There and back again: A circuit extraction tale

Abstract

Translations between the quantum circuit model and the measurement-based one-way model are useful for verification and optimisation of quantum computations. They make crucial use of a property known as gflow. While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane. Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow. The algorithm is efficient and the resulting circuits do not contain ancillae. One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams. In developing this algorithm, we generalise several concepts and results previously known for computations containing only XY-plane measurements. We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus. These rules are used to simplify measurement patterns by reducing the number of qubits while preserving both the semantics and the existence of gflow. The results can be applied to circuit optimisation by translating circuits to patterns and back again.

Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus

“…The most important tool used in this paper is the ZX-calculus, a graphical language for describing and reasoning about quantum processes. The ZX-calculus was developed by Coecke and Duncan in [28,29], which has various applications including quantum circuit synthesis [30][31][32][33], measurement-based quantum computing [34,35], quantum error correction [36,37], condensed matter physics [38], quantum machine learning [39], and quantum natural language processing [40]. In the ZX-calculus, the objects under consideration are ZX-diagrams, which consist of two kinds of tensors: Z-spiders and X-spiders.…”

Relating Measurement Patterns to Circuits via Pauli Flow

“…There exist polynomial-time algorithms for identifying whether or not a labelled open graph admits causal flow or gflow [5,14,27], and for extracting an equivalent unitary circuit for the measurement pattern given either type of flow [5,7].…”

“…Proof. This is just an extension of the planar case given in the proof of [5,Lemma 4.7] to cover Pauli labels by considering the embeddings from Equation 51. The idea is that the Z-basis projections 0| and 1| reduce each of the CZ gates on that qubit to either an identity or a Z gate (modelled as a Z conditioned on the measurement angle), in the same way that they reduce when applied to a |0 or |1 .…”

“…Lemma D.7. (Generalisation of [5,Lemma 3.4]) Let (Γ, α) and (Γ ′ , α ′ ) be measurement patterns related by elimination of a vertex u as in Lemma D.6. If (p, ≺) is a Pauli flow for Γ, then (p ′ , ≺) is a Pauli flow for Γ ′ where for any v ∈ O \ {u}…”

“…Previous work in this area has given algorithms for extracting circuits from measurement patterns that satisfy flow properties which describe how the errors from unwanted measurement outcomes can be corrected using the stabilizers of the graph state [5,7]. The weakest such property that allows for interesting (i.e.…”