Spin shuttling has crystalized as a powerful and promising tool for establishing intermediate-range connectivity in semiconductor spin-qubit devices. Although experimental demonstrations have performed exceptionally well on different materials platforms, the question of how to handle areas of low valley splitting in silicon during shuttling remains unresolved. In this work, we explore the possibility of utilizing the valley degree of freedom, particularly in regions of low valley splitting, to allow mobile spin qubits to be shuttled through an occupied stationary quantum dot, thereby leapfrogging over the stationary electron. This not only grants a more enriched mobility for shuttled electrons, as it opens new possible routing paths, but also enables the implementation of an entangling SWAP$^γ$ two-qubit gate operation in the process. Simulating this process for different sets of parameters, we demonstrate the feasibility of such an operation and offer a unique use case for otherwise precarious regions of a quantum processor chip and propose a possible extension to the set of possible operations for silicon spin qubit devices.
Spin shuttling has crystalized as a powerful and promising tool for establishing intermediate-range connectivity in semiconductor spin-qubit devices. Although experimental demonstrations have performed exceptionally well on different materials platforms, the question of how to handle areas of low valley splitting in silicon during shuttling remains unresolved. In this work, we explore the possibility of utilizing the valley degree of freedom, particularly in regions of low valley splitting, to allow mobile spin qubits to be shuttled through an occupied stationary quantum dot, thereby leapfrogging over the stationary electron. This not only grants a more enriched mobility for shuttled electrons, as it opens new possible routing paths, but also enables the implementation of an entangling SWAPγ two-qubit gate operation in the process. Simulating this process for different sets of parameters, we demonstrate the feasibility of such an operation and offer a unique use case for otherwise precarious regions of a quantum processor chip and propose a possible extension to the set of possible operations for silicon spin qubit devices.
Abstract The scalability and power of quantum computing architectures depend critically on high-fidelity operations and robust and flexible qubit connectivity 1–3 . In this respect, mobile qubits are particularly attractive as they enable dynamic and reconfigurable qubit arrays. This approach allows quantum processors to adapt their connectivity patterns during operation, implement different quantum error correction codes on the same hardware and optimize resource use through dedicated functional zones for specific operations such as measurement or entanglement generation 4–7 . Such flexibility also relieves architectural constraints, as recently demonstrated in atomic systems based on trapped ions 4,5 and neutral atoms manipulated with optical tweezers 6,7 . In solid-state platforms, highly coherent shuttling of electron spins was recently reported 8,9 . A key outstanding question is whether it may be possible to perform quantum gates directly on the mobile spins. Here we demonstrate two-qubit operations between two electron spins carried towards each other in separate travelling potential minima in a semiconductor device. We find that the interaction strength is highly tunable by their spatial separation. When we shuttle the two spins towards the centre by 120 nm each for a total displacement of 240 nm, we achieve an average two-qubit gate fidelity of about 99%. Furthermore, we implement conditional post-selected quantum state teleportation between qubits separated by 320 nm with an average gate fidelity of 87%, showcasing the potential of mobile spin qubits for non-local quantum information processing. We expect that operations on mobile qubits will become a universal feature of future large-scale semiconductor quantum processors.
