Two circuits are considered to be equivalent under noise if the effect of faults on one circuit is no worse than the effect of faults on the other circuit. We call this relationship fault equivalence. Fault equivalence offers a way to transform circuits while provably preserving their fault-tolerant properties, enabling a framework for fault-tolerant circuit synthesis and optimisation that is correct by construction. The ZX calculus offers a diagrammatic way to represent and reason about quantum circuits and is a useful tool for manipulating circuits while preserving fault equivalence. For this, the usual set of ZX rewrites has to be restricted to not only preserve the underlying linear map represented by the diagram, but also fault equivalence. In this work, we provide a set of ZX rewrites that are sound and complete for fault equivalence of Clifford ZX diagrams. This means that any equivalence that can be derived using the proposed rules is certain to be correct, and any correct equivalence can be derived using only these rules. For this, we utilise diagrammatic constructions called fault gadgets to reason about arbitrary, possibly correlated Pauli faults in ZX diagrams. Fault gadgets allow us to separate the diagram into a fault-free part, which captures the noise-free behaviour of a diagram, and a noisy part that enumerates the effects of all possible faults. Using this, we provide a unique normal form for ZX diagrams under noise and show that any diagram can be brought into this normal form using our proposed rule set. Leveraging the constructive nature of our proof of completeness, we implemented a software library to check whether two ZX diagrams with given noise models are fault-equivalent.
Two circuits are considered to be equivalent under noise if the effect of faults on one circuit is no worse than the effect of faults on the other circuit. We call this relationship fault equivalence. Fault equivalence offers a way to transform circuits while provably preserving their fault-tolerant properties, enabling a framework for fault-tolerant circuit synthesis and optimisation that is correct by construction. The ZX calculus offers a diagrammatic way to represent and reason about quantum circuits and is a useful tool for manipulating circuits while preserving fault equivalence. For this, the usual set of ZX rewrites has to be restricted to not only preserve the underlying linear map represented by the diagram, but also fault equivalence. In this work, we provide a set of ZX rewrites that are sound and complete for fault equivalence of Clifford ZX diagrams. This means that any equivalence that can be derived using the proposed rules is certain to be correct, and any correct equivalence can be derived using only these rules. For this, we utilise diagrammatic constructions called fault gadgets to reason about arbitrary, possibly correlated Pauli faults in ZX diagrams. Fault gadgets allow us to separate the diagram into a fault-free part, which captures the noise-free behaviour of a diagram, and a noisy part that enumerates the effects of all possible faults. Using this, we provide a unique normal form for ZX diagrams under noise and show that any diagram can be brought into this normal form using our proposed rule set. Leveraging the constructive nature of our proof of completeness, we implemented a software library to check whether two ZX diagrams with given noise models are fault-equivalent.
Flow criteria are used to efficiently extract computations, either in the form of measurement patterns or quantum circuits, from ZX-diagrams. Existing criteria such as causal flow, generalised flow, and Pauli flow, were all originally formulated for graph states, so they require ZX-diagrams to be in a very particular graph-state-like form. This form is easily broken by applying basic ZX rules and makes establishing some desirable properties very complicated. Here, we introduce a new "ZX-native" flow criterion called ZX-flow, formulated using a new type of decoration of a ZX-diagram we call Pauli semiwebs. These are a generalisation of Pauli webs, which have recently been used extensively in reasoning about fault-tolerant computations in the ZX-calculus. We show that ZX-flow is straightforwardly preserved by all Clifford rewrites and furthermore that a ZX-diagram has ZX-flow if and only if it is Clifford-equivalent to a graph-like ZX-diagram with Pauli flow. Finally, we show that any diagram with ZX-flow can be readily interpreted either as a deterministic measurement-based computation or as a Clifford isometry followed by a sequence of Pauli exponentials. The latter can then be efficiently extracted to a quantum circuit.
