Semiconductor-based electron flying qubits: Review on recent progress accelerated by numerical modelling

The progress of charge manipulation in semiconductor-based nanoscale devices opened up a novel route to realise a flying qubit with a single electron. In the present review, we introduce the concept of these electron flying qubits, discuss their most promising realisations and show how numerical simulations are applicable to accelerate experimental development cycles. Addressing the technological challenges of flying qubits that are currently faced by academia and quantum enterprises, we underline the relevance of interdisciplinary cooperation to move emerging quantum industry forward. The review consists of two main sections: Pathways towards the electron flying qubit: We address three routes of single-electron transport in GaAs-based devices focusing on surface acoustic waves, hot-electron emission from quantum dot pumps and Levitons. For each approach, we discuss latest experimental results and point out how numerical simulations facilitate engineering the electron flying qubit. Numerical modelling of quantum devices: We review the full stack of numerical simulations needed for fabrication of the flying qubits. Choosing appropriate models, examples of basic quantum mechanical simulations are explained in detail. We discuss applications of open-source (KWANT) and the commercial (nextnano) platforms for modelling the flying qubits. The discussion points out the large relevance of software tools to design quantum devices tailored for efficient operation.


Introduction
Flying qubits are originally intended to serve as a communication link within a quantum computer 1 and represent a vital part of global road-maps towards secure data transmission -the so-called quantum internet 2 .
Recently, in-flight manipulations of photon-number states (so-called Fock states) have shown that the flying qubit architecture can also be applied as a stand-alone quantum processing unit 3 . Owing to this progress on photonic quantum-computation approaches 4,5 , flying qubits are typically associated with photons 6 . Employing so-called "time multiplexing" architectures, photonic quantum computing can in principle scale up to millions of qubits. The strongly probabilistic nature of photonic two-qubit gates renders the far-reaching photonic coherence however a double-edged sword making the realisation of photonic quantum computing challenging.
Though, a flying-qubit architecture can also be built on the basis of other quantum systems such as the electron 7 .
The charge of an electron causes Coulomb interaction with its electro-magnetic environment, which exposes its quantum properties to decoherence but enables well-controlled single-particle manipulations and multi-qubit coupling.
Historically, flying qubits stem from the research field of quantum optics which addresses -in opposition to wave optics -the granular nature of light. The need to control the ultimate grain of light -a single photon -led to the advent of the first single-photon source in 1974 8 and over several decades, various promising approaches have been developed 9 . A prominent example are deterministic single-photon sources based on quantum dots (QD) 10,11 . Since these photon emitters allow for efficient coupling to a nanophotonic cavity 12,13 and provide a high degree of indistinguishability 14,15 and brightness [16][17][18] , they turned out as highly suitable sources for quantum communication 6 . Yet, the extraction efficiencies of quantum dot based single-photon sources are not at the level to perform photonic quantum computing. Nevertheless, the demand for photonic components such as on-demand single-photon sources, quantum photonic processors and single-photon detectors fostered the emergence of start-ups such as Quandela 1 , QuiX 2 , or SingleQuantum 3 , to name a few. Photonic quantum computing is currently pursued by the start-up PsiQuantum 4 which uses heralded photons. Here parametric down conversion is used to create from a non-deterministic single photon source a pair of photons serving as signal and idler. Measuring the idler photon with an additional single-photon detector ensures that indeed a single photon (signal) has been generated -however, at the expense of additional hardware.
In order to implement photonic quantum computation, two-qubit gates are a necessity. As photons do not interact, operating two-qubit gates on physical qubits is very difficult. It is possible in theory to use non-linear effects such as Kerr effects, but these effects are so small that it is not possible in practice, at least in known media 5 . Another route is to use photon detectors and the associated projective measurement as a way to entangle photons. While photon detectors can indeed produce entanglement they do so in a probabilistic way, i.e. depending on the result of the measurement, the remaining state will be entangled or not. However, such probabilistic gates cannot be used on a large quantum computer as the probability for the correct circuit to be applied will decay exponentially with the number of two-qubit gates. Protocols have been proposed (such as the Knill-Laflamme-Milburn protocol) 4 trying to mitigate this issue and to enhance the probability for the correct gate to be applied. This approach comes however along with significant cost since many (n 1) ancilla photonic qubits are required to guarantee that the correct gate is applied (with an error probability of the order of 1/n) 5 . From this point of view, the development of non-photonic approaches represents a promising and competitive avenue.
For sake of completeness, let us also point out a different approach to photonic quantum computing based on so-called squeezed states. Here the qubits are generated as a superposition of multiple photons in a light pulse.
This approach is pursued by the start-up Xanadu 5 . It has conceptual similarity to the Leviton single-electron transport which will be presented below in section 2.3.
As for photons, electron quantum optics started by probing the discrete nature of the electrons. The granularity of electrons was undermined by the quantum fluctuations of the current 19 -the so-called shot noisesimilar to photon noise 20,21 . The first experiments mimicking textbook experiments of quantum optics, but with electrons, date back to 1999 with electronic Hanbury Brown and Twiss experiments [22][23][24] . These achievements were succeeded by the realisation of the first electronic Mach-Zehnder interferometer 25 , a quantum device that nicely shows up the wave nature of the electron and which is a key tool for qubit manipulation. All of these pioneering experiments that fostered the idea of electronic quantum control for computing applications have however been performed applying a DC current -meaning a continuous stream of billions of electrons. Only in 2007 the invention of the first single-electron sources 26 opened up the possibility of performing electronquantum-optics experiments at the single-particle level. This achievement triggered tremendous progress on electron quantum optics over the last two decades bringing various single-electron sources [26][27][28][29][30] that reach now emission efficiencies larger than 99% 30-34 -a value far superior to latest single-photon sources 11,16,18,35,36 .
Besides emission, also single-electron detection has significantly progressed. Whenever a single electron is captured on a sufficiently long timescale -of the order of a microsecond or more -also detection efficiencies well above 99% are achieved 31,32 . The efficient control on the single-particle level points out the large potential to exploit single flying electrons for quantum applications.