Quantum physics describes the universe on a microscopic scale. Even though this scale is not accessible without complex experimental setups, quantum effects are respon- sible for the properties of matter and in some instances we have learned to exploit them technologically. A modern computer constituted of transistors passively utilizes the quantum nature of charge carriers in the band structure of semiconductors to encode and process information. Although such a classical computer is a formidable tool already, what could be achieved with active control over the quantum states in the computer is an intriguing question. This is the idea of the quantum computer, which offers an entirely new paradigm of computing by encoding information in quantum binary digits (qubits). Indeed, for several applications it has been shown that a quantum computer is superior to its classical counterpart, even though its realization remains elusive up to date. One candidate to host qubits are gate-defined semiconductor quantum dots, confin- ing single or few charge carriers with spin. The spin of a single confined electron is a quantum two-level system which can be manipulated and measured, allowing the storage and processing of quantum information. This thesis can be viewed as part of the endeavor to give birth to quantum computers based on semiconductor spin qubits. It primarily focuses on the challenge to enable scalable architectures of well-characterized qubits. In the first part, spin shuttling is investigated. This denotes the coherent transport of a single spin between spatially separate nodes of a quantum processor, driven by a time-dependent electric potential. The object of study is a minimal version of a bucket- brigade-like spin shuttling between two quantum dots. The probability of maintaining the spin state is identified as figure of merit and the effect of spin-orbit interaction and the Zeeman effect in an inhomogeneous magnetic field is explored with analytical and numerical methods. Special attention is paid to the repercussions of near-degenerate valleys. The second part of the thesis addresses the challenge to probe the microscopic properties of a quantum dot array, which is a crucial first step before any quantum operation can be performed. The transmission of a microwave resonator coupled to a triple quantum dot occupied by two electrons is investigated, and a scheme to infer the exchange interaction between electrons in adjacent quantum dots is proposed. By means of analytical derivations it is shown that the transmission profile of the resonator directly reveals the value of the exchange coupling strength between two electrons. This holds even if magnetic gradients or a valley splitting comparable to the inter-dot tunnel coupling are included. Subsequently, quantum transport measurements in the Pauli blockade regime are modelled in the presence of a microwave resonator, bringing together two powerful techniques to probe quantum dots. Based on generalized input-output theory, a theoretical framework is derived to investigate how a double quantum dot in a transport setup interacts with a coupled microwave resonator while the current through the double qauntum dot is rectified by Pauli blockade. It is shown that the output field of the resonator can be used to infer the leakage current and thus obtain insight into the blockade mechanisms and to estimate a large number of parameters of the quantum dots. Furthermore, the back-action of the resonator photons on the steady state leakage current is described and quantified. In the context of Pauli blockade, the spin and valley physics of bilayer graphene quantum dots in the few-electron regime is touched upon, describing a situation where spin blockade is observed despite the presence of the valleys. In the final part of the thesis, the attention is shifted towards the readout of the spin qubits. A new paradigm of dispersive readout is devised, which is performed while the qubit dephases, implementing a measurement in the x-basis. Laplace transforming the time dependent cavity response allows a separation of contributions from multiple qubits coupled to the same resonator. Thus, they can be read out simultaneously in a single shot. Modelling silicon spin qubits, as an example, results in in a readout time that compares favourably to conventional dispersive readout with comparable fidelity, even for a single qubit. With an auxiliary qubit the readout scheme can be extended to a quantum non-demolition measurement.
Conveyor-mode shuttling is a key approach for implementing intermediate-range coupling between electron-spin qubits in quantum dots. Initial implementations are encouraging; however, long shuttling trajectories are guaranteed to encounter regions of low conduction-band valley-energy splittings, due to the presence of random-alloy disorder in Si / Si Ge quantum wells. Here, we theoretically explore two schemes for avoiding valley-state excitations at these valley-splitting minima, by allowing the electrons to detour around them. A scheme allows electrons to tunnel between parallel channels, while a provides full omnidirectional control. Using simulations, we estimate shuttling fidelities in these two schemes, obtaining a clear preference for the 2D shuttler. Based on such encouraging results, we propose a modular qubit architecture based on 2D shuttling, which enables all-to-all connectivity within qubit plaquettes and high-fidelity communication between different plaquettes.
