Getting Started

Since quantum mechanical systems are described by operators acting on state vectors living in Hilbert spaces, this library is mainly concerned with providing such operators and functions that manipulate state vectors. In this chapter, you will learn the essentials of using the HQS Quantum Solver software by considering an example computation.

Let us consider the tight-binding Hamiltonian defined on a chain lattice, i.e., the Hamiltonian $H$ is given by $H = - t j = 0 \sum M - 2 (c_{j}^{†} c_{j + 1} + c_{j + 1}^{†} c_{j}),$ where $M$ is the number of lattice sites, $t$ the strength of the hopping term and $c_{j}$ the annihilation operator at site $j$ .

Imports and Parameters

First, we need to import the functions, classes and modules that are required for the script. If you want to follow along, copy the following imports into your script file. We will explain the classes and functions as we go along.

# Title    : HQS Quantum Solver Getting Started Guide
# Filename : getting_started.py

# Python, NumPy and SciPy imports
from math import pi, sqrt
import numpy as np
from scipy.linalg import toeplitz
from scipy.sparse.linalg import eigsh
import matplotlib.pyplot as plt

# HQS Quantum Solver imports
from hqs_quantum_solver.operators import (
    SpinlessFermionOperatorEigen as SpinlessFermionOperator
)
from hqs_quantum_solver.types import (
    SpinlessFermionHamiltonianInput,
    SpinlessFermionAnnihilationInput,
)
from hqs_quantum_solver.interface import scipy_operator_adapter

Next, we define the parameters that we will refer to in the script.

M = 16  # site count
N = 1   # particle count
E = 0   # energy level

With that out of the way, we can start setting up the tight-binding Hamiltonian.

Setting up the Hamiltonian

We want to look at a system given by the tight-binding Hamiltonian. Thus, the first step is to construct said Hamiltonian. Quantum Solver provides a generic interface for constructing Hamiltonians, as well as several convenience functions. Especially, Quantum Solver implements Hamiltonians of the form $H = j = 0 \sum M - 1 k = 0 \sum M - 1 (h_{0})_{jk} c_{j}^{†} c_{k} + additional terms,$ where $h_{0} \in C^{M \times M}$ is the, so called, hopping matrix. Note, that the additional terms default to zero unless we explicitly enable them.

If we define the matrix $h_{0}$ to be $h_{0} = 0 - t - t 0 - t - t ⋱ ⋱ ⋱ ⋱ - t - t 0,$ then the Hamiltonian in the previous formula becomes the tight-binding Hamiltonian defined at the beginning of this chapter.

This matrix can be easily constructed using the toeplitz function from SciPy.

t = 1
c = np.array([-t if j == 1 else 0 for j in range(M)])
h0 = toeplitz(c)

With this matrix at hand, we can construct the Hamiltonian. Quantum Solver supports multiple linear algebra backends. For this reason, the construction of quantum mechanical operators is a two-step process. First, we create an "Input" object that describes the operator to be constructed. Then, we pass this object to an operator class of the desired backend. Note that we have imported the SpinlessFermionOperatorEigen as SpinlessFermionOperator, which means that we are using the Eigen backend, which implements its functionality using the well-known Eigen library.

To create the "Input", we create a SpinlessFermionHamiltonianInput object. The constructor of this object expects four requried arguments and a bunch of additional arguments. The additional arguments define the additional terms mentioned above. The required arguments are M (int), N (int), mod_N (int), and h0 (ndarray).

The parameter M describes the number of sites in the system and the parameter N the number of particles in the system.

The parameter mod_N sets the step size in which the number of particles is allowed to change during the simulation. For example for M = 10, N = 5 and mod_N = 4 the system allows for states with 1, 5, and 9 particles. When mod_N = 0 the number of particles is conserved, i.e., constant.

The parameter h0 is the "hopping matrix" given as a NumPy array.

With all this information, we can create the Hamiltonian by first creating the "Input" object and then handing it over to the linear algebra backend.

H_input = SpinlessFermionHamiltonianInput(M=M, N=N, mod_N=0, h0=h0)
H = SpinlessFermionOperator(H_input)

The Quantum Solver operators primarily provide matrix-vector and matrix-matrix routines. Here, we will be using the SpinlessFermionOperatorEigen. As a first example for using the operator, we want to look at its spectrum. For that, we can use the eigsh function from SciPy. To use this function, however, we need to use an adapter (see scipy_operator_adapter) to make the Quantum Solver operator compatible with SciPy.

k = min(12, H.shape[0] - 1)  # Number of eigenvalue to compute
evals, evecs = eigsh(scipy_operator_adapter(H), k=k, which="SA")

print(evals)

plt.figure("eigenvalues", clear=True)
plt.title(f"Eigenvalues (M = {M}, N = {N})")
plt.plot(evals, ".")

Now, we have enough code to run the script for the first time. Doing so should produce the plot shown below, which shows the ten smallest eigenvalues of the Hamiltonian, and hence, the lowest energy levels of the system.