From measurements on charge qubits in stationary double quantum dots [37][38][39][40] , the coherence time of electron flying qubits is expected to be of the order of a few nanoseconds. Decoherence is, thus, the major obstacle for quantum implementations with single flying electrons. The young research field of electron quantum optics is however constantly producing new findings bringing the coherence properties of flying electrons closer to macroscopic scales [41][42][43][44] . From the quantum-computation perspective, the central figure of merit is the number of operations that can be performed within the coherence time. At present, this number is about 1000 for leading approaches such as superconducting qubits, trapped ions, or spin qubits [45][46][47][48][49] . In order to achieve such operation fidelity with electron flying qubits, a time control at the picosecond scale is required. Such ultrafast in-flight manipulation is presently being pursued in the FET-Open project UltraFastNano 50 .
The EU project UltraFastNano 6 aims to advance ultrafast operations in nanoelectronic devices to demonstrate the first electron flying qubit. The concept is similar to that of a photonic quantum computer but, instead of photons, electrons are used as carrier of quantum information. The project is a concrete example where the academic and the industrial sector are joining forces to develop and benchmark the tools that are required by the value chain of emerging quantum industry. Often, the initial market of tools for sufficiently advanced technologies is too small to be pursued by larger corporations, opening a niche-market opportunity for small and medium-sized enterprises (SME). This demand is satisfied by the industrial partner nextnano GmbH 7an SME based in Munich (Germany) with 12 employees and 300 customers in more than 35 countries -which develops a software tool calculating the quantum mechanical properties of semiconductor nanodevices. The

synergy of academic and industrial partners on numeric simulations and experimental implementations within
UltraFastNano fosters progress on electron flying qubits opening a novel branch of quantum industry.
In this article, we review three promising experimental routes towards electron-flying-qubit implementations and discuss the potential of numerical simulations to speed up experimental development cycles towards quantum-computing applications. The reviewed experimental pathways differ mainly in the way the electron qubit is transported. In particular, we address single-electron transport by means of a surface-acoustic wave, emission form a quantum-dot pump and Levitons. For each transport approach, we point out the specific aspects where numerical simulations are key to unveil efficient routes for follow-up implementations. Identifying these target aspects of cutting-edge electron-quantum-optics experiments, we finally present generic numerical simulations providing insights that are decisive for the development stages towards electron flying qubits. Confronting numerical simulations with latest experimental results, we point out the capability of the numerical simulations to guide experimental implementations faster to success.

Pathways towards the electron flying qubit
Different than for a classical bit where the states 0 and 1 basically correspond to the charging state of a capacitor, the electron flying qubit is defined via the presence of a single electron in two paths of transportation.
The quantum state of the thus-defined flying qubit can be depicted on a Bloch sphere as shown in Fig. 1a. The north and south pole of the sphere represent the classical states of the electron being in one of the two paths of transportation (0 and 1). The probability to end up in one of these states is represented via angular coordinates of the sphere θ and φ that make up the quantum state ψ: The thus-defined quantum state of the flying electron is fully controllable in an electronic Mach-Zehnder interferometry setup as sketched in Fig. 1b. The quantum interferometer hosts two regions where the paths of transportation approach each-other and couple via a narrow potential barrier (see horizontal lines). According to the potential landscape within this coupling-region -see Fig. 1c -, the flying qubit state undergoes periodic oscillations (angle θ) caused by coherent tunneling. In between the two tunnel-coupled regions, the flying qubit picks-up a quantum phase φ that is tunable via the potential along the two paths and the enclosed magnetic flux.
Employing a surface-gate defined quantum interferometer realised in a GaAs-based heterostructure (see Fig. 1d), basic quantum operations of such an electronic flying qubit have already been successfully demonstrated (see In semiconductor devices, a single electron is manipulable in a surface-gate-defined nanoscale structure such as a quantum dot or a waveguide. The majority of such implementations are performed within the two-dimensional electron gas (2DEG) formed near the surface of a GaAs-based heterostructure 7 , which have 7 www.nextnano.com typically coherence lengths of several tens of micrometers 41,42,51,52 . Applying a set of negative voltages on these gates, one can shape the potential landscape in the 2DEG and thus form and control the nanoscale devices. So far, there are two methods to guide the electrons along the desired paths: electrostatic waveguides 42,53 or -at high magnetic field -quantum-Hall edge states 25,54 . The 2DEG quality has been continuously improved 55,56 and combined with clever device design 41 , which allowed to push the phase coherence length up to several hundreds of micrometers in recent experiments 43 .
The availability of quantum dots serving as highly efficient single-electron sources and receivers led to the development of single-electron-transport techniques based on surface acoustic waves 28,29,57 and voltagemodulation pumping 58,59 . Fostered by technological progress and a growing understanding of the underlying physical mechanisms 60 , these approaches now achieve transfer efficiencies well above 99% [31][32][33][34] . The quantumdot-based transport systems represent the electronic counterpart to the deterministic single-photon source we have mentioned earlier. Besides that, different avenues have been explored such as the so-called Leviton that is a well-protected collective single-electron excitation generated by an ultrafast Lorentzian voltage pulse 30,61,62 . As the aforementioned photonic squeezed state that is a special kind of a coherent laser pulse, a Leviton represents a special form of a classical voltage pulse. The progress in these experimental routes -surface acoustic waves, electron pumps and Levitons -opened up the way to realise a flying-qubit platform with electrons instead of photons. In the following sections, we outline these three conceptually different approaches towards the electron flying qubit in more detail and discuss how numerical simulations have played a key role to interpret the experimental results guiding nanodevice design to the next generation.

