arXiv subjects

Robert Salzmann

Publications and source records attributed to Robert Salzmann.

At least 19 recordsLinked to original sources

arXiv2026

Computing the free energy of quantum Coulomb gases and molecules via quantum Gibbs sampling

We develop a quantum algorithm for estimating the free energy as well as the total Gibbs state of interacting quantum Coulomb gases and molecular systems in dimensions $d \in \{2,3\}$ at finite temperature. These systems lie beyond the reach of existing methods due to their singular interactions and infinite-dimensional Hilbert space structure. First, we show that the free energy of the full many-body Hamiltonian can be approximated by that of the same Hamiltonian with a finite-rank low-energy truncation of the interaction, with an explicit error bound polynomial in the particle number. This reduces the problem to a controlled finite-rank perturbation problem. Second, we introduce a quantum Gibbs sampling scheme tailored to this truncated system, based on a class of quantum Markov semigroups. Our main analytical result establishes that the associated generator has a strictly positive spectral gap for every truncation, implying exponential convergence to the target Gibbs state. This provides, to our knowledge, the first rigorous mixing-time guarantee for Gibbs sampling in a Coulomb interacting continuous-variable quantum system. Finally, we give an explicit quantum circuit implementation of the dynamics and derive an end-to-end complexity bound for approximating the free energy and the Gibbs state itself. Our results provide a mathematically rigorous route to quantum algorithms for free energy estimation in interacting quantum systems, without relying on classical approximations such as the Born-Oppenheimer reduction.

quant-ph↗

arXiv2026

Simulating Thermal Properties of Bose-Hubbard Models on a Quantum Computer

While recent advances have established efficient quantum algorithms for preparing Gibbs states of finite-dimensional systems, comparable complexity results for bosonic and other infinite-dimensional models remain unexplored. We introduce the first general rigorous Gibbs sampling framework for bosonic many-body systems, showing that physically relevant bosonic models admit gapped dissipative generators, enabling efficient preparation of thermal states. Although our results hold for broad classes of models, we illustrate them using Bose-Hubbard Hamiltonians, both within and beyond the mean-field regime. In both cases, we show that the associated dissipative generators maintain a positive spectral gap, thereby implying exponential convergence to the thermal state. Our argument in the multi-mode case is based on a finite-rank reduction of the dissipative dynamics, which allows us to control the generator via compact perturbations and deduce the discreteness of the spectrum and the stability of the gap. We apply our results to provide efficient preparation of the corresponding Gibbs state on qubit hardware, and by that a quantum algorithm to compute thermal properties of the associated model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.

quant-ph↗

arXiv2026

Quantum Gibbs Sampling in Infinite Dimensions: Generation, Mixing Times and Circuit Implementation

We develop a rigorous and implementable framework for Gibbs sampling of infinite-dimensional quantum systems governed by unbounded Hamiltonians. Extending dissipative Gibbs samplers beyond finite dimensions raises fundamental obstacles, including ill-defined generators, the absence of spectral gaps on natural Banach spaces, and tensions between implementability and convergence guarantees. We overcome these issues by constructing KMS-symmetric quantum Markov semigroups on separable Hilbert spaces that are both well-posed and efficiently implementable on qubit hardware. Our generation theory is based on the abstract framework of Dirichlet forms, adapted here to the case of algebras of bounded operators over separable Hilbert spaces. Leveraging the spectral properties of our self-adjoint generators, we establish quantitative convergence results in trace distance, including regimes of fast thermalization. In contrast, we also identify Hamiltonians for which a naive choice of generators guaranteeing implementability generally comes at the cost of losing convergence of the associated evolutions, thereby establishing a strong trade-off between implementability and convergence. Our framework applies to a wide class of models, including Schr\"odinger operators, Gaussian systems, and Bose-Hubbard Hamiltonians, and provides a unified approach linking rigorous infinite-dimensional analysis with algorithmic Gibbs state preparation.

