Rust-native automatic differentiation

QMBED defines derivatives at the scientific-operation boundary instead of tracing sparse assembly or iterative solver internals. The native rules use the same QuantumOperator, LinearOperator, and runtime contracts as ordinary execution.

Implemented rules

Operation	Forward/reverse rule	Reliability evidence
A(θ)x	exact JVP and one-shot VJP	dimensions, domains, action counts
isolated ground-state energy	Hellmann--Feynman gradient	residual, spectral gap, solver iterations
ChainRules.rs	optional chainrules-core 0.2 adapters for operator action and ground-state energy	runs the native rules, no duplicate derivative

Rules for eigenvectors, degenerate eigenspaces, time evolution, thermal traces, and response functions remain explicit capability gaps. In particular, QMBED does not assign a derivative to an arbitrary vector in a degenerate eigenspace.

Parameterized operator action

use qmbed::ad::{ParameterValues, apply_vjp};
use qmbed::runtime::{CpuRuntime, Runtime};

# fn example(
#   family: &qmbed::operator::QuantumOperator,
# ) -> Result<(), qmbed::QmbedError> {
let runtime = CpuRuntime::new(1)?;
let parameters = ParameterValues::real(family, [1.0, 0.7])?;
let state = runtime.upload(&vec![qmbed::Complex64::new(1.0, 0.0); family.shape().1])?;
let (value, pullback) = apply_vjp(&runtime, family, &parameters, &state)?;
let cotangents = pullback.backward(&value)?;
assert_eq!(cotangents.parameters.values().len(), 2);
# Ok(())
# }

Enable the protocol adapter only when needed:

qmbed = { version = "0.2", features = ["chainrules"] }

Ground-state energy

ground_state_energy_gradient requests at least two lowest eigenpairs. This is not extra decoration: the second eigenvalue supplies the gap needed to judge whether the scalar eigenvalue derivative is well defined and numerically resolved. For (p) parameters, the analytic rule uses one eigensolve followed by (p) component actions; central finite differences require (2p) additional eigensolves.

Benchmarks report both correctness against finite differences and the separate solver/action counts. A speedup without agreement or without a resolved gap is not accepted as AD evidence.