[Paper Review] Graphical quantum Clifford-encoder compilers from the ZX calculus

Andrey Boris Khesin, Jonathan Z. Lu+1 more|arXiv (Cornell University)|Jan 6, 2023

Quantum Computing Algorithms and Architecture8 citations

TL;DR

This paper introduces a ZX-calculus based compiler that maps Clifford encoders (incomplete stabilizer tableaus) to a unique graphical canonical form (ZXCF), proving canonicity and providing an efficient algorithm for canonicalization.

ABSTRACT

We present a quantum compilation algorithm that maps Clifford encoders, encoding maps for stabilizer quantum codes, to a unique graphical representation in the ZX calculus. Specifically, we develop a canonical form in the ZX calculus and prove canonicity as well as efficient reducibility of any Clifford encoder into the canonical form. The diagrams produced by our compiler visualize information propagation and entanglement structure of the encoder, revealing properties that may be obscured in the circuit or stabilizer-tableau representation. Consequently, our canonical representation may be an informative technique for the design of new stabilizer quantum codes via graph theory analysis.

Motivation & Objective

• Motivate and build a compiler that maps Clifford encoders to a unique ZX-calculus diagrammatic representation.
• Provide a canonical form (ZXCF) that reveals information propagation and entanglement in encoders.
• Prove canonicity: equivalent encoders map to the same ZXCF.
• Develop efficient algorithms to transform any Clifford encoder into ZXCF.
• Discuss how ZXCF can inform design of new stabilizer quantum codes via graph-theoretic analysis.

Proposed method

• Define encoder-respecting ZX diagram form as a semi-bipartite graph linking input and output clusters.
• Impose four rules (Edge, Hadamard, RREF, Clifford) to obtain ZX canonical form (ZXCF).
• Utilize Hu and Khesin ZX encoder framework to convert encoders into ZX-HK form.
• Apply local complementation and pivot-edge removal via ZX-equivalence rules to enforce Clifford rule and remove pivot-pivot edges.
• Prove canonicity by counting arguments equating the number of stabilizer tableaus with ZXCF diagrams and provide an O(n^3) canonicalization algorithm.
• Show how to start from incomplete stabilizer tableaus and produce Clifford encoders and their ZXCFs.

Experimental results

Research questions

• RQ1 Can Clifford encoders be uniquely represented by a ZX-calculus diagram (ZXCF) that is independent of the input encoder’s specific circuit form?
• RQ2 How can one efficiently transform any Clifford encoder into its ZXCF while preserving the encoded information and entanglement structure?
• RQ3 Does the ZXCF visualization provide insights into information propagation and entanglement that are obscured in circuit or tableau representations?
• RQ4 What is the computational complexity of canonicalizing Clifford encoders to ZXCF?
• RQ5 Can ZXCFs be used to analyze or design new stabilizer codes via graph-theoretic properties?

Key findings

• There exists a unique ZXCF for any Clifford encoder satisfying Edge, Hadamard, RREF, and Clifford rules.
• The canonicalization procedure runs in O(n^3) time for an encoder with n inputs/outputs.
• Transforming a ZX encoder into ZX-HK form enables subsequent reductions to ZXCF while preserving encoder structure.
• Pivot-based adjustments (local complementation and phi operations) can remove pivot-pivot edges without violating RREF, achieving canonicity.
• Starting from incomplete stabilizer tableaus, one can obtain Clifford encoders and their ZXCFs, enabling circuit reconstruction in reverse from outputs to inputs.
• For well-known codes (Shor, Steane, and 5-qubit code), the ZXCFs reveal structured, visual representations (e.g., cube geometry for Steane’s code) that reflect stabilizer properties.