arXiv subjects

Christian Bertoni

Publications and source records attributed to Christian Bertoni.

5 recordsLinked to original sources

arXiv2024

Typical thermalization of low-entanglement states

Proving thermalization from the unitary evolution of a closed quantum system is one of the oldest questions that is still nowadays only partially resolved. Several efforts have led to various formulations of what is called the eigenstate thermalization hypothesis, which leads to thermalization under certain conditions on the initial states. These conditions, however, are sensitive to the precise formulation of the hypothesis. In this work, we focus on the important case of low entanglement initial states, which are operationally accessible in many natural physical settings, including experimental schemes for testing thermalization and for quantum simulation. We prove thermalization of these states under precise conditions that have operational significance. More specifically, motivated by arguments of unavoidable finite resolution, we define a random energy smoothing on local Hamiltonians that leads to local thermalization when the initial state has low entanglement. Finally we show that such a transformation affects neither the Gibbs state locally nor, under generic smoothness conditions on the spectrum, the short-time dynamics.

quant-ph↗

arXiv2022

Shallow shadows: Expectation estimation using low-depth random Clifford circuits

We provide practical and powerful schemes for learning many properties of an unknown n-qubit quantum state using a sparing number of copies of the state. Specifically, we present a depth-modulated randomized measurement scheme that interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. These can be seen within our scheme as the special cases of zero and infinite depth, respectively. We focus on the regime where depth scales logarithmically in n and provide evidence that this retains the desirable properties of both extremal schemes whilst, in contrast to the random Clifford scheme, also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We consider observables that can be written as a linear combination of poly(n) Paulis and observables that can be written as a low bond dimension matrix product operator. For the former class of observables both tasks are solved efficiently in n. For the latter class, we do not guarantee efficiency but present a method that works in practice; by variationally computing a heralded approximate inverses of a tensor network that can then be used for efficiently executing both these tasks.

quant-ph↗

arXiv2022

Emergent statistical mechanics from properties of disordered random matrix product states

The study of generic properties of quantum states has led to an abundance of insightful results. A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well approximated by matrix product states. In this work, we introduce a picture of generic states within the trivial phase of matter with respect to their non-equilibrium and entropic properties: We do so by rigorously exploring non-translation-invariant matrix product states drawn from a local i.i.d. Haar-measure. We arrive at these results by exploiting techniques for computing moments of random unitary matrices and by exploiting a mapping to partition functions of classical statistical models, a method that has lead to valuable insights on local random quantum circuits. Specifically, we prove that such disordered random matrix product states equilibrate exponentially well with overwhelming probability under the time evolution of Hamiltonians featuring a non-degenerate spectrum. Moreover, we prove two results about the entanglement Renyi entropy: The entropy with respect to sufficiently disconnected subsystems is generically extensive in the system-size, and for small connected systems the entropy is almost maximal for sufficiently large bond dimensions.

quant-ph↗

arXiv2021

Minimizing couplings in renormalization by preserving short-range mutual information

The connections between renormalization in statistical mechanics and information theory are intuitively evident, but a satisfactory theoretical treatment remains elusive. Recently, Koch-Janusz and Ringel proposed selecting a real-space renormalization map for classical lattice systems by minimizing the loss of long-range mutual information [Nat. Phys. 14, 578 (2018)]. The success of this technique has been related in part to the minimization of long-range couplings in the renormalized Hamiltonian [Lenggenhager et al., Phys. Rev. X 10, 011037 (2020)]. We show that to minimize these couplings the renormalization map should, somewhat counterintuitively, instead be chosen to minimize the loss of short-range mutual information between a block and its boundary. Moreover, the previous minimization is a relaxation of this approach, which indicates that the aims of preserving long-range physics and eliminating short-range couplings are related in a nontrivial way.

cond-mat.stat-mech↗

arXiv2020

Entropic time-energy uncertainty relations: An algebraic approach

We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observable $G$ and the displacement $r$ of the $G$-generated evolution $e^{-ir G}$. We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coles et al.. Our derivation uses as a subroutine a recently proposed novel algebraic method, which can in general be used to derive a wider class of entropic uncertainty principles.

quant-ph↗