quant-ph↗

arXiv2024

Quantitative Quantum Zeno and Strong Damping Limits in Strong Topology

Frequent applications of a mixing quantum operation to a quantum system slow down its time evolution and eventually drive it into the invariant subspace of the named operation. We prove this phenomenon, the quantum Zeno effect, and its continuous variant, strong damping, in a unified way for infinite-dimensional open quantum systems, while merely demanding that the respective mixing convergence holds pointwise for all states. Both results are quantitative in the following sense: Given the speed of convergence for the mixing limits, we can derive bounds on the convergence speed for the corresponding quantum Zeno and strong damping limits. We apply our results to prove quantum Zeno and strong damping limits for the photon loss channel with an explicit bound on the convergence speed.

quant-ph↗

arXiv2024

Optimal Fidelity Estimation from Binary Measurements for Discrete and Continuous Variable Systems

Estimating the fidelity between a desired target quantum state and an actual prepared state is essential for assessing the success of experiments. For pure target states, we use functional representations that can be measured directly and determine the number of copies of the prepared state needed for fidelity estimation. In continuous variable (CV) systems, we utilise the Wigner function, which can be measured via displaced parity measurements. We provide upper and lower bounds on the sample complexity required for fidelity estimation, considering the worst-case scenario across all possible prepared states. For target states of particular interest, such as Fock and Gaussian states, we find that this sample complexity is characterised by the $L^1$-norm of the Wigner function, a measure of Wigner negativity widely studied in the literature, in particular in resource theories of quantum computation. For discrete variable systems consisting of $n$ qubits, we explore fidelity estimation protocols using Pauli string measurements. Similarly to the CV approach, the sample complexity is shown to be characterised by the $L^1$-norm of the characteristic function of the target state for both Haar random states and stabiliser states. Furthermore, in a general black box model, we prove that, for any target state, the optimal sample complexity for fidelity estimation is characterised by the smoothed $L^1$-norm of the target state. To the best of our knowledge, this is the first time the $L^1$-norm of the Wigner function provides a lower bound on the cost of some information processing task.

quant-ph↗

arXiv2024

Convergence Rates for the Trotter Splitting for Unbounded Operators

We study convergence rates of the Trotter splitting $e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$ in the strong operator topology. In the first part, we use complex interpolation theory to treat generators $L$ and $A$ of contraction semigroups on Banach spaces, with $L$ relatively $A$-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schr\"odinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for the Schr\"odinger operator with $V(x)=\pm |x|^{-a}$ potential. In each case, our conditions are fully explicit.

math-ph↗

arXiv2024

Robustness of Fixed Points of Quantum Channels and Application to Approximate Quantum Markov Chains

Given a quantum channel and a state which satisfy a fixed point equation approximately (say, up to an error $\varepsilon$), can one find a new channel and a state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? It is interesting to ask this question for different choices of constraints on the structures of the original channel and state, and requiring that these are also satisfied by the new channel and state. We affirmatively answer the above question, under fairly general assumptions on these structures, through a compactness argument. Additionally, for channels and states satisfying certain specific structures, we find explicit upper bounds on the distances between the pairs of channels (and states) in question. When these distances decay quickly (in a particular, desirable manner) as $\varepsilon\to 0$, we say that the original approximate fixed point equation is rapidly fixable. We establish rapid fixability, not only for general quantum channels, but also when the original and new channels are both required to be unitary, mixed unitary or unital. In contrast, for the case of bipartite quantum systems with channels acting trivially on one subsystem, we prove that approximate fixed point equations are not rapidly fixable. In this case, the distance to the closest channel (and state) which satisfy an exact fixed point equation can depend on the dimension of the quantum system in an undesirable way. We apply our results on approximate fixed point equations to the question of robustness of quantum Markov chains (QMC) and establish the following: For any tripartite quantum state, there exists a dimension-dependent upper bound on its distance to the set of QMCs, which decays to zero as the conditional mutual information of the state vanishes.

