feat(qmc): add Halton quasi-Monte Carlo sampler

[Samuel-Vangu]

Description

Summary of changes

• Adds Halton, a (optionally scrambled) Halton low-discrepancy sampler, shaped exactly like Sobol and RNG — plugs into MonteCarlo.integrate via the existing rng= slot, no changes to MonteCarlo itself.
• torch backend: pure PyTorch implementation (digit extraction, hierarchical digit-shift scrambling, integer Horner reconstruction), generated directly on the target device — no CPU/NumPy round-trip needed, unlike SobolEngine.
• numpy / jax / tensorflow backends: scipy.stats.qmc.Halton (already a hard dependency), converted to the requested backend.
• Unlike Sobol, does not require the sample size to be a power of two.
• Adds tests/halton_test.py, mirroring sobol_test.py's structure, plus Halton-specific regression tests (see Test plan).

Resolved Issues

• Addresses Low Discrepancy Sequences for MonteCarlo and VEGAS #140 (MonteCarlo path for Halton; VEGAS + JIT support out of scope here, same as for Sobol)

How Has This Been Tested?

• halton_test.py, all backends: analytic accuracy against the test-function collection, Halton < plain-MC error at equal N (fixed seed), seed reproducibility, torch gradient flow
• test_digits_needed_is_per_dimension: deterministic, seed/device-independent unit test guarding against a real bug found during development (a shared digit count across dimensions silently broke equidistribution for every dimension except the one with the smallest base)
• test_halton_column_independent_of_other_dimensions: exact-equality check that a dimension's column never depends on which other dimensions are requested alongside it
• test_halton_matches_scipy_when_unscrambled: pure-PyTorch and SciPy backends agree to within float64 machine epsilon (~2e-16) when scrambling is off, validating the digit extraction/reconstruction independently of the scrambling logic
• test_halton_no_power_of_two_warning: confirms no spurious warning for non-power-of-two N
• ruff format --check . clean (57 files)

Related Pull Requests

• feat: add Sobol quasi-Monte Carlo sampler #244 (Sobol) — same issue, same sampler shape/interface, same testing pattern

[Samuel-Vangu]

@gomezzz, this one's ready for CI too whenever you get a chance — thanks!

[Samuel-Vangu]

@gomezzz ,

While waiting on Lattice, I checked Halton against everything you flagged
there and fixed what was necessary.

19/19 tests pass.
Happy to have you take a look whenever you're free.

[sylvesterkaczmarek]

The torch scrambling is truncated to _digits_needed(base, number_of_points), so its claimed unbiased randomisation is not actually uniform: for N=1 in the first dimension, m=1 and the sample can only be 0 or 0.5 across seeds. That biases the QMC estimator for general integrands; add a random tail/continuous shift (or otherwise make the randomisation unbiased) and include an N=1 distribution regression.

[Samuel-Vangu]

@sylvesterkaczmarek

Confirmed and fixed. Added a continuous tail shift after digit
scrambling.

Added the N=1 regression: 200 seeds now span way more than 2 distinct
values, and are no longer confined to {0.0, 0.5}.

[sylvesterkaczmarek]

The scrambled Halton construction now adds continuous tail randomization, and the N=1 multi-seed regression demonstrates that the first coordinate is no longer confined to the two-point set caused by the truncated digit permutation. This resolves the bias issue I raised.