Eigenstate Filtering

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

Overview

Eigenstate filtering: construct a “spike” polynomial block encoding from a block encoding of a matrix \(A\) — components whose spectral variables fall inside the passband around the filter center are preserved (\(|f|\) close to 1), while components outside the passband are suppressed below \(1/T_d(r)\). The target is a Lin–Tong-type filter polynomial (the construction basis cited in the source docstring):

\[ f(x) = \frac{T_d\!\bigl(g(x^2)\bigr)}{T_d(r)}, \qquad g(y) = \frac{2(y - \Delta^2)}{1 - \Delta^2} - 1, \qquad r = \frac{1 + \Delta^2}{1 - \Delta^2}, \]

where \(\Delta\) is the filter width and \(d\) the Chebyshev degree. \(f\) saturates at \(x = 0\) (\(|f(0)| = 1\)), and \(|f(x)| \le 1/T_d(r)\) on \(|x| \ge \Delta\), with the suppression rate decaying exponentially in \(d \cdot \operatorname{arcosh}(r)\). The phase-synthesis framework is the same as QSP phase synthesis (Gilyén et al. 2019, arXiv:1806.01838).

Interface and input model

eigenstate_filter(a, gap, degree, *, center=0.0)

API entry: eigenstate_filter

• a: a BlockEncoding, the block encoding of the filtered matrix (input model BE; the spectral variable is \(x = \lambda/\alpha\), which for the Hermitian case is just the normalized eigenvalue).

• gap: the filter width \(\Delta\), which must lie in \((0, 1)\).

• degree: the Chebyshev degree \(d\), which must be a positive integer (floats are rejected).

• center: the filter center, which must lie in \((-1, 1)\); when nonzero, the spectrum is shifted first (see below).

Returns a BlockEncoding whose zero-signal block is suppressed below suppression outside the passband. Module attributes:

Attribute	Meaning
algorithm	"eigenstate_filter"
be_alpha	normalization of the output BE (the real-part-extraction LCU in equal halves; 1.0)
gap / filter_degree / center	echo of the call parameters
qsp_degree	the synthesized polynomial degree \(2d\)
suppression	the out-of-passband upper bound \(1/T_d(r)\)

Implementation notes

The synthesized degree is \(2d\) (\(f\) is a degree-\(d\) Chebyshev combination of \(x^2\), monomial degree \(2d\)), subject to the synthesis cap of 40 (i.e. \(d \le 20\)); exceeding it raises ValidationError. When center is nonzero, the spectrum is first shifted by the BE linear combination linear_combination(1.0, a, -center, identity(a.width)): a block encoding of \(A - \text{center}\cdot I\) is constructed (the normalization becomes \(\alpha + |\text{center}|\)), the filtering acts on the shifted spectral variable \(x' = (\lambda - \text{center})/(\alpha + |\text{center}|)\), and gap is measured in that variable.

\(f\) saturates at \(x = 0\) but \(|f(\pm 1)| = 1/T_d(r) < 1\) leaves the endpoints unsaturated, so phases can be synthesized only with the imaginary-part completion \(h = \text{sat}\cdot x^2\) (\(\text{sat} = \sqrt{1 - f(1)^2}\)); after synthesis the real part is extracted via \((U_\Phi + U_{-\Phi})/2\) (the mechanism is described in QSP phase synthesis). Applicability boundary: the input must be a block encoding; the shifted spectral values should lie in \([-1, 1]\) — whatever exceeds it loses the upper-bound guarantee.

Validation approach

Category C2 (approximately continuous semantics; the acceptance criteria are in ../../development/validation-plan.md §2): the acting operator must approximate the target continuous function within tolerance. Three layers of evidence:

• Structure: tests/core/test_qsvt.py:TransformWitnessTests.test_transform_input_validation (negative cases such as an out-of-range gap and a non-positive-integer degree) and PhaseSynthesisTests.test_input_validation (the input validation of the underlying synthesis).

• Numerical: tests/core/test_qsvt.py:TransformWitnessTests.test_eigenstate_filter_isolates_eigenvalue — a 2×2 diagonal BE (eigenvalues 0.05 and 0.5, \(\alpha = 0.5\)), gap = 0.2, \(d = 8\), center = 0.1: suppression < 0.08; the passband eigenvalue after shifting is \(\approx -0.083\) (\(|x'| < \Delta\)) with zero-signal amplitude > 0.7; the stopband eigenvalue after shifting is \(\approx 0.667\), with amplitude < 0.1.

• Binding: this algorithm has no independent binding witness (the input already requires a concrete BE).

Known gaps and planned stages

Consistent with the qsvt.py row of the validation coverage matrix: convergence scans are missing (the exponential-decay curve of the suppression rate versus degree is not automated, pending the stage V2 convergence-scan framework); the ill-conditioned negative cases (degree ≳ 16 or too-low precision) are partially covered by the parametrized input_validation, and a separate batch parametrization of “ill-conditioned inputs must raise” is still to be added.

Related links

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

• API reference: QSVT standard transforms

• Same-family pages: QSP phase synthesis, QSVT matrix inversion

• Validation matrix: Validation coverage matrix

Numerical validation

Paper-grade numerical experiments are in tests/verification/verify_hamiltonian.py (real-backend execution, no mock substitutes). Experiment design: first, a 1-qubit diagonal BE with center shift (eigenvalues 0.05 and 0.5, \(\alpha = 0.5\); gap = 0.2, d = 8, center = 0.1; shifted spectral variables \(x' \approx -0.083\) in the passband and \(0.667\) in the stopband); second, a 2-qubit diagonal BE (eigenvalues 0.02, −0.03, 0.4, 0.55; gap = 0.2, d = 6, two passbands and two stopbands). On the three paths reference / rir-pysparq / originir-ext, the diagonal amplitudes of the zero-signal block are read out and cross-checked pointwise against the independent analytic evaluation of the Lin–Tong filter polynomial \(f(x) = T_d(g(x^2))/T_d(r)\) (math.cosh/acosh, not through the in-library helpers).

Case	Scale	Backend paths	Metric	Value
eigenstate-filter-centered-1q	\(2d = 16\), center=0.1	reference, rir-pysparq, originir-ext	max_error	1.32e-7
same as above	—	—	passband amplitude / stopband amplitude (suppression attribute 0.0779)	0.748 / 0.0235
eigenstate-filter-2q	\(2d = 12\), four eigenvalues	reference, rir-pysparq, originir-ext	max_error	2.17e-10
same as above	—	—	passband minimum / stopband maximum (suppression attribute 0.174)	0.914 / 0.174

In both cases the stopband amplitudes do not exceed the suppression attribute bound; the 1.3e-7 deviation of the centered case is numerical noise of layer stripping at degree \(2d = 16\), within the module’s self-check tolerance (1e-5). Measured numerical boundary: \(d = 10\) (\(2d = 20\)) is rejected by the phase-synthesis self-check, and the synthesizable region \(d \le 8\) passes entirely.

Reproduction command:

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

Artifacts: out/verification/hamiltonian.json.