Before continuing, modify the values of $M$ and $N$ , run the script, and look at the result a few times.

Note, for mod_N=0 the number of states Quantum Solver has to store is $(N M)$ . Hence, the number of states can grow rather quickly when increasing N while M is large. This problem gets worse for mod_N != 0. Keep in mind, while Quantum Solver provides you with highly accurate computations, you will have to provide enough computational resources for it to shine.

Below, we have listed the 5 smallest eigenvalues for M = 16 and different values of N.

#	$N = 1$	$N = 2$	$N = 3$
1	-1.97	-3.83	-5.53
2	-1.86	-3.67	-5.31
3	-1.70	-3.57	-5.14
4	-1.48	-3.44	-5.04
5	-1.21	-3.34	-5.04

We can make our first observation: Adding the first two values of the $N = 1$ column gives you the first value in the $N = 2$ column, and adding the first three values of the $N = 1$ column will give you the first value in the $N = 3$ column. The reason for this is that in a two- and three-particle system, the lowest energy to obtain is the one of the particles occupying the lowest two and three energy levels, respectively. Since we are dealing with a fermionic system, the particles are not allowed to occupy the same state, which explains this behavior.

We shall later come back to this observation.

Note how few lines of code were necessary, to produce physically meaningful results.

Site Occupations

To take this example a little bit further, let us look at the computation of site occupation expectation values. More precisely, we want to compute $⟨ n_{j} ⟩$ for $j = 0, \dots, M - 1$ , where $n_{j} = c_{j}^{†} c_{j}$ . Note that $⟨ n_{j} ⟩ = ⟨ ψ ∣ n_{j} ∣ ψ ⟩ = ⟨ ψ ∣ c_{j}^{†} c_{j} ∣ ψ ⟩ = (c_{j} ∣ ψ ⟩)^{†} c_{j} ∣ ψ ⟩ .$ Hence, we just need to compute $c_{j} ∣ ψ ⟩$ and then compute the square norm of this vector.

First, we have to construct the annihilation operator $c_{j}$ . As before, we first have to create an input object that we then hand over to the linear algebra backend. This time we create an SpinlessFermionAnnihilationInput object. The constructor expects the parameters M (int), N (int), and mod_N (int), which describe the domain, i.e., the input, of the operator and have the same meaning as before. Additionally, we need to provide an argument called c_i (int | ndarray), which describes which annihilation operator is to be constructed. When passing an integer j, the annihilation operator for site j is constructed. (See SpinlessFermionAnnihilationInput.) Note that the codomain of the operator has N - 1 particles, so the operator maps to a different space.

Once we have constructed the operator, we can use the dot method, to apply it to a state vector. Below, we define a function site_occupation which computes the expectation value for a given state vector and site. We then plot the expectation value for every site and for the state vector corresponding to the energy level E.

def site_occupation(psi, j):
    annihilation_input = SpinlessFermionAnnihilationInput(M=M, N=N, mod_N=0, c_i=j)
    annihilation = SpinlessFermionOperator(annihilation_input)
    c_psi = annihilation.dot(psi)
    return c_psi.conj() @ c_psi


occupations = np.array([site_occupation(evecs[:, E], j) for j in range(M)])
plt.figure("site_occupation", clear=True)
plt.title(f"Site Occupation (M = {M}, N = {N}, E = {E}")
plt.plot(occupations, "x-")
plt.axis((0, M - 1, 0, 1))
plt.xlabel("j")
plt.ylabel("Expectation Value")

The site occupytion. — The site occupations for `M=16`, `N=1`, `E=0`.

Energy occupations

While looking a site occupations is certainly interesting, for most applications it is more useful to look at energy occupations instead.

Consider the matrix $Q$ given by $Q_{jk} = \frac{2}{M + 1} sin (\frac{( j + 1 ) ( k + 1 ) π}{M + 1}) for j, k = 0, \dots, M - 1 .$ The columns of this matrix are the eigenvectors of the matrix $h_{0}$ . It is a well-known fact that if we define $d_{k} = j = 0 \sum M - 1 Q_{jk}^{*} c_{j}$ we can write the Hamiltonian $H$ as $H = k \sum ε_{k} d_{k}^{†} d_{k}$ where $ε_{k}$ are the energy levels of the system. Consequently, we can measure expectation value of the occupation of the $k$ -th energy level, by computing $⟨ d_{k}^{†} d_{k} ⟩ = ⟨ ψ ∣ d_{k}^{†} d_{k} ∣ ψ ⟩ .$

We can carry out this computation in a similar way to the computation of the site occupation. All we have to do is to set the annihilation variable to a linear combination of annihilation operators, which is given by the definition of $d_{k}$ . Luckily, this is such a common operation that the Quantum Solver includes a convenience method for exactly this case. The c_i parameter of the constructor of the SpinlessFermionAnnihilationInput also accepts an array of coefficients and creates the corresponding linear combination of annihilation operators.

