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Achieving high-precision measurements on near-term quantum devices is critical for advancing quantum computing applications. In this paper, we explore several practical techniques to enhance measurement accuracy using randomized measurements, focusing on minimizing shot overhead, circuit overhead, measurement noise, and time-dependent measurement noise. Our approach leverages locally biased random measurements to reduce shot overhead, in addition to repeated settings and parallel quantum detector tomography to reduce circuit overhead and mitigate measurement noise. Additionally, we employ a blended scheduling technique to mitigate time-dependent measurement noise. We demonstrate the effectiveness of these techniques through a case study on the molecular energy estimation of the BODIPY molecule using the Hartree-Fock state on an IBM Eagle r3 computer, showcasing significant improvements in measurement precision. These strategies pave the way for more reliable and accurate quantum computations, particularly in applications requiring precise molecular energy calculations., Comment: 15 pages, 8 figures

We address the open problem of type safety in Hutchins' pure subtype systems (PSS). PSS (hereafter in the singular) harmoniously mixes terms and types, thus enabling a number of advanced language features that combine dependent types with higher-order subtyping. In PSS terms and types belong to the same kind (everything is a subtype) and the resulting theory is based on subtyping. Since PSS lacks strong normalisation, a type soundness result can only be stated in terms of type safety defined as progress and preservation. Proving type safety rests on the well-known problem of transitivity elimination in higher-order subtyping, where a key inversion lemma fails under the presence of intermediary steps in transitive subtype derivations. Despite his attempts, Hutchins failed to prove PSS type safety. We propose a reformulation of pure subtype systems with a more fine-grained notion of subtyping derivation that enables a direct proof of transitivity elimination, and thus of type safety. We also reformulate Hutchins' practical type-checking algorithm to our system and prove it correct., Comment: 36 pages,6 figures

Federated Learning is a decentralized framework that enables multiple clients to collaboratively train a machine learning model under the orchestration of a central server without sharing their local data. The centrality of this framework represents a point of failure which is addressed in literature by blockchain-based federated learning approaches. While ensuring a fully-decentralized solution with traceability, such approaches still face several challenges about integrity, confidentiality and scalability to be practically deployed. In this paper, we propose Fantastyc, a solution designed to address these challenges that have been never met together in the state of the art.

Accurately assessing tumor removal is paramount in the management of glioblastoma. We developed a pipeline using MRI scans and neural networks to segment tumor subregions and the surgical cavity in postoperative images. Our model excels in accurately classifying the extent of resection, offering a valuable tool for clinicians in assessing treatment effectiveness.

Error mitigation has elevated quantum computing to the scale of hundreds of qubits and tens of layers; however, yet larger scales (deeper circuits) are needed to fully exploit the potential of quantum computing to solve practical problems otherwise intractable. Here we demonstrate three key results that pave the way for the leap from quantum utility to quantum advantage: (1) we present a thorough derivation of random and systematic errors associated to the most advanced error mitigation strategies, including probabilistic error cancellation (PEC), zero noise extrapolation (ZNE) with probabilistic error amplification, and tensor-network error mitigation (TEM); (2) we prove that TEM (i) has the lowest sampling overhead among all three techniques under realistic noise, (ii) is optimal, in the sense that it saturates the universal lower cost bound for error mitigation, and (iii) is therefore the most promising approach to quantum advantage; (3) we propose a concrete notion of practical quantum advantage in terms of the universality of algorithms, stemming from the commercial need for a problem-independent quantum simulation device. We also establish a connection between error mitigation, relying on additional measurements, and error correction, relying on additional qubits, by demonstrating that TEM with a sufficient bond dimension works similarly to an error correcting code of distance 3. We foresee that the interplay and trade-off between the two resources will be the key to a smooth transition between error mitigation and error correction, and hence between near-term and fault-tolerant quantum computers. Meanwhile, we argue that quantum computing with optimal error mitigation, relying on modest classical computer power for tensor network contraction, has the potential to reach larger scales in accurate simulation than classical methods alone., Comment: 36 pages, 14 figures

