New module with stand-alone functions for computation of expected values

[andre-juan]

Resolves to #1715

---

Implementation of a new module with functions for the calculation of the expected value of a Hamiltonian expressed as a linear combination of Pauli words, either in infinite precision from the state (by building a sparse matrix for each Pauli word separately); or from samples (by performing the appropriate measurements, including the grouping of commuting terms).

Here's a simple example executing the new functions for the Heisenberg XXZ model Hamiltonian on n=4 qubits with delta=0.5:

from qibo.models.encodings import hamming_weight_encoder
from qibo.quantum_info import random_statevector
from qibo.hamiltonians.expectation_value import (
    get_expval_from_linear_comb_of_paulis_from_samples,
    get_expval_from_linear_comb_of_paulis_from_statevector,
)
from scipy.special import comb

nqubits = 4
weight = 2
dim = int(comb(nqubits, weight))

seed = 100
np.random.seed(seed)
circuit = hamming_weight_encoder(random_statevector(dim, seed=seed), nqubits, weight)

# XXZ hamiltonian, n=4, delta=0.5
lin_comb_pauli = [
    (1.0, "XXII"),
    (1.0, "IXXI"),
    (1.0, "IIXX"),
    (1.0, "XIIX"),
    (1.0, "YYII"),
    (1.0, "IYYI"),
    (1.0, "IIYY"),
    (1.0, "YIIY"),
    (0.5, "ZZII"),
    (0.5, "IZZI"),
    (0.5, "IIZZ"),
    (0.5, "ZIIZ"),
]

expval_inf_prec = get_expval_from_linear_comb_of_paulis_from_statevector(
    circuit, lin_comb_pauli
)

nshots = int(1e8)
expval_samples, expval_SE_samples, expval_95_CI_samples = (
    get_expval_from_linear_comb_of_paulis_from_samples(circuit, lin_comb_pauli, nshots)
)

print(f"Expval at infinite precision: <H> = {expval_inf_prec:.5f}")
print(
    f"Expval with {nshots:.2e} shots: <H> = {expval_samples:.5f}, 95% CI: {[float(round(x, 5)) for x in expval_95_CI_samples]}"
)

>>> Expval at infinite precision: <H> = 0.51472
>>> Expval with 1.00e+08 shots: <H> = 0.51400, 95% CI: [0.51341, 0.51459]

---

Checklist:

• Reviewers confirm new code works as expected.
• Tests are passing.
• Coverage does not decrease.
• Documentation is updated.

[codecov]

Codecov Report

❌ Patch coverage is 0% with 132 lines in your changes missing coverage. Please review.
✅ Project coverage is 98.14%. Comparing base (79fbb2d) to head (c3afc23).
⚠️ Report is 65 commits behind head on master.

Files with missing lines	Patch %	Lines
src/qibo/hamiltonians/expectation_value.py	0.00%	132 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1716      +/-   ##
==========================================
- Coverage   99.16%   98.14%   -1.02%     
==========================================
  Files          80       81       +1     
  Lines       12751    12883     +132     
==========================================
  Hits        12644    12644              
- Misses        107      239     +132     

Flag	Coverage Δ
unittests	98.14% <0.00%> (-1.02%)	⬇️

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