Mixed-Dimensional ZX Calculus

• Mixed-dimensional ZX calculus is a diagrammatic language that generalizes traditional ZX calculi to accommodate wires of arbitrary dimensions, covering qubits, qutrits, and beyond.
• It features specialized generators like Z-spiders, X-spiders, and dimension split/merge operations along with dimension-aware rewrite rules that ensure universal completeness in FHilb.
• The calculus supports practical applications across quantum circuits, chemistry, condensed matter physics, quantum gravity, and machine learning by providing unique normal forms for linear maps.

The mixed-dimensional ZX calculus is a graphical rewriting language for finite-dimensional quantum theory that subsumes all prior qubit and qudit ZX calculi. It provides a universal and complete system for expressing and deriving any equality in the category FHilb (finite-dimensional Hilbert spaces and linear maps), thereby enabling purely diagrammatic reasoning in quantum information, quantum chemistry, quantum algorithms, and spin-network theory. The calculus generalizes the familiar “spider” generators to accommodate wires of arbitrary dimension, introduces mixed-dimension splitting and merging, and extends the rewrite rules to handle interactions across different Hilbert space dimensions.

1. Generators and Algebraic Interpretation

Objects in the mixed-dimensional ZX calculus are lists of positive integers $d \geq 2$: each wire labeled $d$ carries a Hilbert space $\mathbb{C}^d$. Only wires with the same label can be fused directly, but mixed-dimensional interactions are mediated by special generators. The principal generators are:

Generator	Wire labels (dims)	Matrix interpretation
Z-box (Z-spider)	$d_1,...,d_n$	$\sum_{j=0}^{a-1} a_{j} \ket{j...j}\bra{j...j}$, $a = \min\{d_i\}$
X-spider	$d$	$$\sum_{\sum i+k \equiv \sum j \ (\text{mod } d)} \ket{i_1...i_m}\bra{j_1...j_n}$$
Hadamard ($H_d$)	$d$	$$(1/\sqrt{d}) \sum_{j,k=0}^{d-1} \omega^{jk} \ket{j}\bra{k}$$, $\omega = e^{2\pi i/d}$
Swap (between $m$, $n$)	$m$, $n$	$$\sum_{i=0}^{m-1} \sum_{j=0}^{n-1} \ket{j,i}\bra{i,j}$$
Dimension splitter	$d = mn$, $m$, $n$	$\ket{i n + j} \mapsto \ket{i} \otimes \ket{j}$
Dimension merger	$m$, $n$	$\ket{i} \otimes \ket{j} \mapsto \ket{i n + j}$
Cup / Cap	$d$	Compact structure morphisms
Dualiser (D)	$d$	$\sum_{i=0}^{d-1} \ket{d-i}\bra{i}$
Multiplier edge	$d$	Mod-$d$ mapping on green–pink connections

The mixed-dimensional Z-box can be decomposed into a high-dimensional Z-box plus splitters and mergers, providing a canonical form for diagrams with wires of varying dimensions.

2. Rewrite System and Soundness

The calculus incorporates a set of dimension-aware graphical rewrite rules, enforcing both the algebraic properties of quantum operators and the combinatorics of varying dimensions:

• S1 (Z-fusion): Merges two green spiders (Z-boxes), pointwise multiplying phases, truncating to the minimal dimension across all legs.
• S2 (Identity): Zero-phase spider with one input and one output is equivalent to the identity wire.
• S3 (Snake equations): Cup-cap and their adjoints bend and unbend wires, implementing compact structure.
• S4 (X-fusion): Fuses X-spiders (pink/red), with phase addition modulo dimension.
• B2 (Bialgebra): Green and pink spiders interact as a Hopf algebra, respecting dimension constraints.
• HZ (Colour-change): Hadamard box transforms green to pink spiders and vice versa.
• DD (Digit split): Dimension-splitters transform a wire of dimension $d$ into $m \otimes n$ wires.
• DZX (Dimension-change): Allows an X-spider to change dimension in accordance with its legs.
• Phase-copy laws (K0, K2, PC): Enable copying and commutation of phase parameters among spiders, with truncation to minimal wire dimension.

All rules are validated under the standard matrix interpretation, and mixed-dimensional rules recover the ordinary qudit rules when all labels coincide.

