Scalable quantum circuits for exponential of Pauli strings and Hamiltonian simulations

TL;DR

This paper introduces scalable quantum circuits for exponentiating Pauli strings and simulating Hamiltonian dynamics, demonstrating their effectiveness on low-connected hardware and in noisy quantum environments.

Contribution

The authors develop a scalable circuit model for exponential of Pauli strings applicable to low-connected quantum hardware and extend it to Hamiltonian simulation using Trotterization.

Findings

Circuit models accurately approximate unitary evolution for various Hamiltonians.

Simulations of up to 18 qubits show errors comparable to Trotterization.

Noisy simulations with small gate errors closely match noiseless results.

Abstract

In this paper, we design quantum circuits for the exponential of scaled $n$-qubit Pauli strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity and $X$ gates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with the $n$-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Pauli-string is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for $(n+1)$-qubit systems can be constructed from $n$-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include $2$-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers and observe that the noisy simulation closely resembles the noiseless one when gate and idle errors are on the order of $O(10^{-3})$ or smaller.