Expectation Values

An expectation value ⟨ψ|O|ψ⟩ of a Pauli observable O tells you the average outcome of measuring O on the circuit's output state. MENOS computes it exactly, without sampling, and for many circuits without ever building the state.

From Qiskit

Use MENOSEstimator with a SparsePauliOp:

from qiskit import QuantumCircuit
from qiskit.quantum_info import SparsePauliOp, Statevector

from menos.qiskit import MENOSEstimator

qc = QuantumCircuit(3)
qc.h(0)
qc.cx(0, 1)
qc.ry(0.4, 2)
qc.cx(1, 2)

observable = SparsePauliOp.from_list([("ZZI", 1.0), ("IXX", 0.5), ("YIY", -0.25)])
value = float(MENOSEstimator().run([(qc, observable)]).result()[0].data.evs)
print(value)

assert abs(value - Statevector(qc).expectation_value(observable).real) < 1e-12

From the native API

The native API takes a menos._core.Observable: a list of (PauliString, coefficient) terms.

from menos import Circuit, Opcode, RunOptions, run
from menos._core import Observable, PauliString

circuit = Circuit(2, 0)
circuit.append(Opcode.H, [0])
circuit.append(Opcode.CX, [0, 1])

zz = PauliString()
zz.num_qubits = 2
zz.x_words = [0b00]      # no X component
zz.z_words = [0b11]      # Z on qubits 0 and 1

observable = Observable()
observable.terms = [(zz, 1.0 + 0.0j)]

options = RunOptions()
options.expectation = observable
result = run(circuit, options)

print(result.expectation_value)
assert abs(result.expectation_value - 1.0) < 1e-12

PauliString stores the operator as two bit masks: bit j of x_words set means an X component on qubit j, bit j of z_words set means a Z component. Both set means Y, following Y = i·X·Z. Masks are lists of 64-bit words, so observables on more than 64 qubits use several words. Assign whole lists (observable.terms = [...]); appending to observable.terms modifies a copy and has no effect.

What makes it cheap

An expectation request lets the planner consider engines that cannot produce a statevector at all:

• the Pauli propagation engine pushes the observable backward through the circuit and never holds a state;
• the stabilizer engine evaluates Pauli expectations exactly on Clifford circuits;
• the MPS and tensor-network engines contract networks without a full state;
• cut and recombine splits a circuit into fragments and recombines their expectation values exactly.

And before any of that, light-cone reduction removes every operation outside the observable's causal past. See Simulation Engines.

Conventions

• Bit order. Qubit j is bit j. In a Qiskit label such as "ZZI", the rightmost character is qubit 0.
• Hermiticity is not checked. A non-Hermitian observable has a complex expectation value; the Qiskit estimator returns its real part, exactly as Qiskit's reference StatevectorEstimator does.
• Exact. The value is computed analytically, with no shot noise.

Limits

• Observables must be Pauli sums. Other operator forms are not accepted.
• Circuits must not carry unbound parameters when using the estimator.
• The native API returns a complex; the Qiskit estimator returns a real float.