We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = eiπ/4 and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = eiπ/4 and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.
We propose a non-commutative extension of the Pauli stabilizer for-malism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our frame-work, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI,X, S〉, where α = eipi/4 and S = diag(1, i). We provide techniques to efficiently compute various properties re-lated to bipartite entanglement, expectation values of local observ-ables, preparation by means of quantum circuits, parent Hamiltonians etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons. 1
This work uncovers a fundamental connection between doped stabilizer states, a concept from quantum information theory, and the structure of eigenstates in perturbed many-body quantum systems. We prove that for Hamiltonians consisting of a sum of commuting Pauli operators, i.e., stabilizer Hamiltonians, and a perturbation composed of a limited number of arbitrary Pauli terms, the eigenstates can be represented as doped stabilizer states with small stabilizer nullity. This result enables the application of stabilizer techniques to a broad class of many-body systems, even in highly entangled regimes. Building on this, we develop efficient classical algorithms for tasks such as finding low-energy eigenstates, simulating quench dynamics, preparing Gibbs states, and computing entanglement entropies in these systems. Our work opens up new possibilities for understanding the robustness of topological order and the dynamics of many-body systems under perturbations, paving the way for novel insights into the interplay of quantum information, entanglement, and many-body systems.
We find an interesting relationship between multipartite bound entangled states and the stabilizer formalism. We prove that, if a set of commuting operators from the generalized Pauli group on n qudits satisfy certain constraints, then the maximally mixed state over the subspace stabilized by them is an unlockable bound entangled state. Moreover, the properties of this state, such as symmetry under permutations of parties, undistillability, and unlockability, can be easily explained from the stabilizer formalism without tedious calculation. In particular, the four-qubit Smolin state [Smolin, Phys. Rev. A 63, 032306 (2001)] and its recent generalization to even numbers of qubits [Bandyopadhyay et al., Phys. Rev. A 71, 062317 (2005); Augusiak et al., ibid 73, 012318 (2006)] can be viewed as special examples of our results. Finally, we extend our results to arbitrary multipartite systems in which the dimensions of all parties may be different.
Qudits with local dimension d>2 can have unique structure and uses that qubits ( d=2 ) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, d in several ways. To cover composite values of d , we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.
Quantum computers are expected to have advantages over classical computers in solving a range of high impact problems, but they are highly susceptible to errors due to environmental noise. The Pauli Stabiliser formalism generalises classical error-correction methods and makes use of quantum error correction codes to protect quantum information. In this thesis, we introduce the XP stabiliser formalism, which is a generalisation of the Pauli stabiliser formalism with a number of useful applications. Quantum algorithms are typically written in terms of quantum circuits which involve a series of unitary gates followed by measurements which form the output of the computation. To implement quantum algorithms reliably, we need to perform unitary gates fault-tolerantly so that errors do not propagate in an uncontrolled way. Transversal logical operators are one way of applying unitary gates fault-tolerantly on Pauli stabiliser codes. Identifying transversal logical operators for a given Pauli stabiliser code is challenging, and existing methods have exponential complexity in one or more of the parameters of the code. Making use of the XP formalism, we present efficient algorithms which identify all transversal logical operators that are diagonal in the computational basis for any Pauli stabiliser code. We also show how to construct codes with a transversal implementation of any desired diagonal logical operator. The Pauli stabiliser formalism can also be used to efficiently represent certain quantum states, but many states of interest lie outside the formalism. In the XP formalism, a wider range of states can be represented than in the Pauli stabiliser formalism, including hypergraph states which have interesting non-local properties. The braiding of non-Abelian anyons is a proposed pathway to universal fault-tolerant quantum computation. Certain XP stabiliser codes are known to harbour non-Abelian anyons, and can be studied within the new formalism.
The stabilizer formalism allows the efficient description of a sizeable class of pure as well as mixed quantum states of n -qubit systems. That same formalism has important applications in the field of quantum error correcting codes, where mixed stabilizer states correspond to projectors on subspaces associated with stabilizer codes. In this paper, we derive efficient reduction procedures to obtain various useful normal forms for stabilizer states. We explicitly prove that these procedures will always converge to the correct result and that these procedures are efficient in that they only require a polynomial number of operations on the generators of the stabilizers. On one hand, we obtain two single-party normal forms. The first, the row-reduced echelon form, is obtained using only permutations and multiplications of generators. This form is useful to calculate partial traces of stabilizer states. The second is the fully reduced form, where the reduction procedure invokes single-qubit operations and CNOT operations as well. This normal form allows for the efficient calculation of the overlap between two stabilizer states, as well as of the Uhlmann fidelity between them, and their Bures distance. On the other hand, we also find a reduction procedure of bipartite stabilizer states, where the operations involved are restricted to be local ones. The two-party normal form thus obtained lays bare a very simple bipartite entanglement structure of stabilizer states. To wit, we prove that every bipartite mixed stabilizer state is locally equivalent to a direct product of a number of maximally entangled states and, potentially, a separable state. As a consequence, using this normal form we can efficiently calculate every reasonable bipartite entanglement measure of mixed stabilizer states.