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.
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.
Stabilizer operations (SO) occupy a prominent role in fault-tolerant quantum computing. They are defined operationally by the use of Clifford gates, Pauli measurements, and classical control. These operations can be efficiently simulated on a classical computer, a result which is known as the Gottesman–Knill theorem. However, an additional supply of magic states is enough to promote them to a universal, fault-tolerant model for quantum computing. To quantify the needed resources in terms of magic states, a resource theory of magic has been developed. SO are considered free within this theory; however, they are not the most general class of free operations. From an axiomatic point of view, these are the completely stabilizer-preserving (CSP) channels, defined as those that preserve the convex hull of stabilizer states. It has been an open problem to decide whether these two definitions lead to the same class of operations. In this work, we answer this question in the negative, by constructing an explicit counter-example. This indicates that recently proposed stabilizer-based simulation techniques of CSP maps are strictly more powerful than Gottesman–Knill-like methods. The result is analogous to a well-known fact in entanglement theory, namely, that there is a gap between the operationally defined class of local operations and classical communication and the axiomatically defined class of separable channels.
We give a new algorithm for computing the robustness of magic - a measure of the utility of quantum states as a computational resource. Our work is motivated by the magic state model of fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting non-stabiliser states, which are referred to as magic states in this context. The robustness of magic measures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamous sign problem . Surprisingly, the robustness of magic is sub - multiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states - better than a naive analysis would suggest. However, determining the robustness of n copies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4 n -dimensional space. In this paper, we make use of inherent symmetries to reduce the problem to n dimensions. The total run-time of our algorithm, while still exponential in n , is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certain signed quantum weight enumerators . As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs.
This thesis consists of two parts: Part I deals with properties of stabilizer states and their convex hull, the stabilizer polytope. Stabilizer states, Pauli measurements and Clifford unitaries are the three building blocks of the stabilizer formalism whose computational power is limited by the Gottesman- Knill theorem. This model is usually enriched by a magic state to get a universal model for quantum computation, referred to as quantum computation with magic states (QCM). The first part of this thesis will investigate the role of stabilizer states within QCM from three different angles. The first considered quantity is the stabilizer extent, which provides a tool to measure the non-stabilizerness or magic of a quantum state. It assigns a quantity to each state roughly measuring how many stabilizer states are required to approximate the state. It has been shown that the extent is multiplicative under taking tensor products when the considered state is a product state whose components are composed of maximally three qubits. In Chapter 2, we will prove that this property does not hold in general, more precisely, that the stabilizer extent is strictly submultiplicative. We obtain this result as a consequence of rather general properties of stabilizer states. Informally our result implies that one should not expect a dictionary to be multiplicative under taking tensor products whenever the dictionary size grows subexponentially in the dimension. In Chapter 3, we consider QCM from a resource theoretic perspective. The resource theory of magic is based on two types of quantum channels, completely stabilizer preserving maps and stabilizer operations. Both classes have the property that they cannot generate additional magic resources. We will show that these two classes of quantum channels do not coincide, specifically, that stabilizer operations are a strict subset of the set of completely stabilizer preserving channels. This might have the consequence that certain tasks which are usually realized by stabilizer operations could in principle be performed better by completely stabilizer preserving maps. In Chapter 4, the last one of Part I, we consider QCM via the polar dual stabilizer polytope (also called the Lambda-polytope). This polytope is a superset of the quantum state space and every quantum state can be written as a convex combination of its vertices. A way to classically simulate quantum computing with magic states is based on simulating Pauli measurements and Clifford unitaries on the vertices of the Lambda-polytope. The complexity of classical simulation with respect to the polytope is determined by classically simulating the updates of vertices under Clifford unitaries and Pauli measurements. However, a complete description of this polytope as a convex hull of its vertices is only known in low dimensions (for up to two qubits or one qudit when odd dimensional systems are considered). We make progress on this question by characterizing a certain class of operators that live on the boundary of the Lambda-polytope when the underlying dimension is an odd prime. This class encompasses for instance Wigner operators, which have been shown to be vertices of Lambda. We conjecture that this class contains even more vertices of Lambda. Eventually, we will shortly sketch why applying Clifford unitaries and Pauli measurements to this class of operators can be efficiently classically simulated. Part II of this thesis deals with lattices. Lattices are discrete subgroups of the Euclidean space. They occur in various different areas of mathematics, physics and computer science. We will investigate two types of optimization problems related to lattices. In Chapter 6 we are concerned with optimization within the space of lattices. That is, we want to compare the Gaussian potential energy of different lattices. To make the energy of lattices comparable we focus on lattices with point density one. In particular, we focus on even unimodular lattices and show that, up to dimension 24, they are all critical for the Gaussian potential energy. Furthermore, we find that all n-dimensional even unimodular lattices with n 24 are local minima or saddle points. In contrast in dimension 32, there are even unimodular lattices which are local maxima and others which are not even critical. In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional lattice. A flat torus comes with a metric and our goal is to approximate this metric with a Hilbert space metric. To achieve this, we derive an infinite-dimensional semidefinite optimization program that computes the least distortion embedding of the metric space R^n/L into a Hilbert space. This program allows us to make several interesting statements about the nature of least distortion embeddings of flat tori. In particular, we give a simple proof for a lower bound which gives a constant factor improvement over the previously best lower bound on the minimal distortion of an embedding of an n-dimensional flat torus. Furthermore, we show that there is always an optimal embedding into a finite-dimensional Hilbert space. Finally, we construct optimal least distortion embeddings for the standard torus R^n/Z^n and all 2-dimensional flat tori.
We give a new algorithm for computing the robustness of magic-a measure of the utility of quantum states as a computational resource. Our work is motivated by the magic state model of fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting non-stabiliser states, which are referred to as magic states in this context. The robustness of magic measures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamous sign problem. Surprisingly, the robustness of magic is submultiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states-better than a naive analysis would suggest. However, determining the robustness of n copies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4(n)-dimensional space. In this paper, we make use of inherent symmetries to reduce the problem to n dimensions. The total run-time of our algorithm, while still exponential in n, is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certain signed quantum weight enumerators. As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs.
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.
Non-Clifford gates are essential for avoiding classical simulability under the Gottesman-Knill theorem. An error-corrected circuit with non-Clifford gates will therefore be an important element for realizing useful fault-tolerant quantum algorithms. However, accurately understanding the fault-tolerant performance of such circuits faces challenges due to the classical hardness of simulating these circuits. We avoid simulating logical measurements and develop exact and efficient classical methods for benchmarking circuits composed of X, CNOT, and diagonal gates in the Clifford hierarchy, including state and gate teleportation gadgets with feedforward. For circuits with third-level diagonal gates, our benchmarking algorithm efficiently samples syndrome statistics and estimates the output logical fidelity and expectation values of certain Clifford observables, including all Pauli observables. The evaluation of syndrome expectation values also extends to circuits with fourth-level diagonal gates. Our algorithm scales polynomially in the number of physical qubits n and circuit depth T, independent of the number of logical qubits. The benchmarking applications include IQP sampling and magic state preparation/cultivation. We demonstrate the practical utility of our simulation algorithms by using them to optimize high-rate magic state distillation factories based on three copies of a [[27,3,3]] tricycle code.