The goal in quantum state transfer is to avoid the need to physically transport carriers of quantum information. This is achieved by using a suitably engineered Hamiltonian that induces the transfer of the state of one subsystem to another. A less known generalization of state transfer considers multiple systems such that any pair can exchange quantum information and transfers can take place at any time, starting and stopping independently. This is sometimes called routing of quantum states. State transfer in particular has received a great deal of attention, however the vast majority of results in both state transfer and routing concern qubits transferred in a network of restricted structure. Here we consider routing of single-mode Gaussian states and entanglement through complex networks of quantum harmonic oscillators. We compare a protocol where the transfer is completed in a single step but the effective Hamiltonian only approximately transfers the state with one where the transfer can in principle be perfect but the transfer is done in two steps, and also illustrate the state-dependency of the transfer fidelity. We find that even in a random and homogeneous network, the transfer fidelity still depends on the degree of the nodes for any link density, and that in both random and complex networks it is the community structure that controls the appearance of higher frequency normal modes useful for transfer. Finally, we find that networks of sufficient complexity may have superior routing performance over superficially similar random networks. Our results pave the way for further exploration of the role of community structure in state transfer and related tasks., Comment: 13 pages, 15 figures

Characterization of noise in current near-term quantum devices is of paramount importance to fully use their computational power. However, direct quantum process tomography becomes unfeasible for systems composed of tens of qubits. A promising alternative method based on tensor networks was recently proposed [Nat. Commun. 14, 2858 (2023)]. In this paper, we adapt it for the characterization of noise channels on near-term quantum computers and investigate its performance thoroughly. In particular, we show how experimentally feasible tomographic samples are sufficient to accurately characterize realistic correlated noise models affecting individual layers of quantum circuits, and study its performance on systems composed of up to 20 qubits. Furthermore, we combine this noise characterization method with a recently proposed noise-aware tensor network error mitigation protocol for correcting outcomes in noisy circuits, resulting accurate estimations even on deep circuit instances. This positions the tensor-network-based noise characterization protocol as a valuable tool for practical error characterization and mitigation in the near-term quantum computing era., Comment: Updated version matching publication (minor modifications, added section on SPAM errors)

We present a unified rule format for structural operational semantics with terms as labels that guarantees that the associated labelled transition system has some bounded-nondeterminism property. The properties we consider include finite branching, initials finiteness and image finiteness.

The nominal transition systems (NTSs) of Parrow et al. describe the operational semantics of nominal process calculi. We study NTSs in terms of the nominal residual transition systems (NRTSs) that we introduce. We provide rule formats for the specifications of NRTSs that ensure that the associated NRTS is an NTS and apply them to the operational specification of the early pi-calculus. Our study stems from the recent Nominal SOS of Cimini et al. and from earlier works in nominal sets and nominal logic by Gabbay, Pitts and their collaborators., Comment: Conference version of arXiv:1807.02081

In recent years, informationally complete measurements have attracted considerable attention, especially in the context of classical shadows. In the particular case of informationally over-complete measurements, for which the number of possible outcomes exceeds the dimension of the space of linear operators in Hilbert space, the dual POVM operators used to interpret the measurement outcomes are not uniquely defined. In this work, we propose a method to optimize the dual operators after the measurements have been carried out in order to produce sharper, unbiased estimations of observables of interest. We discuss how this procedure can produce zero-variance estimations in cases where the classical shadows formalism, which relies on so-called canonical duals, incurs exponentially large measurement overheads. We also analyze the algorithm in the context of quantum simulation with randomized Pauli measurements, and show that it can significantly reduce statistical errors with respect to canonical duals on multiple observable estimations.

