Spin Qubits in Silicon and Germanium
Quantum computers based on semiconductor quantum dots are proving promising contenders for large scale quantum information processing. In particular, group IV based semiconductor hosts containing an abundance of nuclear spin-zero isotopes have made considerable headway into fulfilling the requirements of a universal quantum computer. Silicon (Si) and germanium (Ge) are two elements that have played important roles in the history of classical computing, and are now poised to do the same in the
... ture of quantum computation. In this thesis, we advance efforts in developing a quantum computer in both Si, using electron spins in Si-metaloxide-semiconductor (SiMOS) hosted quantum dots, and hole spins in planar germanium quantum wells (Ge/SiGe). In chapter 3, we develop fabrication strategies for quantum dot arrays in SiMOS, Si/SiGe and Ge/SiGe platforms. In each case, a fabrication recipe is developed with considerations according to each of the different platforms challenges and opportunities. Using these recipes, we create quantum dot arrays in silicon and germanium, depletable down to single charge occupancy in all cases. Additionally, we study the cross-capacitance of the electrostatic gates on adjacent quantum dots, finding that SiMOS requires the smallest virtual corrections to gate voltages. Using these fabrication recipes, we then focus on the SiMOS platform, creating devices for quantum information processing. In chapter 4, we show that tunnel couple control is possible between quantum dots in silicon MOS, using dedicated barrier gates. We demonstrate tuneable tunnel rates and tunnnel couplings via a barrier gate, showing tunnel rates can be suppressed to below 1 Hz, and tunnel couplings are tuneable between 3 GHz and 13 GHz, overcoming a longstanding challenge for electrons in silicon-MOS quantum dots. Operation of spin qubits at higher temperatures will permit strategies such as on chip integration of electronics, which may be key for scaling up spin qubits. In chapter 5, we demonstrate a universal set of quantum gates in a pair of SiMOS electron spin qubits, operated at 1.1 K. We find high single qubit fidelities for both qubits of around 99%, and find that dephasing times are not limited by temperature in the measured range, between 300 mK and 1.25 K. Furthermore in chapter 6, we demonstrate a portfolio of two qubit gates in the same device, and construct composite gates that should overcome limitations due to the finite Zeeman energy differences of the qubits. These results mark a milestone in the scale-up of spin qubits. The results using electron spin qubits in SiMOS show that spin qubits can be operated at higher temperatures, where the cooling power of dilution refrigerators is much higher. SiMOS as a material host however still faces several issues in terms of scale-up. First, the interface disorder results in the unwanted formation of quantum dots, as well as the strong localization of their wavefunctions, making reliable qubit interactions non-trivial. This is reflected by the relatively lower mobility, and higher percolation densities when compared to other platforms as measured in chapter 3. We therefore switch gears to hole spins in Ge/SiGe, advancing the platform through a number of experiments. Building on the success of the platform, we utilize the fabrication strategies developed in chapter 3 to fabricate quantum dot arrays in Ge/SiGe, which we deplete to single hole occupancy, and demonstrate the first single hole spin qubit in chapter 7. Exploiting the native spin orbit interaction, we achieve coherent, fast, all-electrical driving via electric dipole spin resonance (EDSR) of a single hole spin qubit, and characterize it's spin dephasing and relaxation times. We also find that xi xii SUMMARY the g -factor is highly dependent on the electric field environment. By tuning the coupling between the hole spin qubit and its charge reservoirs, we find that spin relaxation times can be significantly improved, and we measure single hole spin qubit spin lifetimes of 32 ms. We also find that g -factors are influenced significantly by neighbouring gate potentials, providing both an opportunity for ease of qubit addressability, as well as raising the challenge of enhanced charge noise coupling in the platform. In an effort to reduce charge noise, we fabricated wafers on quantum wells grown deeper below the heterostructure surface. This innovation led to reduced charge noise levels in hole spin qubits and facilitated the work in chapter 9. Here we present a two-by-two array of hole spin qubits, demonstrating universal quantum logic via single qubit manipulations, as well two-qubit CZ and CROT gates between all neighbouring qubit pairs. We further show the expected eight-fold splitting of a qubit resonance frequency as a result of four exchange coupled spins, allowing for both a native Toffoli-like three qubit gate, and even a four qubit controlled rotation gate. To showcase the control of the qubit array, we generate a four-qubit Greenberger-Horne-Zeilinger state in the first ever four qubit algorithm performed on a quantum processor hosted by semiconductor quantum dots. Quantum computers will require very high operational fidelities, and will likely require the simultaneous operation of many qubits in close proximity, leading to qubit cross-talk. In chapter 10, we examine single qubit gate fidelities using Randomized Benchmarking protocols. We find that single hole spin qubits, when operated individually can exhibit native gate fidelities of 99.990(1)%, marking the state of the art for single qubit operational accuracy in spin qubit platforms. Furthermore, we investigate how cross-talk when operating multiple qubits simultaneously affects single qubit gate fidelities. To do this, we develop a new randomized benchmarking protocol called "N -Copy Benchmarking", where the same Random Clifford sequences are applied simultaneously to each of the N qubit in the experiment. This permits extraction of a single qubit gate fidelity that approximates gate fidelity in the absence of qubit-qubit interactions, for example, residual exchange coupling. We find two-copy native gate fidelities above 99.905(8)%, and four-copy fidelities of 99.34(4)%, indicating that classical crosstalk is a major consideration when considering quantum computing scale up. Having explicitly investigated hole spin qubit relaxation times and gate fidelities, we also investigated their dephasing times to better understand the processes that limit coherence times. We found that as a function of magnetic field, a T 2 * ∝ 1/B proportionality described the behaviour of spin dephasing between 0.1 T and 1 T magnetic field strengths. This trend is consistent with the expectation that charge noise coupled to the spin eigenstates via the spin-orbit interaction is the limiting noise process for hole spin qubits in Ge/SiGe. We found also that by implementing refocussing pulses, coherence could be extended as high as 504 µs. To do this, we implemented Carl-Purcell-Meiboom-Gill pulse sequences, of varying numbers of refocussing π-pulses. These pulse sequences exhibit a characteristic filter fucntion based on this number, that effectively reduces low frequency noise effects on qubit coherence, but also enhances certain frequencies noise. We exploit this feature to investigate the sensitivity of hole spins to the 73 Ge nuclei present in the quantum well with an abundance of approximately 7.3%. By modelling the traces resulting from CPMG sequences, we extract a hyperfine coupling strength for the hole spins, and estimate a dephasing time limited by hyperfine interaction. We find this time to be within the same order of magnitude as the charge noise limited dephasing time, at an in-plane magnetic field of 0.25 T. From this we conclude that, in light of the rapidity of advancements in planar germanium material quality, isotopic purification will likely become necessary in the near future in order to extend quantum coherence times of hole spins in Ge/SiGe.