Het gebruik van qudits voor fouttolerante kwantum computatie

Abstract

Qudits can be described by a state vector in a q-dimensional Hilbert space, enabling a more extensive encoding and manipulation of information compared to qubits. This implies that conducting fault-tolerant quantum computations using qudits rather than qubits might entail less overhead. In this work, I investigate the viability of qudits across three key components of a fault-tolerant quantum computer: hardware control, quantum error-correction codes, and logical gate operations. This includes benchmarking physically decomposed qudit gates against qubit-based implementations for a specific donor-spin system, threshold simulations of qudit error-correction codes using tailored simulators and decoders, and simulations of magic state distillation circuits with qudits. At the hardware level, a numerical leakage-aware randomized benchmarking protocol is applied to a native ququart encoding in a donor-spin system and compared against an equivalent pair of encoded qubits. The ququart implementation is found to reduce the lower-bound gate infidelity by 40 – 50%, a consequence of the fewer ESR and EDSR pulses and correspondingly lower cumulative noise exposure. This provides a concrete, platform-specific demonstration of reduced physical-gate overhead prior to the introduction of any error-correction redundancy. At the level of quantum error correction, the viability of qudits is investigated through the construction and simulation of the quantum circuits for the five-qudit perfect code. The code is evaluated under a multidimensional circuit-level noise model using decoders specifically adapted to the qudit setting. This yields the first threshold estimates reported for qutrit and ququint codes under circuit-level noise. After introducing a flag qudit to protect the code from hook errors, comparable error thresholds of the order of 10–4 are obtained for qudits of dimensions 2, 3 and 5. These simulations were subsequently performed using Sdim, a stabilizer tableau simulator that enabled threshold estimation up to dimension q = 7, a regime inaccessible to state-vector simulators such as Cirq. At the level of logical operations, magic state distillation protocols of the Bravyi-Kitaev type are simulated under logical circuit-level noise, to my knowledge for the first time for this protocol family. Once the Clifford operations implementing the distillation circuit are subject to a nonzero logical error rate, the achievable output fidelity saturates within a finite usable distillation range rather than an arbitrarily improvable one: roughly 0.9-15.6% for the |T⟩₂-type protocol, 3.7-5% for the |H⟩₂-type protocol and 5.4-7.4% for the |H+⟩₃-type protocol at a representative logical circuit-level noise level of pCL = 10–3. The qutrit protocol achieves a more favorable distillation threshold but suffers from a substantially lower yield due to its reduced syndrome acceptance probability. This highlights a trade-off between threshold performance and yield that is intrinsic to the higher-dimensional construction. Further investigation with more advanced and larger-scale protocols is required to better understand and characterize this behavior. Overall, these results show that qudits can provide real efficiency benefits: higher-dimensional control can improve physical gate fidelity, and qutrit and ququint error-correction thresholds can match those of qubits when hook errors are properly handled. However, logical circuit-level noise limits the maximum magic state fidelity, showing that the benefits of qudits are not yet fully realized at the logical-operation level. This thesis provides the first broad numerical study of these effects and introduces the simulation tools needed to explore larger code families, decoders, and distillation protocols.