S11 - Quantum Machine Learning I

(pdf file attached Abstract.pdf  for linked sources :)
1 Problem description and relevance
Next to efficiently optimizing a circuit covering an exponentially growing Hilbert space, quantum machine
learning (QML) faces the difficult task of mapping classical data to the quantum computer such that unknown
patterns can be identified with a unitary circuit [13]. Presently, this task is being addressed with research
into encoding schemes [13] and methods based on data-reuploading [13, 4]. However, these methods are not
applicable to quantum data, unless one combines multiple samples, which corresponds to data-reuploading. The
core problem is that separation between two quantum states can not be increased using any quantum channel,
which is known as the quantum data processing inequality [19]. Thus, any encoding that does not sufficiently
separate two classes on the large Hilbert space has limited ability to classify them, especially when partially
tracing to a smaller Hilbert space. The only way to increase the performance is to include non-linear pre-
or post-processing steps, data-reuploading being the most prominent case. With post-selection we propose a
more explainable approach, as compared to other non-linear operations, which directly identifies advantageous
subspaces that offer better separation than the complete Hilbert space.

The use of post-selection in a machine learning context is motivated by the outperformance of classical
tensor networks (CTNs) compared to quantum tensor networks on some problems [10]. Given some non-
linearities in the encoding their optimal linear combination is usually not unitary. The core differences between
CTNs and quantum circuits can be tied to two physical phenomena: First is entanglement, which relates to
the bond dimension [3, 12]; Second is the amount of post-selection, see [11]. Commonly, CTNs reduce the
Hilbert space without partial traces, quantum channels only employ partial traces [9, 18]. In previous work the
authors proposed a hybrid approach between the two models [11], i.e. post-selection can be made trainable.
By controlling a rotations on post-selected ancillas one can implement any diagonal matrix up to a global
scaling factor on a quantum compute, a simple operation that is otherwise inaccessible on quantum computers,
but easily implemented with a CTN. The core idea of implementing classical operations with post-selection is
already known for other quantum algorithms, such as those solving partial differential equations [17, 15].

A big drawback for post-selection are the wasted shots on a quantum computer, which is why it is to be
used sparingly. In this work, we aim to investigate how limited post-selection distributes itself inside a QML
model. Understanding how this resource is distributed in models allows us to understand the structure of the
selected subspace, which can be used as a stepping stone for the design of methods that treat the discarded
subspaces. To this end, we use a hyperparameter from our previous work [11] that controls the amount of overall
permitted post-selection. It controls the normalization of the output after post-selecting and corresponds to
the number of shots kept after the post-selection. Thus, it allocates post-selection to the QML model during
training, improving QML both for classical and quantum data.

Physics-informed machine learning is increasingly used in applied scientific computing. Physics-informed neural networks (PINNs) [Raissi19] embed governing physical laws directly into the training objective or model structure, thus enabling the solution of partial differential equations (PDEs) without reliance on predefined meshes. By operating on scattered collocation points, PINNs offer flexibility in handling complex geometries, boundary conditions and inverse problems that are difficult to address with classical discretisation-based methods such as finite element or finite volume schemes.
Motivated by the expressive properties of parameterized quantum circuits (PQCs), several quantum extensions of PINNs, commonly referred to as quantum PINNs (QPINNs), have been proposed [Kyriienko21, Siegl25]. Although research on QPINNs has expanded, most evaluations have focused on simplified or reduced-scale PDE problems, largely due to the current limitations of quantum hardware and simulation capabilities.  Consequently, it remains challenging for applied researchers to assess the the practical significance, scalability and performance boundaries of QPINNs.
To address this gap, our work benchmarks QPINNs using a subset of PDEs from the PINNacle benchmark [Hao24], a widely used reference for evaluating classical PINNs. Rather than adapting the benchmark to quantum constraints, we evaluate QPINNs in the original benchmark setting. The objective is to characterize the current performance regime of QPINNs in applied contexts. We identify regimes in which QPINNs achieve performance comparable to classical models, as well as PDEs where performance degrades due to algorithmic, optimization or hardware limitations. Additionally, to advance the practical utility of QPINNs, we incorporate a range of enhancement strategies drawn from recent developments in both classical and quantum literature and demonstrate their effects on model performance.

State-of-the-art algorithms in quantum mechanics, computational science and engineering are widely known and established. In this regard, recent advancements in quantum computing promise an improvement by offering a speedup over classical computing methods [1]. To this end, the efficient implementation of quantum algorithms is often hindered by the lack of efficient primitives for operator and state preparation, limiting both the ability of near-term quantum hardware to simulate complex problems and the potential of fault-tolerant algorithms to achieve practical quantum advantage. 
Current efforts in quantum algorithm development follow two main directions: (1) low-level, near-hardware algorithms which mostly employ hybrid classical-quantum approaches [2,3,4], and (2) quantum-inspired techniques based on Tensor-Train (TT) decompositions [5,6]. The latter are implemented on classical hardware but can facilitate translating classical algorithms to quantum hardware by leveraging one dimensional tensor-network representation [7,8], which limit correlations structures of operators and do not take hardware characteristics into account.