arXiv subjects

Cambyse Rouzé

Publications and source records attributed to Cambyse Rouzé.

At least 37 records · Page 2Linked to original sources

arXiv2026

Optimal quantum algorithm for Gibbs state preparation

It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced disispative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then employ our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.

quant-ph↗

arXiv2026

Complexity of mixed Schatten norms of quantum maps

We study the complexity of computing the mixed Schatten $\|Φ\|_{q\to p}$ norms of linear maps $Φ$ between matrix spaces. When $Φ$ is completely positive, we show that $\| Φ\|_{q \to p}$ can be computed efficiently when $q \geq p$. The regime $q \geq p$ is known as the non-hypercontractive regime and is also known to be easy for the mixed vector norms $\ell_{q} \to \ell_{p}$ [Boyd, 1974]. However, even for entanglement-breaking completely-positive trace-preserving maps $Φ$, we show that computing $\| Φ\|_{1 \to p}$ is $\mathsf{NP}$-complete when $p>1$. Moving beyond the completely-positive case and considering $Φ$ to be difference of entanglement breaking completely-positive trace-preserving maps, we prove that computing $\| Φ\|^+_{1 \to 1}$ is $\mathsf{NP}$-complete. In contrast, for the completely-bounded (cb) case, we describe a polynomial-time algorithm to compute $\|Φ\|_{cb,1\to p}$ and $\|Φ\|^+_{cb,1\to p}$ for any linear map $Φ$ and $p\geq1$.

quant-ph↗

arXiv2026

Polynomial-time thermalization and Gibbs sampling from system-bath couplings

Many physical phenomena, including thermalization in open quantum systems and quantum Gibbs sampling, are modeled by Lindbladians approximating a system weakly coupled to a bath. Understanding the convergence speed of these Lindbladians to their steady states is crucial for bounding algorithmic runtimes and thermalization timescales. We study two such families of processes: one characterizing a repeated-interaction Gibbs sampling algorithm, and another modeling open many-body quantum thermalization. We prove that both converge in polynomial time for several non-commuting systems, including high-temperature local lattices, weakly interacting fermions, and 1D spin chains. These results demonstrate that simple dissipative quantum algorithms can prepare complex Gibbs states and that Lindblad dynamics accurately capture thermal relaxation. Our proofs rely on a novel technical result that extrapolates spectral gap lower bounds from quasi-local Lindbladians to the non-local generators governing these dynamics.

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arXiv2025

Modified logarithmic Sobolev inequalities for CSS codes

We consider the class of Davies quantum semigroups modelling thermalization for translation-invariant Calderbank-Shor-Steane (CSS) codes in D dimensions. We prove that conditions of Dobrushin-Shlosman-type on the quantum Gibbs state imply a modified logarithmic Sobolev inequality with a constant that is uniform in the system's size. This is accomplished by generalizing parts of the classical results on thermalization by Stroock, Zegarlinski, Martinelli, and Olivieri to the CSS quantum setting. The results in particular imply the rapid thermalization at any positive temperature of the toric code in 2D and the star part of the toric code in 3D, implying a rapid loss of stored quantum information for these models.

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arXiv2025

Certifying and learning quantum Ising Hamiltonians

In this work, we study the problems of certifying and learning quantum Ising Hamiltonians. Our main contributions are as follows: Certification of Ising Hamiltonians. We show that certifying an Ising Hamiltonian in normalized Frobenius norm via access to its time-evolution operator requires only $\widetilde O(1/\varepsilon)$ time evolution. This matches the Heisenberg-scaling lower bound of $Ω(1/\varepsilon)$ up to logarithmic factors. To our knowledge, this is the first nearly-optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Lemma from Fourier analysis. Learning Ising Gibbs states. We design an algorithm for learning Ising Gibbs states in trace norm that is sample-efficient in all parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state) but suffered from exponential sample complexity in the inverse temperature. Certification of Ising Gibbs states. We give an algorithm for certifying Ising Gibbs states in trace norm that is both sample and time-efficient, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022). Finally, we extend our results on learning and certification of Gibbs states to general $k$-local Hamiltonians for any constant $k$.

quant-ph↗

arXiv2025

Heisenberg-limited Hamiltonian learning continuous variable systems via engineered dissipation

