Daily Summary for 2026-10-25

Introduction to the show: ident: Mathematics. The day in mathematics: new theorems, new proofs, and what they open.

Ada: It's the twenty-fifth of October, twenty twenty-six, and this is the day's research.

Emmy: One new paper came out today.

Ada: I'm Ada.

Emmy: And I'm Emmy, here with Kolya.

Ada: We'll take the day in one pass, then pull out the paper we're staying with.

Emmy: Start us off with Strongly Regular Graph Construction via Quadratic Binary Optimization.

The day's research: Ada: Building strongly regular graphs is important in algebraic graph theory and quantum information because their explicit construction becomes very difficult as you add more vertices.

Emmy: The main problem is that the equations defining these SRGs are quadratic in the edge variables.

Kolya: However, using a sum of squared residuals creates a quartic penalty, which is too complex for standard quadratic unconstrained binary optimization or QUBO.

Ada: The work by Strongly Regular Graph Construction via Quadratic Binary Optimization shows that no simple quadratic function of the edge variables can find all labeled SRGs as its exact global minimizers.

Emmy: This is true except for a few specific degenerate families, such as perfect matchings, cocktail-party graphs, and pentagons.

Kolya: This creates a strong obstruction: the 99-vertex graph from Conway's problem cannot be found using pure QUBO without adding extra variables.

Ada: The first method solves degenerate families exactly within pure QUBO.

Emmy: And it uses a generate-prune-verify pipeline for larger graphs to reduce candidates via a fast classical test.

Kolya: The second approach transforms the quartic penalty into a quadratic one using Rosenberg auxiliary variables.

Ada: Those auxiliary variables increase in size with the number of vertices, which is notable.

Emmy: They allow testing pure-QUBO on small families and the auxiliary route up to sixteen vertices.

Kolya: Ultimately, they tabulated feasible parameters for graphs up to forty-nine vertices using these methods.

Ada: Strongly regular graph construction via quadratic binary optimization proves limitations unless the graph is one of three special types.

Emmy: The authors develop two construction methods: one using pure QUBO and another with auxiliary variables.

Kolya: They test these methods on small graphs up to forty-nine vertices for this paper.

Ada: We can hear Strongly regular graph construction via quadratic binary optimization because it proves limitations unless the graph is one of three special types.

Emmy: This paper shows that you cannot find strongly regular graphs using only a simple quadratic optimization problem unless the graph belongs to one of three special types.

Kolya: The authors develop two methods to construct these graphs, one using pure QUBO and another using auxiliary variables.

Ada: They test these methods on small graphs up to forty-nine vertices for this paper.

Emmy: This paper proves that you cannot find strongly regular graphs using only a simple quadratic optimization problem unless the graph belongs to one of three special types.

Kolya: The authors develop two methods to construct these graphs, one using pure QUBO and another using auxiliary variables.

Ada: They test these methods on small graphs up to forty-nine vertices for this paper.

Lucky paper: 2610.30142: Ada: So we're looking at Strongly Regular Graph Construction via Quadratic Binary Optimization today, and the core idea is that finding these graphs using just a quadratic function of edge variables is really hard.

Emmy: It boils down to this quadratic equation involving binary edge variables, but when you use the sum of squared residuals, it turns into a quartic objective function.

Kolya: Which means standard QUBO methods just can't solve that problem directly because it's not quadratic anymore.

Ada: The authors show there’s a strong obstruction here: no simple quadratic function of the edge variables can yield all labeled strongly regular graphs as its exact set of global minimizers, except for three specific degenerate families.

Emmy: Those three families are perfect matchings, cocktail-party graphs, and pentagons.

Kolya: This finding formally establishes that the ninety-nine-vertex graph from Conway's problem cannot be found using QUBO without introducing extra variables.

Ada: The authors tackle this with two different routes to address this problem. The first route uses pure QUBO to solve those three degenerate families exactly, and for the rest, they use a generate-prune-verify pipeline.

Emmy: That pipeline has two stages: stage one generates highly restricted k-regular candidate graphs using the quadratic penalty Hgen and a symmetry-breaking mechanism.

Kolya: Stage two is QUBO-based spectral postselection of those candidates using maximum clique analysis and the Hoffman bound, followed by an exact algebraic verification against Eq. (one).

Ada: Then there's the second route, which handles the full problem by converting that quartic penalty into a quadratic one using Rosenberg auxiliary variables.

Emmy: That means you enlarge the variable space by introducing these auxiliary variables to cast the whole problem as a single QUBO instance.

Kolya: The benefit of a fully QUBO encoding is that it eliminates all terms of degree higher than two from the original HUBO Hamiltonian, resulting in HSRG = w1 Hgen + w2 H(q)λµ + w3 Haux.

Ada: But this comes at a cost, as the set of auxiliary variables indexed by zero ≤ i < j ≤ N - one and l /∈ i, j has a cardinality of N two(N - one)(N - two).

Emmy: That means the number of these extra variables increases three times as the number of vertices N grows.

Kolya: For Conway’s problem target, that count evaluates to roughly four point seven × one hundred five binary variables before canonical fixing is done.

Ada: The empirical results show they are complementary: the pipeline works on Petersen with seventy-eight verified SRGs in one hundred four SA reads on Hgen, but it fails on Paley(thirteen) and Clebsch within that same budget.

Emmy: On the other hand, the auxiliary-variable formulation succeeds on Clebsch with a success probability of psucc equal to zero point nine five at nreads equal to one hundred five reads.

Kolya: And it also succeeds on Paley(thirteen) with a psucc of zero point six five at nreads equal to two × one hundred five reads, showing they are empirically complementary on the same targets.

Ada: So the central result is this sharp delimitation: a quadratic function of edge variables can only find strongly regular graphs if every k-regular graph is already the target, which limits solutions to those three families.

Emmy: This formally establishes that the ninety-nine-vertex graph of Conway’s one thousand problem cannot be obtained as the exact ground-state set of any QUBO over the four thousand eight hundred fifty-one edge variables.

Kolya: The two routes are not only complementary in scope, but they are empirically complementary on the same targets, as shown by comparing Tables III and IV.