In recent years, quantum Boltzmann methods have gained more and more interest as they might provide a viable path toward solving fluid dynamics problems on quantum computers once this emerging compute technology has matured and fault-tolerant many-qubit systems become available. The major challenge in developing a start-to-end quantum algorithm for the Boltzmann equation consists in encoding relevant data efficiently in quantum bits (qubits) and formulating the streaming, collision and reflection steps as one comprehensive unitary operation. The current literature on quantum Boltzmann methods mostly proposes data encodings and quantum primitives for individual phases of the pipeline, assuming that they can be combined to a full algorithm. In this paper, we disprove this assumption by showing that for encodings commonly discussed in the literature, either the collision or the streaming step cannot be unitary. Building on this landmark result, we propose a novel encoding in which the number of qubits used to encode the velocity depends on the number of time steps one wishes to simulate, with the upper bound depending on the total number of grid points. In light of the non-unitarity result established for existing encodings, our encoding method is to the best of our knowledge the only one currently known that can be used for a start-to-end quantum Boltzmann solver where both the collision and the streaming step are implemented as a unitary operation.

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In this paper we present a scalable algorithm for fault-tolerant quantum computers for solving the transport equation in two and three spatial dimensions for variable grid sizes and discrete velocities, where the object walls are aligned with the Cartesian grid, the relative difference of velocities in each dimension is bounded by 1 and the total simulated time is dependent on the discrete velocities chosen. We provide detailed descriptions and complexity analyses of all steps of our quantum transport method (QTM) and present numerical results for 2D flows generated in Qiskit as a proof of concept. Our QTM is based on a novel streaming approach which leads to a reduction in the amount of CNOT gates required in comparison to state-of-the-art quantum streaming methods. As a second highlight of this paper we present a novel object encoding method, that reduces the complexity of the amount of CNOT gates required to encode walls, which now becomes independent of the size of the wall. Finally we present a novel quantum encoding of the particles' discrete velocities that enables a linear speed-up in the costs of reflecting the velocity of a particle, which now becomes independent of the amount of velocities encoded. Our main contribution consists of a detailed description of a fail-safe implementation of a quantum algorithm for the reflection step of the transport equation that can be readily implemented on a physical quantum computer. This fail-safe implementation allows for a variety of initial conditions and particle velocities and leads to physically correct particle flow behavior around the walls, edges and corners of obstacles. Combining these results we present a novel and fail-safe quantum algorithm for the transport equation that can be used for a multitude of flow configurations and leads to physically correct behavior. We finally show that our approach only requires O(n_wn_g²+dn_t^vn_vmax²) CNOT gates, which is quadratic in the amount of qubits necessary to encode the grid and the amount of qubits necessary to encode the discrete velocities in a single spatial dimension. This complexity result makes our approach superior to state-of-the-art approaches known in the literature.

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Noisy intermediate-scale quantum computers (NISQ) are computing hardware in their childhood, but they are showing high promise and growing quickly. They are based on so-called qubits, which are the quantum equivalents of bits. Any given qubit state results in a given probability of observing a value of zero or one in the readout process. One of the main concerns for NISQ machines is the inherent noisiness of qubits, i.e., the observable frequencies of zeros and ones do not correspond to the theoretically expected probability, as the qubit states are subject to random disturbances over time and with each additional algorithmic operation applied to them. Models to describe the influence of this noise exist.In this study, we conduct extensive experiments on quantum noise. Based on our data, we show that existing noise models lack important aspects. Specifically, they fail to properly capture the aggregation of noise effects over time (or over an algorithm's runtime), and they are underdispersed. With underdispersion, we refer to the fact that observable frequencies scatter much more between repeated experiments than what the standard assumptions of the binomial distribution would allow for. Based on these shortcomings, we develop an extended noise model for the probability distribution of observable frequencies as a function of the number of gate operations. The model roots back to a known continuous random walk on the (Bloch) sphere, where the angular diffusion coefficient can be used to characterize the standard noisiness of gate operations. Here, we superimpose a second random walk at the scale of multiple readouts to account for overdispersion. Further, our model has known, explicit components for noise during state preparation and measurement. The interaction of these two random walks predicts theoretical, runtime-dependent bounds for probabilities. Overall, it is a three-parameter distributional model that fits the data much better than the corresponding one-scale model (without overdispersion), and we demonstrate the better fit and the plausibility of the predicted bounds via Bayesian data-model analysis.

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The past years have seen a surge in quantum algorithms for computational fluid dynamics (CFD). These algorithms have in common that whilst promising a speed-up in the performance of the algorithm, no specific method of measurement has been suggested. This means that while the algorithms presented in the literature may be promising methods for creating the quantum state that represents the final flow field, an efficient measurement strategy is not available. This paper marks the first quantum method proposed to efficiently calculate quantities of interest (QoIs) from a state vector representing the flow field. In particular, we propose a method to calculate the force acting on an object immersed in the fluid using a quantum version of the momentum exchange method (MEM) that is commonly used in lattice Boltzmann methods to determine the drag and lift coefficients. In order to achieve this we furthermore give a scheme that implements bounce back boundary conditions on a quantum computer, as those are the boundary conditions the momentum exchange method is designed for.@en

The rapid growth of quantum technologies requires an increasing number of physicists, computer scientists, and engineers who can work on these technologies. For educating these professionals, quantum mechanics should stop being perceived as incomprehensible. In this paper we contribute to this change by presenting a pedagogical model for explaining Grover's search algorithm, a prominent quantum algorithm. This model visualizes the three main steps of Grover's algorithm and, in addition to explaining the algorithm itself, introduces three key principles of quantum mechanics: superposition, interference, and state collapse at measurement. The pedagogical model, visualized by a video, is called the ant colony maze model. It represents the search problems as finding the exit of a maze, and visualizes Grover's search algorithm as a strategy by which a colony of ants finds that exit.@en

In this paper we present an approach to find quantum circuits suitable to mimic probabilistic and search operations on a physical NISQ device. We present both a gradient based and a non-gradient based machine learning approach to optimize the created quantum circuits. In our optimization procedure we make use of a cost function that differentiates between the vector representing the probabilities of measurement of each basis state after applying our learned circuit and the desired probability vector. As such our quantum circuit generation (QCG) approach leads to thinner quantum circuits which behave better when executed on physical quantum computers. Our approach moreover ensures that the created quantum circuit obeys the restrictions of the chosen hardware. By catering to specific quantum hardware we can avoid unforeseen and potentially unnecessary circuit depth, and we return circuits that need no further transpilation. We present the results of running the created circuits on quantum computers by IBM, Rigetti and Quantum Inspire.

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