J, K = np.meshgrid(np.arange(M), np.arange(M), indexing='ij')
Q = sqrt(2) / sqrt(M+1) * np.sin((J+1) * (K+1) * pi / (M+1))

def energy_occupation(psi, k):
    annihilation_input = SpinlessFermionAnnihilationInput(
        M=M, N=N, mod_N=0, c_i=Q[:,k].conj()
    )
    annihilation = SpinlessFermionOperator(annihilation_input)
    c_psi = annihilation.dot(psi)
    return (c_psi.conj() @ c_psi).real


ks = np.arange(M)
expectation = np.array([energy_occupation(evecs[:, E], k) for k in ks])
plt.figure("energy", clear=True)
plt.title(f"Energy Occupation (M = {M}, N = {N}, E = {E})")
plt.plot(ks, expectation, "x-")
plt.xlabel("p")
plt.ylabel("Expectation Value")

Below, you can find the plots that this script produces for different parameter configurations.

The energy occupytion. — The energy occupations for `M=16`, `N=2`, `E=0`.

In these figures, you can see how the fermions arrange in the different one-particle energy states to achieve the energy levels we have observed in the table in the Setting up the Hamiltonian section.

Note that if you do not know the eigenvectors of the matrix $h_{0}$ , you can easily obtain them by using the eigh function of NumPy.

Summary

To conclude this introduction, let us sum up the main take-aways of this chapter.

Quantum Solver works by providing classes for working with quantum mechanical operators.
Operators are created by constructing an "Input" object and handing it to a linear algebra backend.
The parameters M, N, mod_N describe the domain of the operator, where M is the number of sites, N the number of particles, and mod_N the step size by which the number of particles is allowed to change.
To convert Quantum Solver operators into SciPy linear operators, you can use the scipy_operator_adapter function.
Operators provide the dot method, which applies the operator to a state vector.

Complete Code

# Title    : HQS Quantum Solver Getting Started Guide
# Filename : getting_started.py

# Python, NumPy and SciPy imports
from math import pi, sqrt
import numpy as np
from scipy.linalg import toeplitz
from scipy.sparse.linalg import eigsh
import matplotlib.pyplot as plt

# HQS Quantum Solver imports
from hqs_quantum_solver.operators import (
    SpinlessFermionOperatorEigen as SpinlessFermionOperator
)
from hqs_quantum_solver.types import (
    SpinlessFermionHamiltonianInput,
    SpinlessFermionAnnihilationInput,
)
from hqs_quantum_solver.interface import scipy_operator_adapter

# ===== Parameters =====

M = 16  # site count
N = 1   # particle count
E = 0   # energy level

# ===== Creation of the Hamiltonian =====

t = 1
c = np.array([-t if j == 1 else 0 for j in range(M)])
h0 = toeplitz(c)

H_input = SpinlessFermionHamiltonianInput(M=M, N=N, mod_N=0, h0=h0)
H = SpinlessFermionOperator(H_input)

# ===== Computing the Eigenvalues and Eigenvectors =====

k = min(12, H.shape[0] - 1)  # Number of eigenvalue to compute
evals, evecs = eigsh(scipy_operator_adapter(H), k=k, which="SA")

print(evals)

plt.figure("eigenvalues", clear=True)
plt.title(f"Eigenvalues (M = {M}, N = {N})")
plt.plot(evals, ".")

# ===== Plotting the Site Occupations =====

def site_occupation(psi, j):
    annihilation_input = SpinlessFermionAnnihilationInput(M=M, N=N, mod_N=0, c_i=j)
    annihilation = SpinlessFermionOperator(annihilation_input)
    c_psi = annihilation.dot(psi)
    return c_psi.conj() @ c_psi


occupations = np.array([site_occupation(evecs[:, E], j) for j in range(M)])
plt.figure("site_occupation", clear=True)
plt.title(f"Site Occupation (M = {M}, N = {N}, E = {E}")
plt.plot(occupations, "x-")
plt.axis((0, M - 1, 0, 1))
plt.xlabel("j")
plt.ylabel("Expectation Value")

# ===== Plotting the Energy Occupations =====

J, K = np.meshgrid(np.arange(M), np.arange(M), indexing='ij')
Q = sqrt(2) / sqrt(M+1) * np.sin((J+1) * (K+1) * pi / (M+1))

def energy_occupation(psi, k):
    annihilation_input = SpinlessFermionAnnihilationInput(
        M=M, N=N, mod_N=0, c_i=Q[:,k].conj()
    )
    annihilation = SpinlessFermionOperator(annihilation_input)
    c_psi = annihilation.dot(psi)
    return (c_psi.conj() @ c_psi).real


ks = np.arange(M)
expectation = np.array([energy_occupation(evecs[:, E], k) for k in ks])
plt.figure("energy", clear=True)
plt.title(f"Energy Occupation (M = {M}, N = {N}, E = {E})")
plt.plot(ks, expectation, "x-")
plt.xlabel("p")
plt.ylabel("Expectation Value")