Ansatz

A quantum circuit comprising parameterized gates (e.g. \(RX(\theta)\), \(RY(\theta)\) and \(RZ(\theta)\)), represents a unitary transformation \(U(\theta)\) that transforms some initial quantum state into a parametrized ansatz state \(|\psi(\theta)\rangle\).

Examples of some ansatzes available in Qibochem are described in the subsections below.

Hardware Efficient Ansatz

Qibochem provides a hardware efficient ansatz that simply consists of a layer of single-qubit rotation gates followed by a layer of two-qubit gates that entangle the qubits. For the H₂ case discussed in previous sections, a possible hardware efficient circuit ansatz can be constructed as such:

from qibo import Circuit

from qibochem.ansatz import hardware_efficient

nlayers = 1
nqubits = 4
nfermions = 2

circuit = Circuit(4)
hardware_efficient_ansatz = hardware_efficient.hea(nlayers, nqubits)
circuit.add(hardware_efficient_ansatz)
print(circuit.draw())

q0: ─RY─RZ─o─────Z─
q1: ─RY─RZ─Z─o───|─
q2: ─RY─RZ───Z─o─|─
q3: ─RY─RZ─────Z─o─

The energy of the state generated from the hardware efficient ansatz for the fermionic two-body Hamiltonian can then be estimated, using state vectors or samples.

The following example demonstrates how the energy of the H2 molecule is affected with respect to the rotational parameters:

import numpy as np
from qibo import Circuit

from qibochem.driver.molecule import Molecule
from qibochem.measurement.expectation import expectation
from qibochem.ansatz import hardware_efficient

mol = Molecule([("H", (0.0, 0.0, 0.0)), ("H", (0.0, 0.0, 0.74804))])
mol.run_pyscf()
hamiltonian = mol.hamiltonian()

# Define and build the HEA
nlayers = 1
nqubits = mol.nso
ntheta = 2 * nqubits * nlayers
hea_ansatz = hardware_efficient.hea(nlayers, nqubits)

circuit = Circuit(nqubits)
circuit.add(hea_ansatz)

print("Energy expectation values for thetas: ")
print("-----------------------------")
print("| theta | Electronic energy |")
print("|---------------------------|")
thetas = [-0.2, 0.0, 0.2]
for theta in thetas:
    params = np.full(ntheta, theta)
    circuit.set_parameters(params)
    electronic_energy = expectation(circuit, hamiltonian)
    print(f"| {theta:5.1f} | {electronic_energy:^18.12f}|")
print("-----------------------------")

converged SCF energy = -1.11628373627429

Energy expectation values for thetas:
-----------------------------
| theta | Electronic energy |
|---------------------------|
|  -0.2 |   0.673325849299  |
|   0.0 |   0.707418334474  |
|   0.2 |   0.673325849299  |
-----------------------------

Unitary Coupled Cluster Ansatz

The Unitary Coupled Cluster (UCC) ansatz [1] [2] [3] is a variant of the popular gold standard Coupled Cluster ansatz [3] of quantum chemistry. The UCC wave function is a parameterized unitary transformation of a reference wave function \(\psi_{\mathrm{ref}}\), of which a common choice is the Hartree-Fock wave function.

\[\begin{align*} |\psi_{\mathrm{UCC}}\rangle &= U(\theta)|\psi_{\mathrm{ref}}\rangle \\ &= e^{\hat{T}(\theta) - \hat{T}^\dagger(\theta)}|\psi_{\mathrm{ref}}\rangle \end{align*}\]

Similar to the process for the molecular Hamiltonian, the fermionic excitation operators \(\hat{T}\) and \(\hat{T}^\dagger\) are mapped using e.g. Jordan-Wigner mapping into Pauli operators. This is typically followed by a Suzuki-Trotter decomposition of the exponentials of these Pauli operators, which allows the UCC ansatz to be implemented on quantum computers. [5]

An example of how to build a UCC doubles circuit ansatz for the \(H_2\) molecule is given as:

from qibochem.driver.molecule import Molecule
from qibochem.ansatz.hf_reference import hf_circuit
from qibochem.ansatz.ucc import ucc_circuit

mol = Molecule([("H", (0.0, 0.0, 0.0)), ("H", (0.0, 0.0, 0.74804))])
mol.run_pyscf()
hamiltonian = mol.hamiltonian()

# Set parameters for the rest of the experiment
n_qubits = mol.nso
n_electrons = mol.nelec

# Build UCCD circuit
circuit = hf_circuit(n_qubits, n_electrons) # Start with HF circuit
circuit += ucc_circuit(n_qubits, [0, 1, 2, 3]) # Then add the double excitation circuit ansatz

print(circuit.draw())

q0:     ─X──H─────X─RZ─X─────H──RX─────X─RZ─X─────RX─RX─────X─RZ─X─────RX─H─── ...
q1:     ─X──H───X─o────o─X───H──RX───X─o────o─X───RX─H────X─o────o─X───H──RX── ...
q2:     ─RX───X─o────────o─X─RX─RX─X─o────────o─X─RX─H──X─o────────o─X─H──H──X ...
q3:     ─H────o────────────o─H──H──o────────────o─H──H──o────────────o─H──H──o ...

q0: ... ───X─RZ─X─────H──RX─────X─RZ─X─────RX─H──────X─RZ─X─────H──H──────X─RZ ...
q1: ... ─X─o────o─X───RX─H────X─o────o─X───H──RX───X─o────o─X───RX─H────X─o─── ...
q2: ... ─o────────o─X─H──RX─X─o────────o─X─RX─RX─X─o────────o─X─RX─H──X─o───── ...
q3: ... ────────────o─H──RX─o────────────o─RX─RX─o────────────o─RX─RX─o─────── ...

q0: ... ─X─────H──RX─────X─RZ─X─────RX─
q1: ... ─o─X───H──RX───X─o────o─X───RX─
q2: ... ───o─X─H──H──X─o────────o─X─H──
q3: ... ─────o─RX─RX─o────────────o─RX─

References

[1]

Kutzelnigg, W. (1977). ‘Pair Correlation Theories’, in Schaefer, H.F. (eds) Methods of Electronic Structure Theory. Modern Theoretical Chemistry, vol 3. Springer, Boston, MA.

[2]

Whitfield, J. D. et al., ‘Simulation of electronic structure Hamiltonians using quantum computers’, Mol. Phys. 109 (2011) 735.

[3] (1,2)

Anand. A. et al., ‘A quantum computing view on unitary coupled cluster theory’, Chem. Soc. Rev. 51 (2022) 1659.

[4]

Crawford, T. D. et al., ‘An Introduction to Coupled Cluster Theory for Computational Chemists’, in Reviews in Computational Chemistry 14 (2007) 33.

[5]

Barkoutsos, P. K. et al., ‘Quantum algorithms for electronic structure calculations: Particle-hole Hamiltonian and optimized wave-function expansions’, Phys. Rev. A 98 (2018) 022322.