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.
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.
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 describe a fragment of quantum theory that is known to be efficiently classically simulable, thanks to the Gottesman-Knill theorem. For this reason, nonstabilizer resources such as magic states are necessary for universal quantum computation. Interestingly, the operational and axiomatic approaches to the resource theory of magic differ: the set of free operations in the former, namely, stabilizer operations (SO), is strictly smaller than that in the latter, namely, completely stabilizer preserving operations (CSPO). A simple example showing the separation is given by a three-qubit stabilizer product basis whose states cannot be perfectly discriminated using SO, but which do admit perfect discrimination using CSPO. Such an ensemble of states is said to exhibit nonstabilizerness without magic (NSWM). Here we obtain a principled understanding of this phenomenon, proving necessary and sufficient conditions for its existence. We first derive a simple criterion to decide whether, given a stabilizer basis, its states can be perfectly discriminated using stabilizer operations alone. We then consider the case where the stabilizer basis contains only product states and use its link with process functions---classical models of paradox-free causal loops---to prove the following: the states in a stabilizer product basis require nonstabilizerness for perfect discrimination if and only if the corresponding process function violates a causal inequality. This provides a new operational meaning to causal inequality violations as witnesses of nonstabilizerness, a form of computational nonclassicality.
From both intuitive and physical perspectives, it is generally recognized that within a resource theory framework, free operations cannot broadcast a resource state due to their inability to generate resource from free states. In the stabilizer formalism of fault-tolerant quantum computation, the basic ingredients of the corresponding resource theory consist of stabilizer states as free states and stabilizer operations as free operations. The celebrated Gottesman-Knill theorem shows that quantum advantages over classical computation come from the magic (nonstabilizer) resource, such as magic states or non-Clifford gates. In this work, we prove that broadcasting of any magic state via stabilizer operations is impossible, which is reminiscent of the no-broadcasting theorems for noncommuting states or quantum correlations. We further derive a trade-off relation between the magic resource consumed in the initial system and that gained in the target system. These results characterize magic states in the stabilizer formalism from the broadcasting angle, and may have implications for distributed quantum computation and quantum secret sharing.
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.
In distributed fault-tolerant quantum computing, entanglement and magic are essential resources for quantum communication and universal fault-tolerant computation, respectively. Although they are usually treated as distinct resource currencies, whether they admit a unified resource-theoretic description remains an open question. Here, we introduce the distributed resource theory of entanglement and magic (DREAM). In this framework, the free states are convex mixtures of product local stabilizer states, and the free operations are local stabilizer circuits assisted by classical communication (LSCC). We show that DREAM contains nontrivial resource states that are neither entanglement nor magic, so it is strictly richer than treating the two resources independently. Surprisingly, such resources can enable the teleportation of magic states between distant parties without consuming entanglement, revealing a counterintuitive form of resource teleportation mediated entirely by separable states. We further generalize this result to quantum networks and investigate general quantum-state teleportation under LSCC. We show that the shared resource under DREAM is closely related to the teleportation capability, quantified by the magic of the teleported state. Our work establish a systematic framework for studying distributed quantum resources and uncover intrinsic relations among distinct resources within a unified resource theory.
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.