Pauli observables

Applicable version · ArcQML 0.1.0

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PauliString

PauliString represents a tensor product of PauliOp values, storing only non-I terms in qubit-index order. Construction checks qubit ranges and duplicate qubit s. identity, single, x, y, and z provide shortcuts. Two PauliString objects commute when the number of positions with different, non-I Paulis on the same qubit is even.

P=P_{n-1}\otimes\cdots\otimes P_1\otimes P_0, \qquad P_q\in\{I,X,Y,Z\}.

For two qubits, X(0) corresponds to I ⊗ X, mapping |00⟩ to |01⟩. multiply(rhs) returns (complex phase, resulting PauliString). This supports Pauli algebra; public SparsePauliOp coefficients are restricted to finite real values, so observable expectations are real.

SparsePauliOp and compilation caching

SparsePauliOp is a collection of PauliTerm objects, each term containing a finite real coefficient and a PauliString. Hamiltonian, PauliSum, and PauliObservable are aliases. Construction requires nonzero num_qubits and consistent qubit counts across terms. scale and simplify also reject nonfinite factors or invalid tolerance values.

H=\sum_j c_jP_j, \qquad c_j\in\mathbb{R}.

At the first state-vector evaluation, the public Rust layer prepares a reusable execution representation of SparsePauliOp: diagonal term s can be merged, while off-diagonal Pauli strings become bit masks for expectation and H|ψ⟩ evaluation in the public observable implementation. Adding a term invalidates the cache. clone copies the observable definition but not its execution cache. Reusing one Hamiltonian during training therefore avoids preparing the same execution information each iteration.

rust

let mut h = SparsePauliOp::single(2, 0usize, Pauli::Z, 0.5)?;
h.add_pauli_string(-1.0, PauliString::x(2, 1usize)?)?;
// H = 0.5 Z(0) − 1.0 X(1)

Expectation semantics

For the current pure state, single-state expectation_observable returns an F64 scalar Tensor. The batch version returns a Tensor with shape equal to [B] and dtype F64, one value per initial-state row. The expectation is:

\langle H\rangle =\operatorname{Re}\!\left( \langle\psi\rvert H\lvert\psi\rangle \right).

The differentiable path created by run is reused for adjoint backpropagation, avoiding reconstruction of the same observable-action result.

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