Unifying flavors of fault tolerance with the ZX calculus

TL;DR

This work unifies circuit-based, measurement-based, fusion-based, and Floquet surface-code fault tolerance under the ZX calculus, framing them as manifestations of a common stabilizer fault-tolerance structure and showing local ZX transformations map between flavors. It introduces ZX instrument networks with classical outputs to capture measurements and checks, and Pauli webs as a graphical bookkeeping tool for stabilizers, Clifford gates, and error syndromes. The authors demonstrate that surface-code fault tolerance across these models can be represented in a single canonical ZX form, enabling cross-model translation of techniques, decoders, and logical operations. The ZX framework thus serves as a dictionary for transferring progress across fault-tolerance paradigms and potentially extending to other codes and architectures.

Abstract

There are several models of quantum computation which exhibit shared fundamental fault-tolerance properties. This article makes commonalities explicit by presenting these different models in a unifying framework based on the ZX calculus. We focus on models of topological fault tolerance - specifically surface codes - including circuit-based, measurement-based and fusion-based quantum computation, as well as the recently introduced model of Floquet codes. We find that all of these models can be viewed as different flavors of the same underlying stabilizer fault-tolerance structure, and sustain this through a set of local equivalence transformations which allow mapping between flavors. We anticipate that this unifying perspective will pave the way to transferring progress among the different views of stabilizer fault-tolerance and help researchers familiar with one model easily understand others.

Figures (13)

• Figure 1: (a) $Z$ spiders, $X$ spiders and Hadamards are the elementary building blocks of ZX diagrams. (b) Spider nodes with one leg can be used to describe Pauli eigenstates and their projections. (c) Unitary gates can also be expressed as ZX diagrams, as shown for the CNOT gate and a single-qubit $Z$ rotation. (d) Classical outputs can be added to describe quantum instruments. The classical bit $b$ determines whether the phase of the spider is $\alpha$ or $\alpha + \pi$. (e) Such ZX instruments can be used to describe Pauli measurements.
• Figure 2: (a) The most commonly used ZX-diagrammatic equations. These identities are also valid without classical outputs, i.e., by removing all outputs indicated by thick black lines. The transformation rules can be used to simplify (b) the CZ gate, (c) the circuit of a non-destructive $ZZ$ measurement and (d) the circuit of a destructive two-qubit Bell measurement.
• Figure 3: Pauli webs are a graphical overlay notation that allows us to reason about stabilizer states, Clifford gates and Pauli measurements. Pauli webs can describe the stabilizers of states, as shown for the example of a GHZ state (a/b) and a 6-ring graph state (c). They can also correspond to stabilizers of Clifford unitaries, as shown for the CNOT gate (d) and can specify Pauli projections and measurements, as shown for two-qubit (e) and four-qubit (f) Pauli measurements.
• Figure 4: (a) Pauli webs that are completely internal (i.e., not supported on any output legs) describe checks, as shown for a repetition-code check. (b) Pauli errors can be described by inserting red and green spiders with a phase of $\pi$. (c) The repetition-code check can detect any odd number of $X$ error inserted along the Pauli web. (d) The same is true for $Y$ errors. (e) $Z$ errors cannot be detected, as the green spider has no effect on the red Pauli web.
• Figure 5: (a) A distance-2 surface code patch consisting of four physical qubits and three stabilizers. (b) ZX diagram of a circuit corresponding to three rounds of measurements of the three stabilizers. (c) Pauli web of a $Z$ check. (d) Pauli web of the subsequent $Z$ check. (e) Pauli web of an $X$ check.