On The Stabilizer Formalism And Its Generalization

Abstract

The standard stabilizer formalism provides a setting to show that quantum computation restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman-Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of $SU(d)$ transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results can be used to construct novel Gottesman-Knill-type results and consequently draw a sharper line between quantum and classical computation.

arXiv comment18 pages, 0 figure

References in corpus (29)

• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• Efficient classical simulation of slightly entangled quantum computations
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Stabilizer Codes and Quantum Error Correction
• Qudits and high-dimensional quantum computing
• Instantaneous non-local computation of low T-depth quantum circuits
• The Heisenberg Representation of Quantum Computers
• Quantum measurements and the Abelian Stabilizer Problem
• High-threshold and low-overhead fault-tolerant quantum memory
• Improved classical simulation of quantum circuits dominated by Clifford gates
• Measurement-based quantum computation beyond the one-way model
• Quantum Hypergraph States
• Fast simulation of stabilizer circuits using a graph state representation
• Quantum secret sharing with qudit graph states
• Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic
• Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes
• The elusive source of quantum effectiveness
• Standard Form of Qudit Stabilizer Groups
• Demonstration of Shor's factoring algorithm for N=21 on IBM quantum processors
• Generators and relations for n-qubit Clifford operators
• A Non-Commuting Stabilizer Formalism
• Universal Fault-Tolerant Quantum Computing with Stabiliser Codes
• On the Significance of the Gottesman-Knill Theorem
• Dual frame optimization for informationally complete quantum measurements
• Enhanced observable estimation through classical optimization of informationally over-complete measurement data -- beyond classical shadows
• Deterministic generation of qudit photonic graph states from quantum emitters
• A Herculean task: Classical simulation of quantum computers
• Noisy Stabilizer Formalism
• Efficient entanglement generation and detection of generalized stabilizer states