QUROPE Quantum Information Processing and Communication in Europe

What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires $\Theta(n)$ queries to an NP oracle to compute a witness to a given SAT formula, quantumly $\Theta(\log n)$ queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires $\Omega(\log n)$ queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires $\Omega(\log n)$ queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making $o(log n)$ NP queries, then $\text{BPP}^{\text{NP}[o(n)]}$ contains a $\text{P}^{\text{NP}}$-complete problem if the algorithm is classical and $\text{FBQP}^{\text{NP}[o(n)]}$ contains an $\text{FP}^{\text{NP}}$-complete problem if the algorithm is quantum.

We study the problem of localization in Quantum Field Theory (QFT) from the point of view of inertial and accelerated experimenters. We consider the Newton-Wigner, the Algebraic Quantum Field Theory (AQFT) and the modal localization schemes, which are, respectively, based on the orthogonality condition for states localized in disjoint regions of space, on the algebraic approach to QFT and on the representation of single particles as positive frequency solution of the field equation. We show that only the AQFT scheme obeys causality and physical invariance under diffeomorphisms.

Then, we consider the nonrelativistic limit of quantum fields in the Rindler frame. We demonstrate the convergence between the AQFT and the modal scheme and we show the emergence of the Born notion of localization of states and observables. Also, we study the scenario in which an experimenter prepares states over a background vacuum by means of nonrelativistic local operators and another experimenter carries out nonrelativistic local measurements in a different region. We find that the independence between preparation of states and measurements is not guaranteed when both experimenters are accelerated and the background state is different from Rindler vacuum, or when one of the two experimenters is inertial.

In his book \textit{The Mathematical Foundations of Quantum Mechanics}, published in 1932, J. von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He showed that hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and concluded that if they did exist, the theory would have already failed in situations where it has been successfully applied. von Neumann left open the possibility that the theory is not complete, and his analysis for internal consistency is the best that can be done for a self-referenced logical system (G\"odel's theorem). This analysis had been taken as an ``incorrect proof" against the existence of hidden variables. von Neumann's so-called proof isn't even wrong as such a proof does not exist. One of the earliest attempts at a hidden variable theory was by D. Bohm, and because there were no experimental consequences, W. Pauli referred to it as an ``uncashable check." To our knowledge, a successful hidden variable extension to quantum mechanics with testable consequences has not yet been produced, suggesting that von Neumann's analysis is worthy of rehabilitation, which we attempt to provide in a straightforward manner.

The code-capacity threshold of a scalable quantum error correcting stabilizer code can be expressed as a thermodynamic phase transition of a corresponding Ising model with random bond-disorder. Here we study the XY and XZZX surface codes under phase-biased noise, $p_x=p_y=p_z/(2\eta)$, with $\eta\geq 1/2$, and total error rate $p=p_x+p_y+p_z$. By appropriately formulating the boundary conditions, in the rotated code geometry, we find exact solutions at a special disordered point, $p=\frac{1+\eta^{-1}}{2+\eta^{-1}}\gtrsim 0.5$, for arbitrary odd code-distance $d$. The total logical failure rate is given by $P_{f}=\frac{3}{4}-\frac{1}{4}e^{-2d_Z\,\text{artanh}(1/2\eta)}$, where $d_{Z}=d^2$ and $d$ for the two codes respectively, is the effective code distance for pure phase-flip noise. The large finite size corrections for $d_Z<\eta$ make threshold extractions, using the total logical failure rate for moderate code-distances, unreliable. We show that independently estimating thresholds for the $X_L$ (phase-flip), $Y_L$, and $Z_L$ (bit-flip) logical failure rates can give a more confident threshold estimate. Using this method for the XZZX model with a tensor-network based decoder we find that the thresholds converge with code distance to a single value at moderate bias ($\eta=30, 100$), corresponding to an error rate above the hashing bound. In contrast, for larger bias the thresholds do not converge for practically maximum-likelihood-decodable code distances (up to $d\approx 100$), leaving a large uncertainty in the precise threshold value.

We study the applicability of the Liouvillian exceptional points (LEPs) approach to nanoscale open quantum systems. A generic model of the driven two-level system in a thermal environment is analyzed within the nonequilibrium Green's function (NEGF) and Bloch quantum master equation (QME) formulations. We derive the latter starting from the exact NEGF Dyson equations and highlight the qualitative limitations of the LEP treatment by examining the approximations employed in its derivation. We find that non-Markov character of evolution in open quantum systems does not allow for the introduction of the concept of exceptional points for a description of their dynamics. Theoretical analysis is illustrated with numerical simulations.

One of the most promising nascent technologies, quantum computation faces a major challenge: The need for stable computational building blocks. We present the quantum-optical realization of non-adiabatic holonomies that can be used as single-qubit quantum gates. The hallmark topological protection of non-Abelian geometric phases reduces the need for quantum error correction on a fundamental physical level, while the inherent non-adiabaticity of the structures paves the way for unprecedented miniaturization. To demonstrate their versatility, we realize the Hadamard and Pauli-X gates, experimentally show their non-Abelian nature, and combine them into a single-qubit quantum algorithm, the PQ penny flipover. The planar geometry of such designs enables them to be substituted for the conventional directional coupler meshes currently in wide-spread use in photonic quantum architectures across all platforms.