Discrete and continuous variables oftentimes require different treatments in many learning tasks. Identifying the Hamiltonian governing the evolution of a quantum system is a fundamental task in quantum learning theory. While previous works mostly focused on quantum spin systems, where quantum states can be seen as superpositions of discrete bit-strings, relatively little is known about Hamiltonian learning for continuous-variable quantum systems. In this work we focus on learning the Hamiltonian of a bosonic quantum system, a common type of continuous-variable quantum system. This learning task involves an infinite-dimensional Hilbert space and unbounded operators, making mathematically rigorous treatments challenging. We introduce an analytic framework to study the effects of strong dissipation in such systems, enabling a rigorous analysis of cat qubit stabilization via engineered dissipation. This framework also supports the development of Heisenberg-limited algorithms for learning general bosonic Hamiltonians with higher-order terms of the creation and annihilation operators. Notably, our scheme requires a total Hamiltonian evolution time that scales only logarithmically with the number of modes and inversely with the precision of the reconstructed coefficients. On a theoretical level, we derive a new quantitative adiabatic approximation estimate for general Lindbladian evolutions with unbounded generators. Finally, we discuss possible experimental implementations.

quant-ph↗

arXiv2025

Quantum Gibbs states are locally Markovian

The Markov property entails the conditional independence structure inherent in Gibbs distributions for general classical Hamiltonians, a feature that plays a crucial role in inference, mixing time analysis, and algorithm design. However, much less is known about quantum Gibbs states. In this work, we show that for any Hamiltonian with a bounded interaction degree, the quantum Gibbs state is locally Markov at arbitrary temperature, meaning there exists a quasi-local recovery map for every local region. Notably, this recovery map is obtained by applying a detailed-balanced Lindbladian with jumps acting on the region. Consequently, we prove that (i) the conditional mutual information (CMI) for a shielded small region decays exponentially with the shielding distance, and (ii) under the assumption of uniform clustering of correlations, Gibbs states of general non-commuting Hamiltonians on $D$-dimensional lattices can be prepared by a quantum circuit of depth $e^{O(\log^D(n/ε))}$, which can be further reduced assuming certain local gap condition. Our proofs introduce a regularization scheme for imaginary-time-evolved operators at arbitrarily low temperatures and reveal a connection between the Dirichlet form, a dynamic quantity, and the commutator in the KMS inner product, a static quantity. We believe these tools pave the way for tackling further challenges in quantum thermodynamics and mixing times, particularly in low-temperature regimes.

quant-ph↗

arXiv2025

Learning finitely correlated states: stability of the spectral reconstruction

Matrix product operators allow efficient descriptions (or realizations) of states on a 1D lattice. We consider the task of learning a realization of minimal dimension from copies of an unknown state, such that the resulting operator is close to the density matrix in trace norm. For finitely correlated translation-invariant states on an infinite chain, a realization of minimal dimension can be exactly reconstructed via linear algebra operations from the marginals of a size depending on the representation dimension. We establish a bound on the trace norm error for an algorithm that estimates a candidate realization from estimates of these marginals and outputs a matrix product operator, estimating the state of a chain of arbitrary length $t$. This bound allows us to establish an $O(t^2)$ upper bound on the sample complexity of the learning task, with an explicit dependence on the site dimension, realization dimension and spectral properties of a certain map constructed from the state. A refined error bound can be proven for $C^*$-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators on a finite chain reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of $\tilde{O}(t^3)$. The learning algorithm also works for states that are sufficiently close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states.

quant-ph↗

arXiv2024

Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics

We study the problem of sampling from and preparing quantum Gibbs states of local commuting Hamiltonians on hypercubic lattices of arbitrary dimension. We prove that any such Gibbs state which satisfies a clustering condition that we coin decay of matrix-valued quantum conditional mutual information (MCMI) can be quasi-optimally prepared on a quantum computer. We do this by controlling the mixing time of the corresponding Davies evolution in a normalized quantum Wasserstein distance of order one. To the best of our knowledge, this is the first time that such a non-commutative transport metric has been used in the study of quantum dynamics, and the first time quasi-rapid mixing is implied by solely an explicit clustering condition. Our result is based on a weak approximate tensorization and a weak modified logarithmic Sobolev inequality for such systems, as well as a new general weak transport cost inequality. If we furthermore assume a constraint on the local gap of the thermalizing dynamics, we obtain rapid mixing in trace distance for interactions beyond the range of two, thereby extending the state-of-the-art results that only cover the nearest neighbor case. We conclude by showing that systems that admit effective local Hamiltonians, like quantum CSS codes at high temperature, satisfy this MCMI decay and can thus be efficiently prepared and sampled from.

