Overlap Matrices & Non-Orthogonal Subspaces¶
InQuanto provides the composite computable OverlapMatrixComputable
for calculating overlap matrices:
given a list of ansatzes \(\{|\Psi_0\rangle, |\Psi_1\rangle, \dots\}\). It also supports matrices of overlaps with a Hermitian kernel:
where \(\hat{O}\) is represented by a QubitOperator. Focusing on the latter of these
objects, we first show a statevector calculation with a simple kernel:
from inquanto.ansatzes import TrotterAnsatz
from inquanto.states import QubitState
from inquanto.operators import QubitOperatorList, QubitOperator
from inquanto.computables.composite import OverlapMatrixComputable
states = [
TrotterAnsatz(
QubitOperatorList.from_string("a [(1j, Y0 X1 X2 X3)]"),
QubitState([1, 1, 0, 0])
),
TrotterAnsatz(
QubitOperatorList.from_string("b [(1j, Y0 Z1 Z2 Z3)]"),
QubitState([1, 1, 0, 0])
),
]
kernel = QubitOperator.from_string("(1, Z0 Z1)")
params = {
"a" : 0.5,
"b" : 0.5
}
om_computable = OverlapMatrixComputable(states, kernel)
om_result_statevector = om_computable.default_evaluate(params)
print(om_result_statevector)
[[1. +0.j 0.77+0.j]
[0.77-0.j 0.54+0.j]]
This computable uses ExpectationValue computables for the diagonal
elements of the overlap matrix, and Overlap computables for the
off-diagonal elements. Since the kernel must be Hermitian, it is true that
\(O_{ij} = O_{ji}^\dagger\). The OverlapMatrixComputable uses
this symmetry to this result to reduce the total number of calculations required.
To perform a shot-based experiment, we require two different InQuanto protocols, one for calculating the
ExpectationValue components, and one for the
Overlap components. In this example we choose the
PauliAveraging and HadamardTestOverlap protocols
for these respective tasks. See the Protocols manual page for
more information.
Below we use the build_protocols_from() method to parse the
overlap matrix computable and construct a ProtocolList object. The
ProtocolList class groups shot-based protocols together to measure a
composite computable:
from inquanto.protocols import PauliAveraging, HadamardTestOverlap
from pytket.extensions.qiskit import AerBackend
from pytket.partition import PauliPartitionStrat
shot_backend = AerBackend()
expval_protocols = PauliAveraging.build_protocols_from(
parameters=params,
computable=om_computable,
backend=shot_backend,
shots_per_circuit=10000,
pauli_partition_strategy=PauliPartitionStrat.CommutingSets,
)
overlap_protocols = HadamardTestOverlap.build_protocols_from(
parameters=params,
computable=om_computable,
backend=shot_backend,
shots_per_circuit=10000,
direct=True,
pauli_partition_strategy=PauliPartitionStrat.CommutingSets,
)
protocol_list = expval_protocols + overlap_protocols
print(protocol_list.dataframe_protocol_circuit())
protocol_list.compile_circuits()
Protocol ID Protocol type Qubits Depth2q Shots
0 140298968492624 PauliAveraging 4 0 10000
1 140298967920976 PauliAveraging 4 0 10000
2 140298968480976 HadamardTestOverlap 5 0 10000
3 140298968480976 HadamardTestOverlap 5 0 10000
[<inquanto.protocols.averaging._pauli_averaging.PauliAveraging object at 0x7f99e62dea50>, <inquanto.protocols.averaging._pauli_averaging.PauliAveraging object at 0x7f99e6253150>, <inquanto.protocols.averaging._hadamard_overlap.HadamardTestOverlap object at 0x7f99e62dbcd0>]
The dataframe_protocol_circuit() method shows a breakdown of the
circuits required to measure the overlap matrix. Two circuits come from the
PauliAveraging protocol for measuring the expectation values on the diagonal,
and two more from the HadamardTestOverlap protocol for measuring the real and
imaginary parts of the single, unique off-diagonal element.
Below, we run these circuits using the pytket AerBackend, evaluate the overlap matrix, and
compare to the statevector result obtained above:
protocol_list.run(seed=0)
om_result_shotbased = om_computable.evaluate(protocol_list.get_evaluator())
print(f"Statevector result: \n{om_result_statevector}\n")
print(f"Shot-based result: \n{om_result_shotbased}")
Statevector result:
[[1. +0.j 0.77+0.j]
[0.77-0.j 0.54+0.j]]
Shot-based result:
[[1. +0.j 0.772-0.015j]
[0.772+0.015j 0.53 +0.j ]]
InQuanto also provides the specialized composite computable:
NonOrthogonalMatricesComputable for building the generalized eigenvalue
problem in a non-orthogonal subspace:
where \(H\) is the hamiltonian matrix and \(S\) is the overlap matrix in some subspace \(\{|\Psi_i\rangle\}\):
Given a set of states, the NonOrthogonalMatricesComputable constructs both
such matrices using OverlapMatrixComputables discussed above. Once
evaluated, we may solve the eigenvalue problem straightforwardly with the pd_safe_eigh()
function from inquanto.core. The ground state energy obtained in this approach is bounded from below by the
configuration interaction (CI) energy.
Below we demonstrate a simple calculation of H2 in a minimal basis, using the two Trotter states defined above as a crude, non-orthogonal subspace:
from inquanto.computables.composite import NonOrthogonalMatricesComputable
from inquanto.express import load_h5
from inquanto.core import pd_safe_eigh
h2_data = load_h5("h2_sto3g.h5")
qubit_hamiltonian = h2_data["hamiltonian_operator"].qubit_encode()
no_computable = NonOrthogonalMatricesComputable(
qubit_hamiltonian,
states
)
# Using the same ansatz parameters as above
expval_protocols = PauliAveraging.build_protocols_from(
parameters=params,
computable=no_computable,
backend=shot_backend,
shots_per_circuit=10000,
pauli_partition_strategy=PauliPartitionStrat.CommutingSets,
)
overlap_protocols = HadamardTestOverlap.build_protocols_from(
parameters=params,
computable=no_computable,
backend=shot_backend,
shots_per_circuit=10000,
direct=True,
pauli_partition_strategy=PauliPartitionStrat.CommutingSets,
)
no_protocols = expval_protocols + overlap_protocols
no_protocols.compile_circuits()
no_protocols.run(seed=0)
h, s = no_computable.evaluate(no_protocols.get_evaluator())
e, _, _ = pd_safe_eigh(h, s)
print(f"CASCI energy: {h2_data['energy_casci']}")
print(f"NO shot-based energy: {e[0]}")
CASCI energy: -1.1368465754720547
NO shot-based energy: -0.9825372067689363