quant-ph↗

arXiv2024

An invitation to the sample complexity of quantum hypothesis testing

Quantum hypothesis testing (QHT) has been traditionally studied from the information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of samples of an unknown state. In this paper, we study the sample complexity of QHT, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on QHT, we characterize the sample complexity of binary QHT in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple QHT. In more detail, we prove that the sample complexity of symmetric binary QHT depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary QHT depends logarithmically on the inverse type II error probability and inversely on the quantum relative entropy, provided that the type II error probability is sufficiently small. We then provide lower and upper bounds on the sample complexity of multiple QHT, with it remaining an intriguing open question to improve these bounds. The final part of our paper outlines and reviews how sample complexity of QHT is relevant to a broad swathe of research areas and can enhance understanding of many fundamental concepts, including quantum algorithms for simulation and search, quantum learning and classification, and foundations of quantum mechanics. As such, we view our paper as an invitation to researchers coming from different communities to study and contribute to the problem of sample complexity of QHT, and we outline a number of open directions for future research.

quant-ph↗

arXiv2024

Parallelization of Adaptive Quantum Channel Discrimination in the Non-Asymptotic Regime

We investigate the performance of parallel and adaptive quantum channel discrimination strategies for a finite number of channel uses. It has recently been shown that, in the asymmetric setting with asymptotically vanishing type I error probability, adaptive strategies are asymptotically not more powerful than parallel ones. We extend this result to the non-asymptotic regime with finitely many channel uses, by explicitly constructing a parallel strategy for any given adaptive strategy, and bounding the difference in their performances, measured in terms of the decay rate of the type II error probability per channel use. We further show that all parallel strategies can be optimized over in time polynomial in the number of channel uses, and hence our result can also be used to obtain a poly-time-computable asymptotically tight upper bound on the performance of general adaptive strategies.

quant-ph↗

arXiv2023

Infinite Dimensional Asymmetric Quantum Channel Discrimination

We study asymmetric binary channel discrimination, for qantum channels acting on separable Hilbert spaces. We establish quantum Stein's lemma for channels for both adaptive and parallel strategies, and show that under finiteness of the geometric Rényi divergence between the two channels for some $α> 1$, adaptive strategies offer no asymptotic advantage over parallel ones. One major step in our argument is to demonstrate that the geometric Rényi divergence satisfies a chain rule and is additive for channels also in infinite dimensions. These results may be of independent interest. Furthermore, we not only show asymptotic equivalence of parallel and adaptive strategies, but explicitly construct a parallel strategy which approximates a given adaptive $n$-shot strategy, and give an explicit bound on the difference between the discrimination errors for these two strategies. This extends the finite dimensional result from [B. Bergh et al., arxiv:2206.08350]. Finally, this also allows us to conclude, that the chain rule for the Umegaki relative entropy in infinite dimensions, recently shown in [O. Fawzi, L. Gao, and M. Rahaman, arxiv:2212.14700v2] given finiteness of the max divergence between the two channels, also holds under the weaker condition of finiteness of the geometric Rényi divergence. We give explicit examples of channels which show that these two finiteness conditions are not equivalent.

quant-ph↗

arXiv2023

Composite Classical and Quantum Channel Discrimination

We study the problem of binary composite channel discrimination in the asymmetric setting, where the hypotheses are given by fairly arbitrary sets of channels, and samples do not have to be identically distributed. In the case of quantum channels we prove: (i) a characterization of the Stein exponent for parallel channel discrimination strategies and (ii) an upper bound on the Stein exponent for adaptive channel discrimination strategies. We further show that already for classical channels this upper bound can sometimes be achieved and be strictly larger than what is possible with parallel strategies. Hence, there can be an advantage of adaptive channel discrimination strategies with composite hypotheses for classical channels, unlike in the case of simple hypotheses. Moreover, we show that classically this advantage can only exist if the sets of channels corresponding to the hypotheses are non-convex. As a consequence of our more general treatment, which is not limited to the composite i.i.d. setting, we also obtain a generalization of previous composite state discrimination results.

quant-ph↗

arXiv2023

Learning Quantum Processes with Memory -- Quantum Recurrent Neural Networks

Recurrent neural networks play an important role in both research and industry. With the advent of quantum machine learning, the quantisation of recurrent neural networks has become recently relevant. We propose fully quantum recurrent neural networks, based on dissipative quantum neural networks, capable of learning general causal quantum automata. A quantum training algorithm is proposed and classical simulations for the case of product outputs with the fidelity as cost function are carried out. We thereby demonstrate the potential of these algorithms to learn complex quantum processes with memory in terms of the exemplary delay channel, the time evolution of quantum states governed by a time-dependent Hamiltonian, and high- and low-frequency noise mitigation. Numerical simulations indicate that our quantum recurrent neural networks exhibit a striking ability to generalise from small training sets.

