The Stabilizer Formalism

Keisuke Fujii

János A. Bergou, Mark S. Hillery, Mark E. Saffman

Xiaotong Ni, Oliver Buerschaper, Maarten Van den Nest

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.

Vu Tuan Hai, Bui Cao Doanh, Le Vu Trung Duong +2 more

Simulating Clifford and near-Clifford circuits using the extended stabilizer formalism has become increasingly popular, particularly in quantum error correction. Compared to the state-vector approach, the extended stabilizer formalism can solve the same problems with fewer computational resources, as it operates on stabilizers rather than full state vectors. Most existing studies on near-Clifford circuits focus on balancing the trade-off between the number of ancilla qubits and simulation accuracy, often overlooking performance considerations. Furthermore, in the presence of high-rank stabilizers, performance is limited by the sequential property of the stabilizer formalism. In this work, we introduce a parallelized version of the extended stabilizer formalism, enabling efficient execution on multi-core devices such as GPU. Experimental results demonstrate that, in certain scenarios, our Python-based implementation outperforms state-of-the-art simulators such as Qiskit and Pennylane.

Xiaotong Ni, Oliver Buerschaper, Maarten Van den Nest

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

Walter Greiner, J. Reinhardt