Stabiliser Formalism for Mixed Dimensional Hilbert Spaces

• The stabiliser formalism for mixed dimensional Hilbert spaces is a unified framework that extends quantum error correction to systems with varying local dimensions.
• It utilizes generalized Weyl operators, finite geometry, and modular arithmetic to construct robust stabiliser codes and ensure fault tolerance.
• The framework supports scalable quantum computation, addressing error correction and entanglement measures in non-uniform architectures with practical simulation benefits.

The stabiliser formalism for mixed dimensional Hilbert spaces generalizes standard techniques from quantum error correction and simulation, allowing error correction, entanglement, and fault tolerance in quantum systems where local subsystem dimensions differ. This framework unifies approaches for constant-dimensional qudit systems, systems based on finite fields, boundary and bulk decompositions, as well as infinite-dimensional/bosonic settings, and provides a comprehensive algebraic and geometric description suitable for theoretical and practical analysis.

1. Foundations of Stabiliser Formalism in Mixed Dimensional Spaces

The core necessity for the mixed dimensional formalism arises from the lack of practical motivation for requiring all subsystems of a quantum computer or quantum network to share the same local Hilbert space dimension. Traditional stabiliser codes operate on spaces like $(\mathbb{C}^q)^{\otimes n}$, with stabilisers drawn from the generalized Pauli group, but mixed dimensional systems instead use $$\mathcal{H} = \mathbb{C}^{D_1} \otimes \cdots \otimes \mathbb{C}^{D_n}$$, allowing $D_i$ to vary arbitrarily across different subsystems (Ball et al., 20 Oct 2025). Given this heterogeneity, one constructs stabiliser groups $\mathcal{S}$ as commutative subgroups of unitary operators that permute, possibly up to phase factors, the computational basis states. This relaxation does not require a “nice error basis” or restriction to Pauli/Weyl operators, making the formalism applicable to a broad class of systems and interactions.

A stabiliser code is then the common +1 eigenspace of all $M \in \mathcal{S}$:

$$Q(\mathcal{S}) = \{\,|\psi\rangle \in \mathcal{H}\mid M|\psi\rangle = |\psi\rangle\ \forall\, M\in \mathcal{S}\,\}$$

and its dimension is computed by:

$$\dim Q(\mathcal{S}) = \frac{1}{|\mathcal{S}|} \sum_{M\in\mathcal{S}} \mathrm{tr}(M)$$

This formula recovers the constant-dimension case and is invariant under the choice of error basis.

2. Algebraic and Geometric Structure of Stabiliser Codes

For uniform local dimensions, algebraic correspondences allow one to map stabiliser codes over fields of even order $\mathrm{GF}(2^h)$ (quqit systems) to binary stabiliser codes on $h n$ qubits (Ball et al., 12 Jan 2024), using trace-orthogonal bases to represent each field element. The associated mapping is:

Original System	Mapped System	Structure
$n$ quqits, $q=2^h$	$hn$ qubits	$$U_{n,h}|x\rangle = |x_1\rangle \otimes \cdots \otimes |x_h\rangle$$
Pauli/Weyl over $F_{2^h}$	Tensor products over $\mathbb{Z}_2$	Operators $X(x)$ and $Z(y)$ mapped as $X(x_1) \otimes \cdots \otimes X(x_h)$ etc.

This bijection preserves code parameters (distance, dimension) up to scaling and encodes stabiliser codes as “quantum sets of symplectic polar spaces,” enabling classification by finite geometry techniques. For mixed-dimensional cases, the geometric description becomes more complex and typically involves “quantum sets” of projective subspaces with prescribed intersection and isotropy properties.

In finite plane geometry–Hilbert space correspondences (Revzen et al., 2015), operators associated with points (e.g., MUB projectors) are systematically grouped into “line operators” via sums over geometric lines, yielding orthogonal bases for operator expansion. These mappings provide phase space representations analogous to the finite Wigner function and are crucial for visualizing error syndromes and designing codes.

3. Extension and Linearization of Weyl Operators

Central to the simulation and manipulation of stabiliser codes on mixed or arbitrary dimensions is the generalized Weyl/displacement operator formalism (Beaudrap, 2011). For local dimension $d$, the Weyl operator is defined as:

$W_{a,b} = \tau^{-ab} Z^a X^b$

where $\tau = \exp(\pi i/D)$ and $D=d$ for odd $d$, $D=2d$ for even $d$. The arithmetic is consistently performed modulo $D$, ensuring that commutation relations and phase factors remain linear:

Property	Formula	Comment
Commutation	$W_v W_w = \tau^{\langle v, w \rangle} W_{v + w}$	$\langle v, w \rangle = v^T \sigma_{2n} w$
Clifford conjugation	$U W_v U^\dagger = W_{C_U v}$	$C_U \in \mathrm{Sp}_{2n}(\mathbb{Z}_D)$
Stabiliser state updating	$$[\tilde{\phi}; \tilde{v}] = {}_U [\phi; v]\ (\mathrm{mod}\; D)$$	Tableau transformation under Clifford

This approach eliminates quadratic phase corrections and extends efficiently to mixed-dimensional configurations and even composite dimensions. Tableaus must sometimes be augmented by “phase correction” blocks when no fundamental Weyl vector basis exists, especially for even or composite $d$.