In this letter, we prove the existence of resummed expressions for the diagonal of the heat kernel and the effective action of a quantum field which interacts with a scalar or an electromagnetic background. Working in an arbitrary number of spacetime dimensions, we propose an Ansatz beyond the Schwinger--DeWitt proposal, effectively resumming an infinite number of invariants which can be constructed from powers of the background, as well as its first and second derivatives in the Yukawa case. This provides a proof of the recent conjecture that all terms containing the invariants $F_{\mu\nu}F^{\mu\nu}$ and $\widetilde F_{\mu\nu}F^{\mu\nu}$ in the proper-time series expansion of the SQED effective action can be resummed. Possible generalizations and several applications are also discussed -- in particular, the existence of an analogue of the Schwinger effect for Yukawa couplings., Comment: 14 pages

We recently constructed the G4Jy-3CRE, a catalog of extragalactic radio sources based on the GLEAM 4-Jy (G4Jy) sample, with the aim of increasing the number of powerful radio galaxies and quasars with similar selection criteria to those of the revised release of the Third Cambridge catalog (3CR). The G4Jy-3CRE consists of a total of 264 radio sources mainly visible from the Southern Hemisphere. Here, we present an initial X-ray analysis of 89 G4Jy-3CRE radio sources with archival X- ray observations from the Neil Gehrels Swift Observatory. We reduced a total of 615 Swift observations, for about 0.89 Msec of integrated exposure time, we found X-ray counterparts for 61 radio sources belonging to the G4Jy-3CRE, 11 of them showing extended X-ray emission. The remaining 28 sources do not show any X-ray emission associated with their radio cores. Our analysis demonstrates that X-ray snapshot observations, even if lacking uniform exposure times, as those carried out with Swift, allow us to (i) verify and/or re ne the host galaxy identi cation; (ii) discover the extended X-ray emission around radio galaxies of the intracluster medium when harbored in galaxy clusters, as the case of G4Jy 1518 and G4Jy 1664, and (iii) detect X-ray radiation arising from their radio lobes, as for G4Jy 1863., Comment: 35 pages, 17 figures, 2 tables; second paper of a series, pre-proof version

Since the early sixties, our view of radio galaxies and quasars has been drastically shaped by discoveries made thanks to observations of radio sources listed in the Third Cambridge catalog and its revised version (3CR). However, the largest fraction of data collected to date on 3CR sources was performed with relatively old instruments, rarely repeated and/or updated. Importantly, the 3CR contains only objects located in the Northern Hemisphere thus having limited access to new and innovative astronomical facilities. To mitigate these limitations we present a new catalog of powerful radio sources visible from the Southern Hemisphere, extracted from the GLEAM 4-Jy (G4Jy) catalog and based on equivalent selection criteria as the 3CR. This new catalog, named G4Jy- 3CRE, where the E stands for "equivalent", lists a total of 264 sources at declination below -5 degrees and with 9 Jy limiting sensitivity at ~178 MHz. We explored archival radio maps obtained with different surveys and compared then with optical images available in the Pan-STARRS, DES and DSS databases to search for optical counterparts of their radio cores. We compared mid-infrared counterparts, originally associated in the G4Jy, with the optical ones identified here and we present results of a vast literature search carried out to collect redshift estimates for all G4Jy-3CRE sources resulting in a total of 145 reliable z measurements., Comment: 72 pages, 35 figures, 5 Tables

Until fault-tolerance becomes implementable at scale, quantum computing will heavily rely on noise mitigation techniques. While methods such as zero noise extrapolation with probabilistic error amplification (ZNE-PEA) and probabilistic error cancellation (PEC) have been successfully tested on hardware recently, their scalability to larger circuits may be limited. Here, we introduce the tensor-network error mitigation (TEM) algorithm, which acts in post-processing to correct the noise-induced errors in estimations of physical observables. The method consists of the construction of a tensor network representing the inverse of the global noise channel affecting the state of the quantum processor, and the consequent application of the map to informationally complete measurement outcomes obtained from the noisy state. TEM does therefore not require additional quantum operations other than the implementation of informationally complete POVMs, which can be achieved through randomised local measurements. The key advantage of TEM is that the measurement overhead is quadratically smaller than in PEC. We test TEM extensively in numerical simulations in different regimes. We find that TEM can be applied to circuits of twice the depth compared to what is achievable with PEC under realistic conditions with sparse Pauli-Lindblad noise, such as those in [E. van den Berg et al., Nat. Phys. (2023)]. By using Clifford circuits, we explore the capabilities of the method in wider and deeper circuits with lower noise levels. We find that in the case of 100 qubits and depth 100, both PEC and ZNE fail to produce accurate results by using $\sim 10^5$ shots, while TEM succeeds.

