The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases

Mark A. Webster1,2, Benjamin J. Brown1,3, and Stephen D. Bartlett1

1Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia
2Sydney Quantum Academy, Sydney, NSW, Australia
3Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, University of Copenhagen, 2100 Copenhagen, Denmark

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We propose an extension to the Pauli stabiliser formalism that includes fractional $2\pi/N$ rotations around the $Z$ axis for some integer $N$. The resulting generalised stabiliser formalism – denoted the XP stabiliser formalism – allows for a wider range of states and codespaces to be represented. We describe the states which arise in the formalism, and demonstrate an equivalence between XP stabiliser states and 'weighted hypergraph states' – a generalisation of both hypergraph and weighted graph states. Given an arbitrary set of XP operators, we present algorithms for determining the codespace and logical operators for an XP code. Finally, we consider whether measurements of XP operators on XP codes can be classically simulated.

The Pauli stabiliser formalism allows us to efficiently describe certain quantum states. To describe a state, we specify a list of 'stabiliser generators' which are strings of Pauli $X$, $Y$ and $Z$ operators. The $X$, $Y$ and $Z$ operators can be thought of as half rotations around the $X$, $Y$ and $Z$ axes of the Bloch sphere. The stabiliser generators define a quantum code, and we can perform various calculations efficiently by working with the stabiliser generators rather than the state itself.

In our work, we extend the Pauli stabiliser formalism by defining a $P$ operator which is a $1/N$ rotation around the Z axis. We allow stabiliser generators to be made from $X$ and $P$ operators. This allows us to describe a much wider range of states. We generalise many of the algorithms from the Pauli stabiliser formalism. Unlike the Pauli stabiliser formalism, operations on XP codes cannot always be classically simulated – for instance measurement of arbitrary XP operators on codespaces. The XP formalism allows us to identify a wider range of fault-tolerant logical operators on stabiliser codes – including transversal non-Clifford logical operators which are important in realising universal quantum computation.

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