Complete flow-preserving rewrite rules for MBQC patterns with Pauli measurements

We present a smorgasbord of results on the stabiliser ZX-calculus for odd prime-dimensional qudits (i.e. qupits). We derive a simplified rule set that closely resembles the original rules of qubit ZX-calculus. Using these rules, we demonstrate analogues of the spider-removing local complementation and pivoting rules. This allows for efficient reduction of diagrams to the affine with phases normal form. We also demonstrate a reduction to a unique form, providing an alternative and simpler proof of completeness. Furthermore, we introduce a different reduction to the graph state with local Cliffords normal form, which leads to a novel layered decomposition for qupit Clifford unitaries. Additionally, we propose a new approach to handle scalars formally, closely reflecting their practical usage. Finally, we have implemented many of these findings in DiZX, a new open-source Python library for qudit ZX-diagrammatic reasoning.

Section: The Qupit Clifford Zx-calculusmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification

Poór,

Booth,

Carette

et al. 2023

“…Other future work could involve finding an analogous result to the stabiliser completeness proof of [18] for a more general fragment of the MBQC-form ZX-calculus, using Proposition 5.1 to introduce phases that are not just integer multiples of π 2 .…”

Section: Discussionmentioning

confidence: 99%

Flow-preserving ZX-calculus Rewrite Rules for Optimisation and Obfuscation

McElvanney,

Backens

2023

In the one-way model of measurement-based quantum computation (MBQC), computation proceeds via measurements on a resource state. So-called flow conditions ensure that the overall computation is deterministic in a suitable sense, with Pauli flow being the most general of these. Computations, represented as measurement patterns, may be rewritten to optimise resource use and for other purposes. Such rewrites need to preserve the existence of flow to ensure the new pattern can still be implemented deterministically. The majority of existing work in this area has focused on rewrites that reduce the number of qubits, yet it can be beneficial to increase the number of qubits for certain kinds of optimisation, as well as for obfuscation.In this work, we introduce several ZX-calculus rewrite rules that increase the number of qubits and preserve the existence of Pauli flow. These rules can be used to transform any measurement pattern into a pattern containing only (general or Pauli) measurements within the XY-plane. We also give the first flow-preserving rewrite rule that allows measurement angles to be changed arbitrarily, and use this to prove that the 'neighbour unfusion' rule of Staudacher et al. preserves the existence of Pauli flow. This implies it may be possible to reduce the runtime of their two-qubit-gate optimisation procedure by removing the need to regularly run the costly gflow-finding algorithm.

“…As a language for rigorous diagrammatic reasoning of quantum computation, the ZX calculus consists of ZX diagrams and a set of rewrite rules [16,66]. It has been used to relate stabilizer theory to graphical normal forms: notably, efficient axiomatization of the stabilizer fragments for qubits [4,36,49], qutrits [61,65], and prime-dimensional qudits [8]. This has enabled various applications, such as measurement-based quantum computation [49,56], quantum circuit optimization [19,30] and verification [46], as well as classical simulation [14,40].…”

Section: Introductionmentioning

confidence: 99%

“…It has been used to relate stabilizer theory to graphical normal forms: notably, efficient axiomatization of the stabilizer fragments for qubits [4,36,49], qutrits [61,65], and prime-dimensional qudits [8]. This has enabled various applications, such as measurement-based quantum computation [49,56], quantum circuit optimization [19,30] and verification [46], as well as classical simulation [14,40]. Beyond these, ZX-calculus has been applied to verify QECC [23,26], represent Clifford encoders [38], as well as study various QECC such as tripartite coherent parity check codes [12,13] and surface codes [27,28,29,54].…”

Section: Introductionmentioning

confidence: 99%

Graphical CSS Code Transformation Using ZX Calculus

Huang,

Li,

Yeh

et al. 2023

In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code.We then focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the 15, 1, 3, 3 code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code transforming operations.