Variational Quantum Eigensolver (VQE)

The Variational Quantum Eigensolver (VQE) is a hybrid classical-quantum algorithm designed to find the ground state and ground-state energy of a given Hamiltonian. It is particularly suited for Noisy Intermediate-Scale Quantum (NISQ) devices because it delegates the parameter optimization to a classical computer, keeping the quantum coherence time requirements relatively low.

The Variational Principle

VQE is mathematically anchored on the Ritz variational principle from quantum mechanics. Let \(H\) be a Hermitian operator representing the Hamiltonian of a system. Its ground-state energy \(E_0\) is the lowest eigenvalue of \(H\):

\[H |\psi_0\rangle = E_0 |\psi_0\rangle\]

For any parameterized trial state (or ansatz) \(|\psi(\boldsymbol{\theta})\rangle\), where \(\boldsymbol{\theta} = (\theta_1, \theta_2, \dots, \theta_M)^T\) is a vector of classical parameters, the expectation value of the energy (Rayleigh quotient) satisfies:

\[E(\boldsymbol{\theta}) = \frac{\langle\psi(\boldsymbol{\theta})| H |\psi(\boldsymbol{\theta})\rangle}{\langle\psi(\boldsymbol{\theta})|\psi(\boldsymbol{\theta})\rangle} \ge E_0\]

If the ansatz state is prepared via a unitary quantum circuit acting on a reference state \(|\Phi_{\text{ref}}\rangle\) (such as the Hartree-Fock state), the state is normalized (\(\langle\psi(\boldsymbol{\theta})|\psi(\boldsymbol{\theta})\rangle = 1\)), and the energy expectation simplifies to:

\[E(\boldsymbol{\theta}) = \langle\psi(\boldsymbol{\theta})| H |\psi(\boldsymbol{\theta})\rangle = \langle\Phi_{\text{ref}}| U^\dagger(\boldsymbol{\theta}) H U(\boldsymbol{\theta}) |\Phi_{\text{ref}}\rangle\]

The goal of VQE is to minimize \(E(\boldsymbol{\theta})\) by classically updating the parameter vector \(\boldsymbol{\theta}\) using optimization algorithms (e.g., COBYLA, Nelder-Mead, L-BFGS-B):

\[E_{\text{VQE}} = \min_{\boldsymbol{\theta}} \langle\psi(\boldsymbol{\theta})| H |\psi(\boldsymbol{\theta})\rangle \approx E_0\]

Second-Quantized Electronic Hamiltonian

In quantum chemistry, the electronic Hamiltonian in the Born-Oppenheimer approximation is expressed in second quantization using a set of orthonormal spin-orbitals:

\[H = \sum_{pq} h_{pq} a^\dagger_p a_q + \frac{1}{2} \sum_{pqrs} g_{pqrs} a^\dagger_p a^\dagger_q a_s a_r + E_{\text{nuc}}\]

where:

\(a^\dagger_p\) and \(a_q\) are fermionic creation and annihilation operators satisfying the canonical anticommutation relations:

\[\{a_p, a^\dagger_q\} = \delta_{pq}, \quad \{a_p, a_q\} = \{a^\dagger_p, a^\dagger_q\} = 0\]
\(h_{pq}\) represent the one-body integrals (kinetic energy of electrons and nuclear attraction):

\[h_{pq} = \int \phi^*_p(\mathbf{r}) \left( -\frac{1}{2} \nabla^2 - \sum_A \frac{Z_A}{|\mathbf{r} - \mathbf{R}_A|} \right) \phi_q(\mathbf{r}) \, d\mathbf{r}\]
\(g_{pqrs}\) are the two-body electron-repulsion integrals (ERI) in physicists’ convention:

\[g_{pqrs} = \iint \frac{\phi^*_p(\mathbf{r}_1) \phi^*_q(\mathbf{r}_2) \phi_s(\mathbf{r}_1) \phi_r(\mathbf{r}_2)}{|\mathbf{r}_1 - \mathbf{r}_2|} \, d\mathbf{r}_1 d\mathbf{r}_2\]
\(E_{\text{nuc}}\) is the classical nuclear repulsion energy.

Fermion-to-Qubit Mappings

Since quantum computers operate on qubits (represented by spin-1/2 algebra with Pauli matrices \(I, X, Y, Z\)) rather than fermions, fermionic creation and annihilation operators must be mapped to spin operators. Carcará supports three mapping methods:

1. Jordan-Wigner (JW) Mapping

The Jordan-Wigner mapping maps fermionic occupations directly to qubit states, representing the nonlocal fermionic anticommutation phases using chains of \(Z\) gates:

\[a^\dagger_j = I^{\otimes j-1} \otimes \left( \frac{X - iY}{2} \right) \otimes Z^{\otimes N-j}\]

\[a_j = I^{\otimes j-1} \otimes \left( \frac{X + iY}{2} \right) \otimes Z^{\otimes N-j}\]

2. Parity Mapping

The Parity mapping stores the parity of occupations of preceding spin-orbitals on each qubit. This configuration allows a two-qubit reduction by exploiting the conservation of total electron number and spin parity, which reduces the active qubit count by 2.

3. Bravyi-Kitaev (BK) Mapping

The Bravyi-Kitaev mapping uses a binary tree structure to store partial parities, balancing the locality of update operations and parity calculations. Both JW and BK scale the operator weight logarithmically or linearly.

The UCCSD Ansatz

A common physically motivated trial state is the Unitary Coupled Cluster (UCC) ansatz. The Unitary Coupled Cluster with Singles and Doubles (UCCSD) defines the parameterized state as:

\[|\psi(\boldsymbol{\theta})\rangle = e^{T(\boldsymbol{\theta}) - T^\dagger(\boldsymbol{\theta})} |\text{HF}\rangle\]

where \(|\text{HF}\rangle\) is the Hartree-Fock reference state, and \(T = T_1 + T_2\) represents the single and double excitation operators:

\[T_1 = \sum_{i \in \text{occ}, a \in \text{virt}} \theta_i^a a^\dagger_a a_i\]

\[T_2 = \sum_{i < j \in \text{occ}, a < b \in \text{virt}} \theta_{ij}^{ab} a^\dagger_a a^\dagger_b a_j a_i\]

Because exponentiating a large sum of non-commuting operators is difficult to implement on a quantum circuit, the unitary is approximated using the first-order Trotter-Suzuki decomposition:

\[e^{T(\boldsymbol{\theta}) - T^\dagger(\boldsymbol{\theta})} \approx \prod_k e^{\theta_k (G_k - G_k^\dagger)}\]

where \(G_k\) represents individual single or double excitations.