19 research outputs found
Entropy Production of Doubly Stochastic Quantum Channels
We study the entropy increase of quantum systems evolving under primitive,
doubly stochastic Markovian noise and thus converging to the maximally mixed
state. This entropy increase can be quantified by a logarithmic-Sobolev
constant of the Liouvillian generating the noise. We prove a universal lower
bound on this constant that stays invariant under taking tensor-powers. Our
methods involve a new comparison method to relate logarithmic-Sobolev constants
of different Liouvillians and a technique to compute logarithmic-Sobolev
inequalities of Liouvillians with eigenvectors forming a projective
representation of a finite abelian group. Our bounds improve upon similar
results established before and as an application we prove an upper bound on
continuous-time quantum capacities. In the last part of this work we study
entropy production estimates of discrete-time doubly-stochastic quantum
channels by extending the framework of discrete-time logarithmic-Sobolev
inequalities to the quantum case.Comment: 24 page
Relative Entropy Convergence for Depolarizing Channels
We study the convergence of states under continuous-time depolarizing
channels with full rank fixed points in terms of the relative entropy. The
optimal exponent of an upper bound on the relative entropy in this case is
given by the log-Sobolev-1 constant. Our main result is the computation of this
constant. As an application we use the log-Sobolev-1 constant of the
depolarizing channels to improve the concavity inequality of the von-Neumann
entropy. This result is compared to similar bounds obtained recently by Kim et
al. and we show a version of Pinsker's inequality, which is optimal and tight
if we fix the second argument of the relative entropy. Finally, we consider the
log-Sobolev-1 constant of tensor-powers of the completely depolarizing channel
and use a quantum version of Shearer's inequality to prove a uniform lower
bound.Comment: 21 pages, 3 figure
Limitations of optimization algorithms on noisy quantum devices
Recent technological developments have focused the interest of the quantum
computing community on investigating how near-term devices could outperform
classical computers for practical applications. A central question that remains
open is whether their noise can be overcome or it fundamentally restricts any
potential quantum advantage. We present a transparent way of comparing
classical algorithms to quantum ones running on near-term quantum devices for a
large family of problems that include optimization problems and approximations
to the ground state energy of Hamiltonians. Our approach is based on the
combination of entropic inequalities that determine how fast the quantum
computation state converges to the fixed point of the noise model, together
with established classical methods of Gibbs state sampling. The approach is
extremely versatile and allows for its application to a large variety of
problems, noise models and quantum computing architectures. We use our result
to provide estimates for a variety of problems and architectures that have been
the focus of recent experiments, such as quantum annealers, variational quantum
eigensolvers, and quantum approximate optimization. The bounds we obtain
indicate that substantial quantum advantages are unlikely for classical
optimization unless the current noise rates are decreased by orders of
magnitude or the topology of the problem matches that of the device. This is
the case even if the number of qubits increases substantially. We reach similar
but less stringent conclusions for quantum Hamiltonian problems.Comment: 19 pages, 3 figure
Sandwiched RĂ©nyi Convergence for Quantum Evolutions
We study the speed of convergence of a primitive quantum time evolution
towards its fixed point in the distance of sandwiched R\'enyi divergences. For
each of these distance measures the convergence is typically exponentially fast
and the best exponent is given by a constant (similar to a logarithmic Sobolev
constant) depending only on the generator of the time evolution. We establish
relations between these constants and the logarithmic Sobolev constants as well
as the spectral gap. An important consequence of these relations is the
derivation of mixing time bounds for time evolutions directly from logarithmic
Sobolev inequalities without relying on notions like lp-regularity. We also
derive strong converse bounds for the classical capacity of a quantum time
evolution and apply these to obtain bounds on the classical capacity of some
examples, including stabilizer Hamiltonians under thermal noise.Comment: 35 pages, 4 figures. Version to be published in the Quantum Journa
Optimization at the boundary of the tensor network variety
Tensor network states form a variational ansatz class widely used, both
analytically and numerically, in the study of quantum many-body systems. It is
known that if the underlying graph contains a cycle, e.g. as in projected
entangled pair states (PEPS), then the set of tensor network states of given
bond dimension is not closed. Its closure is the tensor network variety. Recent
work has shown that states on the boundary of this variety can yield more
efficient representations for states of physical interest, but it remained
unclear how to systematically find and optimize over such representations. We
address this issue by defining a new ansatz class of states that includes
states at the boundary of the tensor network variety of given bond dimension.
We show how to optimize over this class in order to find ground states of local
Hamiltonians by only slightly modifying standard algorithms and code for tensor
networks. We apply this new method to a different of models and observe
favorable energies and runtimes when compared with standard tensor network
methods.Comment: 20 pages, 6 figure
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
Limitations of variational quantum algorithms: a quantum optimal transport approach
30 pages, 1 figureThe impressive progress in quantum hardware of the last years has raised the interest of the quantum computing community in harvesting the computational power of such devices. However, in the absence of error correction, these devices can only reliably implement very shallow circuits or comparatively deeper circuits at the expense of a nontrivial density of errors. In this work, we obtain extremely tight limitation bounds for standard NISQ proposals in both the noisy and noiseless regimes, with or without error-mitigation tools. The bounds limit the performance of both circuit model algorithms, such as QAOA, and also continuous-time algorithms, such as quantum annealing. In the noisy regime with local depolarizing noise , we prove that at depths L=\cO(p^{-1}) it is exponentially unlikely that the outcome of a noisy quantum circuit outperforms efficient classical algorithms for combinatorial optimization problems like Max-Cut. Although previous results already showed that classical algorithms outperform noisy quantum circuits at constant depth, these results only held for the expectation value of the output. Our results are based on newly developed quantum entropic and concentration inequalities, which constitute a homogeneous toolkit of theoretical methods from the quantum theory of optimal mass transport whose potential usefulness goes beyond the study of variational quantum algorithms
Information-theoretic generalization bounds for learning from quantum data
48+14 pages, 4 figuresLearning 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