Tensor Hypercontraction

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Category C2 · Module oracq.algorithms.input_model.lowrank · Stage V1

Overview

Tensor hypercontraction (THC) compresses the two-body integral tensor of the electronic Hamiltonian into leaf-operator form

\[ H = \sum_{\mu\nu} \zeta_{\mu\nu}\, L_\mu L_\nu^\dagger , \]

where \(\zeta\) is a real symmetric coefficient matrix and \(L_\mu\) are explicit small-matrix leaf operators. The implementation follows the THC representation of Lee et al. 2021, plugged into this repository’s “low-rank tensor → LCU → BE” pipeline (literature citations in the module docstring): the integral tensor is given as classical input data, this module performs no real chemistry integral computation, and the resulting block encoding can enter qubitization directly.

Interface and input model

THCDecomposition(coefficients, leaves)
thc_encoding(thc)

API entry points: THCDecomposition, thc_encoding

• THCDecomposition: the THC input model (input model CP — ζ and the leaf matrices are given directly as classical parameters, i.e. the “low-rank tensor (HAM) → BE” pipeline in algorithm-coverage). coefficients must be a real symmetric square matrix (symmetry tolerance 1e-12), and leaves are explicit small matrices of consistent dimension (powers of two within 2..32), which need not be unitary or Hermitian. The dataclass attributes width gives the number of target qubits and leaf_count the number of leaf operators.

• thc_encoding: assembles the LCU block encoding of the THC Hamiltonian.

Returns a BlockEncoding with module attributes:

Attribute	Meaning
be_alpha	\(\sum_{\mu\nu}\lvert\zeta_{\mu\nu}\rvert\,\alpha_\mu\alpha_\nu\) (\(\alpha_\mu\) is the Pauli l1 bound of a leaf operator)
be_form	"thc"
thc_leaves / lcu_terms	number of leaf operators / number of nonzero terms of the outer LCU
thc_lambda	a λ report under the same accounting, equal in value to be_alpha

Implementation notes

Each leaf operator is first expanded into a Pauli LCU block encoding via matrix_pauli_encoding (words with Pauli coefficients below 1e-12 are dropped; the explicit expansion supports only small instances of at most 5 bits), \(L_\nu^\dagger\) takes the conjugated direction via adjoint_be, and the \((\mu,\nu)\) term takes the product of the two BEs — signals concatenated, α multiplied. The outer LCU’s PREPARE runs over \((\mu,\nu)\) pairs with weights \(\propto\lvert\zeta_{\mu\nu}\rvert\alpha_\mu\alpha_\nu\), and the signal is the selector (as many bits as the term count requires) concatenated with each leaf encoding’s signal; an all-zero ζ raises ValidationError.

Applicability boundary: leaf matrices of dimension 2..32; the approximation error of the THC decomposition relative to the true Hamiltonian belongs to classical preprocessing, and this module assembles the given ζ and leaves exactly. Large-scale ζ and leaf data should switch to QRAM data binding, and the product can be handed directly to transforms.qubitization_walk.

Validation approach

Category C2 (approximately continuous semantics; the acceptance criteria are in ../development/validation-plan.md §2): the block encoding’s zero-signal block must agree with the target Hamiltonian within tolerance. Three layers of evidence:

• Structure: the construction attribute assertions of tests/core/test_lowrank.py:ThcTests (be_form == "thc", thc_leaves, leaf_count); ThcTests.test_invalid_inputs_fail_at_generation covers the THC negative cases (non-symmetric ζ, non-square ζ, all-zero coefficients).

• Numerical: ThcTests.test_thc_block_equals_hamiltonian — two leaf matrices and a 2×2 ζ are assembled, with the expected matrix \(\sum_{\mu\nu}\zeta_{\mu\nu}L_\mu L_\nu^\dagger\) assembled independently inside the test and cross-checked column by column via assert_block_equals (places = 9).

• Binding: this module’s cross-encoding binding witness is DoubleFactorizationTests.test_matches_pauli_encoding_block in the same file (the DF encoding and the Pauli LCU encoding of the same Hamiltonian cross-checked column by column); THC’s leaf operators are themselves the explicit Pauli LCU encoding path, so no separate abstract declaration is set up.

α tightness witness (added in V1): ThcTests.test_thc_alpha_matches_hand_computed_bound hand-computes the leaf bounds inside the test with an independent pauli_l1 closed form (the coefficients of \(M = c_I I + xX + yY + zZ\) expressed linearly through the matrix elements, without going through matrix_pauli_encoding): \(\alpha_0 = 1.2\), \(\alpha_1 = 1.3\), so α = 0.7·1.44 + 0.1·1.56 + 0.1·1.56 + 0.4·1.69 = 1.996, cross-checked against be.alpha (places = 10).

Known gaps and planned stages

No known gaps; the stage V1 witnesses are complete (block cross-checks + an α cross-check against the independent hand computation).

Related links

• Source: src/oracq/algorithms/input_model/lowrank.py

• Algorithms in the same module: Double factorization block encoding

• API reference: Chemical low-rank decomposition block encodings

• Concepts: Oracles and operator representations

• Validation matrix: Validation coverage matrix

Numerical validation

Paper-grade numerical validation script: tests/verification/verify_blockencoding.py (real-backend execution, no mock substitutes, no skips; all 73 cases passed on 2026-09-16), artifact out/verification/blockencoding.json. This page corresponds to the two cases thc-encoding-leaves2-d2 and thc-encoding-leaves3-d4.

Experiment design: 2×2 dense leaves (2 leaves, general complex matrices, neither unitary nor Hermitian) go through OriginIR-ext + UniQC to_matrix full-unitary extraction, with the (0,0) block multiplied by α and cross-checked element by element against the independently numpy-assembled \(\sum_{\mu\nu}\zeta_{\mu\nu}L_\mu L_\nu^\dagger\), and α compared against the independently hand-computed Pauli l1 accounting \(\sum_{\mu\nu}|\zeta_{\mu\nu}|\alpha_\mu\alpha_\nu\) that bypasses matrix_pauli_encoding; 4×4 with three leaves (diagonal leaves + a fully coupled 3×3 ζ, 7 nonzero LCU terms) goes through reference + rir-pysparq + adapter-pysparq on three backends column by column, exercising multi-leaf, multi-scale assembly.

Case	Scale	Paths	Metric	Value
thc-encoding-leaves2-d2	2 leaves 2×2, α=1.381	originir-ext + to_matrix, three-backend cross	max_error	1.7e-16
ditto	—	—	α − independent l1 accounting	0 (exactly equal)
thc-encoding-leaves3-d4	3 leaves 4×4 (diagonal leaves), α=1.782	three backends	max_error / backend deviation	5.6e-16 / 0

Reproduction command:

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

Artifacts: out/verification/blockencoding.json.