The stabilizer formalism is a central framework in quantum computing and one of the key building blocks for fault-tolerant quantum computers. According to the Gottesman–Knill theorem, we can efficiently simulate any quantum system consisting of stabilizer states, Clifford gates, and Pauli measurements on a classical computer. A key challenge in developing robust quantum technologies is that physical quantum systems are unavoidably affected by noise, which degrades quantum information. Accurate noise modeling is therefore essential. For stabilizer states, the noisy stabilizer formalism has recently been introduced as a framework for efficiently describing the evolution of graph states subject to Pauli-diagonal noise under Clifford operations and Pauli measurements. In this thesis, we explore and extend this formalism in two directions. First, we generalize the method from graph states to arbitrary stabilizer states, thereby extending the Gottesman-Knill theorem to noisy scenarios. Second, we extend the formalism to handle noise models beyond Pauli-diagonal channels. Moreover, we study the applicability of the noisy stabilizer formalism to entanglement purification protocols.
The stabilizer formalism is a central framework in quantum computing and one of the key building blocks for fault-tolerant quantum computers. According to the Gottesman–Knill theorem, we can efficiently simulate any quantum system consisting of stabilizer states, Clifford gates, and Pauli measurements on a classical computer. A key challenge in developing robust quantum technologies is that physical quantum systems are unavoidably affected by noise, which degrades quantum information. Accurate noise modeling is therefore essential. For stabilizer states, the noisy stabilizer formalism has recently been introduced as a framework for efficiently describing the evolution of graph states subject to Pauli-diagonal noise under Clifford operations and Pauli measurements. In this thesis, we explore and extend this formalism in two directions. First, we generalize the method from graph states to arbitrary stabilizer states, thereby extending the Gottesman-Knill theorem to noisy scenarios. Second, we extend the formalism to handle noise models beyond Pauli-diagonal channels. Moreover, we study the applicability of the noisy stabilizer formalism to entanglement purification protocols.
Despite the exponential overhead to describe general multiqubit quantum states and processes, efficient methods for certain state families and operations have been developed and utilized. The stabilizer formalism and the Gottesman-Knill theorem, where pure stabilizer or graph states are manipulated by Clifford operations and Pauli measurements, are prominent examples, and these states play a major role in many applications in quantum technologies. Here we develop a noisy stabilizer formalism, i.e., a method that allows one to efficiently describe and follow not only pure states under Clifford operations and Pauli measurements but also Pauli noise processes acting on such stabilizer states, including uncorrelated and correlated dephasing and single- or multiple-qubit depolarizing noise. The method scales linearly in the number of qubits of the initial state, but exponentially in the size of the target state. Thus, whenever a noisy stabilizer state is manipulated by means of local Pauli measurements such that a multipartite entangled state of a few qubits is generated, one can efficiently describe the resulting state.
We introduce the qudit noisy stabilizer formalism, a framework for efficiently describing the evolution of stabilizer states in prime-power dimensions subject to generalized Pauli-diagonal noise under Clifford operations and generalized Pauli measurements. For arbitrary dimensions, the formalism remains applicable, though restricted to a subset of stabilizer states and operations. The computational complexity scales linearly with the number of qudits in the initial state and exponentially with the number of qudits in the final state. This ensures that when noisy qudit stabilizer states evolve via generalized Pauli measurements and Clifford operations to generate multipartite entangled states of a few qudits, their description remains efficient. We demonstrate this by analyzing the generation of a generalized Bell pair from a noisy linear cluster state subject to two distinct noise sources acting on each of the qudits.
We develop analytical and algorithmic techniques that enable efficient simulation of a broad class of noisy stabilizer circuits. We derive closed-form expressions of expectation values for tensor product of Paulis in circuits with non-deterministic Pauli measurements, yielding an efficient strong simulation method that avoids explicit density matrix construction and enables direct noise parameter sweeps. We introduce a circuit compression framework that reduces the per-sample cost of weak simulation in general noisy stabilizer circuits, including deterministic measurements, by separating parameter-independent preprocessing from sampling. Finally, we extend the analytical framework beyond its standard domain to include a small number of deterministic measurements, general rotations, and non-diagonal noise channels. Our results provide a unified framework for both strong and weak simulation of noisy stabilizer circuits and corresponds to an extension of the noisy stabilizer formalism introduced in \cite{PhysRevA.107.032424}. They offer applications ranging from calculation of the expectation values of entanglement witnesses, determination of reduced states, to energy evaluation.
