Oleksandr Kyriienko

Chukwudubem Umeano, Annie E. Paine, Vincent E. Elfving et al.

Quantum machine learning (QML) shows promise for analyzing quantum data. A notable example is the use of quantum convolutional neural networks (QCNNs), implemented as specific types of quantum circuits, to recognize phases of matter. In this approach, ground states of many-body Hamiltonians are prepared to form a quantum dataset and classified in a supervised manner using only a few labeled examples. However, this type of dataset and model differs fundamentally from typical QML paradigms based on feature maps and parameterized circuits. In this study, we demonstrate how models utilizing quantum data can be interpreted through hidden feature maps, where physical features are implicitly embedded via ground-state feature maps. By analyzing selected examples previously explored with QCNNs, we show that high performance in quantum phase recognition comes from generating a highly effective basis set with sharp features at critical points. The learning process adapts the measurement to create sharp decision boundaries. Our analysis highlights improved generalization when working with quantum data, particularly in the limited-shots regime. Furthermore, translating these insights into the domain of quantum scientific machine learning, we demonstrate that ground-state feature maps can be applied to fluid dynamics problems, expressing shock wave solutions with good generalization and proven trainability.

Chukwudubem Umeano, Stefano Scali, Oleksandr Kyriienko

We consider the problem of distinguishing two vectors (visualized as images or barcodes) and learning if they are related to one another. For this, we develop a geometric quantum machine learning (GQML) approach with embedded symmetries that allows for the classification of similar and dissimilar pairs based on global correlations, and enables generalization from just a few samples. Unlike GQML algorithms developed to date, we propose to focus on symmetry-aware measurement adaptation that outperforms unitary parametrizations. We compare GQML for similarity testing against classical deep neural networks and convolutional neural networks with Siamese architectures. We show that quantum networks largely outperform their classical counterparts. We explain this difference in performance by analyzing correlated distributions used for composing our dataset. We relate the similarity testing with problems that showcase a proven maximal separation between the BQP complexity class and the polynomial hierarchy. While the ability to achieve advantage largely depends on how data are loaded, we discuss how similar problems can benefit from quantum machine learning.

Yuan Wang, Oleksandr Kyriienko

Photonics can offer a hardware-native route for machine learning (ML). However, efficient deployment of photonics-enhanced ML requires hybrid workflows that integrate optical processing with conventional CPU/GPU based neural network architectures. Here, we propose such a workflow that combines photonic positional embeddings (PEs) with advanced graph ML models. We introduce a photonics-based method that augments graph convolutional networks (GCNs) with PEs derived from light propagation on synthetic frequency lattices whose couplings match the input graph. We simulate propagation and readout to obtain internode intensity correlation matrices, which are used as PEs in GCNs to provide global structural information. Evaluated on Long Range Graph Benchmark molecular datasets, the method outperforms baseline GCNs with Laplacian based PEs, achieving $6.3\%$ lower mean absolute error for regression and $2.3\%$ higher average precision for classification tasks using a two-layer GCN as a baseline. When implemented in high repetition rate photonic hardware, correlation measurements can enable fast feature generation by bypassing digital simulation of PEs. Our results show that photonic PEs improve GCN performance and support optical acceleration of graph ML.

Chukwudubem Umeano, Vincent E. Elfving, Oleksandr Kyriienko

Geometric quantum machine learning (GQML) aims to embed problem symmetries for learning efficient solving protocols. However, the question remains if (G)QML can be routinely used for constructing protocols with an exponential separation from classical analogs. In this Letter we consider Simon's problem for learning properties of Boolean functions, and show that this can be related to an unsupervised circuit classification problem. Using the workflow of geometric QML, we learn from first principles Simon's algorithm, thus discovering an example of BQP$^A\neq$BPP protocol with respect to some dataset (oracle $A$). Our key findings include the development of an equivariant feature map for embedding Boolean functions, based on twirling with respect to identified bitflip and permutational symmetries, and measurement based on invariant observables with a sampling advantage. The proposed workflow points to the importance of data embeddings and classical post-processing, while keeping the variational circuit as a trivial identity operator. Next, developing the intuition for the function learning, we visualize instances as directed computational hypergraphs, and observe that the GQML protocol can access their global topological features for distinguishing bijective and surjective functions. Finally, we discuss the prospects for learning other BQP$^A$-type protocols, and conjecture that this depends on the ability of simplifying embeddings-based oracles $A$ applied as a linear combination of unitaries.

Niklas Heim, Atiyo Ghosh, Oleksandr Kyriienko et al.

Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) efficiently. Recently, variational approaches to solving non-linear PDEs also with near-term quantum devices were proposed. One of the most promising general approaches is based on recent developments in the field of scientific machine learning for solving PDEs. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks, including the discovery of differential equations from a dataset of measurements. We use differentiable quantum circuits (DQCs) to solve equations parameterized by a library of operators, and perform regression on a combination of data and equations. Our results show a promising path to Quantum Model Discovery (QMoD), on the interface between classical and quantum machine learning approaches. We demonstrate successful parameter inference and equation discovery using QMoD on different systems including a second-order, ordinary differential equation and a non-linear, partial differential equation.

Annie E. Paine, Vincent E. Elfving, Oleksandr Kyriienko

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.