Single spin quantum dot qubits in silicon are a promising candidate for a scalable quantum processor due to their long coherence times, compact size, and ease of integration into existing fabrication technologies. Realizing a large quantum processor composed of thousands or more logical qubits requires the integration of conventional transistor circuitry \nand wiring interconnects to control each individual dot in the processor. The high density of control wiring required for these processors presents many engineering challenges. In this thesis, we propose a surface code quantum processor for silicon quantum dot qubits based on a node/network architecture. Local nodes consisting of just seven quantum dots are \nspatially separated on the order of microns to facilitate space for the necessary high density wiring. Entanglement is distributed between individual nodes via shuttling of entangled electron pairs throughout the network. X or Z stabilizer operations, necessary for operating the surface code, are realized by distributing three electron spin singlet pairs across four local nodes followed by local gate operations and ancilla measurements. Simulations of electron shuttling indicate that adiabatic transport is possible on timescales that do not bottleneck the processor speed. Phase rotation of the shuttled spin, induced by the Stark shift, can lower the overall shuttling fi delity; however, the error can be mitigated by proper electrostatic tuning of the stationary electron's g-factor. Using realistic noise models, we estimate error thresholds with respect to single and two-qubit gate fidelities as well as \nsinglet dephasing errors during shuttling. Electron shuttling is a key resource of the proposed network architecture. We continue the shuttling simulations by presenting an algorithm for finding constant-adiabatic shuttling control pulses, which enables a more rigorous study of how different conditions impact the shuttling speed and fidelity. These constant adiabatic pulses are used to optimize the physical device geometry to maximize charge shuttling speeds up to 300 nm/ns in the single-valley case. We then switch to an effective Hamiltonian representation where spin and valley degrees of freedom are accounted for during shuttling. Using realistic device and material parameters, shuttle speeds in the range 10-100 nm/ns with high spin entanglement \n fidelities are obtained when the tunneling energy exceeds the Zeeman energy. High fidelity shuttling also requires the inter-dot valley phase difference to be below a threshold determined by the ratio of tunneling and Zeeman energies, so that spin-valley-orbit mixing is weak. In this regime, we find that the primary source of infi delity is a coherent spin rotation that is correctable, in principle, using single spin rotations. Two-qubit gates in the network architecture are mediated by the exchange interaction, \nan interaction that stems from the Coulomb interaction but manifests as a rotation between the |0,1> and |1,0> two qubit states. Realizing fault tolerant two-qubit gates has proven difficult in silicon quantum dots due to charge noise which perturbs the electron orbitals states, causing decoherence. Quantitatively accurate modelling of exchange in general quantum dot networks is important towards realizing fault tolerant gates. Traditional modelling methods, such as a full con figuration interaction approach, are cumbersome due to significant computational overhead required when accounting for the electron-electron interactions in the calculation. We present a modi ed linear combination of harmonic orbitals con figuration interaction (LCHO-CI) approach which signifi cantly reduces the computational time for obtaining quantitatively accurate estimations of exchange. The method works by approximating the single electron orbitals of the quantum dot network using an orthogonal basis of harmonic orbitals. This choice of orthogonal basis allows a precalculated library of matrix elements to be used in evaluating the Coulomb interactions, which speeds up the LCHO-CI calculation. The modi ed LCHO-CI approach is then used to study how the physical device geometry impacts the charge noise sensitivity of a double quantum dot system. We fi nd that, in general, geometries which increase the dot charging energy and decrease the gate lever arms improve the system's sensitivity to charge noise. We conclude this thesis by pivoting away from theoretical studies of silicon quantum dots and towards experimental studies of electron transport in dopant-free GaAs heterostructures. In modulation doped GaAs heterostructures, a doping layer is used to induce a two-dimensional electron gas (2DEG) at the heterojunction. Dopant-free GaAs lack the doping layer which makes fabrication more difficult as local electron reservoirs must be used to bring carriers and induce a 2DEG. The lack of the doping layer offers several advantages over modulation doped heterostructures, such as gate-ability of the 2DEG and ability to make ambipolar devices (both n- and p-type ohmic contacts). Realizing n- and p-type ohmic contacts requires etching a recess pit and depositing the ohmic material at an angle in order to preventing screening of the top gate when inducing a 2DEG or two-dimensional hole gas (2DHG). Magnetotransport experiments are used to characterize an induced 2DEG and 2DHG in a Hall bar fabricated on a 310 nm deep single heterojunction. The carrier mobility and density can be tuned using the top gate, and we achieve peak mobility values of 4.5E6 cm2/Vs and 0.65E6 cm2/Vs for the 2DEG and 2DHG respectively. We then move to a one-dimensional system and study electron transport through a quantum point contact. Conductance quantization is observed, and subband spectroscopy \nmeasurements indicate a 1D subband spacing of 4.5 meV. Finally, we study dynamically driven electron transport in a zero-dimensional system using a tunable-barrier quantum dot acting as a single electron pump. Single electron pumping is observed up to T = 5 K, and fi ts to the electron cascade model suggests pumping errors of 1.87 ppm when operated at a driving frequency of f = 500 MHz.
