Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.30171

Optimal spectrum estimation

Ainesh Bakshi·Apoorv Vikram Singh·Xinyu Tan
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that the spectrum of an unknown $d$-dimensional quantum state can be estimated to error $\varepsilon$ in total variation distance using \[ O\!\left(d^2\min\left\{ \frac{1}{(\varepsilon\log d)^4},\; \frac{1}{(\varepsilon\log d)^2} \right\}\right) \] copies. This matches the recent lower bound of Wang. When restricted to unentangled measurements, we give an algorithm with an additional factor of $d$ in copy complexity, which we conjecture to be optimal. We develop a framework for recovering the small eigenvalues of a quantum state by matching Chebyshev moments. We bound the variance of each Chebyshev moment estimate in terms of scalar derivatives of the corresponding polynomial, using classical and quantum Efron--Stein decompositions. Different rescalings of the Chebyshev polynomials balance approximation error and variance, yielding two regimes in our copy complexity bound.

Explore related subjects

quant-phcs.CCcs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ainesh Bakshi, Apoorv Vikram Singh, Xinyu Tan. 2026-09-24. Optimal spectrum estimation. https://arxiv.org/abs/2609.30171

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantum complexity and generalized area law in fully connected models

The area law for entanglement entropy captures a fundamental constraint on the complexity of quantum many-body ground states and enables their efficient description. While the area law is rigorously established in one dimension, its status in higher-dimensional local systems remains unresolved, and it does not hold in general for geometrically non-local systems. Here, we establish a generalized area law for gapped ground states of fully connected Hamiltonians. We show that the bipartite entanglement entropy grows at most logarithmically with system size despite the absence of geometric locality. In the ground state, each site is only weakly entangled with the rest, while configurations with extensive local fluctuations are strongly suppressed, effectively restricting the accessible Hilbert space. As a consequence, the ground state admits a matrix product state approximation with polynomial bond dimension with respect to the system size at fixed accuracy. In the permutation-invariant setting, we further prove a constant entanglement bound and demonstrate that the gapped ground state can be computed in time polylogarithmic in the system size. These results show that low entanglement complexity need not rely on geometric locality, broadening the conceptual and computational scope of area laws.

quant-ph↗
arXiv2026

Quantum Mechanics as a Reversible Diffusion Theory

This paper proposes an interpretation of quantum mechanics, relying on the time-symmetric stochastic dynamics of quantum particles and on non-classical probability theory. Our main purpose is to demonstrate that the wave function and its complex conjugate can be interpreted as complex probability distributions in two complex diffusion equations related to non-real forward and backward in time stochastic motions respectively. We say non-real because Schroedinger forward and backward diffusions describe both reversible (real trajectories) and irreversible trajectories (non-real trajectories). The reversible trajectories are the only real trajectories and are given by the intersection of those forward and backward processes. It turns out that if we translate this intersection using set-theoretic language, we are led to a reversible diffusion described by Born rule probabilities. This proposal is useful also for explaining more about the role of complex numbers in quantum mechanics that produces this so-called "wave-like" nature of quantum reality. Our perspective also challenges the notion of physical superposition and aims at a derivation of superposition principle not based on the linearity of Schroedinger's equation but relying on pure probability theory. Moreover, it is suggested that, embracing the idea of stochastic processes in quantum theory, explains the reasons for the appearance of classical behavior in large objects, in contrast to the quantum behavior of small ones. In other words, we claim that a combination of a probabilistic and no-ontic view (neither epistemic though) of the wave function with a stochastic hidden-variables approach, may provide some insight into the quantum physical reality and potentially establish the groundwork for a novel interpretation of quantum mechanics.

quant-ph↗
arXiv2026

Flexible Qubit Allocation of Network Resource States

The Quantum Internet is still in its infancy, yet identifying scalable and resilient quantum network resource states is an essential task for realizing it. We explore the use of graph states with flexible, non-trivial qubit-to-node assignments. This flexibility enables adaptable engineering of the entanglement topology of an arbitrary quantum network. In particular, we focus on cluster states with arbitrary allocation as network resource states and as a promising candidate for a \textit{network core}-level entangled resource, due to its intrinsic flexible connectivity properties and resilience to particle losses. We introduce a modeling framework for overlaying entanglement topologies on physical networks and demonstrate how optimized and even random qubit assignment creates shortcuts and improves robustness and memory savings, while reducing the worst-case hop distance between remote network nodes, when compared to conventional approaches.

quant-ph↗