arXiv subjects

David Sutter

Publications and source records attributed to David Sutter.

At least 37 records · Page 2Linked to original sources

arXiv2021

Quantum Legendre-Fenchel Transform

We present a quantum algorithm to compute the discrete Legendre-Fenchel transform. Given access to a convex function evaluated at $N$ points, the algorithm outputs a quantum-mechanical representation of its corresponding discrete Legendre-Fenchel transform evaluated at $K$ points in the transformed space. For a fixed regular discretization of the dual space the expected running time scales as $O(\sqrtκ\,\mathrm{polylog}(N,K))$, where $κ$ is the condition number of the function. If the discretization of the dual space is chosen adaptively with $K$ equal to $N$, the running time reduces to $O(\mathrm{polylog}(N))$. We explain how to extend the presented algorithm to the multivariate setting and prove lower bounds for the query complexity, showing that our quantum algorithm is optimal up to polylogarithmic factors. For multivariate functions with $κ=1$, the quantum algorithm computes a quantum-mechanical representation of the Legendre-Fenchel transform at $K$ points exponentially faster than any classical algorithm can compute it at a single point.

quant-ph↗

arXiv2021

Quantum speedups for convex dynamic programming

We present a quantum algorithm to solve dynamic programming problems with convex value functions. For linear discrete-time systems with a $d$-dimensional state space of size $N$, the proposed algorithm outputs a quantum-mechanical representation of the value function in time $O(T γ^{dT}\mathrm{polylog}(N,(T/\varepsilon)^{d}))$, where $\varepsilon$ is the accuracy of the solution, $T$ is the time horizon, and $γ$ is a problem-specific parameter depending on the condition numbers of the cost functions. This allows us to evaluate the value function at any fixed state in time $O(T γ^{dT}\sqrt{N}\,\mathrm{polylog}(N,(T/\varepsilon)^{d}))$, and the corresponding optimal action can be recovered by solving a convex program. The class of optimization problems to which our algorithm can be applied includes provably hard stochastic dynamic programs. Finally, we show that the algorithm obtains a quadratic speedup (up to polylogarithmic factors) compared to the classical Bellman approach on some dynamic programs with continuous state space that have $γ=1$.

quant-ph↗

arXiv2020

The power of quantum neural networks

Fault-tolerant quantum computers offer the promise of dramatically improving machine learning through speed-ups in computation or improved model scalability. In the near-term, however, the benefits of quantum machine learning are not so clear. Understanding expressibility and trainability of quantum models-and quantum neural networks in particular-requires further investigation. In this work, we use tools from information geometry to define a notion of expressibility for quantum and classical models. The effective dimension, which depends on the Fisher information, is used to prove a novel generalisation bound and establish a robust measure of expressibility. We show that quantum neural networks are able to achieve a significantly better effective dimension than comparable classical neural networks. To then assess the trainability of quantum models, we connect the Fisher information spectrum to barren plateaus, the problem of vanishing gradients. Importantly, certain quantum neural networks can show resilience to this phenomenon and train faster than classical models due to their favourable optimisation landscapes, captured by a more evenly spread Fisher information spectrum. Our work is the first to demonstrate that well-designed quantum neural networks offer an advantage over classical neural networks through a higher effective dimension and faster training ability, which we verify on real quantum hardware.

quant-ph↗

arXiv2020

An information-theoretic treatment of quantum dichotomies

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence.

quant-ph↗

arXiv2020

Bounds on Lyapunov exponents via entropy accumulation

Lyapunov exponents describe the asymptotic behavior of the singular values of large products of random matrices. A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an information theoretic tool called entropy accumulation theorem we derive an upper and a lower bound for the maximal and minimal Lyapunov exponent, respectively. The bounds assume independence of the random matrices, are analytical, and are tight in the commutative case as well as in other scenarios. They can be expressed in terms of an optimization problem that only involves single matrices rather than large products. The upper bound for the maximal Lyapunov exponent can be evaluated efficiently via the theory of convex optimization.

