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Characterizing a quantum system by learning its state or evolution is a fundamental problem in quantum physics and learning theory with a myriad of applications. Recently, as a new approach to this problem, the task of agnostic state tomography was defined, in which one aims to approximate an arbitrary quantum state by a simpler one in a given class. Generalizing this notion to quantum processes, we initiate the study of agnostic process tomography: given query access to an unknown quantum channel $\Phi$ and a known concept class $\mathcal{C}$ of channels, output a quantum channel that approximates $\Phi$ as well as any channel in the concept class $\mathcal{C}$, up to some error. In this work, we propose several natural applications for this new task in quantum machine learning, quantum metrology, classical simulation, and error mitigation. In addition, we give efficient agnostic process tomography algorithms for a wide variety of concept classes, including Pauli strings, Pauli channels, quantum junta channels, low-degree channels, and a class of channels produced by $\mathsf{QAC}^0$ circuits. The main technical tool we use is Pauli spectrum analysis of operators and superoperators. We also prove that, using ancilla qubits, any agnostic state tomography algorithm can be extended to one solving agnostic process tomography for a compatible concept class of unitaries, immediately giving us efficient agnostic learning algorithms for Clifford circuits, Clifford circuits with few T gates, and circuits consisting of a tensor product of single-qubit gates. Together, our results provide insight into the conditions and new algorithms necessary to extend the learnability of a concept class from the standard tomographic setting to the agnostic one., Comment: 11+52 pages, 2 figures, 1 table

This review paper examines state-of-the-art algorithms and techniques in quantum machine learning with potential applications in finance. We discuss QML techniques in supervised learning tasks, such as Quantum Variational Classifiers, Quantum Kernel Estimation, and Quantum Neural Networks (QNNs), along with quantum generative AI techniques like Quantum Transformers and Quantum Graph Neural Networks (QGNNs). The financial applications considered include risk management, credit scoring, fraud detection, and stock price prediction. We also provide an overview of the challenges, potential, and limitations of QML, both in these specific areas and more broadly across the field. We hope that this can serve as a quick guide for data scientists, professionals in the financial sector, and enthusiasts in this area to understand why quantum computing and QML in particular could be interesting to explore in their field of expertise., Comment: 19 pages

A seminal task in quantum information theory is to realize a device able to produce copies of a generic input state with the highest possible output fidelity, thus realizing an \textit{optimal} quantum cloning machine. Recently, the concept of variational quantum cloning was introduced: a quantum machine learning algorithm through which, by exploiting a classical feedback loop informed by the output of a quantum processing unit, the system can self-learn the programming required for an optimal quantum cloning strategy. In this work, we experimentally implement a $1 \rightarrow 2$ variational cloning machine of dual-rail encoded photonic qubits, both for phase-covariant and state-dependent cloning. We exploit a fully programmable 6-mode universal integrated device and classical feedback to reach near-optimal cloning performances. Our results demonstrate the potential of programmable integrated photonic platforms for variational self-learning of quantum algorithms., Comment: 9 pages, 4 figures

Quantum contextuality is a key nonclassical feature essential for understanding advantages in quantum computation and communication. We introduce a new framework to study contextuality based solely on information processing statistics. This simple and intuitive perspective leads to a powerful criterion denoted as rank separation for identifying contextuality in various quantum scenarios. We showcase the power of this technique through several applications, including a new derivation of Hardy's quantum excess-baggage theorem, and a simplified proof of contextuality for minimum error quantum state discrimination. Finally, we show as a prominent example that quantum contextuality provides the resource in optimal phase-covariant and universal cloning schemes, hence establishing it as a fundamental source of nonclassicality in all known optimal quantum cloning scenarios., Comment: 7 pages, comments are encouraged

This work studies the learnability of unknown quantum circuits in the near term. We prove the natural robustness of quantum statistical queries for learning quantum processes and provide an efficient way to benchmark various classes of noise from statistics, which gives us a powerful framework for developing noise-tolerant algorithms. We adapt a learning algorithm for constant-depth quantum circuits to the quantum statistical query setting with a small overhead in the query complexity. We prove average-case lower bounds for learning random quantum circuits of logarithmic and higher depths within diamond distance with statistical queries. Additionally, we show the hardness of the quantum threshold search problem from quantum statistical queries and discuss its implications for the learnability of shallow quantum circuits. Finally, we prove that pseudorandom unitaries (PRUs) cannot be constructed using circuits of constant depth by constructing an efficient distinguisher and proving a new variation of the quantum no-free lunch theorem., Comment: 23+7 pages, 2 figures