quant-ph↗

arXiv2024

Energy preserving evolutions over Bosonic systems

The exponential convergence to invariant subspaces of quantum Markov semigroups plays a crucial role in quantum information theory. One such example is in bosonic error correction schemes, where dissipation is used to drive states back to the code-space - an invariant subspace protected against certain types of errors. In this paper, we investigate perturbations of quantum dynamical semigroups that operate on continuous variable (CV) systems and admit an invariant subspace. First, we prove a generation theorem for quantum Markov semigroups on CV systems under the physical assumptions that (i) the generator is in GKSL form with corresponding jump operators defined as polynomials of annihilation and creation operators; and (ii) the (possibly unbounded) generator increases all moments in a controlled manner. Additionally, we show that the level sets of operators with bounded first moments are admissible subspaces of the evolution, providing the foundations for a perturbative analysis. Our results also extend to time-dependent semigroups and multi-mode systems. We apply our general framework to two settings of interest in continuous variable quantum information processing. First, we provide a new scheme for deriving continuity bounds on the energy-constrained capacities of Markovian perturbations of quantum dynamical semigroups. Second, we provide quantitative perturbation bounds for the steady state of the quantum Ornstein-Uhlenbeck semigroup and the invariant subspace of the photon dissipation used in bosonic error correction.

quant-ph↗

arXiv2024

Sample Complexity of Locally Differentially Private Quantum Hypothesis Testing

Quantum state discrimination is an important problem in many information processing tasks. In this work we are concerned with finding its best possible sample complexity when the states are preprocessed by a quantum channel that is required to be locally differentially private. To that end we provide achievability and converse bounds for different settings. This includes symmetric state discrimination in various regimes and the asymmetric case. On the way, we also prove new sample complexity bounds for the general unconstrained setting. An important tool in this endeavor are new entropy inequalities that we believe to be of independent interest.

quant-ph↗

arXiv2024

Information-theoretic generalization bounds for learning from quantum data

Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.

quant-ph↗

arXiv2024

Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians

Quantum systems typically reach thermal equilibrium rather quickly when coupled to a thermal environment. The usual way of bounding the speed of this process is by estimating the spectral gap of the dissipative generator. However the gap, by itself, does not always yield a reasonable estimate for the thermalization time in many-body systems: without further structure, a uniform lower bound on it only constrains the thermalization time to grow polynomially with system size. Here, instead, we show that for a large class of geometrically-2-local models of Davies generators with commuting Hamiltonians, the thermalization time is much shorter than one would naïvely estimate from the gap: at most logarithmic in the system size. This yields the so-called rapid mixing of dissipative dynamics. The result is particularly relevant for 1D systems, for which we prove rapid thermalization with a system size independent decay rate only from a positive gap in the generator. We also prove that systems in hypercubic lattices of any dimension, and exponential graphs, such as trees, have rapid mixing at high enough temperatures. We do this by introducing a novel notion of clustering which we call "strong local indistinguishability" based on a max-relative entropy, and then proving that it implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour commuting models. This has consequences for the rate of thermalization towards Gibbs states, and also for their relevant Wasserstein distances and transportation cost inequalities. Along the way, we show that several measures of decay of correlations on Gibbs states of commuting Hamiltonians are equivalent, a result of independent interest. At the technical level, we also show a direct relation between properties of Davies and Schmidt dynamics, that allows to transfer results of thermalization between both.

quant-ph↗

arXiv2024

Learning quantum many-body systems from a few copies

Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state from a number of copies that scales polylogarithmically with the system's size and polynomially on the locality of the target observables. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols. We achieve our results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter allows us to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well we approximate the expectation value of a fixed set of few-body observables. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity that are of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states.

quant-ph↗

arXiv2024

Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions

We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL Theorem, Friedgut's Junta Theorem and Talagrand's variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables.