We report on an experiment achieving the dynamical generation of non-Gaussian states of motion of a levitated optomechanical system. We access intrinsic Duffing-like non-linearities by squeezing an oscillator's state of motion through rapidly switching the frequency of its trap. We characterize the experimental non-Gaussian state against expectations from simulations and give prospects for the emergence of genuine non-classical features.

Quantum links between physically separated modules are important for scaling many quantum computing technologies. The key metrics are the generation rate and fidelity of remote Bell pairs. In this work, we propose an experimental protocol for generating remote entanglement between neutral ytterbium atom qubits using an optical cavity. By loading a large number of atoms into a single cavity, and controlling their coupling using only local light shifts, we amortize the cost of transporting and initializing atoms over many entanglement attempts, maximizing the entanglement generation rate. A twisted ring cavity geometry suppresses many sources of error, allowing high fidelity entanglement generation. We estimate a spin-photon entanglement rate of $5 \times 10^5$ s$^{-1}$, and a Bell pair rate of $1.0\times 10^5$ s$^{-1}$, with an average fidelity near $0.999$. Furthermore, we show that the photon detection times provide a significant amount of soft information about the location of errors, which may be used to improve the logical qubit performance. This approach provides a practical path to scalable modular quantum computing using neutral ytterbium atoms.

Quantum chaotic systems with one-dimensional spectra follow spectral correlations of orthogonal (OE), unitary (UE), or symplectic ensembles (SE) of random matrices depending on their invariance under time reversal and rotation. In this letter, we study the non-Hermitian and non-unitary ensembles based on the symmetry of matrix elements, viz. ensemble of complex symmetric, complex asymmetric (Ginibre), and self-dual matrices of complex quaternions. The eigenvalues for these ensembles lie in the two-dimensional plane. We show that the fluctuation statistics of these ensembles are universal and quantum chaotic systems belonging to OE, UE, and SE in the presence of a dissipative environment show similar spectral fluctuations. The short-range correlations are studied using spacing ratio and spacing distribution. For long-range correlations, unfolding at a non-local scale is crucial. We describe a generic method to unfold the two-dimensional spectra with non-uniform density and evaluate correlations using number variance. We find that both short-range and long-range correlations are universal. We verify our results with the quantum kicked top in a dissipative environment that can be tuned to exhibit symmetries of OE, UE, and SE in its conservative limit.

We introduce a transformation of the quantum phase $S'=S+\frac{\hbar}{2}\log\rho$, which converts the deterministic equations of quantum mechanics into the Lagrangian reference frame of stochastic particles. We show that the quantum potential can be removed from the transformed quantum Hamilton-Jacobi equations if they are solved as stochastic Hamilton-Jacobi-Bellman equations. The system of equations provide a local description of quantum mechanics, which is enabled by the inherently retrocausal nature of stochastic Hamilton-Jacobi-Bellman equations. We also investigate the stochastic transformation of the classical system, where is it shown that quantum mechanics with the quantum potential reduced by a factor of $\frac{1}{2}$ has a classical representation, which may have interesting implications. Finally, we discuss the notion of a subsystem correspondence principle, which constrains the ontology of the total quantum system.

The issue of non-locality in quantum mechanics can potentially be resolved by considering relativistically covariant diffusion in four-dimensional spacetime. Stochastic particles described by the Klein-Gordon equation are shown to undergo a classical diffusion process in spacetime coordinates, which is seen by transforming the quantum Cauchy-momentum equations to a Lagrangian frame of reference. Since the quantum potential term is removed under this transformation, the equations for momentum propagation along particle trajectories assume a classical form. A local stochastic de Broglie-Bohm interpretation for the Klein-Gordon system can subsequently be derived. We also introduce the concept of momentum equivariance to replace the second-order Bohm-Newton equations of motion, which break down due to non-linear terms of the stochastic Lagrangian derivative.

We show that energy relaxation causes a point defect in the uniaxial-nematic phase of a spin-2 Bose-Einstein condensate to deform into a spin-Alice ring that exhibits a composite core structure with distinct topology at short and long distances from the singular line. An outer biaxial-nematic core exhibits a spin half-quantum vortex structure with a uniaxial-nematic inner core. By numerical simulation we demonstrate a dynamical oscillation between the spin-Alice ring and a split-core hedgehog configuration via the appearance of ferromagnetic rings with associated vorticity inside an extended core region. We further show that a similar dynamics is exhibited by a spin-Alice ring surrounding a spin-vortex line resulting from the relaxation of a monopole situated on a spin-vortex line in the biaxial-nematic phase. In the cyclic phase similar states are shown instead to form extended phase-mixing cores containing rings with fractional mass circulation or cores whose spatial shape reflect the order-parameter symmetry of cyclic inner core, depending on the initial configuration.