A key challenge in fault-tolerant quantum computing is synthesising and optimising circuits in a noisy environment, as traditional techniques often fail to account for the effect of noise on circuits. In this work, we propose and numerically verify a framework for designing fault-tolerant quantum circuits that are correct by construction. The framework starts with idealised specifications of fault-tolerant gadgets and refines them using provably sound basic transformations. To reason about manipulating circuits while preserving their error correction properties, we define fault equivalence; two circuits are considered fault-equivalent if all undetectable faults on one circuit have a corresponding fault on the other. This guarantees that the effect of undetectable faults on both circuits is the same. We argue that fault equivalence is a concept that is already implicitly present in the literature. Many problems, such as state preparation and syndrome extraction, can be naturally expressed as finding an implementable circuit that is fault-equivalent to an idealized specification. To utilize fault equivalence in a computationally tractable manner, we adapt the ZX calculus, a diagrammatic language for quantum computing. We restrict its rewrite system to not only preserve the underlying linear map but also fault equivalence, i.e. the circuit's behaviour under noise. Enabled by our framework, we verify, optimise, and synthesise new and efficient circuits for syndrome extraction and cat state preparation. We confirm the improved performance of our optimised circuits in simulation. We anticipate that fault equivalence can capture and unify different approaches in fault-tolerant quantum computing, paving the way for an end-to-end circuit compilation framework.
Decoding a quantum error correction code is generally NP-hard, but corrections must be applied at a high frequency to suppress noise successfully. Matchable codes, like the surface code, exhibit a special structure that makes it possible to efficiently, approximately solve the decoding problem through minimum-weight perfect matching (MWPM). However, this efficiency-enabling property can be lost when constructing implementations for fault-tolerant gadgets such as syndrome-extraction circuits or logical operations.
In this work, we take a circuit-centric perspective to formalise how the decoding problem changes when applying ZX rewrites to a ZX diagram with a given detector basis. We demonstrate a set of rewrites that preserve MWPM-decodability of circuits and show that these matchability-preserving rewrites can be used to fault-tolerantly extract quantum circuits from phase-free ZX diagrams. In particular, this allows us to build efficiently decodable, fault-tolerant syndrome-extraction circuits for matchable codes.
Complete sets of graphical rewrite rules enable fully graphical reasoning about quantum computations and have been an area of active research for more than a decade. Many recent applications of the ZX-calculus have made use of the close correspondence between ZX-diagrams and computations in the one-way model of measurement-based quantum computation. In this model, various kinds of flow properties ensure deterministic implementability; for ZX-diagrams, these same properties allow efficient translation into quantum circuits (a problem that is known to be #P-hard in general). Therefore, flow-preserving ZX-calculus rewrite rules are of strong interest.
Here, we extend the set of flow-preserving rules appearing in the literature with a few new rules and extensions of existing rules. We then show that the resulting rule set is complete for all flow-preserving translations between ZX-diagrams of appropriate form. The proof employs a manifestly flow-preserving equivalent of circuit extraction, where a diagram with gflow is transformed, using only flow-preserving rewrite rules, into a diagram with causal flow.
Quantum error-correcting codes (QECC's) are needed to combat the inherent noise affecting quantum processes. Using ZX calculus, we represent QECC's in a form called a ZX diagram, consisting of a tensor network. In this paper, we present canonical forms for CSS codes and CSS states (which are CSS codes with 0 inputs), and we show the resulting canonical forms for the toric code and certain surface codes. Next, we introduce the notion of prime code diagrams, ZX diagrams of codes that have a single connected component with the property that no sequence of rewrite rules can split such a diagram into two connected components. We also show the Fundamental Theorem of Clifford Codes, proving the existence and uniqueness of the prime decomposition of Clifford codes. Next, we tabulate equivalence classes of ZX diagrams under a different definition of equivalence that allows output permutations and any local operations on the outputs. Possible representatives of these equivalence classes are analyzed. This work expands on previous works in exploring the canonical forms of QECC's in their ZX diagram representations.
We present a method for reducing the number of non-Clifford quantum gates, in particularly T-gates, in a circuit, an important task for efficiently implementing fault-tolerant quantum computations. This method matches or beats previous approaches to ancillae-free T-count reduction on the majority of our benchmark circuits, in some cases yielding up to 50% improvement. Our method begins by representing the quantum circuit as a ZX-diagram, a tensor networklike structure that can be transformed and simplified according to the rules of the ZX-calculus. We then extend a recent simplification strategy with a different ingredient, phase gadgetization, which we use to propagate non-Clifford phases through a ZX-diagram to find nonlocal cancellations. Our procedure extends unmodified to arbitrary phase angles and to parameter elimination for variational circuits. Finally, our optimization is self-checking, in the sense that the simplification strategy we propose is powerful enough to independently validate equality of the input circuit and the optimized output circuit. We have implemented the routines of this paper in the open-source library pyzx.