arXiv subjects

Vedran Dunjko

Publications and source records attributed to Vedran Dunjko.

At least 37 records · Page 2Linked to original sources

arXiv2025

Quantum computing and artificial intelligence: status and perspectives

This white paper discusses and explores the various points of intersection between quantum computing and artificial intelligence (AI). It describes how quantum computing could support the development of innovative AI solutions. It also examines use cases of classical AI that can empower research and development in quantum technologies, with a focus on quantum computing and quantum sensing. The purpose of this white paper is to provide a long-term research agenda aimed at addressing foundational questions about how AI and quantum computing interact and benefit one another. It concludes with a set of recommendations and challenges, including how to orchestrate the proposed theoretical work, align quantum AI developments with quantum hardware roadmaps, estimate both classical and quantum resources - especially with the goal of mitigating and optimizing energy consumption - advance this emerging hybrid software engineering discipline, and enhance European industrial competitiveness while considering societal implications.

quant-ph↗

arXiv2025

Quantum Advantage in Learning Quantum Dynamics via Fourier coefficient extraction

One of the key challenges in quantum machine learning is finding relevant machine learning tasks with a provable quantum advantage. A natural candidate for this is learning unknown Hamiltonian dynamics. Here, we tackle the supervised learning version of this problem, where we are given random examples of the inputs to the dynamics as classical data, paired with the expectation values of some observable after the time evolution, as corresponding output labels. The task is to replicate the corresponding input-output function. We prove that this task can yield provable exponential classical-quantum learning advantages under common complexity assumptions in natural settings. To design our quantum learning algorithms, we introduce a new method, which we term \textit{\subroutine}~algorithm for parametrized circuit functions, and which may be of independent interest. Furthermore, we discuss the limitations of generalizing our method to arbitrary quantum dynamics while maintaining provable guarantees. We explain that significant generalizations are impossible under certain complexity-theoretic assumptions, but nonetheless, we provide a heuristic kernel method, where we trade-off provable correctness for broader applicability.

quant-ph↗

arXiv2025

Modern applications of machine learning in quantum sciences

In this book, we provide a comprehensive introduction to the most recent advances in the application of machine learning methods in quantum sciences. We cover the use of deep learning and kernel methods in supervised, unsupervised, and reinforcement learning algorithms for phase classification, representation of many-body quantum states, quantum feedback control, and quantum circuits optimization. Moreover, we introduce and discuss more specialized topics such as differentiable programming, generative models, statistical approach to machine learning, and quantum machine learning.

quant-ph↗

arXiv2025

Note on the Universality of Parameterized IQP Circuits with Hidden Units for Generating Probability Distributions

In a series of recent works, an interesting quantum generative model based on parameterized instantaneous polynomial quantum (IQP) circuits has emerged as they can be trained efficiently classically using any loss function that depends only on the expectation values of observables of the model. The model is proven not to be universal for generating arbitrary distributions, but it is suspected that marginals can be - much like Boltzmann machines achieve universality by utilizing hidden (traced-out in quantum jargon) layers. In this short note, we provide two simple proofs of this fact. The first is near-trivial and asymptotic, and the second shows universality can be achieved with a reasonable number of additional qubits.

quant-ph↗

arXiv2025

Detecting underdetermination in parameterized quantum circuits

A central question in machine learning is how reliable the predictions of a trained model are. Reliability includes the identification of instances for which a model is likely not to be trusted based on an analysis of the learning system itself. Such unreliability for an input may arise from the model family providing a variety of hypotheses consistent with the training data, which can vastly disagree in their predictions on that particular input point. This is called the underdetermination problem, and it is important to develop methods to detect it. With the emergence of quantum machine learning (QML) as a prospective alternative to classical methods for certain learning problems, the question arises to what extent they are subject to underdetermination and whether similar techniques as those developed for classical models can be employed for its detection. In this work, we first provide an overview of concepts from Safe AI and reliability, which in particular received little attention in QML. We then explore the use of a method based on local second-order information for the detection of underdetermination in parameterized quantum circuits through numerical experiments. We further demonstrate that the approach is robust to certain levels of shot noise. Our work contributes to the body of literature on Safe Quantum AI, which is an emerging field of growing importance.

quant-ph↗

arXiv2025

Approximation and Generalization Capacities of Parametrized Quantum Circuits for Functions in Sobolev Spaces