Electron qubits surfing on a sound wave
In III-V semiconductor compounds, such as the presently discussed GaAs-based devices, sound is accompanied by an electric potential wave due to its piezoelectric properties allowing charge displacement 63 . At the first glance, acousto-electric transport seems a rather brute approach to move an electron qubit. A surface acoustic wave (SAW) has proven itself however as an efficient and well-controllable transport medium. Fig. 2a shows a schematic of the single-electron transport approach. A SAW is typically generated with an interdigital transducer (IDT) -a device that is well established in modern consumer electronics products 64 . Applying a finite, resonant input signal on the IDT, a SAW is emitted which then travels relatively slowly with a characteristic speed of about 3 µm ns −1 towards a surface-gate-defined nanoscale device 65 . The circuit is constructed on the basis of fully depleted transport channels whose ends are equipped with quantum dots (QD) -see Fig. 2b serving as highly efficient single-electron source and receiver. Each QD is equipped with an adjacent quantum point contact (QPC) allowing to trace its charge occupation. After loading a single electron at the source QD via a sequence of voltage variations on the corresponding surface gates, a SAW is emitted. The SAW train typically has a duration of tens of nanoseconds and wavelength of 1 µm. When arriving at the depleted transport channel, the potential modulation of the SAW forms a train of moving quantum dots propagating through the surface-gate-defined rail. After loading an electron at the source QD, this SAW train allows to shuttle the electron through the quantum rail to a distant receiver QD 28,29 .
The robustness of a SAW train enables acousto-electric transfer of a single electron in a nanoscale circuit approaching macroscopic dimensions. An experimental investigation of a 22-µm-long SAW-driven single-electron circuit consisting of two tunnel-coupled channels -see single-electron circuit in Fig. 2a -achieved single-shot transfer efficiencies larger than 99% 31 . Here, the exact sending position within the SAW train is controlled via the delay of a picosecond-scale voltage-pulse trigger applied on the source QD. Adjusting the potential landscape in the tunnel-coupled wire (TCW), it is possible to partition the electron wave function via directional coupling at will between the two transport channels. Fig. 2c shows an exemplary measurement of the single-shot transfer probability P as function of potential detuning ∆ in the TCW. The partitioning data shows a constantly high transfer efficiency despite the detuning ∆ of the TCW potential. The shape of the partitioning data bears important information on the time-evolution of the flying quantum state that is of central importance for quantum applications.
To draw the right conclusions from the single-shot partitioning data, numerical simulations are essential. In

Hot-ballistic electrons
On the contrary to the aforementioned acousto-electric transport approach, a lithographically-defined, but highly-tunable, quantum dot can also be employed to emit a single electron at high energy. These hot-electron sources for the controlled emission of single and multiple particles are of high interest from the perspective of higher-temperature operation and isolation from environment. In these devices, electrons can be emitted at an energy ∼ 100 meV above the Fermi energy, hence the cooling of the Fermi sea at millikelvin temperatures may not be necessary. Besides, hot electrons can be transmitted through a depleted channel, eliminating undesirable interactions with the Fermi sea. For the controlled emission of single and multiple particles, hotelectron sources are driven by strong potential modulation determining the timing of electron emission via slow, stochastic tunneling through a barrier. This process has a potential advantage of a high purity, meaning that the energy and time window into which the particles (or wave packets) are emitted fluctuates little between successive emissions. On the other hand, due to a large phase space available, the inelastic scattering rate during propagation can be high, leading to short decoherence time.
This is in contrast to the electrons confined in SAW potential or the Levitons, for which the electrostatic confinement or the limitation in available states (by the filled Fermi sea), respectively, protect the states from scattering processes. The nature of inelastic processes for hot electrons injected in GaAs/AlGaAs heterostructures has been investigated in Ref. [70][71][72][73][74][75][76] . At zero or small magnetic fields, the dominant scattering process is electron-electron interactions by the Fermi sea. For low-energy electrons (a few tens of meV above the Fermi energy), the electron-electron interactions continue to be the dominant process at higher magnetic fields applied in perpendicular to the plane of two-dimensional electron gas. For high-energy electrons (∼ 100 meV above the becomes the dominant process. Generally, the optical-phonon emission rate at high magnetic fields tends to be smaller than the electron-electron scattering rate at low magnetic fields, and therefore the ballistic transport length tends to be larger at higher magnetic fields. The suppression of backscattering processes due to chiral transport also contributes to a longer transport length at higher fields.
The technology to use hot electrons for electron quantum optics experiments is relatively new and not much information has been gathered regarding their suitability for applications in quantum information processing.
In order to explore the potential to use these hot-electron sources for the preparation of flying qubits, it is important to gain very precise knowledge of the relevant properties of the injected particles. These are (i) the time-and energy interval into which particles are emitted (ii) the purity of the wave packets, namely the precision in the time-and energy-interval in which the particle is detected in every driving cycle. At the same time it is crucial to analyse, (iii) how these properties are affected during the propagation of wave packets along depleted channels and to minimise a possible deterioration of the signal. These aspects hence need to be tested and control over them has to be obtained.
The detection of hot-ballistic electrons emitted by a single-electron source was made using a scheme shown in Fig The time-of-flight measurements used a time-gating technique to measure the electron arrival-time distribution 80 . This was later developed into a tomographic measurement of quasi-probability distribution in the energy-time phase space by controlling the ramp speed of gate voltage [81][82][83] . This measurement revealed an energy-time correlation of the distribution imprinted by the ramp speed of source energy state during the emission process (see Fig. 4) 82 . The projection of this distribution onto the time or energy axis gives the arrival-time or energy distribution of the emitted states. The purity of the observed distribution was only 0.04, and therefore the observed state is likely to be a mixed state. We note that the time and energy resolutions of the experiment in Ref. 82 were estimated to be σ t 0.3 ps and σ E 0.8 meV, giving σ t σ E 0.36 , implying that this method is capable of resolving the minimum uncertainty limit ( /2). Therefore the observed low purity is not due to poor measurement resolutions, but is likely due to noises in electron emission process from the source. In this set of experiment, the smallest arrival-time distribution observed was σ t,min 5 ps, and the smallest product of time and energy widths was ∼ 30 times larger than the minimum uncertainty limit (taking into account the energy-time correlation). A method has been proposed in Ref. 84 to emit each electron into Gaussian-shaped minimum uncertainty states. Another important experimental technique is the full counting statistics of the electron number partitioned by a beam splitter (a tunnel barrier). This has been demonstrated using noise measurements 85 or a trap coupled to a single-charge detector 32,86 .
