Quantum spectrum estimation can be done with fewer copies than full tomography

The improvement is small enough to write on a napkin: instead of needing about $d^2$ copies of a quantum state, the new algorithm gets by with $d^2 \cdot (\log\log d / \log d)^2$. That extra factor shrinks as $d$ grows, so for large systems the saving is real. It is also the entire point. For twenty-five years, nobody had managed to beat the standard method by any margin at all, and the question of whether it could be beaten had hardened into a conjecture that it could not.

Angelos Pelecanos, Jack Spilecki, Ewin Tang, and John Wright posted the result to arXiv on July 29, 2026. The paper is a preprint, 58 pages long, and has not yet been through peer review.

To see what is at stake, it helps to separate two things you might want to know about a quantum system. The first is everything: the full description of the state, written as a matrix called the density matrix, with $d$ standing for the dimension of the system. Learning that full description is called tomography, and it is expensive. You cannot measure a single copy of a quantum state and read off its description, because measurement disturbs the state and returns only a sliver of information. So you prepare many identical copies and measure them, and the number of copies you need is the currency in which the difficulty is priced. Full tomography costs $\Theta(d^2)$ copies, meaning that figure is both the floor and the ceiling: nobody can do better, and known methods achieve it.

The second thing you might want is less. The eigenvalues of the density matrix, collectively its spectrum, form a list of $d$ probabilities that captures how mixed or pure the state is, how much entropy it carries, how concentrated it is in a few directions. What the spectrum omits is orientation: which directions in the state space those eigenvalues belong to. It is strictly less information than the full state. Intuitively, less information ought to be cheaper.

It has not been. Michael Keyl and Reinhard Werner gave an algorithm for estimating the spectrum in 2001, and it uses $\Theta(d^2)$ copies, the same order as learning the whole state. They themselves asked whether that could be improved. Fifteen years later, Wright, one of the authors of this new paper, conjectured that it could not, that spectrum estimation was somehow just as hard as full tomography despite asking for less.

The new algorithm settles this, and settles it against the conjecture. It estimates the eigenvalues to constant error in total variation distance, a standard way of measuring how far apart two lists of probabilities are, using strictly fewer copies than $d^2$. So the Keyl-Werner algorithm is not optimal. Whether the new algorithm is optimal is a different question, and this paper does not claim to answer it.

The tool underneath

The authors' main technical device is a new guarantee about tomography itself, and it is arguably the more portable contribution. Ordinary tomography bounds give you a single error figure covering the whole state. The new bound is direction-sensitive: for any direction $|w\rangle$ in the state space, the error in that direction scales with $\langle w | \rho | w \rangle$, roughly, how much of the state actually lives along that direction. Directions where the state has little presence get proportionally small errors. Crucially, this holds for all directions at once, not one chosen in advance.

The authors call this a relative-error bound, and it does more work than the spectrum result alone. They report that two other improvements fall out of it as corollaries: better algorithms for principal component analysis measured in Bures distance, a way of quantifying the gap between quantum states, and for tomography measured in $\chi^2$-divergence, a stricter error measure than the usual ones.