We present a framework for the unification and standardization of quantum network protocols, making their realization easier and expanding their use cases to a broader range of communities interested in quantum technologies. Our framework is available as an open-source repository, the Quantum Protocol Zoo. We follow a modular approach by identifying two key components: Functionality, which connects real-world applications; and Protocol, which is a set of instructions between two or many parties, at least one of which has a quantum device. Based on the different stages of the quantum internet and use-case in the commercialization of quantum communication, our framework classifies quantum cryptographic functionalities and the various protocol designs implementing these functionalities. Towards this classification, we introduce a novel concept of resource visualization for quantum protocols, which includes two interfaces: one to identify the building blocks for implementing a given protocol and another to identify accessible protocols when certain physical resources or functionalities are available. Such classification provides a hierarchy of quantum protocols based on their use-case and resource allocation. We have identified various valuable tools to improve its representation with a range of techniques, from abstract cryptography to graphical visualizations of the resource hierarchy in quantum networks. We elucidate the structure of the zoo and its primary features in this article to a broader class of quantum information scientists, physicists, computer science theorists and end-users. Since its introduction in 2018, the quantum protocol zoo has been a cornerstone in serving the quantum networks community in its ability to establish the use cases of emerging quantum internet networks. In that spirit we also provide some of the applications of our framework from different perspectives., Comment: 12 pages, 6 figures

Learning complex quantum processes is a central challenge in many areas of quantum computing and quantum machine learning, with applications in quantum benchmarking, cryptanalysis, and variational quantum algorithms. This paper introduces the first learning framework for studying quantum process learning within the Quantum Statistical Query (QSQ) model, providing the first formal definition of statistical queries to quantum processes (QPSQs). The framework allows us to propose an efficient QPSQ learner for arbitrary quantum processes accompanied by a provable performance guarantee. We also provide numerical simulations to demonstrate the efficacy of this algorithm. In our new framework, we prove exponential query complexity lower bounds for learning unitary 2-designs, and a doubly exponential lower bound for learning haar-random unitaries. The practical relevance of this framework is exemplified through application in cryptography, highlighting vulnerabilities of a large class of Classical-Readout Quantum Physical Unclonable Functions (CR-QPUFs), addressing an important open question in the field of quantum hardware security. This work marks a significant step towards understanding the learnability of quantum processes and shedding light on their security implications., Comment: 46 pages, 3 figures. v2: Added lower bounds (Section 6)

Differential privacy is a widely used notion of security that enables the processing of sensitive information. In short, differentially private algorithms map "neighbouring" inputs to close output distributions. Prior work proposed several quantum extensions of differential privacy, each of them built on substantially different notions of neighbouring quantum states. In this paper, we propose a novel and general definition of neighbouring quantum states. We demonstrate that this definition captures the underlying structure of quantum encodings and can be used to provide exponentially tighter privacy guarantees for quantum measurements. Our approach combines the addition of classical and quantum noise and is motivated by the noisy nature of near-term quantum devices. Moreover, we also investigate an alternative setting where we are provided with multiple copies of the input state. In this case, differential privacy can be ensured with little loss in accuracy combining concentration of measure and noise-adding mechanisms. En route, we prove the advanced joint convexity of the quantum hockey-stick divergence and we demonstrate how this result can be applied to quantum differential privacy. Finally, we complement our theoretical findings with an empirical estimation of the certified adversarial robustness ensured by differentially private measurements., Comment: This articles supersedes arXiv:2203.03604

We show the security of multi-user key establishment on a single line of quantum communication. More precisely, we consider a quantum communication architecture where the qubit generation and measurement happen at the two ends of the line, whilst intermediate parties are limited to single-qubit unitary transforms. This network topology has been previously introduced to implement quantum-assisted secret-sharing protocols for classical data, as well as the key establishment, and secure computing. This architecture has numerous advantages. The intermediate nodes are only using simplified hardware, which makes them easier to implement. Moreover, key establishment between arbitrary pairs of parties in the network does not require key routing through intermediate nodes. This is in contrast with quantum key distribution (QKD) networks for which non-adjacent nodes need intermediate ones to route keys, thereby revealing these keys to intermediate parties and consuming previously established ones to secure the routing process. Our main result is to show the security of key establishment on quantum line networks. We show the security using the framework of abstract cryptography. This immediately makes the security composable, showing that the keys can be used for encryption or other tasks., Comment: 23 pages, 4 figures, comments are welcome