The time scale of electron emission directly reflects in the width of the emitted wave packets. In quantum optics experiments with electrons, this time scale is important for the visibility of interference effects. In order to obtain these insights, the times at which single or multiple particles are emitted from the hot-electron sources can be studied analytically or numerically taking into account the time-dependent modulation of the single-particle energy levels, as well as of the shape of the tunneling barrier between quantum dot and conductor 80 . Importantly, theoretical studies have recently shown that Coulomb interaction between electrons on the dynamically driven quantum dot have a strong impact on the energies at which the particles are emitted. Most crucially this impact on the electron energies also directly influences the time scale on which the emission of the different particles takes place 87 . This theory furthermore predicts that the separation of time scales becomes particularly relevant for energy-dependent barriers 88 . Different schemes of how to read out these different relevant emission time scales using side-coupled detector dots 89 or nonadiabatic pumping schemes 90 have been suggested. This last scheme in particular also addresses relaxation times due to phonons during the emission process.
Based on the work described above, one can expect that realisations of Mach-Zehnder experiments will become possible with these types of sources, as suggested in Ref. 91,92 , similar to previous proposals for single-and two-particle interferometers for minimal-excitation single-particle sources [93][94][95] . Ref. 92 studied phase-averaging effects, which are particularly important for the temporally-short, high-energy single-electron wave packets.
As a result it becomes necessary to tune asymmetric interference arm lengths and delay time, which could be achieved by tuning the drift velocity. These analytical studies 91-95 assume an emission of pure states and ideal beam splitters, which are over-simplified compared to a realistic experiment. Hence, in order to improve the device characteristics, more realistic numerical modelling of these aspects could be a helpful complement.
While the electron coherence of hot electrons is yet to be demonstrated, the short length of electron wave packet in time domain and the ability to control their emission timing with a picosecond resolution can be useful in ultrafast electronics applications. In-situ voltage sampling under cryogenic environment has been demonstrated with a bandwidth potentially exceeding 100 GHz 96 . This technique was used to determine the precise gate voltage ramp profile for quantum tomography measurements 82 .

Leviton qubits flying over the Fermi sea
A conceptually different approach to realise an electron flying qubit is to generate a single-electron wave packet directly from the Fermi sea. This approach seems counterintuitive as a perturbation of the Fermi sea excites both electrons and holes and does in principle not allow the generation of a pure single-electron wave packet. L. S. Levitov and co-workers came up with an original idea to form a collective electron excitation flying over the Fermi sea without leaving a hole [97][98][99] . It has been shown that a voltage pulse of Lorentzian shape: generates a pure single-electron excitation -the so-called Leviton -if the amplitude V 0 and the full-width at half-maximum τ are chosen to match the quantization condition: where e is the elementary charge and h is Planck's constant. A Lorentzian pulse fulfilling this quantization condition is shown in Fig. 5a. Fig. 5b shows the corresponding excitation spectrum -meaning the occupation of states above and below the Fermi energy. The calculation shows a distribution that is characteristic for Leviton excitation. The collective wave function is only occupying the states right above the Fermi level (zero energy) forming a pure electronic excitation that is robust against relaxation. Single-shot detection represents a major task to realise an electron flying qubit with Levitons. For the aforementioned investigation of quantum oscillations, statistical measurements are sufficient. In order to control single Leviton qubits individually, it will be however necessary to detect the presence for each flying electron via capacitive coupling to an ultra-sensitive quantum detector. One possible implementation of such a quantum detector is a spin qubit that is operated in a regime where it is extremely sensitive to charge fluctuations 125 .
At present, this type of detector is capable of sensing a few electrons and enables a quantum non-demolition measurement 126 . The quantum detector is able to record the presence of a passing flying electron without perturbing its quantum state that can in turn be reused after detection for further quantum manipulations.
This aspect is an important advantage over single-photon detectors where the photon disappears after detection. Another possible implementation that has been put forward recently guides the flying electron through a meander structure which is capacitively coupled to two large metal electrodes. The passage of the flying electron beneath the two surface electrodes generates an oscillating voltage signal. This detector is expected to have a sub-electron sensitivity 127 and, when properly integrated into a quantum circuit, can also be adapted for quantum non-demolition measurements.
Another aspect of major importance is the scalablilty of surface-gate defined quantum-interferometer devices. Electron flying qubits using Levitons could allow one to implement an original architecture to build a universal quantum computer as schematized in Fig. 7. Although the architecture of Fig. 7 is a theorist view at this stage, it has very appealing features, in particular the fact that it is structurally different from the mainstream approach that uses localized qubits. Indeed in the mainstream approach, the hardware corresponds directly to each qubit: for instance for spin qubits, one needs a certain number of electrostatic gates per qubit to confine the electron, address it with microwaves and eventually measure its state. It follows that the hardware footprint is proportional to the number of qubits. In contrast, in this 'synchrotron-like' quantum computer, the flying qubits are stored in a loop and fast quantum routers are used to bring them to single-qubit gates, two-qubit gates, delay lines or measuring apparatus 134 . Hence, the hardware footprint can in principle be extremely small: a few quantum routers (one per type of gates or measurement) are sufficient to control an arbitrarily large number of qubits. The Leviton qubits are created on demand and one only needs a loop, which is large enough to hold Levitons while they go around it.
The second, perhaps more important, advantage of this architecture is the connectivity of the two qubit gates: using the delay line shown in the schematic, one could move the qubits so that any pair of flying qubits could be put next to each other and, hence, one could apply two qubit gates between any 2 qubits. This is again in contrast to the mainstream approach where each qubit being localized, it can only interact with a few other qubits, usually its nearest neighbours. Such a dramatic increase in connectivity could have deep consequences to reduce the overhead of quantum error correction and fault tolerant operations.
