Non-Stabilizer Power in Hybrid Operations

• The paper's main contribution is establishing rigorous monotones, such as mana and relative entropy of magic, to quantify non-stabilizer resources in hybrid operations.
• It leverages resource-theoretic frameworks, simulation theory, and optimized circuit decompositions to benchmark the injection and spread of magic in scalable quantum systems.
• The findings imply practical enhancements in magic state distillation, error-resilient circuit design, and universal quantum computation through structured hybrid architectures.

Non-stabilizer power of hybrid operations refers to the quantifiable ability of quantum operations that go beyond the set of Clifford (stabilizer) dynamics to generate, transmit, and utilize non-stabilizerness—often termed “magic”—as a computational resource necessary for universal quantum computation. Hybrid operations, in this context, combine operations that are classically simulable (e.g. Clifford circuits or Gaussian operations) with non-stabilizer elements, which are essential for surpassing the classical computational boundary. Recent research developments establish rigorous frameworks, operational measures, theoretical limits, and practical algorithms for characterizing, optimizing, and benchmarking the non-stabilizer power of hybrid circuits, channels, and processes.

1. Resource-Theoretic Foundations for Hybrid Operations

The resource theory of stabilizer computation (Veitch et al., 2013) provides a foundational analogy to entanglement theory, declaring stabilizer operations (preparations, measurements in the computational basis, Clifford unitaries, ancilla stabilizer states, and partial traces) as “free.” These operations form a class that is efficiently classically simulable (Gottesman–Knill theorem), and all other states—those lying outside the convex stabilizer polytope—are called “magic” or non-stabilizer states. Hybrid operations are protocols or device architectures that primarily use free stabilizer operations but inject non-stabilizer resources (e.g., via magic states or non-Clifford gates).

Two monotones quantify the non-stabilizer resource:

Monotone	Formula / Principle	Remarks
Relative Entropy of Magic	$$r_m(\rho) = \min_{\sigma \in \text{STAB}} S(\rho \|\sigma)$$	Operational “distance” from stabilizer set
Mana (Sum Negativity)	$$\mathcal{M}(\rho) = \log\left(\sum_u |W_\rho(u)|\right)$$, $W_\rho(u)$: discrete Wigner function	Additive, computable from Wigner negativity

These monotones rigorously bound the efficiency of magic state distillation: to obtain $m$ copies of a target magic state $\sigma$ from $n$ copies of a resource $\rho$ using only stabilizer operations, the minimal cost satisfies $$n \geq m \cdot [\mathcal{M}(\sigma)/\mathcal{M}(\rho)]$$. This establishes a “currency exchange” principle for non-stabilizer power in hybrid operations, imposing absolute constraints on their resource efficiency.

2. Structural and Quantitative Features in Hybrid Quantum Operations

Detailed simulation theory relates the computational cost of simulating hybrid circuits—those with mostly stabilizer components and a sparse number of non-Clifford gates—to the stabilizer rank and the stabilizer extent (Bravyi et al., 2018). For an $n$-qubit state $|\psi\rangle$, the stabilizer rank $\chi$ is the smallest integer such that $$|\psi\rangle = \sum_{\alpha=1}^\chi c_\alpha |\phi_\alpha\rangle$$ with $|\phi_\alpha\rangle$ stabilizer states. The stabilizer extent, $$\xi(\psi) = \min\{\|c\|_1^2 : |\psi\rangle = \sum c_j |\phi_j\rangle\}$$, controls the asymptotic simulation complexity. The main findings include:

• Simulation costs for hybrid circuits with $m$ non-Clifford elements scale exponentially in $m$, but the base depends on the extremality or block structure of the non-Clifford components.
• Optimized decompositions (e.g., for CCZ vs. T gates) yield markedly better scaling.
• Hybrid simulation is tractable up to regimes previously considered classically intractable (e.g., $50$ qubits with $60+$ non-Clifford gates and millions of stabilizer terms).

This provides a quantitative lens for classifying the "hardness" of hybrid operations, measured precisely by non-stabilizerness monotones.

3. Hybrid Power in Indefinite Causal Order and Non-Standard Architectures

The quantum SWITCH (Mo et al., 2023) demonstrates that non-stabilizer power can be “unlocked” via coherent superpositions of causal orders. In this setting, even completely stabilizer-preserving channels (which do not generate magic in any definite order) may yield non-stabilizer output when embedded in a supermap that coherently switches the order of application. The magic resource capacity is introduced as a process-level monotone quantifying maximal magic that a quantum transformation can generate:

• In the quantum SWITCH, the superposition of free (CSPO) channels yields output states with non-zero robustness of magic even when all constituent operations are classically simulable.
• The magic resource capacity is resilient to depolarizing noise, revealing that hybrid operations exploiting indefinite causal order can be more robust to errors in preserving or generating magic than circuits with definite order.
• Random sampling suggests this enhancement is generic, indicating a structural broadening of the non-stabilizer power afforded by nontrivial circuit architectures.

