qDRIFT algorithm, leveraging random compilation to alleviate the reliance on the Hamiltonian’s term count. However, the initial implementation of this algorithm necessitated the involvement of classical computing resources for gate set compilation. Our study presents two novel quantum circuit implementations of the qDRIFT algorithm, eliminating the need for classical computing resources. We employ the Linear Combination of Unitaries (LCU) approach and the Quantum Forking (QF) procedure to construct efficient circuits to implement the qDRIFT algorithm. Furthermore, we conduct a comparative analysis of the gate complexities between our proposed circuits and TS methods, identifying the scenarios in which our circuits exhibit superior performance.