Direction

direction selects which conjugation propagate performs, and the two choices compute different operators. Under "heisenberg" the evolved sum is U†OU; under "forward" it is UOU†. Both are valid, both are cheap or expensive for their own reasons, and neither raises when it is the wrong one for the question being asked — which is why this is the first page of the chapter.

Heisenberg answers “what does this circuit measure?” — you have an observable O and a circuit U, and want ⟨ψ|U†OU|ψ⟩ for a product state |ψ⟩ the circuit acts on. A local observable starts as a handful of terms and only spreads as far as its causal cone, which is why this is the direction almost every example on this site uses.

Forward answers “what does this circuit do to this operator?” — evolving a density matrix or a Hamiltonian in the Schrödinger picture. It starts from a wide operator, so the cost profile is different from the outset.

from paulistrings import Circuit, PauliSum, truncation

observable = PauliSum.from_strings({"XIII": 0.25, "IXII": 0.25, "IIXI": 0.25, "IIIX": 0.25}, num_qubits=4)
circuit = Circuit(4)
circuit.rz(0.3, 0)
circuit.cnot(0, 1)
policy = truncation.coeff(1e-10)

heisenberg = observable.propagate(circuit, policy, direction="heisenberg")
forward = observable.propagate(circuit, policy, direction="forward")

The default is "forward", and most examples on this site do not use it. The Python binding maps direction=None to "forward", so a call that omits the argument silently computes UOU†. Treat the default as a one-off exploration convenience, not an idiom, and pass direction explicitly in anything you keep.

Which state the label is read against

expectation(state) evaluates ⟨s|A|s⟩ for whatever sum A it is handed, so direction decides what the label s refers to:

• under "heisenberg" the evolved sum is U†OU, so expectation(s) is O’s expectation in U|s⟩ — s is the input state, the one the circuit acts on;
• under "forward" the evolved sum is UOU†, so expectation(s) is O’s expectation in U†|s⟩ — s is a state at the circuit’s output end.

Getting it backwards is silent. The same observable, the same circuit and the same state label give two perfectly plausible numbers:

from paulistrings import Circuit, PauliSum, truncation

observable = PauliSum.from_strings({"ZII": 1.0}, num_qubits=3)

both = Circuit(3)
both.h(0)
both.pauli_rotation("ZZ", [0, 1], 0.6)
both.rx(0.9, 0)
both.rx(0.9, 2)
both.cnot(1, 2)

exact = truncation.coeff(1e-12)
heisenberg = observable.propagate(both, exact, direction="heisenberg")
forward = observable.propagate(both, exact, direction="forward")

print("heisenberg", heisenberg.expectation("x+").real)
print("forward   ", forward.expectation("x+").real)

heisenberg 0.6216099682706644
forward    0.8253356149096783

0.6216… is ⟨Z_0⟩ measured after running this circuit on |+++⟩ — the Heisenberg answer, and the one nearly every question of the form “what does this circuit measure?” wants. 0.8253… is the same observable pushed forward through the circuit and then read against |+++⟩ at the far end, which answers a different question. Both were checked against a dense 2³×2³ construction; neither is a bug.

The rule of thumb: decide which end of the circuit your state lives at, and that decides the direction. A state prepared before the circuit is a Heisenberg run; an operator you want to see after the circuit has acted on it is a forward run. Measurements restates this from the measurement side, for readers who arrive there first.

Push order

Under "heisenberg" the engine walks the channel list in reverse and applies each channel’s adjoint; under "forward" it walks the list as written and applies each channel. So pushing ZZ rotations before X rotations builds U = U_X · U_ZZ, and Heisenberg evolution computes U_ZZ† U_X† O U_X U_ZZ — the X rotations act on O first. Build the circuit in the order the gates act on the state; the direction flag does the rest.

Two places this rule surfaces later in the chapter. PropagationStats lists its layers in application order, so under "heisenberg" the first entry is the circuit’s last gate — circuit_index recovers the position as written (Stats). And an incremental propagation extends the evolved operator at the application end, which under "heisenberg" means a second propagate call prepends its circuit rather than appending it — The Heisenberg-ordering trap.

The full two-row table is in the Library. Every cross-engine comparison on this site is run in the Heisenberg direction, because PauliPropagation.jl defines no forward map for several of the gates in the shared vocabulary — Against other tools has the asymmetry. Showcase B5 is a Heisenberg run whose evolved observable is then handed to a shorter front circuit, which only works because the label denotes the input state.