QSVT Resource Model

The resource reports in this repository are proxy reports for small QSVT-style studies. They track the pieces that are visible from a polynomial and a matrix dimension, while deliberately excluding costs that require a specific quantum data-access and hardware model.

Reported Quantities

field

meaning

degree

highest nonzero polynomial degree

coefficient count

number of supplied polynomial coefficients

phase-count proxy

approximate number of QSP/QSVT phase parameters

signal-operator calls

proxy for uses of the block-encoded signal operator

matrix-register width

ceil(log2(matrix_dimension)) when a dimension is supplied

compatibility fields

sampled boundedness, parity, and optional PennyLane synthesis status

These quantities are useful because degree usually dominates the number of signal uses in an idealized QSVT sequence. They are not enough to determine runtime by themselves.

Every resource report includes a truth_contract field with implementation_kind = "polynomial-resource-proxy" and is_end_to_end_quantum_resource_estimate = false. The report is truthful only as a polynomial-level proxy: it can compare degrees, phase-count proxies, signal-call proxies, widths, and sampled compatibility checks. It cannot by itself justify a runtime, hardware, or quantum-advantage claim.

Notation used in resource discussions:

  • degree is the polynomial degree d.

  • matrix_dimension is the logical dimension N of the encoded matrix.

  • encoding_qubits is usually ceil(log2(N)) when inferred from N.

  • signal_operator_calls is a proxy for calls to the block-encoded signal operator, not a measured runtime.

  • For linear systems, gamma is the scaled positive lower spectral bound and 1 / gamma is a condition-number-style proxy when the scaled maximum eigenvalue is one.

Omitted Costs

The proxy model does not include:

  • block-encoding construction,

  • state preparation or right-hand-side loading,

  • measurement and amplitude estimation,

  • amplitude amplification,

  • sparse-oracle query implementation,

  • Hamiltonian simulation subroutines used to build a block encoding,

  • fault-tolerant synthesis and error correction,

  • hardware compilation,

  • memory movement or classical pre-processing.

For many realistic workflows, these omitted costs dominate the polynomial degree. The reports are therefore best read as polynomial-resource summaries, not full algorithmic complexity estimates.

For a finite dense construction that verifies an actual top-left block for one small matrix, see Block encodings. That page covers what the package can validate directly and what still requires a scalable oracle or problem-specific circuit.

How To Use The Proxy

Use the proxy when comparing candidate polynomial designs:

  1. hold the problem normalization fixed,

  2. compare degrees needed to reach similar approximation error,

  3. inspect boundedness and parity compatibility,

  4. compare signal-call proxies across designs,

  5. separately document how the block encoding and input state would be prepared.

This separation keeps the notebook examples honest: a low-degree polynomial is promising only if the surrounding access model is also favorable.

Linear Systems

For inverse-like linear-system workflows, polynomial degree is tied to the scaled minimum eigenvalue and condition number. Classical dense solves and conjugate gradients remain important baselines because they make residuals, conditioning, and iteration behavior visible.

The proxy does not include quantum state preparation for b, nor the cost of extracting a classical solution vector from a quantum state.

Spectral Projectors And Filters

For thresholding, band-pass filters, and low-rank projection, degree depends on transition width. Sharper spectral edges need higher degree, and leakage outside the selected interval should be reported alongside any rank or retained weight proxy.

The proxy does not include how the selected state or projected subspace would be read out.

Matrix Functions

For exponentials, square roots, Gibbs weights, and resolvents, degree depends on the function smoothness over the scaled spectral domain. Singularities, small broadening parameters, and sharp windows usually increase degree.

Dense spectral matrix functions remain the exact small-system reference.