math-ph↗

arXiv2020

Exact and practical pattern matching for quantum circuit optimization

Quantum computations are typically compiled into a circuit of basic quantum gates. Just like for classical circuits, a quantum compiler should optimize the quantum circuit, e.g. by minimizing the number of required gates. Optimizing quantum circuits is not only relevant for improving the runtime of quantum algorithms in the long term, but is also particularly important for near-term quantum devices that can only implement a small number of quantum gates before noise renders the computation useless. An important building block for many quantum circuit optimization techniques is pattern matching, where given a large and a small quantum circuit, we are interested in finding all maximal matches of the small circuit, called pattern, in the large circuit, considering pairwise commutation of quantum gates. In this work, we present a classical algorithm for pattern matching that provably finds all maximal matches in time polynomial in the circuit size (for a fixed pattern size). Our algorithm works for both quantum and reversible classical circuits. We demonstrate numerically that our algorithm, implemented in the open-source library Qiskit, scales considerably better than suggested by the theoretical worst-case complexity and is practical to use for circuit sizes typical for near-term quantum devices. Using our pattern matching algorithm as the basis for known circuit optimization techniques such as template matching and peephole optimization, we demonstrate a significant (~30%) reduction in gate count for random quantum circuits, and are able to further improve practically relevant quantum circuits that were already optimized with state-of-the-art techniques.

quant-ph↗

arXiv2020

Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities

The Lyapunov exponent characterizes the asymptotic behavior of long matrix products. Recognizing scenarios where the Lyapunov exponent is strictly positive is a fundamental challenge that is relevant in many applications. In this work we establish a novel tool for this task by deriving a quantitative lower bound on the Lyapunov exponent in terms of a matrix sum which is efficiently computable in ergodic situations. Our approach combines two deep results from matrix analysis --- the $n$-matrix extension of the Golden-Thompson inequality and the Avalanche-Principle. We apply these bounds to the Lyapunov exponents of Schrödinger cocycles with certain ergodic potentials of polymer type and arbitrary correlation structure. We also derive related quantitative stability results for the Lyapunov exponent near aligned diagonal matrices and a bound for almost-commuting matrices.

math.DS↗

arXiv2019

A chain rule for the quantum relative entropy

The chain rule for the classical relative entropy ensures that the relative entropy between probability distributions on multipartite systems can be decomposed into a sum of relative entropies of suitably chosen conditional distributions on the individual systems. Here, we prove a similar chain rule inequality for the quantum relative entropy in terms of channel relative entropies. The new chain rule allows us to solve an open problem in the context of asymptotic quantum channel discrimination: surprisingly, adaptive protocols cannot improve the error rate for asymmetric channel discrimination compared to non-adaptive strategies. In addition, we give examples of quantum channels showing that the channel relative entropy is not additive under the tensor product.

quant-ph↗

arXiv2019

Generalized maximum entropy estimation

We consider the problem of estimating a probability distribution that maximizes the entropy while satisfying a finite number of moment constraints, possibly corrupted by noise. Based on duality of convex programming, we present a novel approximation scheme using a smoothed fast gradient method that is equipped with explicit bounds on the approximation error. We further demonstrate how the presented scheme can be used for approximating the chemical master equation through the zero-information moment closure method, and for an approximate dynamic programming approach in the context of constrained Markov decision processes with uncountable state and action spaces.

math.OC↗

arXiv2018

Necessary criterion for approximate recoverability

A tripartite state $ρ_{ABC}$ forms a Markov chain if there exists a recovery map $\mathcal{R}_{B \to BC}$ acting only on the $B$-part that perfectly reconstructs $ρ_{ABC}$ from $ρ_{AB}$. To achieve an approximate reconstruction, it suffices that the conditional mutual information $I(A:C|B)_ρ$ is small, as shown recently. Here we ask what conditions are necessary for approximate state reconstruction. This is answered by a lower bound on the relative entropy between $ρ_{ABC}$ and the recovered state $\mathcal{R}_{B\to BC}(ρ_{AB})$. The bound consists of the conditional mutual information and an entropic correction term that quantifies the disturbance of the $B$-part by the recovery map.