4. Non-Stabilizer Power in Many-Body and Hybrid Physical Systems

Matrix product state (MPS) representations in the Pauli basis (Tarabunga et al., 29 Jan 2024, Smith et al., 20 Jun 2024) enable scalable evaluation of non-stabilizerness (e.g., stabilizer Rényi entropy $M_n$, stabilizer nullity $\nu$, Bell magic $\mathcal{B}_a$) in large systems. These tools prove invaluable for analyzing and benchmarking hybrid operations in many-body architectures, including:

• Rydberg atom arrays naturally serve as reservoirs of “many-body magic” via the Rydberg blockade, accessible experimentally through either quench protocols or adiabatic ground-state preparation.
• Analytical circuit decompositions of many-body MPS (with Clifford and sparse non-Clifford elements) identify the direct sources of non-stabilizerness, clarifying how hybrid gates catalyze magic propagation.
• Hybrid measurement and simulation techniques (tensor networks, Pauli basis contraction) provide resource estimates for designing and benchmarking logical qubits and hybrid processors.

In summary, these frameworks allow precise characterization and harvesting of non-stabilizer resources in complex, scalable physical systems, directly informing the construction of hybrid quantum processors.

5. Operational Amplification and Control via Entanglement and Circuit Structure

Entanglement (specifically stabilizer entanglement) acts as a “magic highway,” efficiently enabling the spread of locally injected magic throughout a system (Hou et al., 26 Mar 2025). Main findings include:

• The post-injection global magic (linear stabilizer entropy $Y$) obeys $\overline{Y} \propto 2^{-|A| - E}$ after a Haar random (magical) unitary is applied on a small subsystem $A$ of a stabilizer-entangled state (with bipartite entanglement $E$).
• For highly entangled initial states, product applications of local Haar unitaries result in almost global Haar-random magic content, even though only small, local non-stabilizer resources are applied—an effect robust to shallow brickwork circuit architectures.
• This amplification is independent of system size for a fixed (modest) entanglement content.

This mechanism underpins hybrid operation design where efficient Clifford circuits build the entangled substrate, while a small number of non-Clifford gates or resource states inject the required non-stabilizerness to achieve quantum advantage.

6. Fundamental Limits and No-Broadcasting of Non-Stabilizerness

Rigorous no-go theorems (Gupta et al., 27 Jan 2025) establish severe limitations on the broadcastability and clonability of non-stabilizerness under both stabilizer and arbitrary (hybrid) operations:

• It is fundamentally impossible to clone or broadcast the “magic” of generic quantum states, even with unrestricted operations, unless the magic content of the input does not exceed the reference states for which the transformation is designed.
• Traditional state cloning (Buzek–Hillery, Wootters–Zurek) may afford high-fidelity state copying, but these schemes do not optimize for non-stabilizerness; in fact, their non-stabilizer power is strictly lower than optimal magic generators.
• The resource theory is thus nonreproducible in the sense that one cannot circumvent distillation overhead or resource limitations via universal broadcasting or cloning—an essential structural difference from classical or entanglement resources.

This limits the scope of hybrid operations for scalable magic generation and has direct consequences for algorithmic design and the architecture of resource-aware quantum processors.

7. Extensions to Hybrid CV Systems and Unified Phase-Space Framework

Unified phase-space frameworks incorporating both bosonic (CV) and fermionic systems (Sarkis et al., 5 Sep 2025) extend non-stabilizer power quantification to hybrid operations in realistic platforms (e.g., polaron models, boson-fermion circuit QED):

• Hybrid Wigner functions and their $L_p$ norms serve as operational proxies for hybrid magic, smoothly interpolating between conventional mana and stabilizer Rényi entropy.
• The non-stabilizer power of hybrid gates, such as the conditional displacement gate, is quantified via averaged magic yield over “free” Gaussian–Majorana inputs, with explicit closed-form scaling formulas.
• The interplay of electron–phonon coupling or Jaynes–Cummings dynamics amplifies magic growth relative to purely bosonic states, making hybrid interactions a potent route for generating non-stabilizer resources.

These methodologies allow the resource theory of magic to capture the operational power and limits of hybridized platforms combining CV, qudit, and fermionic degrees of freedom.

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In summary, the non-stabilizer power of hybrid operations is a rigorously defined, quantitatively bounded resource that underpins the quantum advantage in universal computation. It is governed by resource monotones (e.g., mana, stabilizer Rényi entropy, robustness), facilitated and amplified by circuit structure (entanglement, causal superpositions), and fundamentally constrained in terms of reproducibility and broadcasting. Recent advances unify its quantification across discrete and hybrid CV systems, establish tight operational lower bounds on resource costs (e.g., $T$-count), and clarify the ultimate efficiency and architecture of quantum computational protocols predicated upon hybrid operations.