Parametrized quantum circuits (PQC) are quantum circuits which consist of both fixed and parametrized gates. In recent approaches to quantum machine learning (QML), PQCs are essentially ubiquitous and play the role analogous to classical neural networks. They are used to learn various types of data, with an underlying expectation that if the PQC is made sufficiently deep, and the data plentiful, the generalization error will vanish, and the model will capture the essential features of the distribution. While there exist results proving the approximability of square-integrable functions by PQCs under the $L^2$ distance, the approximation for other function spaces and under other distances has been less explored. In this work we show that PQCs can approximate the space of continuous functions, $p$-integrable functions and the $H^k$ Sobolev spaces under specific distances. Moreover, we develop generalization bounds that connect different function spaces and distances. These results provide a theoretical basis for different applications of PQCs, for example for solving differential equations. Furthermore, they provide us with new insight on the role of the data normalization in PQCs and of loss functions which better suit the specific needs of the users.

quant-ph↗

arXiv2025

Multiple-basis representation of quantum states

Classical simulation of quantum physics is a central approach to investigating physical phenomena. Quantum computers enhance computational capabilities beyond those of classical resources, but it remains unclear to what extent existing limited quantum computers can contribute to this enhancement. In this work, we explore a new hybrid, efficient quantum-classical representation of quantum states, the multiple-basis representation. This representation consists of a linear combination of states that are sparse in some given and different bases, specified by quantum circuits. Such representation is particularly appealing when considering depth-limited quantum circuits within reach of current hardware. We analyze the expressivity of multiple-basis representation states depending on the classical simulability of their quantum circuits. In particular, we show that multiple-basis representation states include, but are not restricted to, both matrix-product states and stabilizer states. Furthermore, we find cases in which this representation can be used, namely approximation of ground states, simulation of deeper computations by specifying bases with shallow circuits, and a tomographical protocol to describe states as multiple-basis representations. We envision this work to open the path of simultaneous use of several hardware-friendly bases, a natural description of hybrid computational methods accessible for near-term hardware.

quant-ph↗

arXiv2024

Exponential quantum advantages in learning quantum observables from classical data

Quantum computers are believed to bring computational advantages in simulating quantum many body systems. However, recent works have shown that classical machine learning algorithms are able to predict numerous properties of quantum systems with classical data. Despite various examples of learning tasks with provable quantum advantages being proposed, they all involve cryptographic functions and do not represent any physical scenarios encountered in laboratory settings. In this paper we prove quantum advantages for the physically relevant task of learning quantum observables from classical (measured out) data. We consider two types of observables: first we prove a learning advantage for linear combinations of Pauli strings, then we extend the result for a broader case of unitarily parametrized observables. For each type of observable we delineate the boundaries that separate physically relevant tasks which classical computers can solve using data from quantum measurements, from those where a quantum computer is still necessary for data analysis. Differently from previous works, we base our classical hardness results on the weaker assumption that $\mathsf{BQP}$ hard processes cannot be simulated by polynomial-size classical circuits and provide a non-trivial quantum learning algorithm. Our results shed light on the utility of quantum computers for machine learning problems in the domain of quantum many body physics, thereby suggesting new directions where quantum learning improvements may emerge.

quant-ph↗

arXiv2024

Testing the presence of balanced and bipartite components in a sparse graph is QMA1-hard

Determining whether an abstract simplicial complex, a discrete object often approximating a manifold, contains multi-dimensional holes is a task deeply connected to quantum mechanics and proven to be QMA1-hard by Crichigno and Kohler. This task can be expressed in linear algebraic terms, equivalent to testing the non-triviality of the kernel of an operator known as the Combinatorial Laplacian. In this work, we explore the similarities between abstract simplicial complexes and signed or unsigned graphs, using them to map the spectral properties of the Combinatorial Laplacian to those of signed and unsigned graph Laplacians. We prove that our transformations preserve efficient sparse access to these Laplacian operators. Consequently, we show that key spectral properties, such as testing the presence of balanced components in signed graphs and the bipartite components in unsigned graphs, are QMA1-hard. These properties play a paramount role in network science. The hardness of the bipartite test is relevant in quantum Hamiltonian complexity, as another example of testing properties related to the eigenspace of a stoquastic Hamiltonians are quantumly hard in the sparse input model for the graph.

quant-ph↗

arXiv2024

Challenges and Opportunities in Quantum Optimization

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of different approaches for major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions. This work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory - highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains - finance and sustainability - as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization.

quant-ph↗

arXiv2024

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.

quant-ph↗

arXiv2024

Improved separation between quantum and classical computers for sampling and functional tasks

This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with postselection are as powerful as classical computers with postselection ($\mathsf{PostBQP=PostBPP}$), (ii) any one of several quantum sampling experiments ($\mathsf{BosonSampling}$, $\mathsf{IQP}$, $\mathsf{DQC1}$) can be approximately performed by a classical computer (contingent on existing assumptions). This last result implies that if any of these experiment's hardness conjectures hold, then quantum computers can implement functions classical computers cannot ($\mathsf{FBQP\neq FBPP}$) unless the polynomial hierarchy collapses to its 2nd level. These results are an improvement over previous work which either achieved a collapse to the third level or were concerned with exact sampling, a physically impractical case. The workhorse of these results is a new technical complexity-theoretic result which we believe could have value beyond quantum computation. In particular, we prove that if there exists an equivalence between problems solvable with an exact counting oracle and problems solvable with an approximate counting oracle, then the polynomial hierarchy collapses to its second level, indeed to $\mathsf{ZPP^{NP}}$.

