Energy Spectrums of Qbits

In this turotial we will explore how to set up arbitrary lattice configurations to house qbits and assign various interactions between qbits to simulate qbit operations. Our results on energy spectrums could be benchmarks of initial qbit setups for quantum computing theories/experiments.

Mixed 4-site Qbits

Introduction

We first use the 4-site mixed graph in the lattice configuration file: lattices.xml

<GRAPH name="4-site mixed" vertices="4"> 
  <VERTEX id="1" type="0"/>
  <VERTEX id="2" type="1"/>
  <VERTEX id="3" type="0"/>
  <VERTEX id="4" type="1"/>
  <EDGE type="0" source="1" target="2"/>
  <EDGE type="0" source="2" target="3"/>
  <EDGE type="0" source="3" target="4"/>
  <EDGE type="0" source="4" target="1"/>
  <EDGE type="1" source="1" target="3"/>
  <EDGE type="1" source="2" target="4"/>
</GRAPH> 

The lattice configuration is illustrated in the following diagram:

The same graph, with the Hamiltonian couplings labeled on each bond and the transverse field $\Gamma$ on each site:

Γ   1 ---J1--- 2   Γ
    |  \     / |
    J1  J2 J2  J1
    |  /     \ |
Γ   4 ---J1--- 3   Γ

In this lattice configuration there are two types of vertices, labeled as “0” for sites 1 and 3 and “1” for sites 2 and 4. For each qbit site there is a transverse magnetic field with strength Gamma. There are also two types of bonds, labeled as “0” for bonds between sites (1,2), (2,3), (3,4), and (4,1), and “1” for bonds between sites (1,3) and (2,4). For bond type “0”, we will assign an interaction J1 for bond type “0” and J2 for bond type “1”. All these is done in the model configuration file: models.xml

<HAMILTONIAN name="qbit operation">
  <PARAMETER name="J1" default="1"/>
  <PARAMETER name="J2" default="0.5"/>
  <BASIS ref="spin"/>
  <SITETERM site="i">
    -Gamma*Sx(i)
  </SITETERM>
  <BONDTERM source="1" target="2">
    J1*Sz(1)*Sz(2)
  </BONDTERM>
  <BONDTERM source="2" target="3">
    J1*Sz(2)*Sz(3)
  </BONDTERM>
  <BONDTERM source="3" target="4">
    J1*Sz(3)*Sz(4)
  </BONDTERM>
  <BONDTERM source="4" target="1">
    J1*Sz(4)*Sz(1)
  </BONDTERM>
  <BONDTERM source="1" target="3">
    J2*Sz(1)*Sz(3)
  </BONDTERM>
  <BONDTERM source="2" target="4">
    J2*Sz(2)*Sz(4)
  </BONDTERM>
</HAMILTONIAN>

With the above setups, the Hamiltonian for the 4-site qbits is given by

This is a small, pedagogical model (not tied to a specific published qbit device) used here to demonstrate how to define a custom lattice graph and Hamiltonian in ALPS from scratch, rather than using one of the built-in lattices/models.

Parameters

Method Choice

With only 4 sites the Hilbert space is $2^4=16$-dimensional, so any diagonalization method would work instantly; sparsediag’s Lanczos algorithm is used here purely for consistency with the other exact-diagonalization tutorials, and to illustrate the custom-lattice/custom-model workflow that scales to larger systems.

Simulation

We first import some modules:

import pyalps
import numpy as np
import matplotlib.pyplot as plt

We then set up parameters for the system and loop over the second coupling constants J2.

parms = []
# Loop over second coupling constant
for J2 in [0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6]:
    parms.append({
        'GRAPH'      : "4-site mixed",
        'MODEL'      : "qbit operation",
        'local_S'    : 0.5,
        'Gamma'      : 0.5,
        'J2'         : J2,
        'NUMBER_EIGENVALUES' : 5
    })

Now we set up the input files and run the simulations.

prefix = 'qbitenergy'
input_file = pyalps.writeInputFiles(prefix,parms)
res = pyalps.runApplication('sparsediag', input_file)
data = pyalps.loadEigenstateMeasurements(pyalps.getResultFiles(prefix=prefix))

We then iterate through parameter J2 and plot the lowest energy level for each J2.

x = []
E0 = []
for Lsets in data:
    J2 = pyalps.flatten(Lsets)[0].props['J2']
    x.append(J2)
    lowestE = pyalps.flatten(Lsets)[0].y[0]
    E0.append(lowestE)
    
# Set the scatter plot label
lbl="J1=1.0, Gamma=0.5"
plt.scatter(x,E0, label=lbl)
plt.legend()
plt.xlabel("J2")
plt.ylabel("E")
plt.title("4-site Mixed Graph")
plt.show()

The resulting energy spectrums for the lowest energies for various coupling constants J2 are shown in the following diagram:

Results

Ground-state energy $E_0$ as a function of $J_2$, from running the code above with $J_1=1$, $\Gamma=0.5$:

At $J_2=0$ (only the square-edge bonds active), $E_0=-1$ exactly, matching the isolated 4-site ring with a weak transverse field. As $J_2$ grows, the diagonal bonds add further antiferromagnetic frustration and the ground-state energy decreases monotonically.

Summary and Outlook

Diagonalizing this custom 4-site mixed-graph Hamiltonian shows the ground-state energy decreasing smoothly and monotonically as the diagonal coupling $J_2$ is turned on, with no sign of a level crossing in this parameter range.

1. What happens to the gap between the ground state and first excited state as $J_2$ is increased — does it close anywhere?
2. How would you extend the lattices.xml/models.xml pair here to simulate an 8-site version of the same mixed graph?
3. What is the ground state energy in the two limits $J_2 \to 0$ and $J_2 = J_1$, and can you explain them from the bond structure alone?