The stabiliser formalism is a widely used and successful subtheory of quantum mechanics consisting of stabiliser states, Clifford unitaries and Pauli measurements. The power of the formalism comes from the description of its elements via simple group theory. Although the origins of the formalism lie in quantum error correction and fault-tolerant quantum computing, the utility of the formalism goes beyond that. The Gottesman-Knill theorem states that the dynamics of stabiliser states under Clifford unitaries and Pauli measurements can be efficiently simulated on a classical computer. This algorithm can be extended to arbitrary states and unitaries in multiple ways at the cost of an increased runtime. This runtime can be seen as a quantification of the non-stabiliser resources needed to implement a quantum circuit. Moreover, as non-stabiliser elements are necessary for universal quantum computing, the runtime provides a way to measure the “non-classicality” of a computation. This is particularly pronounced in the magic state model of quantum computing where the only non-stabiliser elements are given by magic states. Hence, in the resource theory of magic, resources are measured through magic monotones which are operationally linked to runtimes of classical simulation algorithms. In this thesis, I discuss different aspects of the resource theory of magic. The mentioned classical simulation algorithms require the computation of magic monotones which is in general a computationally intractable problem. However, I show that the computational complexity can be exponentially reduced for certain classes of symmetric states, such as copies of magic states. To this end, the symmetries of the convex hull of stabiliser states are characterised and linked to their properties as so-called designs. In addition, I study the recently introduced class of completely stabiliser-preserving channels (CSP), which is the class of quantum channels unable to generate magic resources. It is shown that this class is strictly larger than the class of stabiliser operations, composed of Clifford unitaries and Pauli measurements. This finding could have several interesting consequences. First, it implies that there is a class of efficiently simulable quantum channels beyond the Gottesman-Knill theorem. Second, it is possible that optimal magic state distillation rates cannot be achieved via stabiliser operations and this gap is in fact significant. Further applications of the stabiliser formalism come through design theory. A unitary t-design is an ensemble of unitaries which reproduce the first t moments of the Haar measure on the unitary group. Randomness in the form of Haar-random unitaries is an essential building block in many quantum information protocols. Implementing such Haar-random unitaries is however often impractical. Here, designs can exhibit considerably lower resource requirements while still being random enough for most applications. Some of the most prominent examples of such protocols concern the certification and characterisation of quantum systems, such as randomised benchmarking. Interestingly, the group of Clifford unitaries forms a unitary 3-design and is often the prime choice thanks to efficient group operations. In this thesis, I summarise my recent results obtained with collaborators in constructing approximate unitary t-designs from the Clifford group supplemented by only few non-Clifford gates. Intriguingly, this construction uses only O(t^4) single qubit non-Clifford gates and is independent of the number of qubits n. Overall, this yields a gate count which is a significant improvement over for the Brandao-Harrow- Horodecki construction based on local random circuits. To provide context for this result, I review the representation theory of the Clifford group and define the Clifford semigroup. In an attempt to generalise approximations results for the unitary group to the Clifford group, the Clifford semigroup is investigated for suitable approximations of the Clifford twirl.
Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.
The preparation of high-fidelity logical magic states is a crucial subroutine for universal fault-tolerant quantum computation (FTQC). Predicting the performance of FTQC and developing improved protocols rely on numerical methods to classically simulate logical magic state preparation in the presence of noise. Clifford logic on Pauli-stabilizer codes with circuit-level Pauli errors can be efficiently simulated using Pauli-stabilizer formalism, but the non-Clifford operations required to prepare logical magic states render generic simulation inefficient. We introduce Clifford-stabilizer simulation, an exact and efficient algorithm based on updating a Clifford-stabilizer group to simulate noisy preparation protocols for a broad class of logical magic states used to implement non-Clifford gates in the third level of the Clifford hierarchy under circuit-level Pauli errors. Clifford-stabilizer simulation applies to a range of operations that commonly appear in preparation protocols for such logical magic states, including Pauli-stabilizer measurements, logical Clifford measurements, and transversal non-Clifford gates. Our algorithm for Clifford-stabilizer simulation maps a non-Clifford circuit with sampled circuit-level Pauli errors to a Clifford circuit that exactly reproduces its measurement outcome distribution, achieving time and space complexities polynomial in relevant protocol parameters. We perform exact simulation of magic state cultivation up to fault distance 7 by Clifford-stabilizer simulation. Our method provides a route to perform exact benchmarking of large-scale logical magic state preparation protocols required for useful FTQC.