Another advantage of the flying qubit architecture for quantum computing is that qubits can easily be recycled: old qubits can be expelled from the loop and fresh ones incorporated while ancilla ones can be used to calibrate or test the various parts of the circuit in order to isolate and retune sections that are not performing correctly. This flexibility could again be very instrumental in quantum error correction in order to get rid of rare lethal errors. Indeed, in quantum error correction, not all errors are equal; some, even if rare, are lethal to the calculation 135 . In this respect, a long-term advantage of the flying qubit architecture is the possibility to correct these rare errors. Altogether a functional flying qubit technology could make quantum error correction affordable, bringing the millions of qubits which are required to build a fault tolerant quantum computer down to tens of thousands. Alternatively, the electron-flying-qubit approach could be used to complement other approaches by providing a 'quantum bus' that implements the missing long-range coupling between distant localised qubits. Experimentally, we are still in the early stage of the development of such a electron-flyingqubit platform. Yet, it is very interesting and appealing to see that it leads to a conceptually very distinct object from the localized qubit approach. This means in return that there is a lot of room for a new architecture to be invented to bypass the intrinsic limitations of the ones that are pursued so far.
To end this section on the experimental progress on electron flying qubits realised in semiconductor devices, we would also like to point out promising approaches to manipulating single electrons on other unique platforms.
Alternative to the here-described semiconductor devices, single electrons can be confined on the surface of liquid helium [136][137][138][139] or rare earth atoms such as neon, argon or krypton 140 . These systems provide a two- Once the static properties are well understood, one can proceed to simulate the propagation of the electron flying qubits, including voltage pulses and the associated Levitons, in real time. The TKWANT extension 144 of KWANT provides the necessary environment for such simulations (e.g., to study the role of Coulomb repulsion at the time-dependent mean field level). The next level is a proper treatment of many-body effects aiming to account for e.g. interactions between different Levitons or various relaxation and dephasing mechanisms (such as one electron decaying into two electrons and one hole). We note that there are no general purpose simulation approaches that can handle this problem in a "blackbox" way. At the top of the stack are "pure" quantum computer simulators where the actual underlying physics has been hidden and one simulates only the effective dynamics of the computational degrees of freedom (potentially with some extra noise or dissipation terms to account for the actual limitations of the devices). As indicated by the arrows on the schematic, the different parts of the stack provide parameters to calibrate the other levels. As one goes up the stack, one usually must give up some microscopic details in order for the computations to remain affordable. Therefore, the calibrations must be done with care for the errors not to accumulate. Below, we focus at first on the static simulation part of the stack with a special emphasis on the calibration of the simulations with respect to the experiments and on the modelling of real nanodevices. At the end of this section, we briefly address time-resolved and many-body simulations.

Static quantum mechanical simulations
Tuning a single qubit into optimal operation is so far a tedious task. An attempt to find such conditions trying various setups at random is time and resource consuming. In order to go easier beyond experimental proof of concept (also known as Technology Readiness Level (TRL) 3), it is thus crucial being able to predict the viability of a certain sample design prior to its physical realisation. As outlined above, precise potential calculations combined with dynamic quantum mechanical simulations are playing a key role in this regard enabling validation of electron-flying-qubit technology in the lab (TRL 4). Being able to predict the reliability of a certain sample geometry paves the way to implement and setup electron flying qubits in a reproducible manner -enabling validation of the technology in a demonstrative or even commercial setting (TRL 5 and 6).
Since the basic elements of the electron flying qubits (interconnects, TCWs and interferometers) are exploiting to a great extent single-particle physics, they require high-quality quantum mechanical simulations for one electron in complicated electrostatic potentials. The necessary information is obtained from a numerical solution of the stationary Schrödinger equation, see Eq. (4) below. The precise solution can be found by using a platform such as the nextnano software. Its advantages include the possibility to adapt the numerical procedure for different materials, various geometries of the nanoconductors and shapes of the gate-induced potentials.
Let us review the basics of the static quantum mechanical simulations, some features of the nextnano tools, and provide examples of how these tools can be used for calibration of the experiments and engineering the nanodevices.

Basic equations and methods of static single-particle simulations
Basic targets of the static quantum mechanical simulations include the study of the shape of electron wave functions and the energy-dependent transmission through one nanounit and through the entire circuit [145][146][147] .
The transverse profile (i.e. along the direction being perpendicular to the propagation direction) allows one to judge whether the quantum wires are close to the desirable setup and to control, e.g., the absence (presence) of tunnelling between two isolated (coupled) wires. Ballistic transmission of the electron through the circuit is even more important. When various units are connected, there are always spatial inhomogeneities which can result in reflection of the propagating electron. The reflection hinders the flying qubits from their normal operation and must be minimised as much as possible. To this end, one can numerically find an energy corresponding to transparency windows for a realistic circuit and to work further in a vicinity of this special energy range.
The quantum mechanical system can be modeled at different levels of approximations that range from a semiclassical description to an effective mass approximation to a multi-band k·p model. Considering conduction-band electrons within the single-band approximation, the envelope wave functions, ψ n , are solutions of the stationary Schrödinger equationĤ whereĤ is the Hamiltonian operator of the closed quantum system, E n are the energy levels defining the energy spectrum of the system, n are quantum numbers marking different single electron quantum states, and x = (x, y, z) is the space coordinate. The Hamiltonian operator is the sum of a kinetic energy operator (p) and a potential energy V (x), where the electron momentum operator is defined in the standard way asp = −i ∇. Here, (p, x) is the (possibly position-dependent) dispersion relation describing the momentum dependence of the electron energy which accounts for all effects governed by the crystalline lattice, and V (x) is the inhomogeneous potential in which the electron propagates.
