We propose several algorithms for learning unitary operators from quantum
statistical queries (QSQs) with respect to their Choi-Jamiolkowski state.
Quantum statistical queries capture the capabilities of a learner with limited
quantum resources, which receives as input only noisy estimates of expected
values of measurements. Our methods hinge on a novel technique for estimating
the Fourier mass of a unitary on a subset of Pauli strings with a single
quantum statistical query, generalizing a previous result for uniform quantum
examples. Exploiting this insight, we show that the quantum Goldreich-Levin
algorithm can be implemented with quantum statistical queries, whereas the
prior version of the algorithm involves oracle access to the unitary and its
inverse. Moreover, we prove that $\mathcal{O}(\log n)$-juntas and quantum
Boolean functions with constant total influence are efficiently learnable in
our model, and constant-depth circuits are learnable sample-efficiently with
quantum statistical queries. On the other hand, all previous algorithms for
these tasks require direct access to the Choi-Jamiolkowski state or oracle
access to the unitary. In addition, our upper bounds imply that the actions of
those classes of unitaries on locally scrambled ensembles can be efficiently
learned. We also demonstrate that, despite these positive results, quantum
statistical queries lead to an exponentially larger sample complexity for
certain tasks, compared to separable measurements to the Choi-Jamiolkowski
state. In particular, we show an exponential lower bound for learning a class
of phase-oracle unitaries and a double exponential lower bound for testing the
unitarity of channels, adapting to our setting previous arguments for quantum
states. Finally, we propose a new definition of average-case surrogate models,
showing a potential application of our results to hybrid quantum machine
learning.

我们提出了几种算法，用于从量子统计查询（QSQs）中学习幺正算符，与其 Choi-Jamiolkowski 状态相关。

学习量子统计查询的幺正变换

Learning unitaries with quantum statistical queries

We propose a reinforcement learning (RL) scheme for feedback quantum control
within the quan-tum approximate optimization algorithm (QAOA). QAOA requires a
variational minimization for states constructed by applying a sequence of
unitary operators, depending on parameters living ina highly dimensional space.
We reformulate such a minimum search as a learning task, where a RL agent
chooses the control parameters for the unitaries, given partial information on
the system. We show that our RL scheme finds a policy converging to the optimal
adiabatic solution for QAOA found by Mbeng et al. arXiv:1906.08948 for the
translationally invariant quantum Ising chain. In presence of disorder, we show
that our RL scheme allows the training part to be performed on small samples,
and transferred successfully on larger systems.

我们提出了一个基于强化学习的算法，用于量子近似优化算法（QAOA）内的量子反馈控制，能够基于局部信息选择控制参数，并实现了训练数据的小样本转移学习。

强化学习辅助的量子优化

Reinforcement Learning assisted Quantum Optimization

A major challenge in the training of recurrent neural networks is the
so-called vanishing or exploding gradient problem. The use of a norm-preserving
transition operator can address this issue, but parametrization is challenging.
In this work we focus on unitary operators and describe a parametrization using
the Lie algebra $\mathfrak{u}(n)$ associated with the Lie group $U(n)$ of $n
\times n$ unitary matrices. The exponential map provides a correspondence
between these spaces, and allows us to define a unitary matrix using $n^2$ real
coefficients relative to a basis of the Lie algebra. The parametrization is
closed under additive updates of these coefficients, and thus provides a simple
space in which to do gradient descent. We demonstrate the effectiveness of this
parametrization on the problem of learning arbitrary unitary operators,
comparing to several baselines and outperforming a recently-proposed
lower-dimensional parametrization. We additionally use our parametrization to
generalize a recently-proposed unitary recurrent neural network to arbitrary
unitary matrices, using it to solve standard long-memory tasks.

本论文研究了使用可定义为 Lie 代数与 Lie 群相关联的单位矩阵组 U (n) 来描述参数化单元操作符，并通过 Lie 代数的基础定义了一个由 n^2 个实系数组成的单位矩阵空间，以解决循环神经网络训练中的梯度消失或爆炸问题。研究表明，这种参数化方法在学习任意单位操作符和解决长期记忆问题方面具有高效性。