Stage 4: Staggered magnetization

At this stage we turn to studying the ferromagnetic properties of our spin chain. We evaluate the performance of the QTD circuits we’ve implemented so far.

Concretely, we wish to plot the chain’s staggered magnetization, which is given by

(1)\[M_{\mathrm{stag}}(t) = \frac{1}{N} \sum_{j=1}^N (-1)^j \, \langle \psi(t) \mid \sigma_j^Z \mid \psi(t) \rangle,\]

as a function of time. Here \(\vert \psi(t) \rangle = U(t) \vert \psi_0 \rangle\) denotes the chain’s time-dependent quantum state.

Given a time value \(t \geq 0\) and an initial state \(\vert \psi_0 \rangle\), we compute \(M_{\mathrm{stag}}(t)\) using the time evolution circuits implemented in Stage 3: prepare and measure \(\vert \psi(t) \rangle\) using a quantum backend and then compute the expected value of the observable

\[\frac{1}{N} \sum_{j=1}^N (-1)^j \, \sigma_j^Z\]

with respect to the observed measurement distribution on a classical device.

Exercise 4.1

Suppose \(\mathcal{O}\) is a Hermitian observable whose matrix with respect to the computational basis is diagonal. Let \(\lambda_x\) denote the eigenvalue of \(\mathcal{O}\) corresponding to the computational basis state \(\vert x \rangle\). Let \(\vert \psi \rangle = \sum_x \alpha_x \vert x \rangle\) denote any quantum state, written as a superposition over the computational basis, and let \(p_x = \vert \alpha_x \vert^2\) denote the probability of observing \(\vert \psi \rangle\) in the state \(\vert x \rangle\).

Show that the expectation value

\[\langle \psi \vert\, \mathcal{O} \,\vert \psi \rangle = \sum_{x} p_x \lambda_x\]

is simply a weighted average of the eigenvalues of \(\mathcal{O}\).

Expectation value of magnetization operator

Now use the last result to implement the StaggeredMagnetization.energies() method defined in src.staggered_magnetization. This will allow you to compute compute \(M_{\mathrm{stag}}(t)\) by calling an instance of the StaggeredMagnetization class, as exemplified below.

class src.staggered_magnetization.StaggeredMagnetization(num_qubits)

Bases: object

Calculator for the time-dependent staggered magnetization of a spin chain evolving under the action of a Heisenberg Hamiltonian.

In particular, this class implements a __call__ dunder method that evaluates the expectation value defining \(M_{\mathrm{stag}}(t)\) with respect to the quantum state described by the measurements dictionary of (state, counts) key-value pairs.

EXAMPLES:

>>> from qiskit_aer import Aer
>>> from src.staggered_magnetization import StaggeredMagnetization
>>> from src.xyz_evolution import XYZEvolutionCircuit
>>> num_qubits, t = 3, 1
>>> qc = XYZEvolutionCircuit(num_qubits, [1, 1, 1], final_time=t, trotter_num=10)
>>> backend = Aer.get_backend("aer_simulator")
>>> qc = qc.decompose(["Uxz", "Uxyz"])
>>> qc.measure_all()
>>> psi = backend.run(qc).result().get_counts()
>>> m_stag = StaggeredMagnetization(num_qubits)
>>> m_stag(psi)
0.3333333333333333

energies(states)

Quickly obtain eigenvalues of an Ising Hamiltonian corresponding to the given states.

EXAMPLES:

>>> import numpy as np
>>> from src.staggered_magnetization import StaggeredMagnetization
>>> m_stag = StaggeredMagnetization(5)
>>> states = np.array([[0, 1, 1, 0, 1], [1, 1, 0, 0, 0]])
>>> m_stag.energies(states)
array([-0.2,  0.2])

Putting it all together: testing the (poor) performance of uncompressed QTD circuits

Head over to the xyz_evolution_challenge.ipynb notebook and plot the staggered magnetization \(M_{\mathrm{stag}}(t)\) of the spin chain as a function of time, for \(0 \leq t \leq 2.5\) and using \(n = 100\) Trotter steps. Use the coupling constants \((J_x, J_y, J_z) = (-2.4, 0, -0.6)\) and no external magnetic field (set \(h = 0\)). Repeat this exercise using num_qubits=3 and num_qubits=4 and include your plots with your submission.