Classical Baseline Details¶

The benchmark helpers in qsvt.benchmarks are intentionally small classical references. They make the comparison target explicit before attaching QSVT resource proxies such as polynomial degree, signal-operator calls, and matrix register width.

These baselines are not tuned benchmark suites. They are reproducible small-matrix measurements for asking whether a QSVT-style polynomial transform is being compared against the right classical operation.

Baseline Summary¶

helper	classical operation	typical cost model	QSVT comparison role
`dense_eigendecomposition_benchmark`	`numpy.linalg.eigh` on a dense Hermitian matrix	cubic dense linear algebra	exact spectral reference for matrix functions
`dense_linear_solve_benchmark`	`numpy.linalg.solve` on a dense system	cubic dense linear algebra	strong small-system baseline for inverse-polynomial workflows
`conjugate_gradient_benchmark`	pure-NumPy conjugate gradients	iteration count times matrix-vector cost	sparse positive-definite baseline for linear systems
`spectral_matrix_function_benchmark`	eigendecompose then apply a scalar function to eigenvalues	eigendecomposition dominated	exact small-system reference for non-polynomial functions
`polynomial_matrix_function_benchmark`	apply a polynomial through spectral functional calculus	eigendecomposition dominated in this implementation	classical reference for the same polynomial used by QSVT

Dense Eigensolvers¶

Dense eigendecomposition is the cleanest exact reference for the examples in this repository. It exposes the spectrum, eigenvectors, reconstruction error, and spectral width without requiring sparse matrix infrastructure.

What it measures: : Wall-clock time for dense Hermitian eigendecomposition and reconstruction diagnostics.

What it assumes: : The full matrix is already available classically in dense memory.

Why it matters: : Many QSVT demonstrations reduce to applying a function to eigenvalues. Dense eigendecomposition is the most direct small-scale reference for that action.

Where it is unfair to QSVT: : It assumes dense classical access to all matrix entries and does not model large sparse or oracle-access settings where block encodings could be natural.

Dense Linear Solves¶

Dense direct solves are strong baselines for the toy linear systems in the notebooks. They avoid iterative convergence questions and provide residual norms and condition-number metadata.

What it measures: : Wall-clock time for numpy.linalg.solve, residual norm, relative residual, solution norm, and 2-norm condition number.

What it assumes: : A dense matrix and right-hand side are already available in memory.

QSVT link: : The associated proxy usually comes from an inverse-like polynomial whose degree is related to the spectral gap or condition number after rescaling.

Conjugate Gradients¶

Conjugate gradients are the relevant classical baseline for positive-definite linear systems when matrix-vector products are cheaper than dense factorization.

This repository’s implementation is pure NumPy and uses dense matrix-vector products for transparency. The reported matvec_count is still useful because it separates iteration behavior from the dense test harness.

What it measures: : Iterations, convergence status, residuals, condition number, and repeated-run timing.

What it assumes: : Hermitian positive-definite input and a tolerance target.

QSVT link: : Both CG and QSVT-style inverse transforms are sensitive to spectral gaps and condition number. The benchmark table keeps those quantities visible.

Dense Spectral Matrix Functions¶

The spectral matrix-function baseline diagonalizes the Hermitian input and evaluates a scalar function on its eigenvalues. It is used for functions such as exponentials before comparing with polynomial approximations.

What it measures: : Dense spectral functional calculus for a named scalar function.

What it assumes: : Full dense access and an exactly evaluable scalar function.

QSVT link: : QSVT applies bounded polynomials. This baseline supplies the exact target matrix function that polynomial designs approximate.

Polynomial Matrix Evaluation¶

The polynomial baseline applies the same polynomial transform that a QSVT sequence would implement, but classically through eigendecomposition.

What it measures: : Output norm and timing for applying a supplied polynomial to a Hermitian matrix.

What it assumes: : Dense eigendecomposition rather than a quantum block encoding.

QSVT link: : This is the closest apples-to-apples functional comparison for a fixed polynomial. The remaining difference is the implementation model, not the mathematical transform.

Interpreting Timings¶

Microbenchmarks in this repository are intentionally small and should not be read as hardware-performance claims. The useful release artefacts are:

problem type and matrix dimension,
residual or approximation diagnostics,
condition-number and degree metadata,
QSVT proxy fields when a polynomial is supplied,
rough timing scale for local reproducibility.

The benchmark notebooks should be used to identify regimes and missing costs, not to claim end-to-end quantum speedups.