V (x) contains the electrostatic potential and conduction band offsets at material interfaces. For example, in the simple case of a homogeneous isotropic material where the electrons move almost freely, one can use the effective mass approximation which yields (p) = (p 2 x + p 2 y + p 2 z )/2m * , where m * is the effective mass of the electron in the material.
The potential φ(x) describes the electrostatics within the system and is the solution of the Poisson equation where ε(x) is the material-dependent permittivity, and ρ(x) is the charge density throughout the system. This charge density is given by where n(x) and p(x) are the electron and hole densities, and N + D (x) and N − A (x) are the ionized donor and acceptor concentrations, respectively, e is the (positive) elementary charge, and ρ fixed (x) contains immobile space or surface charges.
Here, the electron density n(x) explicitly depends on the energy levels E n and envelope wave functions ψ n from Schrödinger's equation, Eqs. (4,5). For a finite system at equilibrium, the electron density is given by where E F is the Fermi level (or chemical potential), T is the temperature, and k B is the Boltzmann constant.
Thus, the electrostatic potential φ(x) depends on the energy levels E n and wave functions ψ n , but also enters the Schrödinger equation, Eqs. (4,5), as part of the potential energy operator V (x). This shows that the Schrödinger equation and Poisson equation (Eq. (6)) are coupled and need to be solved self-consistently.
The self-consistently obtained spectrum and wave functions can be used further to calculate quantities which explain and describe quantum transport through various nanodevices connected to external leads. Two such quantities are (i) the partial local density of states (pLDoS), n(x, y, E), and (ii) the energy dependent transmission, T ij (E). The pLDoS is the probability to find in a given space-point the propagating electron which has a given energy, E, and has been injected in a given lead. We note in passing that the local density of states is the sum of the pLDoS over all leads. Hence, the coordinate dependence of pLDoS illustrates how the electron with a given energy propagates through the device 148 . The energy dependent transmission, T ij (E), is determined by the probability for the electron which is injected into lead i to reach lead j.
The pLDoS and the transmission from one lead to another can be found by using the retarded Green's  The main focus is on the simulation of the quantum mechanical properties of such devices. The nextnano's core product, the nextnano++ software 142 , is a 3D Schrödinger-Poisson-Current/CBR solver for nanotransistors, LEDs, laser diodes, photodetectors, quantum dots, nanowires, solar cells and qubits. The second product, nextnano.NEGF 152 , is a quantum transport solver targeting quantum cascade lasers and resonant tunneling diodes. The nextnano software (including its early versions) has been successfully used to optimise the design of semiconductor-based (charge and spin) qubits [156][157][158][159][160][161] . Below, we focus on several important applications of this software for engineering the electron flying qubits.
Appropriate models for quantitative simulations: Quantum devices made from GaAs semiconductor heterostructures can easily be engineered by proper design of the gate geometry. To ensure the best performance of the electronic device -that is to find the most suitable gate geometry -it is crucial to know the exact electrostatic potential landscape generated by the electrostatic gates. This requires to take into account the material parameters such as 2DEG density and mobility, dopants concentration, induced surface charges, etc.
Traditionally, the workflow to determine the optimum gate geometries for a given heterostructure has been an iterative process between device fabrication in clean room facilities and low-temperature characterisations. This is immensely time consuming and resource demanding. The ideal workflow is presented in Fig. 9 where the iterative process takes place mainly at the modelling stage, before the device fabrication.
To find an accurate model for quantitative simulations, Chatzikyriakou et al. 162   In the same spirit, one can use such simulations to calculate the potential variations seen by the electrons within the 2DEG. An example of a complex quantum device with a tunnel-coupled wire from Takada et al. 31 is shown in Fig. 11. Using the exact gate geometries and voltages from the experiment, the electrostatic simulations reveal the variation of the potential along the path which an electron would follow before entering the tunnel-coupled region (black line). In these experiments, the single electron is excited to higher energy states which were attributed to the abruptness in the potential landscape at this location. This undesired excitation could be mitigated by optimising the device geometries thanks to quantitative modellings.
The shape of the electrostatic potential in the 2DEG plane is input to further calculations of the energydependent transmission function which corresponds to the probability that an electron is reflected or transmitted along the different paths in the flying qubit structure. One-dimensional cuts through this potential in the uncoupled wires (blue) and within (red) the tunnel-coupled regions are shown in Fig. 11b. One can see parabolically shaped double-well confinement potentials, cf. Fig. 1c. Such potentials and the interplay between symmetric and anti-symmetric states with respect to the direction perpendicular to the propagation direction have been analyzed numerically in Ref. 145 where detailed features of the transport measurements such as in-phase and anti-phase oscillations of the two output currents as well as a smooth phase shift when sweeping a side gate have been reproduced. By injecting an electron into the upper rail |0 , the wave function will evolve into a superposition of symmetric and anti-symmetric states. While travelling through the interaction region, the wave function of the electron will then pick up a phase and will evolve into a superposition of |0 and |1 7 as shown in Fig. 12 by a simulation example.
Simulations of pLDoS and transmission through nanodevices: Eq. (1) describes how the ideal electron flying qubit is expected to operate: The electron is injected in either the upper or lower incoming channel (see Fig. 1b) and propagates without reflection through the quantum device, where the electron state is rotated in the Hilbert space. This rotation can be illustrated with the help of the Bloch sphere, Fig. 1a. Angles θ and φ are generated by the tunneling regions and interferometers, respectively. The output state is a coherent superposition of the states |0 and |1 . It is controlled by the tunneling barriers and either by the magnetic field or by the asymmetric bias of the interferometer. Since using the magnetic field is technologically inconvenient, we focus in this section on the asymmetrically biased interferometers.
There are two points which have fundamental importance for engineering the electron flying qubits. Namely, one needs an experimental setup where, on one hand, the reflection is reduced to a minimum and, on the other hand, the sensitivity of the electron state to the respective gate voltages is high. Let us explain how using the nextnano software helps to find such a setup.
