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The connection between Europe and Asia has always been important for the flow of materials and intellectual goods. Hungary is in an important position in Central Europe to connect different key ports and logistics hubs. Therefore, this article aims to analyse the opportunities and impacts of the New Silk Road initiative on the supply chain and transportation network in Hungary. This result showed that the existence of the New Silk Road gives huge opportunities for different supply-chain-related activities and initiatives, especially for the Hungarian automotive industry. Specifically, the New Silk Road has the potential to enable the advantages and positive impact of rail transport activities in Hungarian automotive supply chains by providing a long-term competitive solution as an alternative to maritime transport while mitigating several related risks and increasing the domestic supply chain’s resilience. Consequentially, the New Silk Road initiative may affect the supply chains and transportation route networks of several European and Asian nations, including Hungary.

In our increasingly globalized world, access to the right products and food is a central issue for all consumers. An alternative form of supply are local products, which can meet all or part of consumers’ needs. These local products are local specialties and non-mass products, which are tied to local features, technologies, culture, and identity. Supply chains can also help to improve the viability and sustainability of producers. In order for these short supply chains to work well, there is a need for continuous feedback and evaluation, therefore supplier evaluation forms could act as the starting point of these chains. The supplier evaluation form can serve to support and develop these supply chains at the local level. The aim of this study is to present a short supply chain from an intermediary’s point of view based on a case study. This method enables to gain a broader insight into the operation of local food chains. As the case study reveals, it is recommended for the members of these chains to apply supplier evaluation forms in order to maintain a better relationship between the partners. According to the experiences and conclusions a supplier evaluation system is defined by the authors.

Realizing the potential of quantum computing requires sufficiently low logical error rates 1 . Many applications call for error rates as low as 10 −15 (refs. 2 – 9 ), but state-of-the-art quantum platforms typically have physical error rates near 10 −3 (refs. 10 – 14 ). Quantum error correction 15 – 17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device 18 , 19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits. Repetition codes running many cycles of quantum error correction achieve exponential suppression of errors with increasing numbers of qubits.

A fault-tolerant error correction (FTEC) protocol with a high error suppression rate and low overhead is very desirable for the near-term implementation of quantum computers. In this work, we develop a distance-preserving flag FTEC protocol for the [[49,1,9]] concatenated Steane code, which requires only two ancilla qubits per generator and can be implemented on a planar layout. We generalize the weight-parity error correction (WPEC) technique from [Phys. Rev. A 104, 042410 (2021)] and find a gate ordering of flag circuits for the concatenated Steane code which makes syndrome extraction with two ancilla qubits per generator possible. The FTEC protocol is constructed using the optimization tools for flag FTEC developed in [PRX Quantum 5, 020336 (2024)] and is simulated under the circuit-level noise model without idling noise. Our simulations give a pseudothreshold of \\(1.64 \\times 10^{-3}\\) for the [[49,1,9]] concatenated Steane code, which is better than a pseudothreshold of \\(1.43 \\times 10^{-3}\\) for the [[61,1,9]] 6.6.6 color code simulated under the same settings. This is in contrast to the code capacity model where the [[61,1,9]] code performs better.

The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance \\(d\\) that can correct up to \\(t=\\lfloor(d-1)/2\\rfloor\\) errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated \\(t+1\\) times in a row; in the worst-case scenario, \\((t+1)^2\\) rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than \\((t+3)^2/4-1\\) rounds and \\((t+3)^2/4-2\\) rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance \\(d\\), our FTEC protocols with the adaptive schemes require no more than \\(d\\) rounds on average.

Lookup table decoding is fast and distance-preserving, making it attractive for near-term quantum computer architectures with small-distance quantum error-correcting codes. In this work, we develop several optimization tools that can potentially reduce the space and time overhead required for flag fault-tolerant quantum error correction (FTQEC) with lookup table decoding on Calderbank-Shor-Steane (CSS) codes. Our techniques include the compact lookup table construction, the Meet-in-the-Middle technique, the adaptive time decoding for flag FTQEC, the classical processing technique for flag information, and the separated \\(X\\) and \\(Z\\) counting technique. We evaluate the performance of our tools using numerical simulation of hexagonal color codes of distances 3, 5, 7, and 9 under circuit-level noise. Combining all tools can result in more than an order of magnitude increase in pseudothreshold for the hexagonal color code of distance 9, from \\((1.34 \\pm 0.01) \\times 10^{-4}\\) to \\((1.42 \\pm 0.12) \\times 10^{-3}\\).

2D compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the X-Shor and Z-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance \\(L\\). However, the current analytical lower bound on the threshold for the rotation angle \\(\\theta\\) is \\(|\\sin(\\theta)| < 1/L\\), which linearly vanishes in \\(L\\) instead of being constant. We show that this lower bound is achievable by the Z-Shor code which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families with thresholds near the rotated surface code's numerically established coherence threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the Z-Shor codes and the X-Shor codes.

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC). Concretely, dynamical phases can be defined in periodically driven many-body localized systems via the concept of eigenstate order. In eigenstate-ordered phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, wherein few select states can mask typical behavior. Here we implement a continuous family of tunable CPHASE gates on an array of superconducting qubits to experimentally observe an eigenstate-ordered DTC. We demonstrate the characteristic spatiotemporal response of a DTC for generic initial states. Our work employs a time-reversal protocol that discriminates external decoherence from intrinsic thermalization, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. In addition, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to study non-equilibrium phases of matter on current quantum processors.

Interaction in quantum systems can spread initially localized quantum information into the many degrees of freedom of the entire system. Understanding this process, known as quantum scrambling, is the key to resolving various conundrums in physics. Here, by measuring the time-dependent evolution and fluctuation of out-of-time-order correlators, we experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We engineer quantum circuits that distinguish the two mechanisms associated with quantum scrambling, operator spreading and operator entanglement, and experimentally observe their respective signatures. We show that while operator spreading is captured by an efficient classical model, operator entanglement requires exponentially scaled computational resources to simulate. These results open the path to studying complex and practically relevant physical observables with near-term quantum processors.

Purine bases and their bioisosteric analogs are widely used as building blocks in combinatorial chemistry. Recently a great number of fused pyrimidine derivatives became known as potential drug molecules against various types of proliferative diseases, caused by over-expression of protein kinases [1]. One of the most important compound