Pauli Observables

MENOS computes expectation values of Pauli observables: real or complex weighted sums of Pauli strings.

O  =  c₁ · P₁  +  c₂ · P₂  +  …        each Pᵢ a tensor product of I, X, Y, Z

Representation

A Pauli string on n qubits is stored as two bit masks:

Mask	Bit j set means
x_words	an X component on qubit j
z_words	a Z component on qubit j

Both set means Y, following Y = i·X·Z. Neither set means identity on that qubit. Masks are lists of 64-bit words, so observables on more than 64 qubits use several words. This is exactly the convention Qiskit's SparsePauliOp uses internally, which is why conversion is exact, with no string parsing.

Pauli on qubit j	x bit	z bit
I	0	0
X	1	0
Z	0	1
Y	1	1

Building one natively

from menos._core import Observable, PauliString

def pauli(n, x_qubits=(), z_qubits=()):
    words = (n + 63) // 64
    x = [0] * words
    z = [0] * words
    for q in x_qubits:
        x[q // 64] |= 1 << (q % 64)
    for q in z_qubits:
        z[q // 64] |= 1 << (q % 64)
    p = PauliString()
    p.num_qubits = n
    p.x_words = x        # assign the whole list
    p.z_words = z
    return p

# Z0 Z1 + 0.5 * X2 on three qubits, and Y on qubit 1 (x and z both set).
observable = Observable()
observable.terms = [
    (pauli(3, z_qubits=[0, 1]), 1.0 + 0j),
    (pauli(3, x_qubits=[2]), 0.5 + 0j),
    (pauli(3, x_qubits=[1], z_qubits=[1]), 1.0 + 0j),
]
assert len(observable.terms) == 3

Assign whole lists

PauliString.x_words, PauliString.z_words and Observable.terms are copied across the Python and C++ boundary each time you read them. Modifying an element in place, for example p.z_words[0] |= 1 or observable.terms.append(...), changes a temporary copy and silently does nothing. Build the list first and assign it, as above.

from menos._core import PauliString

p = PauliString()
p.num_qubits = 2
p.x_words = [0]
p.z_words = [0]
p.z_words[0] |= 1          # modifies a temporary copy
assert p.z_words == [0]    # the assignment did not stick

Supported observables

Only Pauli sums. See Supported Observables.

An all-identity observable is not supported

An observable in which every term is the identity is a constant, ⟨I⟩ = 1, but MENOS does not currently handle it: planning reduces the circuit to zero qubits and the run raises InternalError (Tableau: num_qubits must be at least 1). An identity term alongside any other term works and is exact.

If you need a constant offset, add it yourself after the run:

from qiskit import QuantumCircuit
from qiskit.quantum_info import SparsePauliOp

from menos.qiskit import MENOSEstimator

qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)

constant = 2.5                                  # the coefficient of the identity term
value = constant + float(
    MENOSEstimator().run([(qc, SparsePauliOp("ZZ", coeffs=[0.5]))]).result()[0].data.evs
)
assert abs(value - 3.0) < 1e-12

Hermiticity

MENOS does not check that an observable is Hermitian. A non-Hermitian sum has a complex expectation value; the native API returns it as a complex, and the Qiskit estimator returns the real part, as Qiskit's reference estimator does.

Related

SparsePauliOp · Expectation Values · Conventions