LISA - QUBO

The LISA global fit must decide, jointly, how many galactic binaries a frequency segment contains
and what their parameters are. We isolate the discrete half of that problem — which subset of a
candidate template dictionary best explains the segment — and map it exactly onto a quadratic
unconstrained binary optimisation (QUBO) problem, in which the linear coefficients are matched
filters and the quadratic couplings are template–template overlaps, i.e. source confusion. Two
formulations are derived: a sparsity-penalised one whose penalty is fixed by a Poisson-process prior
rather than by hand, and a cardinality-constrained one whose penalty weight is derived from a
single-flip bound. The mapping is validated twice: the underlying waveform agrees with the LISA
Data Challenge fastGB reference to 2.220 ×10−16 in raw network overlap with no time or phase
maximisation, and exhaustive enumeration of all 2 097 152 bitstrings of a three-source instance
reproduces the direct likelihood to 1.455×10−11 with the global minimum sitting on the true source
set. The results are, deliberately, led by their negative findings. On all three test instances —
two synthetic and one real LDC2a Sangria segment — the QUBO recovers exactly every source
its dictionary is able to represent, so the dictionary, not the formulation and not the solver, is the
binding constraint; the recovery gate fails on two of the three for that reason. Orthogonal matching
pursuit matches or beats the QUBO on every instance measured, at 1807 times lower cost on the
easiest one. An offline minor-embedding study measures the hardware wall: a fully dense instance of
180 logical variables consumes 3000 of a Pegasus P16 processor’s 5640 qubits, and the Gram density
of a galactic-binary dictionary is flat in dictionary size, so a coarser dictionary does not escape that
wall. The structure that makes the problem embeddable at all, however, is the structure that makes
it classically easy, and that is the paper’s strongest finding. Compact template support makes the
Gram matrix banded with half-bandwidth b≃ρGM/2; a band graph has treewidth exactly b; and
the ground state of an Ising model of treewidth t follows exactly in O(n2t). On the same instances
the embedder was given, exact dynamic programming returns a certified ground state at M = 400
in 0.033 s, where minor embedding costs 2251 physical qubits, and at M = 1000 in 1.338 s, where
embedding fails outright. Over the region measured there is no (M,ρG) at which an annealer is both
applicable and the best tool available. We claim no quantum advantage and measure no speed-up.
What we contribute is a correct and independently checkable formulation, its cost measured on
both sides of that comparison, three negative results, and a published benchmark instance family
in standard optimisation formats.