VQE

Hamiltonian

Potential Energy

The potential energy of a system of electrons and nuclei arises from the electrostatic interactions between the charged particles. There are 3 components, electron-electron repulsion, nuclear-nuclear repulsion, and electron-nuclear attraction. Coulomb's law $V = \frac{q_1q_2}{4\pi\epsilon_0 r}$ can be used to calculate their potential energy:

$$ \hat{V} = \underbrace{\sum_{i,j}^{\text{electrons}} \frac{e^2}{4\pi\epsilon_0 |r_i - r_j|}}_{\text{electron-electron repulsion}} + \underbrace{\sum_{i,j}^{\text{nuclei}} \frac{Z_i Z_j e^2}{4\pi\epsilon_0 |R_i - R_j|}}_{\text{nuclear-nuclear repulsion}} - \underbrace{\sum_{i}^{\text{electrons}} \sum_{j}^{\text{nuclei}} \frac{Z_j e^2}{4\pi\epsilon_0 |r_i - R_j|}}_{\text{electron-nuclear attraction}} $$

Kinetic Energy

Since classically, $T_x = \frac{P_x^2}{2m}$, and the quantum momentum operator $\hat{P}_x = -i\hbar\frac{\partial}{\partial x}$, so

$$ \hat{T}_x = \frac{\hat{P}_x^2}{2m} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} $$

If the particle is moving in 3 dimensions, then $P = \vec{x}P_x + \vec{y}P_y + \vec{z}P_z$ and the total momentum operator would be

$$ \hat{P} = -i\hbar(\vec{x}\frac{\partial}{\partial x} + \vec{y}\frac{\partial}{\partial y} + \vec{z}\frac{\partial}{\partial z}) = -i\hbar\nabla$$

$$ \therefore \hat{T} = -\frac{\hbar^2}{2m} \nabla^2 $$

The kinetic energy of the system is divided into kinetic energy of the nuclei and the electrons.

$$ \hat{T} = -\sum_{i}^{\text{nuclei}} \frac{\hbar^2}{2m_i} \nabla_i^2 - \sum_{i}^{\text{electrons}} \frac{\hbar^2}{2m_e} \nabla_i^2 $$

Simplified Molecular Hamiltonian

Several approximations are made to simplify the molecular Hamiltonian:

$$ \hat{H} \approx -\sum_{i}^{\text{electrons}} \frac{1}{2} \nabla_i^2 + \sum_{i,j}^{\text{electrons}} \frac{1}{|r_i - r_j|} - \sum_{i}^{\text{electrons}} \sum_{j}^{\text{nuclei}} \frac{Z_j}{|r_i - R_j|} + C'_n $$

Born-Oppenheimer Approximation: This approximation assumes that the nuclei are much heavier than the electrons and can be treated as fixed in space. This allows separating the electronic and nuclear motions, effectively removing the kinetic energy term for the nuclei.
Atomic Units: By using a system of units where certain fundamental constants (such as the Planck constant $\hbar$, the electron mass $m_e$, and the electron charge $e$) are set to 1, the equations can be simplified.
Constant Term: The term $C'_n$ is a constant that accounts for the nuclear-nuclear repulsion energy, as the positions of the nuclei are considered fixed in this approximation.

The simplified Hamiltonian includes the electronic kinetic energy term, the electron-electron repulsion term, and the electron-nuclear attraction term.

Second Quantization

Spin-Orbital Basis Functions

In quantum chemistry, wavefunctions are often represented using spin-orbital basis functions, which are products of spatial orbitals and spin functions. These basis functions are used to construct the many-electron wavefunction as a linear combination.

Spin-orbital basis functions are typically denoted as $\psi_p$, $\psi_q$, $\psi_r$, $\psi_s$, and so on. Each spin-orbital function is a product of a spatial orbital $\phi_p(r)$ and a spin function $\alpha(\omega)$ or $\beta(\omega)$, representing the spin up or spin down state, respectively:

$$\psi_p(x) = \phi_p(r) \alpha(\omega)$$ $$\psi_q(x) = \phi_q(r) \beta(\omega)$$

The spatial orbitals $\phi_p(r)$ and $\phi_q(r)$ are typically constructed as linear combinations of atomic orbitals (LCAO) centered on the nuclei. Common basis sets used in quantum chemistry include Slater-type orbitals (STOs) and Gaussian-type orbitals (GTOs).

