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Product · Pauli-string propagation

Lumen Trace 1.0

Lumen Trace answers expectation values by pushing Pauli strings backward through the circuit. It never builds an amplitude vector. If the backward cone stays a short list of Pauli terms, Trace is the cheapest exact method in the family. If the list explodes, Trace stops and names the engine that should take over.

The object

A Pauli string is a phase in {+1, −1, +i, −i} times one of I, X, Y, Z on every wire, with identity omitted in the printed form. The expectation of P in a state is what callers usually want from a variational layer, a witness, or a risk factor that was encoded as a Hamiltonian term.

Heisenberg pullback: a gate U sends P to U† P U. For Clifford gates the image is one Pauli string. For a single RZ or RX the image is two strings. For a non-Clifford rotation the branch factor is two per gate on the support. Trace multiplies those branches and sums the coefficients against the initial |0…0⟩, where only the all-identity term survives.

Truncation policy

A coefficient below the caller’s cutoff is dropped and added to a discarded-mass counter. The response includes that mass. Data-T does not round it into the expectation silently. If discarded mass exceeds the tolerance, the status is approximate and the suggested fallback is Lumen State or Lumen Weave, whichever the width allows.

There is a hard cap on the number of live strings. Hitting the cap is an error, not a truncation. Truncation happens only through the coefficient cutoff the caller set.

  • Clifford prefix is folded exactly before any branching rotation.
  • Identical strings are merged by summing coefficients.
  • Commutation through a CX uses the standard Pauli conjugation table published in the reference page, not an external header.

Observables that are sums

A Hamiltonian is a list of weighted Pauli strings. Trace evaluates each string, or batches strings that share a cone, and returns the weighted sum plus a per-term discarded mass. Correlations are not assumed. The caller who wants ⟨AB⟩ − ⟨A⟩⟨B⟩ must ask for the three expectations and assemble them. Trace will not invent a cumulant.

Noise

A Pauli channel conjugates cleanly: strings pick up a damping factor from the probabilities of the errors that anticommute with them. Trace applies that factor and stays exact for Pauli noise. Amplitude damping does not stay inside the Pauli basis at the same cost. Those layers force a hand-off note to Lumen Drift.

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