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EMS Press eBooks

Spectral gaps in quantum computing

David Pérez-Garcı́a

Published Oct 14, 2026• 0 citations
Publisher

Abstract

How does the spectral gap of a quantum many-body system scale with the system size?This turns out to be a very relevant question both in condensed matter physics and in quantum computing.However, there are very few available techniques to give bounds on that scaling.In the first part of this paper, we will review recent results which explain why it is undecidable to know if the spectral gap will vanish or not with the system size.In the second part, we will review complementary results which show that, despite this general impossibility result, one can still prove spectral gap estimates in a rather wide family of systems.As a consequence, one obtains new bounds on the lifetime of quantum memories.

DOI
10.4171/9ecm/28

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How does the spectral gap of a quantum many-body system scale with the system size?This turns out to be a very relevant question both in condensed matter physics and in quantum computing.However, there are very few available techniques to give bounds on that scaling.In the first part of this paper, we will review recent results which explain why it is undecidable to know if the spectral gap will vanish or not with the system size.In the second part, we will review complementary results which show that, despite this general impossibility result, one can still prove spectral gap estimates in a rather wide family of systems.As a consequence, one obtains new bounds on the lifetime of quantum memories.
The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] or more dimensions [T. S. Cubitt et al., "Undecidability of the spectral gap", Nature 528 (2015) and Forum of Mathematics Pi, 10 (2022)]. In this work, we focus on the 1-dimensional result, showing that the constructed family with undecidable behavior is extremely sensitive to perturbations. In particular, for any ε>0\varepsilon > 0ε>0, there exists a 1-local, rank 1, perturbation with norm O(ε)O(\varepsilon)O(ε), such that the spectral gap problem for the family in [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] now becomes decidable.
This paper provides a self-contained conceptual overview of one of the most striking results in contemporary quantum many-body physics: the undecidability of the spectral gap problem. The spectral gap—the energy difference between a system's ground state and its first excited state—is a fundamental property that distinguishes insulating from conducting behavior and marks quantum phase transitions. In 2015, Cubitt, Pérez-García, and Wolf proved that deciding whether a translationally invariant quantum spin system with finite-range interactions is gapped or gapless is algorithmically undecidable. This result was extended to one-dimensional systems by Bausch and colleagues in 2020, overturning the intuition that lower-dimensional systems are inherently more tractable. The proof architecture rests on two main ingredients: aperiodic tilings and history-state constructions. Aperiodic tilings encode undecidable combinatorial properties into the spatial structure of the Hamiltonian, while history states embed the evolution of a Turing machine into the ground state. By combining these elements, the authors construct explicit Hamiltonians whose spectral gap depends directly on whether a simulated Turing machine halts—a problem known to be undecidable. Thus, no algorithm can classify every instance correctly. The paper also distinguishes two forms of undecidability: algorithmic undecidability, which asserts that no Turing machine can solve the problem for all instances, and axiomatic independence, which shows that for some Hamiltonians, the gap status is not determined by the standard axioms of mathematics. These results impose fundamental limits on numerical simulation and phase classification, but they do not render all systems intractable. They demarcate a boundary between decidable and undecidable behavior, revealing deep connections between physics, computation, and logic.
This work provides an overview of gapped quantum spin systems, including concepts, techniques, properties, and results. The basic framework and objects of interest for quantum spin systems are introduced, and the main ideas behind methods for proving spectral gaps for frustration-free models are outlined. After reviewing recent progress on several spectral gap conjectures, we discuss quasi-locality of the Heisenberg dynamics and its utility in proving properties of gapped quantum spin systems. Lieb-Robinson bounds have played a central role in establishing exponential decay of ground state correlations, an area law for one-dimensional systems, a many-body adiabatic theorem, and spectral gap stability. They also aided in the development of the quasi-adiabatic continuation, which is a useful for investigating gapped ground state phases, both of which are also discussed.
Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.