3. Normal Forms and Completeness

Every mixed-dimensional ZX diagram expresses a linear map in FHilb and can be systematically rewritten to a unique normal form, constructed as follows:

Given a tensor $A: \bigotimes_i \mathbb{C}^{m_i} \to \mathbb{C}$, flatten $A$ to a vector $\vec{a}$ of size $M = \prod_i m_i$, where each basis index $k$ is uniquely mapped to its digit expansion $(e_{k,0}, ..., e_{k,s-1})$ with $0 \leq e_{k,i} < m_i$. The corresponding normal-form diagram is:

$\tikzfig{qufit-normal-form/qufitnormalform-full}$

which, under map-state duality, represents universality—any finite-dimensional quantum map is equivalently expressible as a ZXW diagram in normal form.

Completeness is established by showing every diagram rewrites to normal form via:

1. Reducing generators to normal forms.
2. Closure under tensor product and partial trace.
3. Faithfulness: diagrams are equal in FHilb iff their ZXW normal forms coincide.

4. Mixed-Dimensional Interactions and Spin Networks

The calculus elegantly captures interactions between subsystems of distinct dimensions, crucial in quantum information and theoretical physics.

• Spin networks (SU(2) coupling): The spin-$j$ irrep on $(\mathbb{C}^2)^{\otimes n}$ is the symmetrizer $$\mathcal{S}_n = \frac{1}{n!} \sum_{\sigma \in S_n} \sigma$$. In ZXW, this is represented as an $n$-leg Z-spider with symmetric phase-vector.
• Clebsch-Gordan, 3jm-symbols: Coupling of spins, embedding of spin-$j$ spaces, and decompositions via explicitly constructed spiders and dimension-splitters. The standard SU(2) coupling objects (isometries, projectors, intertwiners) become simple composite spiders under the calculus.
• Permutation and tensor contraction: Mixed-dimensional rewrites—digit-splitting ($6 \to 2 \otimes 3$ wires), mixed Z-fusion, dimension-change for X-spiders—illustrate the full flexibility for building quantum circuits involving qubits, qutrits, and higher dimensional systems.

The calculus thus subsumes both the algebraic content of Penrose and Yutsis diagrams and the quantum information-theoretic representation of spin systems and networks.

5. Translation to Other Calculi and Structural Properties

The mixed-dimensional ZX calculus is tightly integrated with the finite-dimensional ZW calculus via explicit functors:

• $\iXW: ZX_f \to ZW_f$, mapping objects and generators with dimension shifts and vector correspondences.
• $\iWX: ZW_f \to ZX_f$, mapping back, preserving semantics.

Every equality provable in ZW lifts to a ZXW derivation, and vice versa. This mutual translation establishes universal completeness for the mixed-dimensional ZX calculus:

$$ZX_f \vdash D_1 = D_2 \leftrightarrow \llbracket D_1 \rrbracket_{ZX} = \llbracket D_2 \rrbracket_{ZX}$$

Thus, the calculus forms an expressive, sound, and complete rewriting system for finite-dimensional quantum theory.

6. Applications Across Quantum Theory

Representative applications include:

• Quantum circuits: Mixed-dimensional CNOT gates, efficient representation of quantum algorithms (e.g., QFT as large spiders).
• Quantum chemistry: Interaction of multi-level systems; diagrammatic expression of Jaynes–Cummings Hamiltonians.
• Condensed matter physics: Construction and analysis of AKLT states, parent Hamiltonians, and topological invariants, expressed and proved diagrammatically.
• Quantum gravity: Loop quantum gravity operators—e.g., diagrammatic calculation of quantized volume eigenvalues and explicit expression of intertwiner spaces.
• Quantum machine learning: Circuit gradients and variances for SU(2)-equivariant architectures, computed symbolically using the calculus—avoiding sampling-induced barren plateaus.

A plausible implication is that the mixed-dimensional ZX calculus provides a unifying diagrammatic layer underlying diverse quantum formalism, enabling new algorithmic and symbolic reasoning methods across theoretical physics and quantum computing.

7. Impact and Further Directions

By establishing a complete and universal graphical language for FHilb, the mixed-dimensional ZX calculus bridges the gap between diagrammatic quantum information theory and representation-theoretic approaches in physics and chemistry. This comprehensive encoding subsumes dynamics, symmetry, entanglement, and measurement in a single rewrite system. The embedding of spin-calculus fragments enables direct application to SU(2)-based models (quantum gravity, condensed matter, and quantum machine learning). Future directions may include further integration with categorical frameworks, extension to infinite-dimensional cases, and software implementations leveraging the calculus for automated reasoning and circuit optimization.