# Degeneracy Counting Quantum Algorithm using Decoherence

arXiv: https://arxiv.org/abs/2608.14941 · HTML: https://mlsanity.com/paper/2608.14941/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 52 references that are in the map.
- First posted on arXiv 14 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm](https://mlsanity.com/paper/1910.02071/) (2019)
- [Quantum-enhanced Markov chain Monte Carlo](https://mlsanity.com/paper/2203.12497/) (2022)
- [Quantum Algorithm for Higher-Order Unconstrained Binary Optimization and MIMO Maximum Likelihood Detection](https://mlsanity.com/paper/2205.15478/) (2022)
- [Fast counting with tensor networks](https://mlsanity.com/paper/1805.00475/) (2018)
- [Counting with the quantum alternating operator ansatz](https://mlsanity.com/paper/2503.07720/) (2025)

## Abstract (arXiv)

Counting the global optima of a classical optimization problem is a #P-hard task. We develop the canonical thermal pure quantum (CTPQ) state-based degeneracy counting (CTPQsd#) algorithm that determines the number of global optima of a classical optimization problem P by measuring only a small probe S, without finding individual minima. The method exploits a perturbative relation between the decoherence measure of S and the degeneracy of P when S and P are together in a CTPQ state. We provide the first numerical demonstration that this relation can be used to count the global minima, applying it to problems encoded by diagonal random-energy Hamiltonians as a maximally unstructured testbed for classical binary optimization problems. Classical simulations of up to 20 problem qubits quantify the algorithm's sensitivity to variations in the temperature of the CTPQ state, the Hamiltonian energy range, the problem size, and degeneracy. We establish the temperature threshold for determining the exact degeneracy and identify a second, lower threshold that provides a temperature window to count near-degenerate minima within a user-defined energy tolerance. By confining measurement to S, the protocol replaces tomography over the exponentially large problem Hilbert space with tomography over a small probe represented by only four qubits.