9 research outputs found
Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series
We propose a quantum algorithm for sampling from a solution of stochastic
differential equations (SDEs). Using differentiable quantum circuits (DQCs)
with a feature map encoding of latent variables, we represent the quantile
function for an underlying probability distribution and extract samples as DQC
expectation values. Using quantile mechanics we propagate the system in time,
thereby allowing for time-series generation. We test the method by simulating
the Ornstein-Uhlenbeck process and sampling at times different from the initial
point, as required in financial analysis and dataset augmentation.
Additionally, we analyse continuous quantum generative adversarial networks
(qGANs), and show that they represent quantile functions with a modified
(reordered) shape that impedes their efficient time-propagation. Our results
shed light on the connection between quantum quantile mechanics (QQM) and qGANs
for SDE-based distributions, and point the importance of differential
constraints for model training, analogously with the recent success of physics
informed neural networks.Comment: v3, minor updat
What can we learn from quantum convolutional neural networks?
We can learn from analyzing quantum convolutional neural networks (QCNNs)
that: 1) working with quantum data can be perceived as embedding physical
system parameters through a hidden feature map; 2) their high performance for
quantum phase recognition can be attributed to generation of a very suitable
basis set during the ground state embedding, where quantum criticality of spin
models leads to basis functions with rapidly changing features; 3) pooling
layers of QCNNs are responsible for picking those basis functions that can
contribute to forming a high-performing decision boundary, and the learning
process corresponds to adapting the measurement such that few-qubit operators
are mapped to full-register observables; 4) generalization of QCNN models
strongly depends on the embedding type, and that rotation-based feature maps
with the Fourier basis require careful feature engineering; 5) accuracy and
generalization of QCNNs with readout based on a limited number of shots favor
the ground state embeddings and associated physics-informed models. We
demonstrate these points in simulation, where our results shed light on
classification for physical processes, relevant for applications in sensing.
Finally, we show that QCNNs with properly chosen ground state embeddings can be
used for fluid dynamics problems, expressing shock wave solutions with good
generalization and proven trainability.Comment: 13 pages, 7 figure
Quantum Chebyshev Transform: Mapping, Embedding, Learning and Sampling Distributions
We develop a paradigm for building quantum models in the orthonormal space of
Chebyshev polynomials. We show how to encode data into quantum states with
amplitudes being Chebyshev polynomials with degree growing exponentially in the
system size. Similar to the quantum Fourier transform which maps computational
basis space into the phase (Fourier) basis, we describe the quantum circuit for
the mapping between computational and Chebyshev spaces. We propose an embedding
circuit for generating the orthonormal Chebyshev basis of exponential capacity,
represented by a continuously-parameterized shallow isometry. This enables
automatic quantum model differentiation, and opens a route to solving
stochastic differential equations. We apply the developed paradigm to
generative modeling from physically- and financially-motivated distributions,
and use the quantum Chebyshev transform for efficient sampling of these
distributions in extended computational basis.Comment: 6 pages (+ references), 3 figure
Quantum Kernel Methods for Solving Differential Equations
We propose several approaches for solving differential equations (DEs) with
quantum kernel methods. We compose quantum models as weighted sums of kernel
functions, where variables are encoded using feature maps and model derivatives
are represented using automatic differentiation of quantum circuits. While
previously quantum kernel methods primarily targeted classification tasks, here
we consider their applicability to regression tasks, based on available data
and differential constraints. We use two strategies to approach these problems.
First, we devise a mixed model regression with a trial solution represented by
kernel-based functions, which is trained to minimize a loss for specific
differential constraints or datasets. Second, we use support vector regression
that accounts for the structure of differential equations. The developed
methods are capable of solving both linear and nonlinear systems. Contrary to
prevailing hybrid variational approaches for parametrized quantum circuits, we
perform training of the weights of the model classically. Under certain
conditions this corresponds to a convex optimization problem, which can be
solved with provable convergence to global optimum of the model. The proposed
approaches also favor hardware implementations, as optimization only uses
evaluated Gram matrices, but require quadratic number of function evaluations.
We highlight trade-offs when comparing our methods to those based on
variational quantum circuits such as the recently proposed differentiable
quantum circuits (DQC) approach. The proposed methods offer potential quantum
enhancement through the rich kernel representations using the power of quantum
feature maps, and start the quest towards provably trainable quantum DE
solvers.Comment: 13 pages, 6 figures, v