Valley excitations are considered one of the biggest challenges for a coherent spin shuttling in Si/SiGe heterostructures. In this work we theoretically investigate the valley dynamics during spin shuttling in a Si/SiGe heterostructure by considering a two-level system given by the two low-lying valley states of the mobile quantum dot. We assume that the electron is initially in the valley ground state and analyze the evolution of this state during shuttling. We find that, depending on the shuttling velocity, the system can be in three distinct regimes. In one of these regimes, the quantum information is preserved even in the presence of the valley excitations, allowing for coherent spin shuttling independently of the shuttling distance. This regime is obtained for a shuttling velocity of tens of m/s, a desired regime where dephasing mechanisms, such as spin-valley hotspots and spatially varying spin splitting, can be suppressed. This result is in accordance with recent experimental work that obtained a high spin shuttling fidelity after a fast and long shuttling. We also propose strategies based on the Partial Landau-Zener model to enhance the valley shuttling fidelity, which was found to be much more efficient than previously proposed strategies based on the standard Landau-Zener model.
This thesis addresses the concept of quantum computing with semiconductor quantum dots. The basic unit of a quantum computer is a quantum mechanical two-level system, the so-called quantum bit (qubit). The qubit can be defined as the spin of an electron confined in a quantum dot or as a two-dimensional subspace of the Hilbert space for several spins. Some semiconductors have several minima in their conduction band, socalled valleys. A two-dimensional valley degree of freedom can also be considered as a qubit. In this thesis, quantum registers storing spin as well as valley qubits are described theoretically. Virtual hopping yields an exchange interaction between neighboring quantum dots, which reduces the energy of antisymmetric spin-valley states according to the Pauli principle. Considering the spin and the valley degrees of freedom of two electrons in neighboring quantum dots as qubits, the exchange interaction is a four-qubit interaction. In this thesis, it will be shown that it is, nevertheless, possible to generate a universal two-qubit gate for spin qubits or for valley qubits by combining the exchange interaction and individual single-qubit gates. The exchange interaction in this two-electron double quantum dot directly provides a universal two-qubit gate for one spin and one valley qubit as well. For this, the qubits are defined as suitable subspaces, spanned by the singlet and one triplet state of the spin and the valley states, respectively, Using those gates, arbitrary quantum operations can be performed for single-spin and single-valley qubits in the same quantum register, where in one double quantum dot the quantum states are restricted to the singlet-triplet subspace. While in this register single-spin and single-valley rotations are required, these gates are dispensable in a quantum register storing spin and valley singlet-triplet qubits. The single-qubit operations, in this case, can be provided by the exchange interaction and a gradient in the energy splitting of spin and valley states when the spin qubit is stored in a double quantum dot with spin degrees of freedom only and when the spin states in the double quantum dot which stores the valley qubit are polarized. For providing the two-qubit gate, the spin and the valley qubits need to be stored in the same double quantum dot. The crucial task for this register is to interchange the spin of a spin-only and a spin-valley quantum dot in order to switch between the single-qubit and the two-qubit operation mode. This interchanging of the spin can be achieved by a valley-depending virtual hopping between the quantum dots. Furthermore, echo sequences for a so-called exchange-only qubit are considered. An exchange-only qubit is defined as a subspace in a system of three electron spins in a triple quantum dot. An inhomogeneous magnetic field, which may be caused by nuclear spins in the semiconductor, can lead to decoherence and leakage, i.e., the state might leave the qubit subspace. Suppressing this effect by an echo sequence which consists of pulses interchanging neighboring spin states, so-called SWAP gates, was proposed. Controllability over these quantum operations is provided via the exchange interaction. In this thesis, the performance of different such sequences is investigated. The analysis shows that optimization strategies developed for single spin echoes can be adapted for the three spin system. The echo sequences can be different in their order of applied SWAP gates interchanging the spins in the first and second dots or in the second and third dots. The noise suppression performance depends slightly on this order of SWAP gates. For two spins in a double quantum dot a scheme for quantum state tomography is presented. The corresponding measurement operators are constructed from a set of mutually unbiased bases of the four-dimensional two-qubit Hilbert space. These measurements are represented by short sequences of no more than three elementary quantum gates and projective measurements by the well-established spin-to-charge conversion.