These spin-orbital basis functions form the basis set in which the molecular orbitals and Hamiltonian are represented.

One-Electron Part

In quantum chemistry, it is often convenient to express the Hamiltonian and wavefunctions in a second quantized form, which involves creation and annihilation operators acting on spin-orbital basis functions.

In this equation:

$$ \sum_i h_1(x_i) = \sum_{p,q} \langle p | \hat{h}_1 | q \rangle a^\dagger_p a_q $$

$a^\dagger_p$ and $a_q$ are the creation and annihilation operators, respectively, which create or annihilate an electron in the spin-orbital basis function $ψ_p$ or $ψ_q$.
$h_{pq} = \langle p | \hat{h}_1 | q \rangle$ represents the one-electron integrals, obtained by integrating the one-electron part of the Hamiltonian ($\hat{h}_1$) over the spin-orbital basis functions $ψ_p$ and $ψ_q$.

This second quantized form provides a compact way to represent the one-electron part of the Hamiltonian, including both the kinetic energy and the electron-nuclear attraction terms, in terms of creation and annihilation operators acting on the spin-orbital basis.

By expressing the Hamiltonian in this form, it becomes easier to perform calculations and manipulations using the tools of second quantization, which are particularly useful in quantum chemistry and condensed matter physics.

Two-Electron Part

Building on equation 15, which represents the one-electron part of the Hamiltonian in second quantized form, equation 16 introduces the two-electron part:

$$\sum_{i,j} h_2(x_i, x_j) = \sum_{p,q,r,s} \langle pq | \hat{H} | rs \rangle a^\dagger_p a^\dagger_q a_r a_s$$

Here, the two-electron integrals $\langle pq | \hat{H} | rs \rangle$ represent the electron-electron repulsion term, integrated over the spin-orbital basis functions $\psi_p$, $\psi_q$, $\psi_r$, and $\psi_s$.

The creation and annihilation operators $a^\dagger_p$, $a^\dagger_q$, $a_r$, and $a_s$ act on the spin-orbital basis functions to create or annihilate electrons in the corresponding orbitals.

This equation captures the contribution of the electron-electron repulsion term to the Hamiltonian in the second quantized form, allowing for efficient calculations and manipulations.

Full Molecular Hamiltonian

Equation 17 combines the one-electron and two-electron parts from equations 15 and 16, along with a constant term $h_0$, to give the full molecular Hamiltonian in second quantized form:

$$\hat{H} = \sum_{p,q} h_{pq} a^\dagger_p a_q + \frac{1}{2} \sum_{p,q,r,s} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s + h_0$$

The first term represents the one-electron part, the second term represents the two-electron part, and $h_0$ is a constant shift factor that accounts for the nuclear-nuclear repulsion energy and any other constant shift needed to align with the desired reference energy.

This equation provides a compact representation of the full molecular Hamiltonian in second quantized form, expressed in terms of creation and annihilation operators acting on the spin-orbital basis functions.

Mapping

Jordan-Wigner Transform

The Jordan-Wigner (JW) transform is a mathematical technique used to map the fermionic creation and annihilation operators in second quantization onto a system of spin operators, enabling their representation on a quantum computer using qubit states.

The JW transform maps the fermionic operators as follows:

$$a^\dagger_n \rightarrow \frac{1}{2} \left(\prod_{j=0}^{n-1} -Z_j\right) (X_n - iY_n)$$ $$a_n \rightarrow \frac{1}{2} \left(\prod_{j=0}^{n-1} -Z_j\right) (X_n + iY_n)$$

Here, $X_n$, $Y_n$, and $Z_n$ are the Pauli spin operators acting on the n-th qubit, and the product $\prod_{j=0}^{n-1} -Z_j$ introduces a phase factor to account for the fermionic anti-commutation relations.

The JW transform provides a way to map the fermionic operators onto a system of qubits, enabling the simulation of fermionic systems, such as molecules, on a quantum computer. However, it has some limitations, such as the non-local nature of the operator representations, which can limit scalability for larger systems.