The geometric renormalization technique for complex networks has successfully revealed the multiscale self-similarity of real network topologies and can be applied to generate replicas at different length scales. In this letter, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our findings demonstrate that weights in real networks exhibit multiscale self-similarity under a renormalization protocol that selects the connections with the maximum weight across increasingly longer length scales. We present a theory that elucidates this symmetry, and that sustains the selection of the maximum weight as a meaningful procedure. Based on our results, scaled-down replicas of weighted networks can be straightforwardly derived, facilitating the investigation of various size-dependent phenomena in downstream applications., Comment: 5 pages, 3 figures

We present a self consistent field approach (SCF) within the Adaptive Derivative-Assembled Problem-Tailored Ansatz Variational Quantum Eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression which is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition metal complex ferrocene (Fe$\rm (C_5H_5)_2$), indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to CASSCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way towards a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets., Comment: 21 pages, 5 figures

We propose an estimation method for quantum measurement tomography (QMT) based on semidefinite programming (SDP), and discuss how it may be employed to detect experimental imperfections, such as shot noise and/or faulty preparation of the input states on near-term quantum computers. Moreover, if the positive operator-valued measure (POVM) we aim to characterize is informationally complete, we put forward a method for self-consistent tomography, i.e., for recovering a set of input states and POVM effects that is consistent with the experimental outcomes and does not assume any a priori knowledge about the input states of the tomography. Contrary to many methods that have been discussed in the literature, our approach does not rely on additional assumptions such as low noise or the existence of a reliable subset of input states., Comment: Accepted version

Quantum processing units boost entanglement at the level of hardware and enable physical simulations of highly correlated electron states in molecules and intermolecular chemical bonds. The variational quantum eigensolver provides a hardware-efficient toolbox for ground state simulation; however, with limitations in precision. Even in the absence of noise, the algorithm may result into a biased energy estimation, particularly with some shallower ansatz types. Noise additionally degrades entanglement and hinders the ground state energy estimation (especially if the noise is not fully characterized). Here we develop a method to exploit the quantum-classical interface provided by informationally complete measurements to use classical software on top of the hardware entanglement booster for ansatz- and noise-related error reduction. We use the tensor network representation of a quantum channel that drives the noisy state toward the ground one. The tensor network is a completely positive map by construction, but we elaborate on making the trace preservation condition local so as to activate the sweeping variational optimization. This method brings into reach energies below the noiseless ansatz by creating additional correlations among the qubits and denoising them. Analyzing the example of the stretched water molecule with a tangible entanglement, we argue that a hybrid strategy of using the quantum hardware together with the classical software outperforms a purely classical strategy provided the classical parts have the same bond dimension. The proposed optimization algorithm extends the variety of noise mitigation methods and facilitates the more accurate study of the energy landscape for deformed molecules. The algorithm can be applied as the final postprocessing step in the quantum hardware simulation of protein-ligand complexes in the context of drug design., Comment: 11 pages, 7 figures

Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations. In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees. We discuss in an intuitive manner the connection between the generating trees' structure and certain properties of the resulting mapping, such as Pauli weight and the delocalisation of mode occupation. Moreover, we introduce a recipe that guarantees Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing. Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the SWAP overhead with respect to other paradigmatic mappings. We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers. The resulting mappings have a favourable Pauli weight scaling $\mathcal{O}(\sqrt{N})$ on this connectivity, while ensuring that no SWAP gates are necessary for single excitation operations.