To this end, nextnano enables the calculation of the pLDoS and the transmission of the nanodevices using the CBR method 149,153 . In the following, we demonstrate such calculation via two cases that are the central building blocks of the electron-flying-qubit architecture. Firstly (see left columns of Fig. 12 and Fig. 13), we address electron propagation through a tunnel-coupled wire (TCW). We remind readers that here at y = 0 a narrow potential barrier is present between two transport channels that allows for coherent tunneling of the electron. Secondly (see right columns of Fig. 12 and Fig. 13), we investigate electron propagation through a quantum interferometer (compare Fig. 1b,d) where two TCWs embrace an AB island enabling full control of the quantum state via magnetic and geometric phase manipulations. The potential profile of both models is shown in Fig. 13a,d. For such study of a nanoscale device, materials properties have to be specified in the nextnano input file such that the potential energies are properly set. We have used GaAs for the device and leads, and adjusted the potential energy in different regions representing high insulating barriers (red lines), tunneling barriers (light blue lines), and gates (green regions). The potential energy at the gates can be tuned by applying gate voltages and the strengths of the barriers are given in Fig. 13 and its caption.
Several examples of the pLDoS are shown in Fig. 12 which have been generated using the nextnano software.
In these simulations, the electron has been injected into lead no. 1 with a given energy that is E = 9.2 meV for TCW and E = 7.5 meV for the quantum interferometer (TCW-AB-TCW). Slices of the pLDoS are shown at these energies.
Let us first discuss the TCW (left column of Fig. 12) for different voltages (V T ) applied on the tunnel-barrier gate. The TCW is able to change only the angle θ -see Eq.
(1). Thus, the output state can be written as |Ψ TCW ∝ cos(θ/2)|0 + sin(θ/2)|1 . When the tunneling barrier is absent (Fig. 13a), one observes at the output an equal superposition of |0 and |1 which can occur at either θ = π/2 or θ = 3π/2, both corresponding to two points on the equator of the Bloch sphere (Fig. 1). Since a small increase of V T drives the output state to |0 ( Fig. 13b) corresponding to the north pole of the Bloch sphere, we conclude that in barrier-less setup θ = 3π/2.
With further increasing V T , the output state becomes |1 , i.e. the south pole of the Bloch sphere (Fig. 13c, θ = π) and returns to the equator (Fig. 13d, θ = π/2). The latter point is the opposite equator point to that of the barrier-less setup. When the tunneling barrier becomes high (Fig. 12d) one enters the regime of two fully decoupled transport channels with output state |0 . Clearly, these coherent tunnel oscillations of the electron wave function manifest itself in the quantum oscillations of the transmission via TCW (see Fig. 13c).
Secondly, let us focus on electron propagation through the quantum interferometer ( Fig. 12f-j). Increasing the potential on a side gate of the lower transport channel (V g ) modifies the geometric phase of the electron's quantum state and changes the second angle φ in Eq. (1). This, in turn, causes coherent oscillations between the output terminals 3 and 4, i.e. oscillations between output states |0 and |1 , and results in quantum oscillations of the TCW-AB-TCW transmission (see Fig. 13f). The tunnel regions in the example of Figs. 12f-j are the same and each of them changes the angle θ by π/2. This is apparent from Fig. 12a where V g = 0 and φ = 0: the output state after successive rotation in two TCW regions is |1 , i.e. the total change of θ in the TCWs is π. Therefore, each individual connection changes θ by π/2. This allows one to approximate the output state as |Ψ TCW−AB−TCW ∝ (e iφ − 1)|0 + (e iφ + 1)|1 7 . Similar to the analysis of the pLDoS in the TCW, we can now trace rotations of the electron state with increasing V g . Two output states shown in Fig. 12g,i correspond to two opposite points on the equator of the Bloch sphere with φ = π/2 and φ = 3π/2. The south pole of the Bloch sphere is reached in Fig. 12j despite a substantial blockage of the lower transport path by the strong gate potential of the side gate.
To conclude the discussion of the pLDoS, we note that a detailed analysis of the flying qubit geometry in relation with experiments has also been performed using a combination of the theoretical approach and the KWANT software, see Ref. 145 .
Having discussed electron propagation at a qualitative level via the pLDoS, let us now investigate electron propagation in a more quantitative way employing energy dependent transmission, T ij (E) from the first lead to the output leads no. 3 and no. 4. First, we study the dependence on the injection energy (Fig. 13b,e). The energy of the incoming electron is counted from the potential energy of the lead no. 1. All device parameters are fixed at this stage. Transmission is zero if the electron energy is smaller than the energy of propagating states of the device, that is below 6.2 meV for the chosen parameters. Above this threshold, transmission starts to grow. However, it is first accompanied by substantial reflection of the electron to the input leads no. 1 and no. 2 which are described by T 11 and T 12 (not shown). This is apparent in the central panels of Fig. 13, up to an energy of about 7 meV, where the total transmission, T total = T 13 + T 14 (magenta lines), is smaller than the ideal value, T total < T ideal = 1. In a second step, we have identified the energy at which T total 1 (i.e. reflection is minimised) but T 13 , T 14 = 0, 1. In this regime, we expect strong sensitivity of the electron state |ψ , Eq. (1), to the gates and the barriers of the quantum device. Two examples of these energies are shown by red dashed vertical lines and are studied at the second stage of the simulations. Fixing these two energies for the TCW and the quantum interferometer, we have studied the dependence of the transmission on the potential of the tunnel barrier, V T (Fig. 13c), and the side gate, V g (Fig. 13f). The goal of this stage is twofold: We identify regions of the parameters where T total (magenta lines) is close to the ideal value of 1 and, simultaneously, the rotation of the electron states is pronounced. The latter condition is fulfilled by the crossover between regimes T 13 < T 14 and T 13 > T 14 . The vicinity of the crossover can be chosen as an operation range of the qubit (or of the qubit element), provided the reflection is almost absent. In order to find optimal parameters which provide both ideal transmission and range for manipulation of the quantum phase, simulations such as the here-discussed case are helpful to identify experimentally relevant voltage ranges. The here-discussed simulations indicate that sufficient control is obtained in TCW and AB regions with the length ≥ 500 nm and further allow to identify optimal operation voltages. Let us discuss these aspects in more detail for the TCW and the quantum interferometer.