We have analyzed in detail the angular polarization properties in the center of mass reference frame of Landau's two-photon spherical states in momentum space. The angular distributions for fixed values of $J$ and $M$ do not depend on the parity but are defined by two different functions of the polar angle between the relative momentum and the quantization axes. The two-photon polarization density matrices are derived for each values of $J$, $M$, and $P$. The linear polarization correlations of individual photons are analyzed in detail. We find, besides the usual correlation laws for $J\geq 2$ in terms of $sin$ and $cos$ of the angle between the orientation of the analyzers, correlations in terms of the sum of the orientation angles of the analyzers.

Quantum support vector machines employ quantum circuits to define the kernel function. It has been shown that this approach offers a provable exponential speedup compared to any known classical algorithm for certain data sets. The training of such models corresponds to solving a convex optimization problem either via its primal or dual formulation. Due to the probabilistic nature of quantum mechanics, the training algorithms are affected by statistical uncertainty, which has a major impact on their complexity. We show that the dual problem can be solved in $O(M^{4.67}/\varepsilon^2)$ quantum circuit evaluations, where $M$ denotes the size of the data set and $\varepsilon$ the solution accuracy compared to the ideal result from exact expectation values, which is only obtainable in theory. We prove under an empirically motivated assumption that the kernelized primal problem can alternatively be solved in $O(\min \{ M^2/\varepsilon^6, \, 1/\varepsilon^{10} \})$ evaluations by employing a generalization of a known classical algorithm called Pegasos. Accompanying empirical results demonstrate these analytical complexities to be essentially tight. In addition, we investigate a variational approximation to quantum support vector machines and show that their heuristic training achieves considerably better scaling in our experiments.

The experimental observation of quantum phase transitions predicted by the quantum Rabi model in quantum critical systems is usually challenging due to the lack of signature experimental observables associated with them. Here, we describe a method to identify the dynamical critical phenomenon in the quantum Rabi model consisting of a three-level atom and a cavity at the quantum phase transition. Such a critical phenomenon manifests itself as a sudden change of steady-state output photons in the system driven by two classical fields, when both the atom and the cavity are initially unexcited. The process occurs as the high-frequency pump field is converted into the low-frequency Stokes field and multiple cavity photons in the normal phase, while this conversion cannot occur in the superradiant phase. The sudden change of steady-state output photons is an experimentally accessible measure to probe quantum phase transitions, as it does not require preparing the equilibrium state.

Quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem. In this paper, we give complexity-theoretic evidence that the central task of TDA -- estimating Betti numbers -- is intractable even for quantum computers. Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard. Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best. Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case. We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone and Zanardi achieves a quadratic speedup over the best known classical approach in asymptotically almost all cases. Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.

The performance of quantum key distribution (QKD) is severely limited by multiphoton emissions, due to the photon-number-splitting attack. The most efficient solution, the decoy-state method, requires that the phases of all transmitted pulses are independent and uniformly random. In practice, however, these phases are often correlated, especially in high-speed systems, which opens a security loophole. Here, we address this pressing problem by providing a security proof for decoy-state QKD with correlated phases that offers key rates close to the ideal scenario. Our work paves the way towards high-performance secure QKD with practical laser sources, and may have applications beyond QKD.

A quantum-light source that delivers photons with a high brightness and a high degree of entanglement is fundamental for the development of efficient entanglement-based quantum-key distribution systems. Among all possible candidates, epitaxial quantum dots are currently emerging as one of the brightest sources of highly entangled photons. However, the optimization of both brightness and entanglement currently requires different technologies that are difficult to combine in a scalable manner. In this work, we overcome this challenge by developing a novel device consisting of a quantum dot embedded in a circular Bragg resonator, in turn, integrated onto a micromachined piezoelectric actuator. The resonator engineers the light-matter interaction to empower extraction efficiencies up to 0.69(4). Simultaneously, the actuator manipulates strain fields that tune the quantum dot for the generation of entangled photons with fidelities up to 0.96(1). This hybrid technology has the potential to overcome the limitations of the key rates that plague current approaches to entanglement-based quantum key distribution and entanglement-based quantum networks. Introduction

One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Mu et al, we show that if certain monotones - namely multivariate extensions of R\'{e}nyi divergences - are strictly ordered between the two tuples, then for sufficiently large $n$, there exists a stochastic matrix taking the $n$-fold Kronecker power of each input distribution to the $n$-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples.

Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization.

We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via block encoding circuits, the input model for the quantum singular value transform and related algorithms. We demonstrate how to construct block encoding circuits based on an arithmetic description of the sparsity and pattern of repeated values of a matrix. We present schemes yielding different subnormalisations of the block encoding; a comparison shows that the best choice depends on the specific matrix. The resulting circuits reduce flag qubit number according to sparsity, and data loading cost according to repeated values, leading to an exponential improvement for certain matrices. We give examples of applying our block encoding schemes to a few families of matrices, including Toeplitz and tridiagonal matrices.