Magic state distillation is a crucial component in the leading approaches to implementing universal fault-tolerant quantum computation, with existing protocols for both qubit and higher dimensional systems. Early work focused on determining the region of distillable states for qubit protocols; yet comparatively little is known about which states can be distilled and with what distillable region for d>2. Here we focus on d=3 and present new four-qutrit distillation schemes that improve upon the known distillable region, and achieve distillation tight to the boundary of undistillable states for some classes of state. As a consequence of recent results, this implies that there is a family of quantum states that enable universality if and only if they exhibit contextuality with respect to stabilizer measurements. We also identify a new routine whose fixed point is a magic state with maximal sum negativity; i.e., it is maximally nonstablizer in a specific sense.
Magic state distillation is a crucial component in the leading approaches to implementing universal fault-tolerant quantum computation, with existing protocols for both qubit and higher dimensional systems. Early work focused on determining the region of distillable states for qubit protocols; yet comparatively little is known about which states can be distilled and with what distillable region for d>2. Here we focus on d=3 and present new four-qutrit distillation schemes that improve upon the known distillable region, and achieve distillation tight to the boundary of undistillable states for some classes of state. As a consequence of recent results, this implies that there is a family of quantum states that enable universality if and only if they exhibit contextuality with respect to stabilizer measurements. We also identify a new routine whose fixed point is a magic state with maximal sum negativity; i.e., it is maximally nonstablizer in a specific sense.
This thesis contains the results of research projects broadly related to the pursuit of universal fault-tolerant quantum computation. We are interested in questions such as which resources are required to achieve a quantum computational advantage, how to utilize such resources in practice, and how to characterize and control noise throughout a computation. \n \nIn part 1, we cover results pertaining to magic state distillation, one of the leading approaches to implementing universal fault-tolerant quantum computation. Early work in this area focused on determining the region of distillable states for qubit protocols, yet comparatively little is known about which states can be distilled and with what distillable region for d>2. In the first project, we focus on d=3 and present new four-qutrit distillation schemes that improve upon the known distillable region, and achieve distillation tight to the boundary of undistillable states for some classes of states. As a consequence of recent results, this implies that there is a family of quantum states that enable universality if and only if they exhibit contextuality with respect to stabilizer measurements. We also identify a new routine whose fixed point is a magic state with maximal sum-negativity i.e., it is maximally non-stabilizer in a specific sense. In the second project, we return to d=2 and present a number of interesting new distillation routines based on small codes. Many of these distill noisy states right up to the boundary of the known undististillable region, while some distill toward non-stabilizer states that have not previously been considered. \n \nIn part 2, we address the issue of noise characterization. The characterization of noise in a quantum system serves as a foundation towards meeting two important goals. Firstly, we may wish to know about specific sources of noise in order to adapt or correct errors as much as possible. Secondly, we would like a way to verify that noise is below some threshold value in order to meet the criteria of threshold theorems. We consider the case where noise is approximately known to coincide with the generalized damping channel (encompassing the common intrinsic processes of amplitude damping and dephasing), but may contain additional unknown noise sources. We provide methods to accurately obtain the noise parameters, and compare with the results of a randomized benchmarking experiment. Using this information, we show how to make meaningful statements about fault-tolerance threshold theorems by considering the diamond distance.
Magic state distillation (MSD) is a purification protocol that plays an important role in fault-tolerant quantum computation. Repeated iteration of the steps of an MSD protocol generates pure single non-stabilizer states, or magic states, from multiple copies of a mixed resource state using stabilizer operations only. Thus mixed resource states promote the stabilizer operations to full universality. MSD was introduced for qubit-based quantum computation, but little has been known concerning MSD in higher-dimensional qudit-based computation. Here, we describe a general approach for studying MSD in higher dimensions. We use it to investigate the features of a qutrit MSD protocol based on the five-qutrit stabilizer code. We show that this protocol distils non-stabilizer magic states, and identify two types of states that are attractors of this iteration map. Finally, we show how these states may be converted, via stabilizer circuits alone, into a state suitable for state-injected implementation of a non-Clifford phase gate, enabling non-Clifford unitary computation.
Magic state distillation is a critical component in leading proposals for fault-tolerant quantum computation. Relatively little is known, however, about how to construct a magic state distillation routine or, more specifically, which stabilizer codes are suitable for the task. While transversality of a non-Clifford gate within a code often leads to efficient distillation routines, it appears to not be a necessary condition. Here we have examined a number of small stabilizer codes and highlight a handful of which displaying interesting, albeit inefficient, distillation behaviour. Many of these distill noisy states right up to the boundary of the known undististillable region, while some distill toward non-stabilizer states that have not previously been considered.
Magic states enable universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness supplies the resource needed to bypass the Eastin-Knill theorem while allowing fault-tolerant distillation. Although many measures of magic exist, not every nonzero-magic state is known to be distillable, and many of currently known distillable states are special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite Clifford subgroups. We develop a general framework for group-covariant functionals on Hermitian operators, introducing the notions of G-stabilizer spaces, states, and codes for arbitrary finite subgroups G⊂U(H). We define analytic families of G-covariant functionals and prove that any G-invariant pure state is extremal for a broad class of derived functionals, including symmetric, max-type, and Rényi-type functionals, whenever the underlying family is G-covariant. This extremality holds for purity-preserving variations orthogonal to the stabilized subspace. Specializing to the Pauli and Clifford groups, we recover the extremality structure of canonical magic measures such as mana, stabilizer Rényi entropies, generalized Rényi entropies, and stabilizer fidelity, and show that Clifford-stabilizer states extremize them. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding standard benchmarks, and conjecture that SIC-POVM fiducial states are Clifford-stabilizer states.
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all [[n,1]]3 qutrit stabilizer codes with a complete set of transversal Clifford gates for n≤23, and all [[n,1]]3 stabilizer codes with a transversal H2 gate with n≤9 qudits. For n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.
Magic plays a dual role in quantum computation: it promotes stabilizer dynamics from efficient classical simulability to computational universality, but also challenges fault-tolerant architectures, since non-stabilizer operations are harder to protect against noise. Magic state distillation addresses this issue, yet existing protocols remain largely tailored to specific assumptions about the input or noise model and, more fundamentally, no general information-theoretic theory of optimal magic-state conversion currently exists. Here we develop such a characterization for pure qubit states through universal magic state concentration: a fixed stabilizer protocol that converts a few copies of an unknown pure non-stabilizer qubit state into an exact target magic state. Motivated by the impossibility of exact T-state concentration, we build six- and eight-copy protocols producing an exact CCZ state, with six copies being minimal. Their success probabilities are governed by the linearized order-three stabilizer Rényi entropy M3lin. We show that this connection is structural: for up to nine input copies, M3lin fully determines the success probability of every universal Clifford-invariant stabilizer protocol. Remarkably, this characterization persists asymptotically: our constructions achieve optimal rates among universal single-output protocols and remain optimal up to logarithmic factors among arbitrary stabilizer protocols. As a corollary, we show that any unknown pure non-stabilizer state suffices for universal quantum computation via probabilistic CCZ-state injection. Together, these results identify the stabilizer Rényi entropy as a fundamental operational quantity in magic state distillation.