Simulation of a Spin-Chain System

Motivation

This section demonstrates the practical considerations for perfoming dynamical evolution of an Hamiltonian written in the basis of Pauli matrices using a qubit-based quantum computer framework. The chosen Hamiltonian can be used to describe the chemical process of electron transfer across a chromophore chain, with the onsite parameters describing the strength of the electronic affinity at each chromophore and off-site couplings describing the barrier for transferring an electron between adjacent sites.

electron-transfer

A problem of practical interest that could be modelled by this method would be a functionalized graphene nanoribbon, where alternating sites contain radical character (Nano Lett. 2022, 22, 1, 164–171).

radical-figure

The stability of the radical character at each site can be described by the on-site parameter (\(\Omega _n\)) and the coupling between sites (\(J _{n,n+1}\)) governed by the properties of the linker regions containing the diketone groups. These parameters can be tuned by synthetic design of each component part of the nanoribbon.

\[\begin{align} H= \sum_{n=0}^{N-1} \hbar \Omega _n \sigma_n ^z - \dfrac{1}{2} \sum_{n=0}^{N-2} \big(J_{n,n+1}^x \hat{\sigma}_{n}^x \hat{\sigma}_{n+1}^x + J_{n,n+1}^y \hat{\sigma}_{n}^y \hat{\sigma}_{n+1}^y + J_{n,n+1}^z \hat{\sigma}_{n}^z \hat{\sigma}_{n+1}^z \big) \end{align}\]

Protocol for Hamiltonian evolution in a quantum computer framework:

Construct Hamiltonian operator (sum of tensor product of Pauli matrices)
Construct Trotter approximation of time-evolution operator (\(e^{-i\hat{H}t}\))
Construct circuit corresponding to time-evolution operator
Construct circuit for initial state
Combine the initial state and time-evolution circuits
Execute combined circuit
Measure final state
Computing observables with Hadamard test (correlation function and observables)

Model Hamiltonian and Parameters

Here we use as an example the Hamiltonian for the Heisenberg model, defined as follows:

\[\begin{align} H= \sum_{n=0}^{N-1} \hbar \Omega _n \sigma_n ^z - \dfrac{1}{2} \sum_{n=0}^{N-2} \big(J_{n,n+1}^x \hat{\sigma}_{n}^x \hat{\sigma}_{n+1}^x + J_{n,n+1}^y \hat{\sigma}_{n}^y \hat{\sigma}_{n+1}^y + J_{n,n+1}^z \hat{\sigma}_{n}^z \hat{\sigma}_{n+1}^z \big) \end{align}\]

where the coupling elements are described in terms of the \(\sigma _x, \sigma _y, \sigma _z\) (Pauli X, Y and Z) matrices with a coupling associated with each type of interaction term for each site/pair of sites. We consider the following parameters, as used in the publication Non-Markovian decay beyond the Fermi Golden Rule: Survival Collapse of the polarization in spin chains.(but with reduced number of spin sites):

\[\begin{align} N &= 3 \\ \hbar \Omega _0 &= 0.65 \\ \hbar \Omega _n &= 1.0, \; n > 0 \\ J_{0,1}^x = J_{1,0}^y &= 0.75 \\ J_{n,n+1}^x = J_{n,n+1}^y &= 1.0, \; n > 0 \\ J_{n,n+1}^z &= 0, \; \forall n \\ \end{align}\]

for a chain of \(N=3\) spins and an initial state with the first spin up, \(| \uparrow \rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\), and the remaining spins down, \(| \downarrow \rangle = \begin{bmatrix} 0 \\ 1\end{bmatrix}\). Although the dynamics of this Hamiltonian can be simulated in a classical computer, we could also use quantum computer to simulate this same problem. One way to do so is by harnessing Qiskit, a python library containing functions that ease the simulation of the problem in quantum device framework.

Statevector Simulation

from qflux.closed_systems.spin_propagators import * 
from qflux.closed_systems.hamiltonians import * 
from qflux.closed_systems.spin_dynamics_oo import * 

num_q = 3
evolution_timestep = 0.1
n_trotter_steps = 1
hamiltonian_coefficients = [[0.75 / 2, 0.75 / 2, 0.0, 0.65]] + [[0.5, 0.5, 0.0, 1.0]
                            for _ in range(num_q - 1)]
initial_state = "011"  # Specify the initial state as a binary string

csimulation = SpinDynamicsS(
                            num_q,
                            evolution_timestep,
                            n_trotter_steps,
                            hamiltonian_coefficients
                            )
csimulation.run_dynamics(nsteps=250, state_string=initial_state)
csimulation.save_results(f"{num_q}_spin_chain")
csimulation.plot_results(f"{num_q}_spin_chain_statevector")

Quantum Circuit Simulation with Hadamard Test

num_q = 3
evolution_timestep = 0.1
n_trotter_steps = 1
hamiltonian_coefficients = [[0.75 / 2, 0.75 / 2, 0.0, 0.65]] + [[0.5, 0.5, 0.0, 1.0]
                            for _ in range(num_q - 1)]
initial_state = "011"  # Specify the initial state as a binary string

qsimulation = SpinDynamicsH(
                            num_q,
                            evolution_timestep,
                            n_trotter_steps,
                            hamiltonian_coefficients
                            )
qsimulation.run_simulation(state_string=initial_state, total_time=25, num_shots=100)
qsimulation.save_results('hadamard_test')
qsimulation.plot_results('hadamard_test')