TCW: A simple phenomenological scattering theory predicts that T 13 cos 2 (δφ TCW /2), T 14 sin 2 (δφ TCW /2), δφ TCW ≡ δkL TCW , where δk = k 1 − k 2 is the difference of wave vectors of two modes with the lowest energy, which support the transmission, and L TCW is the effective spatial scale of the region, where the tunneling takes place 7 . The quantum phase δφ TCW is expected to grow with increasing the length of the tunneling barrier, L tun , which is equal to 300 nm in the example of the left column of Fig. 13. We distinguish L TCW and L tun since the former depends on the shape of the electrostatic potential inside the device. Therefore, one may expect L TCW L tun . This inequality has been confirmed by a comparison of the scattering theory (green and cyan dots in Fig. 13c) with the outcome of the true 2D simulations (solid lines in the same figure). L TCW is the only adjustable parameter of this comparison. The value of δk has been found by using the dispersion relation of almost free electrons propagating in a semiconductor, E 1,2 = ( k 1,2 ) 2 /2m * . Energy levels E 1,2 have been obtained from 1D simulations of the spectrum at the 1D transverse cross section in the center of the device (dashed line in Fig. 13a). Interestingly, the ratio L TCW /L tun is almost insensitive to the transverse size of the device. An excellent agreement between the 2D simulations and the scattering theory suggests that the latter can be used as a compact model of TCW in simulations of more complicated circuits. Such a simplification will allow one to minimize computer resources needed for the simulations. The inset in Fig. 13c shows that the range of V T , where the quantum oscillations occur, shrinks with making L tun , and correspondingly the space for the quantum interference, smaller. Since T total is very close to 1 (no reflection) in the entire range 0 < V T < 1 eV, such an idealized qubit would operate properly in a vicinity of any crossover point where T 13 T 14 , e.g. V T ∼ 0.1 eV or V T ∼ 0.33 eV.
Interferometer: When the electron modes propagate through the upper and lower arms of the electrostatic version of the Aharanov-Bohm interferometer, they acquire a relative phase which governs the quantum interference. If the interferometer is connected directly to the leads, the transmission through the device can be estimated as T AB = cos 2 (e δV τ AB /2 ) 165 . Here, τ AB is the flight time of the electron through the unit.
In the ballistic case, it is the ratio of the interferometer length over the electron velocity, τ AB = L AB /v. We have introduced the relative total potential, δV = V u − V l (integrated over the upper, V u , or lower, V l , arm), which the electron feels inside the interferometer. The interference oscillations are more pronounced in longer devices, compare results presented in Fig. 13f and its inset. To avoid complexity, we do not discuss here semiphenomenological analytical calculation of transmission for the composite TCW-AB-TCW device and do not compare the scattering theory with the 2D simulations. Similar to the TCW device, it is useful to tune V g to the vicinity of the crossover point where T 13 T 14 , i.e. either V g ∼ 0.1 eV or V g ∼ 0.5 eV in the example of Fig. 13f. Note, however, that T total < 1 in both cases. Clearly, the preference should be given to the regime with smaller reflection, i.e. the second crossover point in the simulated example.
The above predictions of numerical simulations serve as important input for experimental realisations of the flying qubit. Integrating the simulation additionally in a feedback loop of the workflow (Fig. 9) would enable to find optimised device geometries tailored to the different approaches such as Levitons or SAW. To conclude this section, we would like to mention that the applications of the nextnano software can be very broad since it can straightforwardly be adapted for modelling devices made from other semiconductor materials (e.g., industrially highly relevant SiGe), for including effects of the magnetic field on the interferometer, for mimicking dephasing and decoherence with the help of the artificially connected leads, to name just a few.

Low energy time-resolved and many-body simulations
In Note that there are no general purpose simulation approaches that can handle many-body problems in an exact and systematic way except in very particular cases. Most approaches rely on some approximation scheme whose validity must be checked a posteriori. A promising route followed by some of us to design a systematic method with a controlled accuracy uses calculations of high order processes (i.e. processes where electrons interact strongly) made possible by the use of a machine learning approach to evaluate the corresponding high dimensional integrals 168 .

Conclusion and outlook
The realisation of flying qubits with single electrons opens a novel, viable route of quantum technology with considerable potential for quantum-computation applications. In this review we introduced the novel electron-flying-qubit approach and discussed three equally promising transport techniques -surface acoustic waves (SAW), hot-electron emission from quantum-dot pumps and Levitons -which are rapidly advancing.
Owing to similarities between the different approaches -such as emission from a gate-defined quantum dot in SAW-transport and the electron pump -we suspect that progress in one-field will also drive the others.
Based on latest progress and relevant simulation cases, we showed that numerical modelling of quantum devices is decisive to speed up experimental deployment cycles towards the first implementation of an electron flying qubit. We anticipate that automatised optimisation of the device design via numerical modelling will enable nanofabrication tailored for efficient quantum operations.
In order to make the electron flying qubit competitive with cutting-edge approaches in the field of quantum         Equipotential lines are shown as continuous lines. Vertical cuts (blue and red) show the double-well potential before and within the tunnel-coupled region. The black line represents the path which follows the minimum in the potential landscape. Figure 12: nextnano simulations of the electron partial local density of states in the tunnel-coupled wire (TCW: panels (a-e)) and the TCW -Aharonov-Bohm interferometer -TCW nanodevice (TCW-AB-TCW: panels (f-j)). Both devices are connected to four terminals (marked by white numbers). The background shows the potential landscape defined by the voltage on the surface gates, cf. Fig. 13(a,d). The electron with a given energy (E = 9.2 meV for TCW and E