Eight years ago, three physicists made a guess about a number. Antonio Garcia-Garcia, Yiyang Jia and Jacobus Verbaarschot predicted how large the extreme energies of a particular quantum system should grow as the system gets bigger. The formula was clean: something close to the square root of twice the number of particles, divided by how many particles interact at once. Now Arpon Basu, Pravesh K. Kothari and Siddhant Midha say they can prove it.
Their preprint, posted to arXiv on July 29, concerns the Sachdev-Ye-Kitaev model, usually shortened to SYK. It is a mathematical description of a swarm of particles called Majorana modes, and its defining feature is randomness. Instead of a tidy rule saying which particles push and pull on which others, SYK draws its interaction strengths at random. Every group of k particles interacts, and the strength of each interaction is a coin flip of sorts, a random number drawn fresh each time you build the model. Physicists call these random ingredients the disorder.
What the authors compute is the expected operator norm of the SYK Hamiltonian, written in the paper as the expectation of the largest eigenvalue's magnitude. In plainer terms: averaged over all the random draws, how far from zero do the most extreme energy levels sit? Because the SYK spectrum is symmetric, that same quantity fixes the ground state energy, the lowest rung on the system's energy ladder. Their answer is that it equals the square root of 2n divided by k, up to a correction that shrinks away as the system grows. The result holds when k grows with n but stays well below the square root of n.
That qualifier matters, and the paper is careful about it. The proof does not cover the case physicists study most often, SYK with k fixed at four, where every interaction involves exactly four particles. It covers the regime where k itself grows, slowly, alongside the number of modes. Within that window, though, the bound is sharp in both directions: not merely an upper limit on how large the energy can be, but a matching lower limit showing it really does get that large.
How you prove a thing about randomness
The method is the interesting part, and it is unusual for a physics-adjacent problem. Rather than wrestling with the random Hamiltonian directly, the team found a single fixed, non-random operator, which they call x, whose behavior encodes the random model's statistics exactly. Specifically, a particular quadratic form built from even powers of x reproduces the expected trace moments of the SYK Hamiltonian, the averaged quantities that describe the shape of its energy spectrum. Not approximately. Exactly, for every n and every k.
That swap turns a probabilistic question into a deterministic one. Once you have x, you no longer need to reason about coin flips; you need to find the edge of x's spectrum, the largest value it can produce. The authors describe x as a twisted model of bosons living on the hyperedges of a hypergraph, which is a way of saying that the combinatorial object doing the bookkeeping is a network whose connections join more than two nodes at a time. They then show that x's spectral edge is controlled by a much more familiar object: a matrix of dimension n-choose-k from the Johnson scheme, a classical structure in combinatorics whose eigenvalues are already known. At that point the calculation becomes, in their words, straightforward.
Proving the matching lower bound took a separate construction. The team built an explicit witness state, a specific configuration that makes x's quadratic form large, then translated it into a certificate that the SYK Hamiltonian's own extreme energy is at least as big. The bound holds from both sides, so the value is pinned rather than merely bracketed.
The results also extend to sparse SYK, a stripped-down variant in which most of the possible interactions are simply deleted. And the paper draws out one concrete consequence for computing. Basso, Chen and Dalzell proposed a dissipative quantum algorithm in 2024 for finding SYK ground state energies. Basu, Kothari and Midha show that this algorithm provably lands within a constant multiplicative factor of the true answer for every k below the square root of n divided by four. Before, that guarantee was not available.
Why it matters
SYK occupies an odd position in physics. It is simple enough to analyze yet rich enough to behave like a strongly interacting quantum system, which is why it turns up in work on black holes, on quantum chaos, and on the search for tractable models of matter that resists ordinary approximation. When a quantity as basic as its ground state energy rests on a prediction rather than a proof, everything built on top inherits that softness.
For quantum computing the payoff is more immediate. SYK ground state energy has become a benchmark problem, a target that new quantum algorithms aim at to demonstrate they can do something hard. Judging whether an algorithm succeeded requires knowing the right answer, or at least knowing it within a controlled factor. This paper supplies exactly that, and in doing so converts a plausible claim about the Basso-Chen-Dalzell algorithm into a theorem.
It remains a preprint, not yet peer reviewed, and the authors invite comments. The gap they leave open is visible: constant k, including the much-studied k equals 4, sits outside their range. Whether the same operator trick reaches down into that regime is a question the paper does not answer.