Related papers: On the uncomputability of the spectral gap
We prove nonintegrability of a model Hamiltonian system defined on the Lie algebra $\mathfrak{su}_3$ suitable for investigation of connections between classical and quantum characteristics of chaos.
Universal quantum computation is usually associated with interaction among two-level quantum subsystems, as this interaction is commonly viewed as a necessity to achieve universal quantum computation. In this work, we show that, contrary to…
In this paper we investigate the linear and nonlinear models of optical quantum computation and discuss their scalability and efficiency. We show how there are significantly different scaling properties in single photon computation when…
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is…
We relate the notions of spectral gap for unitary representations and subfactors with definability of certain important sets in the corresponding structures. We give several applications of this relationship.
Two quantum channels are called compatible if they can be obtained as marginals from a single broadcasting channel; otherwise they are incompatible. We derive a characterization of the compatibility relation in terms of concatenation and…
In this paper, we develop spectral analysis of a discrete non-Hermitian quantum system that is a discrete counterpart of some continuous quantum systems on a complex contour. In particular, simple conditions for discreteness of the spectrum…
This is not a disproof of the quantum PCP conjecture! In this note we use perturbation on the commuting Hamiltonian problem on a graph, based on results by Bravyi and Vyalyi, to provide a partial no-go theorem for quantum PCP. Specifically,…
We give a reformulation of Salem's conjecture about the absence of Salem numbers near one in terms of a uniform spectral gap for certain arithmetic hyperbolic surfaces.
We show that all scaling quantum graphs are explicitly integrable, i.e. any one of their spectral eigenvalues $E_n$ is computable analytically, explicitly, and individually for any given $n$. This is surprising, since quantum graphs are…
We consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. For instance, this is the case a global Kepler system, non-autonomous integrable Hamiltonian systems and integrable systems…
Some formulas and speculations are presented relative to integrable systems and quantum mechanics.
As inductive inference and machine learning methods in computer science see continued success, researchers are aiming to describe ever more complex probabilistic models and inference algorithms. It is natural to ask whether there is a…
The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of $\mathcal N \leq 2$ quantum mechanical systems is $\#P$-complete and therefore…
We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded…
It is well established numerically that spectral statistics of pseudo-integrable models differs considerably from the reference statistics of integrable and chaotic systems. In [PRL,93 (2004) 254102] statistical properties of a certain…
We examine the effect of previous history on starting a computation on a quantum computer. Specifically, we assume that the quantum register has some unknown state on it, and it is required that this state be cleared and replaced by a…
I assess the potential of quantum computation. Broad and important applications must be found to justify construction of a quantum computer; I review some of the known quantum algorithms and consider the prospects for finding new ones.…
In this paper we present the notion of globally integrable quantum system that we introduced in [BL22]: we motivate it using the spectral theory of pseudodifferential operators and then we give some results on linear and nonlinear…
Identifying the Hamiltonian of a quantum system from experimental data is considered. General limits on the identifiability of model parameters with limited experimental resources are investigated, and a specific Bayesian estimation…