Trotter Product-Formula Simulation

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Category C2 · Module oracq.algorithms.common.hamiltonian · Stage V2

Overview

Given a Pauli decomposition \(H = \sum_j c_j P_j\) (each \(P_j\) a same-width Pauli word), simulate \(e^{-iHt}\) with the first-order product formula:

\[ e^{-iHt} \approx \left(\prod_j e^{-i c_j P_j\, t / r}\right)^{r}, \]

The larger the repetition count \(r\), the smaller the product-formula error. A single term \(e^{-i\theta P}\) is turned by basis changes into a phase rotation on the computational basis: apply basis changes to the non-identity positions of \(P\) (H on X positions; on Y positions a \(-\pi/2\) phase gate followed by H), fold the parity of the active positions onto the last position with a CNOT chain, apply the phase rotation to the last position, and then restore in reverse order.

Interface and input model

trotter_hamsim(terms, final_time, *, steps=2)

API entry: trotter_hamsim

• terms: a ((coefficient, word), ...) sequence, with coefficient real (internally converted through float) and word a same-width I/X/Y/Z string; the input model is HAM (Pauli-term decomposition).

• final_time: the total evolution time \(t\) (a finite real number).

• steps: the repetition count \(r\), a positive integer, default 2.

Returns an Operation (a bare operation, not a BlockEncoding), with registers target = Bits(width) and signal = Bits(0) — the evolution is an exact unitary that requires no post-selection. Module attributes:

Attribute	Meaning
algorithm	"trotter_hamsim"
validation_stage	"paradigm"

Generation-time validation: raises ValidationError when terms is empty, steps < 1, or the Pauli words have inconsistent widths. An all-I term consumes no qubits and degenerates to the global phase \(e^{-i c_j t / r}\).

Implementation notes

The circuit body is a single Repeat(steps) block whose interior evolves term by term in input order; it is not expanded during the generation stage, and text serialization and backend lowering keep the Repeat structure. Each term is generated as “basis change → CNOT chain → phase rotation on the last position → restore”: the CNOT chain covers only the non-I positions, and the last-position rotation angle is \(2 c_j t / r\) (rz(θ) follows the \(e^{-i\theta Z/2}\) convention, and after the factor of 2 it realizes exactly a phase of the \(e^{-i\theta Z^{\otimes k}}\) type).

Applicability boundary: only Pauli-word decompositions are accepted, and no higher-order Suzuki ordering is performed (higher-order formulas require the caller to reorder terms itself and use a larger steps); coefficients go through float() conversion, and complex coefficients raise an error outright. For the protocol-layer entry point see Hamiltonian simulation protocol, which routes the per-term evolution of any Trotterizable operator back to this function (PauliOperator.evolution calls exactly this function with steps=1).

Validation approach

Category C2 (approximately continuous semantics; the acceptance criteria are in ../development/validation-plan.md §2). Consistent with the hamiltonian.py row of the validation coverage matrix. Three layers of evidence:

• Structure: tests/core/test_algorithm_protocols.py:AlgorithmProtocolTests.test_trotter_protocol_keeps_phase_and_repeat pins down the Repeat structure (the first block of the module body has count == steps); test_trotter_validates_each_term covers term-level validation (TrotterTerm rejects non-Evolvable operators, and a PauliHamiltonian with inconsistent widths is rejected at construction).

• Numerical: test_trotter_protocol_keeps_phase_and_repeat cross-checks \(H = 0.3I + 0.7X\), \(t = 0.4\), steps = 3 amplitude by amplitude against \(e^{-0.12i}\bigl(\cos 0.28\,|0\rangle - i\sin 0.28\,|1\rangle\bigr)\) (places = 11), indirectly covering the per-term stage order generated by this function; test_trotter_only_input_does_not_need_block_encoding witnesses a single-term Z evolution (coefficient 0.5, \(t = 0.2\)) outputting \(e^{-0.1i}\).

• Binding: no independent binding witness (the input is an explicit Pauli decomposition with no abstract slots).

Known gaps and planned stages

The Trotter order-error-rate sweep is missing: batch curves of the error versus steps / \(\Delta t\) are not automated and will be wired in once the stage V2 convergence-scan framework (validation-plan §5) lands; the current numerical witnesses cross-check analytic values at single parameter points only.

Related links

• Source: src/oracq/algorithms/common/hamiltonian.py

• Tutorial: Supplying your own Hamiltonian decomposition

• Same-family pages: Hamiltonian simulation protocol, Truncated Taylor block encoding

• API reference: Hamiltonian evolution

• Validation matrix: Validation coverage matrix

Numerical validation

Paper-grade numerical experiments are in tests/verification/verify_hamiltonian.py (real-backend execution, no mock substitutes; all 27 cases PASS, total runtime about 28 s). Experiment design: single-term Pauli evolutions (10 instances, covering Y basis changes, 2–4-qubit CNOT chains, global phases of identity terms, and negative times) are exported via OriginIR-ext, the full unitary is extracted by UniQC Circuit.to_matrix, and the result is cross-checked element by element against the independent closed form \(\cos(ct)I - i\sin(ct)P\); many-term non-commuting decompositions (1–4 qubits, steps ∈ {1,2,3,5}, t ∈ {0.4, −0.7, 0.9, 1.3}) are cross-checked against the classical product-formula matrix \(\bigl(\prod_j e^{-ic_jP_jt/r}\bigr)^r\); a 3-qubit TFIM on a uniform-superposition input is cross-checked across four paths; the convergence order comes from sweeping steps ∈ {1,…,64} at \(t = 1.0\) for 2- and 3-qubit TFIMs, taking the spectral-norm error against scipy.linalg.expm as an independent oracle and fitting log–log. UniQC to_matrix drops trailing gate-free qubits (the case of a Pauli word ending in I); the verification script embeds them back as identity factors before cross-checking (see the embed_unitary comment in the script).

Case	Scale	Backend paths	Metric	Value
trotter-single-term-exact	10 instances, 1–4 qubits	originir-ext + UniQC to_matrix	max_error	4.4e-16
trotter-product-formula-1q	3 terms (including an identity term), steps×t grid	originir-ext + UniQC to_matrix	max_error	5.7e-16
trotter-product-formula-2q	4 terms, steps ∈ {1,2,5}	originir-ext + UniQC to_matrix	max_error	1.5e-15
trotter-product-formula-3q	5-term TFIM, steps ∈ {2,3}	originir-ext + UniQC to_matrix	max_error	1.4e-15
trotter-product-formula-4q	5 terms, steps=3	originir-ext + UniQC to_matrix	max_error	1.7e-15
trotter-superposition-cross-3q	5-term TFIM, steps=3, superposition exhaustive	reference, rir-pysparq, adapter-pysparq, originir-ext	max_deviation	1.2e-15
trotter-convergence-2q	\(t = 1.0\), \(r\) = 1…64	originir-ext + UniQC to_matrix	fitted convergence order (theory 1)	1.008 (error 0.7627 → 0.01049)
trotter-convergence-3q	\(t = 1.0\), \(r\) = 1…32	originir-ext + UniQC to_matrix	fitted convergence order (theory 1)	1.013 (error 1.0680 → 0.02809)

The product-formula semantics are validated to machine precision (≤ 1.7e-15), and the fitted convergence orders 1.008 / 1.013 of first-order Lie–Trotter agree with the theoretical value 1, closing the gap in this section previously recorded as “missing Trotter order-error-rate sweep”.

Reproduction command:

PYTHONPATH=src /path/to/backend/python tests/verification/verify_hamiltonian.py

Artifacts: out/verification/hamiltonian.json (including the error sequences over all step counts and the per-word errors).