The impossibility of creating perfect identical copies of unknown quantum systems is a fundamental concept in quantum theory and one of the main non-classical properties of quantum information. This limitation imposed by quantum mechanics, famously known as the no-cloning theorem, has played a central role in quantum cryptography as a key component in the security of quantum protocols. In this thesis, we look at Unclonability in a broader context in physics and computer science and more specifically through the lens of cryptography, learnability and hardware assumptions. We introduce new notions of unclonability in the quantum world, namely quantum physical unclonability, and study the relationship with cryptographic properties and assumptions such as unforgeability, and quantum pseudorandomness. The purpose of this study is to bring new insights into the field of quantum cryptanalysis and into the notion of unclonability itself. We also discuss several applications of this new type of unclonability as a cryptographic resource for designing provably secure quantum protocols. Furthermore, we present a new practical cryptanalysis technique concerning the problem of approximate cloning of quantum states. We design a quantum machine learning-based cryptanalysis algorithm to demonstrate the power of quantum learning tools as both attack strategies and powerful tools for the practical study of quantum unclonability., Comment: PhD Thesis. Defended 17th August 2022, University of Edinburgh

Differential privacy provides a theoretical framework for processing a dataset about $n$ users, in a way that the output reveals a minimal information about any single user. Such notion of privacy is usually ensured by noise-adding mechanisms and amplified by several processes, including subsampling, shuffling, iteration, mixing and diffusion. In this work, we provide privacy amplification bounds for quantum and quantum-inspired algorithms. In particular, we show for the first time, that algorithms running on quantum encoding of a classical dataset or the outcomes of quantum-inspired classical sampling, amplify differential privacy. Moreover, we prove that a quantum version of differential privacy is amplified by the composition of quantum channels, provided that they satisfy some mixing conditions., Comment: This article is superseded by arXiv:2307.04733

This paper, for the first time, addresses the questions related to the connections between the quantum pseudorandomness and quantum hardware assumptions, specifically quantum physical unclonable functions (qPUFs). Our results show that the efficient pseudorandom quantum states (PRS) are sufficient to construct the challenge set for the universally unforgeable qPUF, improving the previous existing constructions that are based on the Haar-random states. We also show that both the qPUFs and the quantum pseudorandom unitaries (PRUs) can be constructed from each other, providing new ways to obtain PRS from the hardware assumptions. Moreover, we provide a sufficient condition (in terms of the diamond norm) that a set of unitaries should have to be a PRU in order to construct a universally unforgeable qPUF, giving yet another novel insight into the properties of the PRUs. Later, as an application of our results, we show that the efficiency of an existing qPUF-based client-server identification protocol can be improved without losing the security requirements of the protocol., Comment: 33 pages, 4 figures

Physical unclonable functions(PUFs) provide a unique fingerprint to a physical entity by exploiting the inherent physical randomness. Gao et al. discussed the vulnerability of most current-day PUFs to sophisticated machine learning-based attacks. We address this problem by integrating classical PUFs and existing quantum communication technology. Specifically, this paper proposes a generic design of provably secure PUFs, called hybrid locked PUFs(HLPUFs), providing a practical solution for securing classical PUFs. An HLPUF uses a classical PUF(CPUF), and encodes the output into non-orthogonal quantum states to hide the outcomes of the underlying CPUF from any adversary. Here we introduce a quantum lock to protect the HLPUFs from any general adversaries. The indistinguishability property of the non-orthogonal quantum states, together with the quantum lockdown technique prevents the adversary from accessing the outcome of the CPUFs. Moreover, we show that by exploiting non-classical properties of quantum states, the HLPUF allows the server to reuse the challenge-response pairs for further client authentication. This result provides an efficient solution for running PUF-based client authentication for an extended period while maintaining a small-sized challenge-response pairs database on the server side. Later, we support our theoretical contributions by instantiating the HLPUFs design using accessible real-world CPUFs. We use the optimal classical machine-learning attacks to forge both the CPUFs and HLPUFs, and we certify the security gap in our numerical simulation for construction which is ready for implementation., Comment: 47 pages, 13 figures

