Real Basis Rotation Ansatzes¶

An orbital basis transformation can be represented on a quantum circuit with Givens rotation gates [43]. This circuit representation allows us to define the set of classes:

These classes apply a unitary transformation to a reference state, in the form:

(101)¶

| Ψ (θ) ⟩ = \exp [\sum_{i j} θ_{i j} a_{i}^{†} a_{j}] | R e f ⟩

where the relationship between a real basis transformation matrix $R$ and the variational parameters is $θ_{i j} = [\ln R]_{i j}$ , according to the Thouless theorem [42]. This ansatz can be used in variatonal algorithms to find for example the mean-field solution of a chemistry Hamiltonian on quantum computer:

from pytket.extensions.qiskit import AerStateBackend

from inquanto.ansatzes import RealGeneralizedBasisRotationAnsatz
from inquanto.express import load_h5, run_vqe
from inquanto.states import QubitState

reference = QubitState([1, 1, 0, 0])
ra = RealGeneralizedBasisRotationAnsatz(reference=reference)

h2_sto3g = load_h5("h2_sto3g.h5", as_tuple=True)

hamiltonian_lowdin = h2_sto3g.hamiltonian_operator_lowdin.qubit_encode()

print("HF energy (ref):  ", h2_sto3g.energy_hf)
print("<REF|H|REF>:      ", reference.vdot(hamiltonian_lowdin.dot_state(reference)))

vqe = run_vqe(ra, hamiltonian_lowdin, AerStateBackend(), initial_parameters=ra.state_symbols.construct_random() )

print("VQE energy:       ", vqe.final_value)

HF energy (ref):   -1.1175058842043306
<REF|H|REF>:       -0.12440620192256865
# TIMER BLOCK-0 BEGINS AT 2025-05-28 20:24:02.740408

# TIMER BLOCK-0 ENDS - DURATION (s):  5.2603768 [0:00:05.260377]
VQE energy:        -1.117505884195969

Given a real unitary matrix $R$ , we can also compute the corresponding ansatz parameters:

import numpy

# unitary matrix for which the QR gives R with diagonals [1,1,1,-1]
R = numpy.array(
    [
        [0.04443313, -0.95103332, 0.1990091, -0.2322858],
        [-0.14642999, -0.16713913, -0.96063852, -0.16672251],
        [0.98783978, 0.02520736, -0.14785981, -0.04092234],
        [-0.02750505, 0.25877542, 0.1253255, -0.95737781],
    ]
)

p = ra.ansatz_parameters_from_unitary(R)
print(f"Rotation parameters:\n{p}")

Rotation parameters:
{phi_0: np.float64(0.0), phi_1: np.float64(0.0), phi_2: np.float64(0.0), phi_3: np.float64(3.141592653589793), theta_1_0: np.float64(-1.526348563129208), theta_2_0: np.float64(-1.7179010586215773), theta_2_1: np.float64(-0.3111780679781251), theta_3_0: np.float64(0.027836442642734202), theta_3_1: np.float64(-1.0105744336026967), theta_3_2: np.float64(0.8624009177850533)}

The RealGeneralizedBasisRotationAnsatz supports generalized real rotations, which may couple spin channels. For restricted rotations i.e. both $α$ and $β$ spins are rotated by the same matrix independently of one another, one should use the RealRestrictedBasisRotationAnsatz. For unrestricted rotations, where $α$ and $β$ spins are rotated by different matrices, one should use the RealUnrestrictedBasisRotationAnsatz.