Graphical description of the action of Clifford operators on stabilizer states

Elliott, Matthew; Eastin, Bryan; Caves, Carlton M.

doi:10.1103/physreva.77.042307

Cited by 17 publications

(42 citation statements)

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“…there is a non-unique normal form for stabilizer state diagrams consisting of a graph state diagram and local Clifford operators. Based on work by Elliott et al [11], we then show that even though this normal form is not unique, there is a straightforward algorithm for testing equality of diagrams given in this form. In particular, this algorithm shows that two diagrams are equal if and only if they correspond to the same quantum mechanical state.…”

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confidence: 92%

The ZX-calculus is complete for stabilizer quantum mechanics

2014

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The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer QM, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations.are both represented in one dimension, namely a line of text. Yet the quantum circuit notation has one big disadvantage: there are no widely-taught transformation rules for quantum circuit diagrams. Thus papers about quantum circuits often contain long appendices full of circuit identities, which have to be proved (or at least stated) anew each time.Unlike quantum circuit notation, the ZX-calculus developed in [5,6] is not just a graphical notation: it has built-in rewrite rules, which transform one diagram into a different diagram representing the same overall process. These rewrite rules make the ZX-calculus into a formal system with non-trivial equalities between diagrams. In the following, we will thus distinguish between diagrams which are identical-i.e. they consist of the same elements combined in the same way-and diagrams which are equal, meaning one can be rewritten into the other. Two identical diagrams are necessarily equal to each other, but two equal diagrams may not be identical.As a formal system modeling pure state qubit quantum mechanics (QM), there are several properties the ZX-calculus must have to be useful. One of these is universality: the question of whether any pure state, unitary operator, or post-selected measurement can be represented by a ZX-calculus diagram. The ZX-calculus is indeed universal [6]. A second important property is soundness: Can any equality which can be derived in the ZX-calculus also be derived using other formalisms, such as matrix mechanics? By considering the rewrite rules one-by-one, it is not too difficult to show that the ZX-calculus is sound [6]. As a result of this, the ZX-calculus can be used to analyze a variety of questions, e.g. quantum non-locality [7] and the verification of measurement-based quantum computations [6,10,12].The converse of the soundness property is completeness: the ZX-calculus is complete if any equality that can be derived using matrices can also be derived graphically. It has been conjectured that the ZX-calculus is not complete for general pure state qubit QM, but in this paper we show that it is complete for qubit stabilizer QM. Stabilizer QM is an extensively studied part of quantum theory, which can be operationally described as the fragment of pure state QM where the only allowed operations are preparations or measurements in the computational basis and unit...

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confidence: 92%

The ZX-calculus is complete for stabilizer quantum mechanics

2014

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“…A recent paper [10], independent to ours, also extends the graphical notation to deal with the action of the local Clifford group on stabilizer states. Reference [10] also implicitly utilises a bipartite splitting of the graph (via 'hollow' and 'filled-in' nodes), and also requires graph loops.…”

Section: Main Aims Of This Papermentioning

confidence: 95%

“…Reference [10] also implicitly utilises a bipartite splitting of the graph (via 'hollow' and 'filled-in' nodes), and also requires graph loops. Reference [10] describes the action of H , S and Z on their graph, whereas we describe the action of H and N . Their model and our model must be equivalent in terms of characterising the action of the local Clifford group on stabilizer states.…”

Section: Main Aims Of This Papermentioning

confidence: 99%

“…In contrast, we focus, primarily, on modelling the action of the local transform group, T n , and/or {I, H, N } ⊗n as we are more interested in evaluating spectral metrics for the graph state as efficiently as possible, up to as many qubits as possible, although a secondary result of our work is that the action of the complete local Clifford group is also modelled. Secondly, [10] implicitly considers the stabilizer state as a joint eigenstate, and does not therefore have to consider an explicit basis for the state. In contrast, in our paper we consider an explicit computational basis for the state, and this allows us to distinguish between magnitude and phase properties of the stabilizer state.…”

Section: Main Aims Of This Papermentioning

confidence: 99%

“…However one can list some differences in approach between the papers as follows. Firstly, [10] focusses, primarily, on modelling the action of the local Clifford group. In contrast, we focus, primarily, on modelling the action of the local transform group, T n , and/or {I, H, N } ⊗n as we are more interested in evaluating spectral metrics for the graph state as efficiently as possible, up to as many qubits as possible, although a secondary result of our work is that the action of the complete local Clifford group is also modelled.…”

Section: Main Aims Of This Papermentioning

confidence: 99%

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From graph states to two-graph states

Riera

Jacob²,

Parker

2008

Des. Codes Cryptogr.

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The name 'graph state' is used to describe a certain class of pure quantum state which models a physical structure on which one can perform measurement-based quantum computing, and which has a natural graphical description. We present the two-graph state, this being a generalisation of the graph state and a two-graph representation of a stabilizer state. Mathematically, the two-graph state can be viewed as a simultaneous generalisation of a binary linear code and quadratic Boolean function. It describes precisely the coefficients of the pure quantum state vector resulting from the action of a member of the local Clifford group on a graph state, and comprises a graph which encodes the magnitude properties of the state, and a graph encoding its phase properties. This description facilitates a computationally efficient spectral analysis of the graph state with respect to operations from the local Clifford group on the state, as all operations can be realised graphically. By focusing on the so-called local transform group, which is a size 3 cyclic subgroup of the local Clifford group over one qubit, and over n qubits is of size 3 n , we can efficiently compute spectral properties of the graph state.

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Graphical description of Pauli measurements on stabilizer states

Elliott

Eastin

Caves

2009

J. Phys. A: Math. Theor.

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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 [Phys. Rev. A 77, 042307 (2008)], which described in graphical terms the action of Clifford operations on stabilizer states.PACS numbers: 03.67.-a * Electronic address: mabellio@unm.edu 1 See also Ref.[3] for a single-qubit measurement rule applied to a very different graphical representation of stabilizer states.2 C Z is often called the controlled-phase or controlled-Z gate.

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Graphical description of the action of Clifford operators on stabilizer states

Cited by 17 publications

References 11 publications

The ZX-calculus is complete for stabilizer quantum mechanics

The ZX-calculus is complete for stabilizer quantum mechanics

From graph states to two-graph states

Graphical description of Pauli measurements on stabilizer states

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