One of the most important properties of classical neural networks is how
surprisingly trainable they are, though their training algorithms typically
rely on optimizing complicated, nonconvex loss functions. Previous results have
shown that unlike the case in classical neural networks, variational quantum
models are often not trainable. The most studied phenomenon is the onset of
barren plateaus in the training landscape of these quantum models, typically
when the models are very deep. This focus on barren plateaus has made the
phenomenon almost synonymous with the trainability of quantum models. Here, we
show that barren plateaus are only a part of the story. We prove that a wide
class of variational quantum models -- which are shallow, and exhibit no barren
plateaus -- have only a superpolynomially small fraction of local minima within
any constant energy from the global minimum, rendering these models untrainable
if no good initial guess of the optimal parameters is known. We also study the
trainability of variational quantum algorithms from a statistical query
framework, and show that noisy optimization of a wide variety of quantum models
is impossible with a sub-exponential number of queries. Finally, we numerically
confirm our results on a variety of problem instances. Though we exclude a wide
variety of quantum algorithms here, we give reason for optimism for certain
classes of variational algorithms and discuss potential ways forward in showing
the practical utility of such algorithms.

通过研究局部极小值和 Barren 高原现象，证明在全局最小能量附近未知最优参数的情况下，一类浅而无 barren 高原的变分量子模型具有很少的局部极小值，以至于在没有好的初值参数的情况下无法进行训练。此外，通过统计查询框架研究变分量子算法的可训练性，并表明大多数量子模型的噪声优化都需要指数级的查询数，虽然有希望通过研究某些类别的变分算法来解决这个问题。

超越贫瘠高原：量子变分算法被陷阱所淹没

Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With  Traps

Physicists use quantum models to describe the behavior of physical systems.
Quantum models owe their success to their interpretability, to their relation
to probabilistic models (quantization of classical models) and to their high
predictive power. Beyond physics, these properties are valuable in general data
science. This motivates the use of quantum models to analyze general
nonphysical datasets. Here we provide both empirical and theoretical insights
into the application of quantum models in data science. In the theoretical part
of this paper, we firstly show that quantum models can be exponentially more
efficient than probabilistic models because there exist datasets that admit
low-dimensional quantum models and only exponentially high-dimensional
probabilistic models. Secondly, we explain in what sense quantum models realize
a useful relaxation of compressed probabilistic models. Thirdly, we show that
sparse datasets admit low-dimensional quantum models and finally, we introduce
a method to compute hierarchical orderings of properties of users (e.g.,
personality traits) and items (e.g., genres of movies). In the empirical part
of the paper, we evaluate quantum models in item recommendation and observe
that the predictive power of quantum-inspired recommender systems can compete
with state-of-the-art recommender systems like SVD++ and PureSVD. Furthermore,
we make use of the interpretability of quantum models by computing hierarchical
orderings of properties of users and items. This work establishes a connection
between data science (item recommendation), information theory (communication
complexity), mathematical programming (positive semidefinite factorizations)
and physics (quantum models).

本研究提供了量子模型在数据科学中的应用，论文通过理论和实证相结合的方式，展示了量子模型相较于概率模型的高效性、可压缩性以及稀疏数据在低维度量子模型中的表现。此外，本文还利用量子模型的解释性和阳性半定因式分解方法，进行了物品的建议和用户属性的计算等实验。

受量子理论结构启发的推荐系统

Recommender systems inspired by the structure of quantum theory

One of the main applications of future quantum computers will be the
simulation of quantum models. While the evolution of a quantum state under a
Hamiltonian is straightforward (if sometimes expensive), using quantum
computers to determine the ground state phase diagram of a quantum model and
the properties of its phases is more involved. Using the Hubbard model as a
prototypical example, we here show all the steps necessary to determine its
phase diagram and ground state properties on a quantum computer. In particular,
we discuss strategies for efficiently determining and preparing the ground
state of the Hubbard model starting from various mean-field states with broken
symmetry. We present an efficient procedure to prepare arbitrary Slater
determinants as initial states and present the complete set of quantum circuits
needed to evolve from these to the ground state of the Hubbard model. We show
that, using efficient nesting of the various terms each time step in the
evolution can be performed with just $\mathcal{O}(N)$ gates and
$\mathcal{O}(\log N)$ circuit depth. We give explicit circuits to measure
arbitrary local observables and static and dynamic correlation functions, both
in the time and frequency domain. We further present efficient non-destructive
approaches to measurement that avoid the need to re-prepare the ground state
after each measurement and that quadratically reduce the measurement error.

利用量子计算机演化调控 Hubbard 模型，采用有效嵌套策略实现指数级降低影响因素，设计高效测量和非破坏式的实验方法，实现对模型的精确刻画和参数量子模拟。