The ZX-calculus is complete for stabilizer quantum mechanics

Abstract

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...

ZX-Rules for 2-Qubit Clifford+T Quantum Circuits

“…Recently ZX-calculus has been completed by Ng and Wang [24], that is, provided with sufficient additional rules so that any equation between matrices in Hilbert space can be derived in ZX-calculus. This followed earlier completions by Backens for stabiliser theory [2] and one-qubit Clifford+T circuits [4], and by Jeandel, Perdrix and Vilmart for general Clifford+T theory [20]. In Section 3 we present Backens' two theorems.…”

“…Stabiliser ZX-calculus is the restriction of ZX-calculus to α ∈ { nπ 2 | n ∈ N}. As shown in [2], the following rules make ZX-calculus complete for this fragment of quantum theory:…”

Optimising Clifford Circuits with Quantomatic

“…Various axiomatisations have been proposed ( [12,13,5,23,16,22]) with various advantages and drawbacks. Here we adopt the scheme of Backens [3] which is clean, concise, and adequate for the treatment of the Clifford group. These are shown in Figure 2.…”

Generalised Compositional Theories and Diagrammatic Reasoning

“…One example is the ZX-calculus, which is a graphical calculus for the interaction of the Pauli-Z and Pauli-X observable structures. In addition to the usual rules (complementarity, strong complementarity), several other rules are added, which turn out to be complete for stabiliser quantum mechanics [38]. This calculus has been applied to the study of measurement-based quantum computing [15], topological MBQC [16], and quantum protocols [39].…”