Quantum computing and persistence in topological data analysis

Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA -- determining whether a given hole persists across different length scales -- is $\mathsf{BQP}_1$-hard and contained in $\mathsf{BQP}$. This result implies an exponential quantum speedup for this problem under standard complexity-theoretic assumptions. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.

On the relation between trainability and dequantization of variational quantum learning models

The quest for successful variational quantum machine learning (QML) relies on the design of suitable parametrized quantum circuits (PQCs), as analogues to neural networks in classical machine learning. Successful QML models must fulfill the properties of trainability and non-dequantization, among others. Recent works have highlighted an intricate interplay between trainability and dequantization of such models, which is still unresolved. In this work we contribute to this debate from the perspective of machine learning, proving a number of results identifying, among others when trainability and non-dequantization are not mutually exclusive. We begin by providing a number of new somewhat broader definitions of the relevant concepts, compared to what is found in other literature, which are operationally motivated, and consistent with prior art. With these precise definitions given and motivated, we then study the relation between trainability and dequantization of variational QML. Next, we also discuss the degrees of "variationalness" of QML models, where we distinguish between models like the hardware efficient ansatz and quantum kernel methods. Finally, we introduce recipes for building PQC-based QML models which are both trainable and nondequantizable, and corresponding to different degrees of variationalness. We do not address the practical utility for such models. Our work however does point toward a way forward for finding more general constructions, for which finding applications may become feasible.

Curriculum reinforcement learning for quantum architecture search under hardware errors

The key challenge in the noisy intermediate-scale quantum era is finding useful circuits compatible with current device limitations. Variational quantum algorithms (VQAs) offer a potential solution by fixing the circuit architecture and optimizing individual gate parameters in an external loop. However, parameter optimization can become intractable, and the overall performance of the algorithm depends heavily on the initially chosen circuit architecture. Several quantum architecture search (QAS) algorithms have been developed to design useful circuit architectures automatically. In the case of parameter optimization alone, noise effects have been observed to dramatically influence the performance of the optimizer and final outcomes, which is a key line of study. However, the effects of noise on the architecture search, which could be just as critical, are poorly understood. This work addresses this gap by introducing a curriculum-based reinforcement learning QAS (CRLQAS) algorithm designed to tackle challenges in realistic VQA deployment. The algorithm incorporates (i) a 3D architecture encoding and restrictions on environment dynamics to explore the search space of possible circuits efficiently, (ii) an episode halting scheme to steer the agent to find shorter circuits, and (iii) a novel variant of simultaneous perturbation stochastic approximation as an optimizer for faster convergence. To facilitate studies, we developed an optimized simulator for our algorithm, significantly improving computational efficiency in simulating noisy quantum circuits by employing the Pauli-transfer matrix formalism in the Pauli-Liouville basis. Numerical experiments focusing on quantum chemistry tasks demonstrate that CRLQAS outperforms existing QAS algorithms across several metrics in both noiseless and noisy environments.

Compilation of product-formula Hamiltonian simulation via reinforcement learning

Hamiltonian simulation is believed to be one of the first tasks where quantum computers can yield a quantum advantage. One of the most popular methods of Hamiltonian simulation is Trotterization, which makes use of the approximation $e^{i\sum_jA_j}\sim \prod_je^{iA_j}$ and higher-order corrections thereto. However, this leaves open the question of the order of operations (i.e. the order of the product over $j$, which is known to affect the quality of approximation). In some cases this order is fixed by the desire to minimise the error of approximation; when it is not the case, we propose that the order can be chosen to optimize compilation to a native quantum architecture. This presents a new compilation problem -- order-agnostic quantum circuit compilation -- which we prove is NP-hard in the worst case. In lieu of an easily-computable exact solution, we turn to methods of heuristic optimization of compilation. We focus on reinforcement learning due to the sequential nature of the compilation task, comparing it to simulated annealing and Monte Carlo tree search. While two of the methods outperform a naive heuristic, reinforcement learning clearly outperforms all others, with a gain of around 12% with respect to the second-best method and of around 50% compared to the naive heuristic in terms of the gate count. We further test the ability of RL to generalize across instances of the compilation problem, and find that a single learner is able to solve entire problem families. This demonstrates the ability of machine learning techniques to provide assistance in an order-agnostic quantum compilation task.

On the expressivity of embedding quantum kernels

One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Kernel methods rely on kernels, which are inner products of feature vectors living in large feature spaces. Quantum kernels are typically evaluated by explicitly constructing quantum feature states and then taking their inner product, here called embedding quantum kernels. Since classical kernels are usually evaluated without using the feature vectors explicitly, we wonder how expressive embedding quantum kernels are. In this work, we raise the fundamental question: can all quantum kernels be expressed as the inner product of quantum feature states? Our first result is positive: Invoking computational universality, we find that for any kernel function there always exists a corresponding quantum feature map and an embedding quantum kernel. The more operational reading of the question is concerned with efficient constructions, however. In a second part, we formalize the question of universality of efficient embedding quantum kernels. For shift-invariant kernels, we use the technique of random Fourier features to show that they are universal within the broad class of all kernels which allow a variant of efficient Fourier sampling. We then extend this result to a new class of so-called composition kernels, which we show also contains projected quantum kernels introduced in recent works. After proving the universality of embedding quantum kernels for both shift-invariant and composition kernels, we identify the directions towards new, more exotic, and unexplored quantum kernel families, for which it still remains open whether they correspond to efficient embedding quantum kernels.

