Rigorous Maximum Likelihood Estimation for Quantum States
Kuchibhotla Aditi, Stephen Becker
TL;DR
This work addresses the exponential challenge of quantum state tomography by recasting maximum likelihood estimation through a Burer–Monteiro factorization, writing $\rho = UU^{\dagger}$ to enforce PSD structure and enable low-rank modeling. By analytically eliminating the trace constraint and choosing a priori $\lambda = \sum_i f_i$, the authors obtain a fully unconstrained BM–MLE objective $J_\lambda(U)$ and develop scalable first-order optimization methods (notably L-BFGS) along with a low-memory BM scheme that avoids forming $\rho$ explicitly. They provide a rigorous a posteriori error bound and demonstrate competitive performance on moderate systems, plus a substantial advance in scalability by solving a 20-qubit state tomography problem on a laptop within hours using Pauli and tetrahedral POVMs. Overall, the framework delivers a principled, scalable approach to QST that does not rely on tensor-network structure, enabling more practical and rigorous tomography for large quantum devices and facilitating uncertainty quantification and benchmarking.
Abstract
Existing quantum state tomography methods are limited in scalability due to their high computation and memory demands, making them impractical for recovery of large quantum states. In this work, we address these limitations by reformulating the maximum likelihood estimation (MLE) problem using the Burer-Monteiro factorization, resulting in a non-convex but low-rank parameterization of the density matrix. We derive a fully unconstrained formulation by analytically eliminating the trace-one and positive semidefinite constraints, thereby avoiding the need for projection steps during optimization. Furthermore, we determine the Lagrange multiplier associated with the unit-trace constraint a priori, reducing computational overhead. The resulting formulation is amenable to scalable first-order optimization, and we demonstrate its tractability using limited-memory BFGS (L-BFGS). Importantly, we also propose a low-memory version of the above algorithm to fully recover certain large quantum states with Pauli-based POVM measurements. Our low-memory algorithm avoids explicitly forming any density matrix, and does not require the density matrix to have a matrix product state (MPS) or other tensor structure. For a fixed number of measurements and fixed rank, our algorithm requires just $\mathcal{O}(d \log d)$ complexity per iteration to recover a $d \times d$ density matrix. Additionally, we derive a useful error bound that can be used to give a rigorous termination criterion. We numerically demonstrate that our method is competitive with state-of-the-art algorithms for moderately sized problems, and then demonstrate that our method can solve a 20-qubit problem on a laptop in under 5 hours.