quant-ph↗

arXiv2022

Total insecurity of communication via strong converse for quantum privacy amplification

Quantum privacy amplification is a central task in quantum cryptography. Given shared randomness, which is initially correlated with a quantum system held by an eavesdropper, the goal is to extract uniform randomness which is decoupled from the latter. The optimal rate for this task is known to satisfy the strong converse property and we provide a lower bound on the corresponding strong converse exponent. In the strong converse region, the distance of the final state of the protocol from the desired decoupled state converges exponentially fast to its maximal value, in the asymptotic limit. We show that this necessarily leads to totally insecure communication by establishing that the eavesdropper can infer any sent messages with certainty, when given very limited extra information. In fact, we prove that in the strong converse region, the eavesdropper has an exponential advantage in inferring the sent message correctly, compared to the achievability region. Additionally we establish the following technical result, which is central to our proofs, and is of independent interest: the smoothing parameter for the smoothed max-relative entropy satisfies the strong converse property.

quant-ph↗

arXiv2021

Symmetric distinguishability as a quantum resource

We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system $XA$, corresponding to an ensemble of two quantum states, with $X$ being classical and $A$ being quantum. We study the resource theory for two different classes of free operations: $(i)$ ${\rm{CPTP}}_A$, which consists of quantum channels acting only on $A$, and $(ii)$ conditional doubly stochastic (CDS) maps acting on $XA$. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.

quant-ph↗

arXiv2021

Interpolating between symmetric and asymmetric hypothesis testing

The task of binary quantum hypothesis testing is to determine the state of a quantum system via measurements on it, given the side information that it is in one of two possible states, say $ρ$ and $σ$. This task is generally studied in either the symmetric setting, in which the two possible errors incurred in the task (the so-called type I and type II errors) are treated on an equal footing, or the asymmetric setting in which one minimizes the type II error probability under the constraint that the corresponding type I error probability is below a given threshold. Here we define a one-parameter family of binary quantum hypothesis testing tasks, which we call $s$-hypothesis testing, and in which the relative significance of the two errors are weighted by a parameter $s$. In particular, $s$-hypothesis testing interpolates continuously between the regimes of symmetric and asymmetric hypothesis testing. Moreover, if arbitrarily many identical copies of the system are assumed to be available, then the minimal error probability of $s$-hypothesis testing is shown to decay exponentially in the number of copies, with a decay rate given by a quantum divergence which we denote as $ξ_s(ρ\|σ)$, and which satisfies a host of interesting properties. Moreover, this one-parameter family of divergences interpolates continuously between the corresponding decay rates for symmetric hypothesis testing (the quantum Chernoff divergence) for $s = 1$, and asymmetric hypothesis testing (the Umegaki relative entropy) for $s = 0$.

quant-ph↗

arXiv2021

The $α\to 1$ Limit of the Sharp Quantum Rényi Divergence

Fawzi and Fawzi recently defined the sharp Rényi divergence, $D_α^\#$, for $α\in (1, \infty)$, as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit $α \to 1$ of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin-Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences, and for which we prove various properties including the data-processing inequality.

quant-ph↗

arXiv2020

Quantum Zeno effect for open quantum systems

We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence. In addition, we establish, for the first time, the existence of a quantum Zeno limit for the case of unbounded generators. We also provide a variety of physically interesting examples of quantum operations to which our results apply.

quant-ph↗

arXiv2019

Derivation of 3D energy-critical nonlinear Schrödinger equation and Bogoliubov excitations for Bose gases

We derive the 3D quintic NLS as the mean field limit of a Bose gas with three-body interactions. The quintic NLS is energy-critical, leading to several new difficulties in comparison with the cubic NLS which emerges from Bose gases with pair-interactions. Our method is based on Bogoliubov's approximation, which also provides the information on the fluctuations around the condensate in terms of a norm approximation for the N-body wave function.

math-ph↗