Comparing quantum computing algorithms for physics simulations

Abstract

We find the Boltzmann probability distribution function (PDF) of the quantum Ising model through a variational quantum algorithm (VQA) using state-vector, shots, and noisy simulations and compare their respective fidelity with the exact PDF at different temperatures and magnetic field strengths. The Rudolph and Grover quantum circuit was used alongside the classical optimizers, Powell for ‘state-vector’ and COBYLA for ‘shots’ and ‘noisy’ modes. The state vector result of a quantum circuit represents the optimal theoretical quantum state of the system. A real quantum computer can never output the optimal state vector result and instead utilises ‘shots’. ’Shots’ rely on repeatedly running a quantum circuit to obtain statistical results as a measure. Given enough shots it was shown that the results approximate state-vector results for small qubit numbers. Because real quantum computers are noisy, a noisy simulation was created using one of the most common types of noise in quantum computers, depolarizing noise. The results were then compared across the three types of simulations. One of the major findings was that even a small degree of depolarizing noise randomly implemented on the gates of a quantum circuit reduced the fidelity for different temperatures and magnetic field strengths tested. Future research analysing the fidelity of the Boltzmann distribution in noisy intermediate scale quantum (NISQ) computers would benefit from more noise-resilient qubits or more efficient noise reduction schemes.