Exponential separations between classical and quantum learners

Despite significant effort, the quantum machine learning community has only demonstrated quantum learning advantages for artificial cryptography-inspired datasets when dealing with classical data. In this paper we address the challenge of finding learning problems where quantum learning algorithms can achieve a provable exponential speedup over classical learning algorithms. We reflect on computational learning theory concepts related to this question and discuss how subtle differences in definitions can result in significantly different requirements and tasks for the learner to meet and solve. We examine existing learning problems with provable quantum speedups and find that they largely rely on the classical hardness of evaluating the function that generates the data, rather than identifying it. To address this, we present two new learning separations where the classical difficulty primarily lies in identifying the function generating the data. Furthermore, we explore computational hardness assumptions that can be leveraged to prove quantum speedups in scenarios where data is quantum-generated, which implies likely quantum advantages in a plethora of more natural settings (e.g., in condensed matter and high energy physics). We also discuss the limitations of the classical shadow paradigm in the context of learning separations, and how physically-motivated settings such as characterizing phases of matter and Hamiltonian learning fit in the computational learning framework.

Enhancing variational quantum state diagonalization using reinforcement learning techniques

The development of variational quantum algorithms is crucial for the application of NISQ computers. Such algorithms require short quantum circuits, which are more amenable to implementation on near-term hardware, and many such methods have been developed. One of particular interest is the so-called the variational diagonalization method, which constitutes an important algorithmic subroutine, and it can be used directly for working with data encoded in quantum states. In particular, it can be applied to discern the features of quantum states, such as entanglement properties of a system, or in quantum machine learning algorithms. In this work, we tackle the problem of designing a very shallow quantum circuit, required in the quantum state diagonalization task, by utilizing reinforcement learning. To achieve this, we utilize a novel encoding method that can be used to tackle the problem of circuit depth optimization using a reinforcement learning approach. We demonstrate that our approach provides a solid approximation to the diagonalization task while using a small number of gates. The circuits proposed by the reinforcement learning methods are shallower than the standard variational quantum state diagonalization algorithm, and thus can be used in situations where the depth of quantum circuits is limited by the hardware capabilities.

Shadows of quantum machine learning

Quantum machine learning is often highlighted as one of the most promising uses for a quantum computer to solve practical problems. However, a major obstacle to the widespread use of quantum machine learning models in practice is that these models, even once trained, still require access to a quantum computer in order to be evaluated on new data. To solve this issue, we suggest that following the training phase of a quantum model, a quantum computer could be used to generate what we call a classical shadow of this model, i.e., a classically computable approximation of the learned function. While recent works already explore this idea and suggest approaches to construct such shadow models, they also raise the possibility that a completely classical model could be trained instead, thus circumventing the need for a quantum computer in the first place. In this work, we take a novel approach to define shadow models based on the frameworks of quantum linear models and classical shadow tomography. This approach allows us to show that there exist shadow models which can solve certain learning tasks that are intractable for fully classical models, based on widely-believed cryptography assumptions. We also discuss the (un)likeliness that all quantum models could be shadowfiable, based on common assumptions in complexity theory.

Quantum policy gradient algorithms

Understanding the power and limitations of quantum access to data in machine learning tasks is primordial to assess the potential of quantum computing in artificial intelligence. Previous works have already shown that speed-ups in learning are possible when given quantum access to reinforcement learning environments. Yet, the applicability of quantum algorithms in this setting remains very limited, notably in environments with large state and action spaces. In this work, we design quantum algorithms to train state-of-the-art reinforcement learning policies by exploiting quantum interactions with an environment. However, these algorithms only offer full quadratic speed-ups in sample complexity over their classical analogs when the trained policies satisfy some regularity conditions. Interestingly, we find that reinforcement learning policies derived from parametrized quantum circuits are well-behaved with respect to these conditions, which showcases the benefit of a fully-quantum reinforcement learning framework.

On establishing learning separations between classical and quantum machine learning with classical data

Despite years of effort, the quantum machine learning community has only been able to show quantum learning advantages for certain contrived cryptography-inspired datasets in the case of classical data. In this note, we discuss the challenges of finding learning problems that quantum learning algorithms can learn much faster than any classical learning algorithm, and we study how to identify such learning problems. Specifically, we reflect on the main concepts in computational learning theory pertaining to this question, and we discuss how subtle changes in definitions can mean conceptually significantly different tasks, which can either lead to a separation or no separation at all. Moreover, we study existing learning problems with a provable quantum speedup to distill sets of more general and sufficient conditions (i.e., ``checklists'') for a learning problem to exhibit a separation between classical and quantum learners. These checklists are intended to streamline one's approach to proving quantum speedups for learning problems, or to elucidate bottlenecks. Finally, to illustrate its application, we analyze examples of potential separations (i.e., when the learning problem is build from computational separations, or when the data comes from a quantum experiment) through the lens of our approach.