ADAPT-VQE stands out as a robust algorithm for constructing compact ans\"atze for molecular simulation. It enables to significantly reduce the circuit depth with respect to other methods, such as UCCSD, while achieving higher accuracy and not suffering from so-called barren plateaus that hinder the variational optimisation of many hardware-efficient ans\"atze. In its standard implementation, however, it introduces a considerable measurement overhead in the form of gradient evaluations trough estimations of many commutator operators. In this work, we mitigate this measurement overhead by exploiting a recently introduced method for energy evaluation relying on Adaptive Informationally complete generalised Measurements (AIM). Besides offering an efficient way to measure the energy itself, Informationally Complete (IC) measurement data can be reused to estimate all the commutators of the operators in the operator pool of ADAPT-VQE, using only classically efficient post-processing. We present the AIM-ADAPT-VQE scheme in detail, and investigate its performance with several H4 Hamiltonians and operator pools. Our numerical simulations indicate that the measurement data obtained to evaluate the energy can be reused to implement ADAPT-VQE with no additional measurement overhead for the systems considered here. In addition, we show that, if the energy is measured within chemical precision, the CNOT count in the resulting circuits is close to the ideal one. With scarce measurement data, AIM-ADAPT-VQE still converges to the ground state with high probability, albeit with an increased circuit depth in some cases.

We study the equivalence between eval-readback and eval-apply big-step evaluators in the general setting of the pure lambda calculus. We study `one-step' equivalence (same strategy) and also discuss `big-step' equivalence (same final result). One-step equivalence extends for free to evaluators in other settings (calculi, programming languages, proof assistants, etc.) by restricting the terms (closed, convergent) while maintaining the strategy. We present a proof that one-step equivalence holds when (1) the `readback' stage satisfies straightforward well-formedness provisos, (2) the `eval' stage implements a `uniform' strategy, and (3) the eval-apply evaluator implements a `balanced hybrid' strategy that has `eval' as a subsidiary strategy. The proof proceeds by application of the `lightweight fusion by fixed-point promotion' program transformation on evaluator implementations to fuse readback and eval into the balanced hybrid. The proof can be followed with no previous knowledge of the transformation. We use Haskell 2010 as the implementation language, with all evaluators written in monadic style to guarantee semantics (strategy) preservation, but the choice of implementation language is immaterial. To illustrate the large scope of the equivalence, we provide an extensive survey of the strategy space using canonical eval-apply evaluators in code and big-step `natural' operational semantics. We discuss the strategies' properties, some of their uses, and their abstract machines. We improve the formal definition of uniform and hybrid strategy, use it to structure and classify the strategy space, and to obtain generic higher-order evaluators which are used in the equivalence proof. We introduce a systematic notation for both evaluator styles and use it to summarise strategy and evaluator equivalences, including (non-)equivalences., Comment: 76 pages, 15 figures

Turing patterns, arising from the interplay between competing species of diffusive particles, has long been an important concept for describing non-equilibrium self-organization in nature, and has been extensively investigated in many chemical and biological systems. Historically, these patterns have been studied in extended systems and lattices. Recently, the Turing instability was found to produce topological patterns in networks with scale-free degree distributions and the small world property, although with an apparent absence of geometric organization. While hints of explicitly geometric patterns in simple network models (e.g Watts-Strogatz) have been found, the question of the exact nature and morphology of geometric Turing patterns in heterogeneous complex networks remains unresolved. In this work, we study the Turing instability in the framework of geometric random graph models, where the network topology is explained by an underlying geometric space. We demonstrate that not only can geometric patterns be observed, their wavelength can also be estimated by studying the eigenvectors of the annealed graph Laplacian. Finally, we show that Turing patterns can be found in geometric embeddings of real networks. These results indicate that there is a profound connection between the function of a network and its hidden geometry, even when the associated dynamical processes are exclusively determined by the network topology., Comment: 16 pages, 8 figures, 1 table