4. Error Correction, Singleton Bound, and Dimensional Weight

Error correction in mixed-dimensional stabiliser codes is quantified by a generalization of the Singleton bound (Ball et al., 20 Oct 2025). For a QECC on $$\mathcal{H} = \mathbb{C}^{D_1}\otimes\cdots\otimes\mathbb{C}^{D_n}$$ with code dimension $K$ and distance $D$, the bound asserts:

$$\prod_{i=1}^s D_i < D,\quad \prod_{i=s+1}^{r+s} D_i < D\implies \prod_{i=r+s+1}^n D_i \geq K$$

where the “dimensional weight” of an error operator is $\mathrm{dimwt}(E) = \prod_{i\in S} D_i$ with $S$ its support.

This generalizes the familiar $n \geq k + 2(d-1)$ result for uniform dimension and, when used with the code dimension formula above, provides an explicit guide for code construction and resource allocation across heterogeneous quantum architectures.

5. Entanglement Measures and Absolutely Maximally Entangled States

The formalism incorporates entanglement via a generalized measure for mixed-dimensional subsystems. Given $S\subset \{1,\ldots,n\}$ with total dimension $r=\prod_{i\in S} D_i$, the average entanglement is:

$$EM_r(|\psi\rangle) = \frac{1}{f} \sum_{\dim(S)=r} \frac{r}{r-1} (1-\mathrm{tr}(\rho_S^2))$$

where $f$ is the number of subsystems of dimension $r$ and $\rho_S$ is the reduced density matrix on $S$.

A state is defined to be absolutely maximally entangled (AME) if for all $S$ with $\dim(S) \leq \Delta$ (where $\Delta \approx \sqrt{\dim(\mathcal{H})}$) the reduced $\rho_S$ is maximally mixed. AME states coincide with one-dimensional stabiliser codes of minimum dimensional distance $\lfloor\Delta\rfloor + 1$. Examples include constructions on $\mathbb{C}^2 \otimes (\mathbb{C}^3)^{\otimes 3}$ and other mixed configurations (Ball et al., 20 Oct 2025).

6. Simulation Complexity and Algorithmic Implications

Simulating mixed-dimensional stabiliser circuits is provably complete for the complexity class $\mathrm{coMod}_d L$; that is, the problem reduces to solving systems of linear equations modulo $d$ (Beaudrap, 2011). Simulations—including evolution, measurement updates, and code dimension calculation—thus admit classical Boolean circuit implementations of depth $O(\log(n)^2)$ for constant local dimensions.

In cases where coprime or composite local dimensions are present, tableau extension and augmented phase correction steps maintain the linear structure. These results delineate classical simulability for the Clifford group and stabiliser circuits even in non-uniform settings, defining boundaries between efficient simulation and regimes where quantum advantage may be manifest.

7. Theoretical Frameworks and Extensions

The formalism interfaces with broader frameworks addressing mixed representations in condensed matter physics and quantum field theory (Buot et al., 2022). Universal operator expansions employ displacement operators $Y(u,v)$ over mixed Hilbert spaces, extending the scope to infinite-dimensional, bosonic, and fermionic systems. These representations, together with categorical extensions in higher Hilbert spaces (Chen et al., 7 Oct 2024), guarantee that error-correcting and stabiliser maps respecting physical unitarity are fully encoded at all algebraic levels.

Further, diagrammatic approaches, as in complete ZX-calculi for odd prime dimensions (Booth et al., 2022), demonstrate completeness for rewrite rules and mixed-state fragment stabiliser mechanics. These results suggest universal methods for code construction, verification, and equivalence testing across mixed and composite-dimensional systems.

8. Applications and Future Directions

Mixed-dimensional stabiliser codes provide a theoretical and practical foundation for heterogeneous quantum computation, fault tolerance with non-uniform quantum network nodes, generalized quantum secret sharing, and entanglement-based protocols. The formalism supports quantum error correction where subsystem physical realizations (e.g., ions, photonic modes, superconducting circuits with varying level structure) differ, and directly informs the design of codes attaining generalized Singleton bounds.

A plausible implication is the potential extension of these methods to codes that reach the Hamming bound for mixed alphabets, as well as constructions of new classes of AME states leveraging the flexible stabiliser definitions. Research in this direction can clarify the impact of dimensional weight on practical error models and the integration of nonlocal stabilisers in scalable architectures.

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In summary, the stabiliser formalism for mixed dimensional Hilbert spaces generalizes core quantum error-correcting structures to settings with heterogeneous subsystems by employing a uniform group-theoretic and operator-theoretic approach grounded in linear algebra over modular arithmetic, geometric classification via symplectic polar spaces, and entanglement theory. This provides a unifying methodology applicable to quantum computation, condensed matter, and beyond, with robust simulation, error correction, and entanglement quantification capabilities.