VQE Pauli Measurements

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Category C4 · Module oracq.algorithms.optimization.variational · Stage V3

Overview

The variational quantum eigensolver (VQE) uses a parametrized state \(\lvert\psi\rangle\) to approximate the ground-state energy of a Hamiltonian \(H=\sum_k c_k P_k\), where \(P_k\) are Pauli words (bitwise tensor products of I/X/Y/Z) and \(c_k\) are real coefficients. The quantum side only needs to repeatedly measure \(\langle P_k\rangle\) for each \(P_k\); the classical side forms the weighted sum \(E=\sum_k c_k\langle P_k\rangle\) for the energy and drives the parameter optimization.

This module implements the measurement-circuit family: pauli_measurement rotates the prepared state into the requested Pauli measurement basis, and vqe_measurements batch-generates the circuits for a whole Hamiltonian term list. Energy aggregation and parameter optimization happen on the classical side; the module itself contains no optimizer.

Interface and input model

vqe_measurements(preparation, terms)
pauli_measurement(preparation, word)

API entry points: vqe_measurements, pauli_measurement

• preparation: an ansatz with parameters already instantiated, or another state preparation (StatePreparation); the input model is SP; it must satisfy the zero-input, clean-workspace contract (checked by checked_state_preparation; violations raise ContractError).

• terms: a list of (real coefficient, Pauli word); the coefficients must be finite real numbers, the Pauli words must be as wide as the state (I/X/Y/Z), and an empty list raises ValidationError.

• word: an I/X/Y/Z string of the same width; the first character corresponds to the least significant bit.

vqe_measurements returns a tuple whose elements are (coefficient, measurement Operation). Each measurement circuit’s registers are target (the state width) and work (the preparation’s workspace). Module attributes:

Attribute	Meaning
algorithm	"pauli_measurement"
pauli_word	the Pauli word being measured
readout_register	"target"

Implementation notes

Generation chain: the preparation oracle is first invoked to fill target (the work semantics are inherited from the preparation), then the bit-by-bit basis change is applied — a Y bit first receives a phase gate of angle \(-\pi/2\) (\(S^\dagger\)) followed by a Hadamard, an X bit only a Hadamard, an I bit nothing. After the change, target is measured in the standard Z basis; reading out the Z parity of the non-I bits suffices to estimate the expectation of that Pauli word.

Each Pauli word generates an independent circuit; repeated measurement and statistics are organized by the caller; same-name terms can reuse samples from the same circuit, and the module itself performs no qubit-wise commuting grouping. Applicability boundary: only the measurement circuits are provided; confidence intervals for expectation values and parameter-optimization strategies are not part of this module.

Validation approach

Category C4 (heuristic/optimization semantics; acceptance criteria in ../../development/validation-plan.md §2). Three layers of evidence:

• Structure: construction and attribute assertions in tests/core/test_algorithm_expansion.py:AlgorithmExpansionTests; the same family’s test_bad_inputs_fail_at_generation covers generation-time violations of this module’s entry points (Pauli-word width/character checks are executed at generation time in the source; there is no separate negative test).

• Numerical: AlgorithmExpansionTests.test_vqe_measurement_of_y_eigenstate — prepare the Y eigenstate \(\lvert y_+\rangle\) (Hadamard followed by a \(\pi/2\) phase gate), generate the measurement circuit for the term (2, "Y"), the coefficient is echoed back as 2, and target reads 0 with probability 1, i.e. \(\langle Y\rangle=+1\) is recovered exactly.

• Binding: this module has no separate binding witness (the input is already a concrete state preparation, with no abstract slots).

Known gaps and planned stages

Consistent with the variational.py row of validation-coverage.md: the only gap registered on that row is a strong witness for “reaching the optimal cut on small instances” (it lands on the QAOA side, to be wired in at V3); the VQE measurement circuits themselves have no registered gap.

Related links

• Source: src/oracq/algorithms/optimization/variational.py

• Same-module pages: MaxCut QAOA, Hardware-Efficient Ansatz

• API reference: variational algorithm circuits

• Validation matrix: Validation coverage matrix

Numerical validation

Paper-grade numerical experiments are in tests/verification/verify_misc_algorithms.py (misc_algorithms group), all executed on real backends.

Experimental design: a 2-qubit hardware-efficient ansatz is adapted into a preparation oracle via StatePreparation.from_unitary; Pauli measurement circuits are generated for a five-term Hamiltonian (ZI, IZ, XX, YY, ZZ with coefficients 0.5/−0.3/0.7/0.2/−0.1); the Z-parity expectation of the non-I bits of each measurement circuit is computed from the exact state vector (no sampling) and independently compared against \(\langle\psi\lvert P\rvert\psi\rangle\) of the numpy tensor-product operators, then aggregated into the weighted total energy. Backend paths: reference (also matching the numpy oracle within machine precision).

Key metrics:

Case	Scale	Path	Max per-term error	Total energy (measured / exact)	Energy error
vqe-pauli-expectations	2 qubits, 5 terms	reference	3.3e-16	0.4863695277137 / 0.4863695277137	1.1e-16

Reproduce:

PYTHONPATH=src /home/agony/projects/qcfd-dev/quantum-cfd-software/.venv/bin/python tests/verification/verify_misc_algorithms.py

Artifact: out/verification/misc_algorithms.json (all 24 cases pass; this page corresponds to the vqe-pauli-expectations case).