quant-ph↗

arXiv2018

Universal recovery maps and approximate sufficiency of quantum relative entropy

The data processing inequality states that the quantum relative entropy between two states $ρ$ and $σ$ can never increase by applying the same quantum channel $\mathcal{N}$ to both states. This inequality can be strengthened with a remainder term in the form of a distance between $ρ$ and the closest recovered state $(\mathcal{R} \circ \mathcal{N})(ρ)$, where $\mathcal{R}$ is a recovery map with the property that $σ= (\mathcal{R} \circ \mathcal{N})(σ)$. We show the existence of an explicit recovery map that is universal in the sense that it depends only on $σ$ and the quantum channel $\mathcal{N}$ to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.

quant-ph↗

arXiv2018

Approximate quantum Markov chains

This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen the data-processing inequality such that it reveals a statement about the reconstruction of lost information. The main difficulty in order to understand the behavior of quantum Markov chains arises from the fact that quantum mechanical operators do not commute in general. As a result we start by explaining two techniques of how to deal with non-commuting matrices: the spectral pinching method and complex interpolation theory. Once the reader is familiar with these techniques a novel inequality is presented that extends the celebrated Golden-Thompson inequality to arbitrarily many matrices. This inequality is the key ingredient in understanding approximate quantum Markov chains and it answers a question from matrix analysis that was open since 1973, i.e., if Lieb's triple matrix inequality can be extended to more than three matrices. Finally, we carefully discuss the properties of approximate quantum Markov chains and their implications.

quant-ph↗

arXiv2017

Approximate Degradable Quantum Channels

Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that a complementary channel can be obtained from the channel by applying a degrading channel. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite $\varepsilon\geq0$. That is, there exists a degrading channel which upon composition with the channel is $\varepsilon$-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such $\varepsilon$ can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.

quant-ph↗

arXiv2016

Pretty good measures in quantum information theory

Quantum generalizations of Renyi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Renyi divergence $\bar{D}_α$ and the minimal quantum Renyi divergence $\tilde{D}_α$. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely that $α\bar{D}_α(ρ\| σ) \leq \tilde{D}_α(ρ\| σ)$ for $α\in [0,1]$ and where $ρ$ and $σ$ are density operators. This bound suggests defining a "pretty good fidelity", whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction. We also find a new necessary and sufficient condition for optimality of the pretty good measurement and singlet fraction.

quant-ph↗

arXiv2016

Multivariate Trace Inequalities

We prove several trace inequalities that extend the Golden-Thompson and the Araki-Lieb-Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb's triple matrix inequality. As an example application of our four matrix extension of the Golden-Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.

math-ph↗

arXiv2016

Capacity of Random Channels with Large Alphabets

We consider discrete memoryless channels with input alphabet size $n$ and output alphabet size $m$, where $m=$ceil$(γn)$ for some constant $γ>0$. The channel transition matrix consists of entries that, before being normalised, are independent and identically distributed nonnegative random variables $V$ and such that $E[(V \log V)^2]<\infty$. We prove that in the limit as $n\to \infty$ the capacity of such a channel converges to $Ent(V) / E[V]$ almost surely and in $L^2$, where $Ent(V):= E[V\log V]-E[V] \log E[V]$ denotes the entropy of $V$. We further show that, under slightly different model assumptions, the capacity of these random channels converges to this asymptotic value exponentially in $n$. Finally, we present an application in the context of Bayesian optimal experiment design.

cs.IT↗

arXiv2016

Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map

The quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a "recovery map" that exactly reverses the effects of the quantum channel on both states. In this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination of rotated Petz recovery maps and perfectly reverses the quantum channel on the second state. As a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved in [Fawzi and Renner, Commun. Math. Phys., 2015]. Our proof only relies on elementary properties of pinching maps and the operator logarithm.

quant-ph↗

arXiv2015

Efficient Quantum Polar Codes Requiring No Preshared Entanglement

We construct an explicit quantum coding scheme which achieves a communication rate not less than the coherent information when used to transmit quantum information over a noisy quantum channel. For Pauli and erasure channels we also present efficient encoding and decoding algorithms for this communication scheme based on polar codes (essentially linear in the blocklength), but which do not require the sender and receiver to share any entanglement before the protocol begins. Due to the existence of degeneracies in the involved error-correcting codes it is indeed possible that the rate of the scheme exceeds the coherent information. We provide a simple criterion which indicates such performance. Finally we discuss how the scheme can be used for secret key distillation as well as private channel coding.

quant-ph↗