quant-ph↗

arXiv2024

Trainability maximization using estimation of distribution algorithms assisted by surrogate modelling for quantum architecture search

Quantum architecture search (QAS) involves optimizing both the quantum parametric circuit configuration but also its parameters for a variational quantum algorithm. Thus, the problem is known to be multi-level as the performance of a given architecture is unknown until its parameters are tuned using classical routines. Moreover, the task becomes even more complicated since well-known trainability issues, e.g., barren plateaus (BPs), can occur. In this paper, we aim to achieve two improvements in QAS: (1) to reduce the number of measurements by an online surrogate model of the evaluation process that aggressively discards architectures of poor performance; (2) to avoid training the circuits when BPs are present. To detect the presence of the BPs, we employed a recently developed metric, information content, which only requires measuring the energy values of a small set of parameters to estimate the magnitude of cost function's gradient. The main idea of this proposal is to leverage a recently developed metric which can be used to detect the onset of vanishing gradients to ensure the overall search avoids such unfavorable regions. We experimentally validate our proposal for the variational quantum eigensolver and showcase that our algorithm is able to find solutions that have been previously proposed in the literature for the Hamiltonians; but also to outperform the state of the art when initializing the method from the set of architectures proposed in the literature. The results suggest that the proposed methodology could be used in environments where it is desired to improve the trainability of known architectures while maintaining good performance.

quant-ph↗

arXiv2024

Shadows of quantum machine learning

Quantum machine learning is often highlighted as one of the most promising practical applications for which quantum computers could provide a computational advantage. 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 introduce a new class of quantum models where quantum resources are only required during training, while the deployment of the trained model is classical. Specifically, the training phase of our models ends with the generation of a 'shadow model' from which the classical deployment becomes possible. We prove that: i) this class of models is universal for classically-deployed quantum machine learning; ii) it does have restricted learning capacities compared to 'fully quantum' models, but nonetheless iii) it achieves a provable learning advantage over fully classical learners, contingent on widely-believed assumptions in complexity theory. These results provide compelling evidence that quantum machine learning can confer learning advantages across a substantially broader range of scenarios, where quantum computers are exclusively employed during the training phase. By enabling classical deployment, our approach facilitates the implementation of quantum machine learning models in various practical contexts.

quant-ph↗

arXiv2024

Parameterized quantum circuits as universal generative models for continuous multivariate distributions

Parameterized quantum circuits have been extensively used as the basis for machine learning models in regression, classification, and generative tasks. For supervised learning, their expressivity has been thoroughly investigated and several universality properties have been proven. However, in the case of quantum generative modelling, much less is known, especially when the task is to model distributions over continuous variables. In this work, we elucidate expectation value sampling-based models. Such models output the expectation values of a set of fixed observables from a quantum circuit into which classical random data has been uploaded. We prove the universality of such variational quantum algorithms for the generation of multivariate distributions. We explore various architectures which allow universality and prove tight bounds connecting the minimal required qubit number, and the minimal required number of measurements needed. Our results may help guide the design of future quantum circuits in generative modelling tasks.

quant-ph↗

arXiv2024

On Bounded Advice Classes

Advice classes in computational complexity have frequently been used to model real-world scenarios encountered in cryptography, quantum computing and machine learning, where some computational task may be broken down into a preprocessing and deployment phase, each associated with a different complexity. However, in these scenarios, the advice given by the preprocessing phase must still be generated by some (albeit more powerful) bounded machine, which is not the case in conventional advice classes. To better model these cases we develop `bounded advice classes', where a more powerful Turing machine generates advice for another, less powerful, Turing machine. We then focus on the question of when various classes generate useful advice, to answer this we connect bounded advice to unary languages. This connection allows us to state various conditional and unconditional results on the utility of advice generated by $\mathsf{EXP}$, $\mathsf{NP}$, $\mathsf{BQP}$, $\mathsf{PSPACE}$, and more. We study the relations between bounded advice classes, quantum bounded advice classes, and randomised bounded advice. We also examine how each of these concepts interact with recently introduced classes, like $\mathsf{BPP/samp}$. Our results also improve the state of the art in existing research on the complexity of advice functions.

cs.CC↗

arXiv2024

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.

quant-ph↗

arXiv2024

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.

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