We describe the symmetry group of the stabilizer polytope for any number n of systems and any prime local dimension d. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. In the case of qudits, the structure is somewhat richer: For n = 1, it is a wreath product of permutations of bases and permutations of the elements within each basis. For n > 1, the symmetries are given by affine symplectic similitudes. These are the affine maps that preserve the symplectic form of the underlying discrete phase space up to a non-zero multiplier. We phrase these results with respect to a number of a priori different notions of “symmetry,” including Kadison symmetries (bijections that are compatible with convex combinations), Wigner symmetries (bijections that preserve inner products), and symmetries realized by an action on Hilbert space. Going beyond stabilizer states, we extend an observation of Heinrich and Gross [Quantum 3, 132 (2019)] and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments. In particular: the symmetries of a set that behaves like a 3-design preserve Jordan products and are therefore realized by conjugation with unitaries or anti-unitaries. (The structure constants of the Jordan algebra are encoded in an order-three tensor, which we connect to the third moments of a design). This generalizes Kadison’s formulation of the classic Wigner theorem on quantum mechanical symmetries.
We describe the symmetry group of the stabilizer polytope for any number n of systems and any prime local dimension d. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. In the case of qudits, the structure is somewhat richer: For n = 1, it is a wreath product of permutations of bases and permutations of the elements within each basis. For n > 1, the symmetries are given by affine symplectic similitudes. These are the affine maps that preserve the symplectic form of the underlying discrete phase space up to a non-zero multiplier. We phrase these results with respect to a number of a priori different notions of “symmetry,” including Kadison symmetries (bijections that are compatible with convex combinations), Wigner symmetries (bijections that preserve inner products), and symmetries realized by an action on Hilbert space. Going beyond stabilizer states, we extend an observation of Heinrich and Gross [Quantum 3, 132 (2019)] and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments. In particular: the symmetries of a set that behaves like a 3-design preserve Jordan products and are therefore realized by conjugation with unitaries or anti-unitaries. (The structure constants of the Jordan algebra are encoded in an order-three tensor, which we connect to the third moments of a design). This generalizes Kadison’s formulation of the classic Wigner theorem on quantum mechanical symmetries.
Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan–Wigner form in which they can act on multi-qubit states. Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian. They are, however, masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits. To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations. They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices. We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations for many-body systems.
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.
Unitary t-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary t-designs with $t\ensuremath{\ge}3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension 4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful in studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest in studying quantum computation.
The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group-a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with k -fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of k , constrained by the number of qubits n . In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary n and k . We also introduce an alternative and easy-to-manipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations, which generate the unitary group commutant, and at most three other operators. Additionally, we develop a graphical calculus allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all measurable magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension.
There is a bijection between odd prime dimensional qudit pure stabilizer states modulo invertible scalars and affine Lagrangian subspaces of finite dimensional symplectic Fp-vector spaces. In the language of the stabilizer formalism, full rank stabilizer tableaux are exactly the bases for affine Lagrangian subspaces. This correspondence extends to an isomorphism of props: the composition of stabilizer circuits corresponds to the relational composition of affine subspaces spanned by the tableaux, the tensor product corresponds to the direct sum. In this paper, we extend this correspondence between stabilizer circuits and tableaux to the mixed setting; regarding stabilizer codes as affine coisotropic subspaces (again only in odd prime qudit dimension/for qubit CSS codes). We show that by splitting the projector for a stabilizer code we recover the error detection protocol and the error correction protocol with affine classical processing power.
Absolutely stabilizer states are those that remain convex mixtures of stabilizer states after conjugation by any unitary. Here we give a characterization of such states for multiple qudits of all prime dimensions by introducing a polytope of their allowed spectra. We illustrate this through the examples of one qubit, two qubits, and one qutrit. In particular, the set of absolutely stabilizer states for a single qubit is a ball inscribed in the stabilizer octahedron, but for higher dimensions the geometry is more complicated. For odd-prime-dimensional qudits, we also give a complete characterization of absolutely Wigner-positive states, i.e., states whose Wigner function remains nonnegative after conjugation by any unitary. In so doing, we show there are absolutely Wigner-positive states that are not absolutely stabilizer, which can be seen as a unitarily invariant version of bound magic. We then study the radii of the largest balls contained in the sets of absolutely stabilizer states and absolutely Wigner-positive states. These radii respectively tell us the lowest possible purity of nonstabilizer and Wigner-negative states. Conversely, we also find the radius of the smallest ball containing the set of absolutely Wigner-positive states, giving a tight purity-based necessary condition thereof. Finally, we apply our methods to magic witnessing and find simple sufficient conditions for determining if an n-qubit state is reducible to an m-qubit magic state for any m<n.