math.FA↗

arXiv2023

Provably Efficient Learning of Phases of Matter via Dissipative Evolutions

The combination of quantum many-body and machine learning techniques has recently proved to be a fertile ground for new developments in quantum computing. Several works have shown that it is possible to classically efficiently predict the expectation values of local observables on all states within a phase of matter using a machine learning algorithm after learning from data obtained from other states in the same phase. However, existing results are restricted to phases of matter such as ground states of gapped Hamiltonians and Gibbs states that exhibit exponential decay of correlations. In this work, we drop this requirement and show how it is possible to learn local expectation values for all states in a phase, where we adopt the Lindbladian phase definition by Coser \& Pérez-García [Coser \& Pérez-García, Quantum 3, 174 (2019)], which defines states to be in the same phase if we can drive one to other rapidly with a local Lindbladian. This definition encompasses the better-known Hamiltonian definition of phase of matter for gapped ground state phases, and further applies to any family of states connected by short unitary circuits, as well as non-equilibrium phases of matter, and those stable under external dissipative interactions. Under this definition, we show that $N = O(\log(n/δ)2^{polylog(1/ε)})$ samples suffice to learn local expectation values within a phase for a system with $n$ qubits, to error $ε$ with failure probability $δ$. This sample complexity is comparable to previous results on learning gapped and thermal phases, and it encompasses previous results of this nature in a unified way. Furthermore, we also show that we can learn families of states which go beyond the Lindbladian definition of phase, and we derive bounds on the sample complexity which are dependent on the mixing time between states under a Lindbladian evolution.

quant-ph↗

arXiv2023

Limitations of local update recovery in stabilizer-GKP codes: a quantum optimal transport approach

Local update recovery seeks to maintain quantum information by applying local correction maps alternating with and compensating for the action of noise. Motivated by recent constructions based on quantum LDPC codes in the finite-dimensional setting, we establish an analytic upper bound on the fault-tolerance threshold for concatenated GKP-stabilizer codes with local update recovery. Our bound applies to noise channels that are tensor products of one-mode beamsplitters with arbitrary environment states, capturing, in particular, photon loss occurring independently in each mode. It shows that for loss rates above a threshold given explicitly as a function of the locality of the recovery maps, encoded information is lost at an exponential rate. This extends an early result by Razborov from discrete to continuous variable (CV) quantum systems. To prove our result, we study a metric on bosonic states akin to the Wasserstein distance between two CV density functions, which we call the bosonic Wasserstein distance. It can be thought of as a CV extension of a quantum Wasserstein distance of order 1 recently introduced by De Palma et al. in the context of qudit systems, in the sense that it captures the notion of locality in a CV setting. We establish several basic properties, including a relation to the trace distance and diameter bounds for states with finite average photon number. We then study its contraction properties under quantum channels, including tensorization, locality and strict contraction under beamsplitter-type noise channels. Due to the simplicity of its formulation, and the established wide applicability of its finite-dimensional counterpart, we believe that the bosonic Wasserstein distance will become a versatile tool in the study of CV quantum systems.

quant-ph↗

arXiv2023

Efficient learning of the structure and parameters of local Pauli noise channels

The unavoidable presence of noise is a crucial roadblock for the development of large-scale quantum computers and the ability to characterize quantum noise reliably and efficiently with high precision is essential to scale quantum technologies further. Although estimating an arbitrary quantum channel requires exponential resources, it is expected that physically relevant noise has some underlying local structure, for instance that errors across different qubits have a conditional independence structure. Previous works showed how it is possible to estimate Pauli noise channels with an efficient number of samples in a way that is robust to state preparation and measurement errors, albeit departing from a known conditional independence structure. We present a novel approach for learning Pauli noise channels over n qubits that addresses this shortcoming. Unlike previous works that focused on learning coefficients with a known conditional independence structure, our method learns both the coefficients and the underlying structure. We achieve our results by leveraging a groundbreaking result by Bresler for efficiently learning Gibbs measures and obtain an optimal sample complexity of O(log(n)) to learn the unknown structure of the noise acting on n qubits. This information can then be leveraged to obtain a description of the channel that is close in diamond distance from O(poly(n)) samples. Furthermore, our method is efficient both in the number of samples and postprocessing without giving up on other desirable features such as SPAM-robustness, and only requires the implementation of single qubit Cliffords. In light of this, our novel approach enables the large-scale characterization of Pauli noise in quantum devices under minimal experimental requirements and assumptions.

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