By applying the JW transform to the second quantized Hamiltonian (equation 17), the molecular Hamiltonian can be expressed as a linear combination of tensor products of Pauli operators acting on the qubit states, as shown in equation 22 of the paper for the hydrogen molecule example.

This mapping is a crucial step in performing quantum simulations of molecular systems on quantum computers using algorithms like the Variational Quantum Eigensolver (VQE).

Slater Determinant

The slater determinant is a way to construct an antisymmetric many-particle wavefunction that satisfies the requirements of fermi-dirac statistics. This ensures that the wavefunction is antisymmetric with respect to the interchange of any two fermionic particles, such as electrons, which are indistinguishable from each other. Essentially, the wave function must simply change sign when any two electrons are swapped.

Let's first consider a single-particle wavefunction $\psi_x$, which depends on the spatial coordinates x. for a system with n indistinguishable particles, we need to construct an n-particle wavefunction $\Psi(x_1, x_2, \ldots, x_n)$ that is antisymmetric with respect to the interchange of any two particles. We can construct such an antisymmetric n-particle wavefunction by taking the determinant of an n x n matrix, whose entries are the single-particle wavefunctions:

$$ \Psi(x_1, x_2, \ldots, x_n) = \det \begin{bmatrix} \psi(x_1) & \psi(x_2) & \cdots & \psi(x_n) \end{bmatrix} $$

The key property of the determinant that ensures antisymmetry is that it changes sign when any two rows (or columns) are interchanged. Mathematically, for any n x n matrix $A$:

$$ det(A) = -det(A') $$

where $A'$ is the matrix obtained by interchanging any two rows (or columns) of $A$.

Now, let's consider the Slater determinant for an n-fermion system:

$$\psi(x_1, x_2, \dots, x_n) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_i(x_1) & \chi_j(x_1) & \dots & \chi_k(x_1) \\ \chi_i(x_2) & \chi_j(x_2) & \dots & \chi_k(x_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_i(x_n) & \chi_j(x_n) & \dots & \chi_k(x_n) \end{vmatrix}$$

Here, $\chi_m(x_l)$ represents the the molecular orbital wavefunctions, for the $m$th orbital of the $l$th electron. The $1/\sqrt{N!}$ factor is a normalization constant.

If we interchange the coordinates of two particles, say $x_1$ and $x_2$, it corresponds to interchanging two rows of the matrix. According to the property of determinants mentioned above, this operation changes the sign of the determinant:

$$\Psi(x_2, x_1, x_3, \dots, x_n) = -\Psi(x_1, x_2, x_3, \dots, x_n)$$

This sign change satisfies the antisymmetry principle for fermions, which states that the many-particle wavefunction must change sign when any two particles are interchanged.

Occupation Number Representation

In the occupation number representation, we specify the many-fermion wavefunction by indicating the occupation of each single-particle state (spin-orbital) by either 0 (unoccupied) or 1 (occupied). For a system with n fermions and m spin-orbitals ${\phi_1, \phi_2, \dots, \phi_m}$, the occupation number representation is:

$$|\Psi\rangle = \sum_{i} C_i |n_1^i n_2^i \dots n_m^i\rangle$$

Here, $|n_1^i n_2^i \dots n_m^i\rangle$ represents a specific configuration where $n_j^i$ is either 0 or 1, indicating whether the j-th spin-orbital is occupied or unoccupied in the i-th configuration. The coefficients $C_i$ are the amplitudes of each configuration in the overall wavefunction.

The occupation number representation provides a compact way to express the many-fermion wavefunction as a linear combination of Slater determinants, where each configuration represents a specific occupation pattern of the spin-orbitals, automatically satisfying the antisymmetry principle.

This representation is particularly useful for computational purposes, as it allows us to work with the occupation numbers directly, rather than dealing with the explicit form of the Slater determinants, which can become increasingly complex for larger systems.

Solving

Sure, let me explain in detail how the Variational Quantum Eigensolver (VQE), Unitary Coupled Cluster Singles and Doubles (UCCSD), and the Hartree-Fock (HF) method are used to solve the problem of finding the ground-state energy of a molecular system, such as the hydrogen molecule.

