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Journal of Physics A Mathematical and Theoretical

Graphical description of Pauli measurements on stabilizer states

Matthew Elliott, Bryan Eastin, Carlton M. Caves

Published Dec 10, 2009• 13 citations
Publisher

Abstract

We use a graphical representation of stabilizer states to describe, simply and efficiently, the effect of measurements of Pauli products on stabilizer states. This work complements our earlier work (2008 Phys. Rev. A 77 042307), which described in graphical terms the action of Clifford operations on stabilizer states.
DOI
10.1088/1751-8113/43/2/025301

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Stabilizer states hold a special place in quantum information science due to their connection with quantum error correction and quantum circuit simulation. In the context of classical simulations of many-body physics, they are an example of states that can be both highly entangled and efficiently represented and transformed under Clifford operators. Recently, Clifford operators have been discussed in the context of fermionic quantum computation through their extension, the Majorana Clifford group. Here, we document the phase-sensitive form of the corresponding Majorana stabilizer states, as well as the algorithms for computing their amplitudes, their inner products, and update rules for transforming Majorana stabilizer states under Majorana Clifford gates.
We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of solid and hollow nodes, with (undirected) edges between nodes and with loops and signs attached to individual nodes. We find that local Clifford transformations are completely described in terms of local complementation on nodes and along edges, loop complementation, and change of node type or sign. Additionally, we show that a small set of equivalence rules generates all graphs corresponding to a given stabilizer state; we do this by constructing an efficient procedure for testing the equality of any two stabilizer graphs.
Stabilizer states, along with Clifford manipulations (unitary transformations and measurements) thereof—despite being efficiently simulable on a classical computer—are an important tool in quantum information processing, with applications to quantum computing, error correction, and networking. Graph states, defined on a graph, are a special class of stabilizer states that are central to measurement-based quantum computing, all-photonic quantum repeaters, distributed quantum computing, and entanglement distribution in a network. All stabilizer states are local-Clifford equivalent to graph states. In this paper, we review the stabilizer framework and extend it by incorporating general stabilizer measurements such as multi-qubit joint projections. We provide an explicit procedure—using Karnaugh maps from Boolean algebra—for converting arbitrary stabilizer gates into tableau operations of the CHP formalism for efficient stabilizer manipulations. We derive graphical rules for arbitrary stabilizer manipulations of graph states, including multi-qubit stabilizer projections and unitaries. We implement the graphical rulebook resulting from above into a MATLAB simulator with a GUI. A user of this tool, e.g., for research in quantum networks, will not require any background in quantum information or the stabilizer framework.