We investigate the Λ-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, Λn, for every number n of qubits. We establish two properties of the family {Λn,n∈N}, namely (i) Any extremal point (vertex) Aα∈Λm can be used to construct vertices in Λn, for all n>m. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage Aα. In addition, we describe a new class of vertices in Λ2 which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of Λn, the above results extend efficient classical simulation of quantum computations beyond the presently known range.
We investigate the Λ-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, Λn, for every number n of qubits. We establish two properties of the family {Λn,n∈N}, namely (i) Any extremal point (vertex) Aα∈Λm can be used to construct vertices in Λn, for all n>m. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage Aα. In addition, we describe a new class of vertices in Λ2 which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of Λn, the above results extend efficient classical simulation of quantum computations beyond the presently known range.
We investigate the Λ-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, Λn, for every number n of qubits. We establish two properties of the family {Λn,n∈N}, namely (i) Any extremal point (vertex) Aα∈Λm can be used to construct vertices in Λn, for all n>m. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage Aα. In addition, we describe a new class of vertices in Λ2 which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of Λn, the above results extend efficient classical simulation of quantum computations beyond the presently known range.
The degenerate vertices of the 2-qubit Λ-polytope and their update rules
Recently, a class of objects, known as Λ-polytopes, were introduced for classically simulating universal quantum computation with magic states. In Λ-simulation, the probabilistic update of Λ vertices under Pauli measurement yields dynamics consistent with quantum mechanics. Thus, an important open problem in the study of Λ-polytopes is characterizing its vertices and determining their update rules. In this paper, we obtain and describe the update of all degenerate vertices of Λ2, the 2-qubit Λ polytope. Our approach exploits the fact that Λ2 projects to a well-understood polytope MP consisting of distributions on the Mermin square scenario. More precisely, we study the ``classical" polytope MP, which is MP intersected by the polytope defined by a set of Clauser-Horne-Shimony-Holt (CHSH) inequalities. Owing to a duality between CHSH inequalities and vertices of MP we utilize a streamlined version of the double-description method for vertex enumeration to obtain certain vertices of MP.
Identifying the boundary between classical and quantum computation is a central challenge in quantum information. In multi-qubit systems, entanglement and magic are the key resources underlying genuinely quantum behaviour. While entanglement is well understood, magic - essential for universal quantum computation - remains relatively poorly characterised. Here we show that determining membership in the stabilizer polytope, which defines the free states of magic-state resource theory, requires super-exponential time $\class{exp} ( n^2)$ in the number of qubits n, even approximately. We reduce the problem to solving a 3-\class{SAT} instance on n2 variables and, by invoking the exponential time hypothesis, the result follows. As a consequence, both quantifying and certifying magic are fundamentally intractable: any magic monotone for general states must be super-exponentially hard to compute, and deciding whether an operator is a valid magic witness is equally difficult. As a corollary, we establish the robustness of magic as computationally optimal among monotones. This barrier extends even to classically simulable regimes: deciding whether a state lies in the convex hull of states generated by a logarithmic number of non-Clifford gates is also super-exponentially hard. Together, these results reveal intrinsic computational limits on assessing classical simulability, distilling pathological magic states, and ultimately probing and exploiting magic as a quantum resource.
Negativity in certain quasiprobability representations is a necessary condition for a quantum computational advantage. Here we define a quasiprobability representation exhibiting this property with respect to quantum computations in the magic state model. It is based on generalized Jordan-Wigner transformations, and it has a close connection to the probability representation of universal quantum computation based on the $\mathrm{\ensuremath{\Lambda}}$ polytopes. For each number of qubits, it defines a polytope contained in the $\mathrm{\ensuremath{\Lambda}}$ polytope with some shared vertices. It leads to an efficient classical simulation algorithm for magic state quantum circuits for which the input state is positively represented, and it outperforms previous representations in terms of the states that can be positively represented.
We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides probability estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.
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.