Hartree-Fock Method

The Hartree-Fock (HF) method is a mean-field approximation used in quantum chemistry to obtain an initial estimate of the many-electron wavefunction and energy for a molecular system. It is a crucial starting point for more advanced methods like VQE and UCCSD.

In the HF method, the many-electron wavefunction is approximated as a single Slater determinant constructed from a set of spin-orbitals. The HF equations are solved iteratively to find the optimal set of spin-orbitals that minimize the energy of this Slater determinant, subject to the constraint of being an antisymmetric product of single-particle wavefunctions.

The HF method provides a good initial approximation to the ground-state wavefunction and energy, but it does not account for the instantaneous electron-electron interactions (correlation effects) beyond a mean-field treatment.

In the context of the VQE algorithm, the HF method is often used to generate an initial guess for the wavefunction, which is then refined by the variational optimization procedure.

Unitary Coupled Cluster Singles and Doubles (UCCSD)

UCCSD is a specific form of the Ansatz (trial wavefunction) used in the VQE algorithm. It is based on the coupled cluster (CC) method, which is a widely used technique in quantum chemistry for including electron correlation effects beyond the HF approximation.

The UCCSD Ansatz is defined as:

$$|\Psi_{\text{UCCSD}}(\vec{\theta})\rangle = e^{\hat{T}(\vec{\theta})} |\Phi_0\rangle$$

Where $|\Phi_0\rangle$ is the HF reference wavefunction (a single Slater determinant), and $\hat{T}(\vec{\theta})$ is the cluster operator, which is a sum of excitation operators that generate singly and doubly excited Slater determinants from the reference:

$$\hat{T}(\vec{\theta}) = \hat{T}_1(\vec{\theta}) + \hat{T}_2(\vec{\theta})$$

The excitation operators $\hat{T}_1$ and $\hat{T}_2$ are parameterized by the variational parameters $\vec{\theta}$, which are optimized during the VQE procedure.

The UCCSD Ansatz is a compact and efficient way to capture essential electron correlation effects, making it a popular choice for the VQE algorithm in quantum chemistry applications.

Variational Quantum Eigensolver (VQE)

The VQE algorithm is a hybrid quantum-classical algorithm that leverages both quantum and classical resources to find the ground-state energy and wavefunction of a molecular system.

The key steps in the VQE algorithm are:

Prepare the Ansatz Wavefunction: An initial trial wavefunction, called the Ansatz, is prepared on a quantum computer. This Ansatz is a parameterized quantum circuit, often based on a form like UCCSD, acting on the qubit register.
Encode the Molecular Hamiltonian: The molecular Hamiltonian, expressed as a linear combination of Pauli operators (e.g., equation 22 for the hydrogen molecule), is encoded onto the quantum computer using appropriate quantum gates.
Measure the Energy Expectation Value: The quantum computer is used to measure the expectation value of the Hamiltonian with respect to the current Ansatz wavefunction: $\langle\Psi(\vec{\theta})|\hat{H}|\Psi(\vec{\theta})\rangle$, where $\vec{\theta}$ represents the parameters of the Ansatz circuit.
Classical Optimization: A classical optimization algorithm, such as the L-BFGS algorithm used in the paper, is employed to iteratively update the parameters $\vec{\theta}$ of the Ansatz circuit, minimizing the energy expectation value measured on the quantum computer. This step is repeated until convergence is achieved.

The VQE algorithm starts with an initial guess for the wavefunction, typically obtained from the HF method, and then iteratively refines it by optimizing the Ansatz parameters to minimize the energy expectation value. The UCCSD form is often used as the Ansatz due to its ability to capture essential electron correlation effects.

By combining the computational power of quantum computers with classical optimization techniques, the VQE algorithm can approximate the ground-state energy and wavefunction of molecular systems with high accuracy, provided that the Ansatz circuit is expressive enough and the quantum hardware has sufficiently low noise and error rates.

In summary, the HF method provides an initial guess for the wavefunction, the UCCSD form is used as the parameterized Ansatz circuit, and the VQE algorithm iteratively optimizes the Ansatz parameters to find the ground-state energy and wavefunction of the molecular system, leveraging both quantum and classical resources.