In this paper, we continue the line of work initiated by Boneh and Zhandry at CRYPTO 2013 and EUROCRYPT 2013 in which they formally define the notion of unforgeability against quantum adversaries specifically, for classical message authentication codes and classical digital signatures schemes. We develop a general and parameterised quantum game-based security model unifying unforgeability for both classical and quantum constructions allowing us for the first time to present a complete quantum cryptanalysis framework for unforgeability. In particular, we prove how our definitions subsume previous ones while considering more fine-grained adversarial models, capturing the full spectrum of superposition attacks. The subtlety here resides in the characterisation of a forgery. We show that the strongest level of unforgeability, namely existential unforgeability, can only be achieved if only orthogonal to previously queried messages are considered to be forgeries. In particular, we present a non-trivial attack if any overlap between the forged message and previously queried ones is allowed. We further show that deterministic constructions can only achieve the weaker notion of unforgeability, that is selective unforgeability, against such restricted adversaries, but that selective unforgeability breaks if general quantum adversaries (capable of general superposition attacks) are considered. On the other hand, we show that PRF is sufficient for constructing a selective unforgeable classical primitive against full quantum adversaries. Moreover, we show similar positive results relying on Pseudorandom Unitaries (PRU) for quantum primitives. These results demonstrate the generality of our framework that could be applicable to other primitives beyond the cases analysed in this paper., Comment: 50 pages, 5 figures

Cryptanalysis on standard quantum cryptographic systems generally involves finding optimal adversarial attack strategies on the underlying protocols. The core principle of modelling quantum attacks in many cases reduces to the adversary's ability to clone unknown quantum states which facilitates the extraction of some meaningful secret information. Explicit optimal attack strategies typically require high computational resources due to large circuit depths or, in many cases, are unknown. In this work, we propose variational quantum cloning (VQC), a quantum machine learning based cryptanalysis algorithm which allows an adversary to obtain optimal (approximate) cloning strategies with short depth quantum circuits, trained using hybrid classical-quantum techniques. The algorithm contains operationally meaningful cost functions with theoretical guarantees, quantum circuit structure learning and gradient descent based optimisation. Our approach enables the end-to-end discovery of hardware efficient quantum circuits to clone specific families of quantum states, which in turn leads to an improvement in cloning fidelites when implemented on quantum hardware: the Rigetti Aspen chip. Finally, we connect these results to quantum cryptographic primitives, in particular quantum coin flipping. We derive attacks on two protocols as examples, based on quantum cloning and facilitated by VQC. As a result, our algorithm can improve near term attacks on these protocols, using approximate quantum cloning as a resource., Comment: 16 pages main text, 25 pages supplementary material, 20 figures. Comments welcome

Recently, major progress has been made towards the realisation of quantum internet to enable a broad range of classically intractable applications. These applications such as delegated quantum computation require running a secure identification protocol between a low-resource and a high-resource party to provide secure communication. In this work, we propose two identification protocols based on the emerging hardware secure solutions, the quantum Physical Unclonable Functions (qPUFs). The first protocol allows a low-resource party to prove its identity to a high-resource party and in the second protocol, it is vice-versa. Unlike existing identification protocols based on Quantum Read-out PUFs which rely on the security against a specific family of attacks, our protocols provide provable exponential security against any Quantum Polynomial-Time adversary with resource-efficient parties. We provide a comprehensive comparison between the two proposed protocols in terms of resources such as quantum memory and computing ability required in both parties as well as the communication overhead between them., Comment: 40 pages, 11 figures

A Physical Unclonable Function (PUF) is a device with unique behaviour that is hard to clone hence providing a secure fingerprint. A variety of PUF structures and PUF-based applications have been explored theoretically as well as being implemented in practical settings. Recently, the inherent unclonability of quantum states has been exploited to derive the quantum analogue of PUF as well as new proposals for the implementation of PUF. We present the first comprehensive study of quantum Physical Unclonable Functions (qPUFs) with quantum cryptographic tools. We formally define qPUFs, encapsulating all requirements of classical PUFs as well as introducing a new testability feature inherent to the quantum setting only. We use a quantum game-based framework to define different levels of security for qPUFs: quantum exponential unforgeability, quantum existential unforgeability and quantum selective unforgeability. We introduce a new quantum attack technique based on the universal quantum emulator algorithm of Marvin and Lloyd to prove no qPUF can provide quantum existential unforgeability. On the other hand, we prove that a large family of qPUFs (called unitary PUFs) can provide quantum selective unforgeability which is the desired level of security for most PUF-based applications., Comment: 32 pages including the appendix

It is known that no quantum process can produce a predetermined superposition of unknown arbitrary states. It has already been shown that with some partial information about the states, one can produce with some probability such superpositions. Here we show that there are universal machines which can produce superpositions of unknown orthogonal states with unit probability. Our construction unravels the relation between the no-cloning theorem and the no-superposition theorem, that is we show that if a perfect cloning machine exists, then a universal superposition machine can also exist., Comment: 13 pages, 2 figures, Accepted for publication in Physical Review A