Magic Capacity in Quantum Systems

• Magic capacity is a measure of non-stabilizerness, defined as the variance of the estimator for the von-Neumann Stabilizer Rényi Entropy, revealing deviations from a flat Pauli spectrum.
• It enables scalable computation in quantum many-body systems using efficient MPS-based algorithms that reduce simulation costs and accurately capture complex state transitions.
• Applications include detecting quantum phase transitions in models like the TFIM and benchmarking non-Clifford resources in quantum circuits for enhanced error correction.

Magic capacity is a quantitative probe of non-stabilizerness—alternatively, "magic"—in quantum many-body systems and quantum information contexts. In the framework introduced by (Tarabunga et al., 9 Apr 2025), magic capacity is specifically defined as the variance of the estimator for the von-Neumann Stabilizer Rényi Entropy (SRE), yielding a direct measure of the anti-flatness of the Pauli spectrum and characterizing the complexity inherent in magic estimation. Efficient algorithms now allow for scalable computation of magic capacity and related mutual SRE across large systems, particularly those represented as matrix product states (MPS).

1. Formal Definition and Conceptual Significance

Magic capacity $C_M(|\psi\rangle)$ for a pure quantum state $|\psi\rangle$ is defined as the variance of the estimator for the von-Neumann SRE, obtained by sampling over Pauli strings $P$ with probability $p(P) = 2^{-N} \langle \psi | P | \psi \rangle^2$:

$$C_M(|\psi\rangle) = \operatorname{var}(\hat{M}_1) = \mathbb{E}_P\left[(\ln (\langle \psi | P | \psi \rangle^2))^2\right] - M_1^2$$

where $$M_1 = \mathbb{E}_P[\ln (\langle \psi | P | \psi \rangle^2)]$$ is the von-Neumann SRE. Only pure stabilizer states have a perfectly flat Pauli spectrum (which makes $C_M=0$); any deviation signals the presence of magic.

Magic capacity generalizes the notion of entanglement capacity to the resource theory of magic and has broad interpretational value:

• It quantifies not only the total magic but the spread (or "variance") of magic present in the Pauli spectrum.
• It controls the sample complexity for SRE estimation: higher $C_M$ means more samples are required for a fixed precision.
• It serves as an intrinsic diagnostic of non-stabilizerness, necessary for quantum computational advantage.

2. Computational Efficiency and Algorithmic Advances

Efficient computation of magic capacity is realized for MPS of bond dimension $\chi$, yielding an overall cost of $O(N\chi^3)$ for both the SRE and its variance, up to a factor $\epsilon^{-2}$ required to reach additive error $\epsilon$. This efficiency stands in contrast to exponential scaling methods.

• Individual Pauli expectation values $\langle \psi | P | \psi \rangle^2$ can be computed in $O(N\chi^3)$.
• The estimator error follows $\epsilon \sim \sqrt{C_M/K}$, with $K$ samples.
• The second derivative of $(1-\alpha)M_\alpha$ at $\alpha=1$ connects magic capacity to Rényi SREs:

$$C_M = \frac{d^2}{d\alpha^2}\left[(1-\alpha) M_\alpha\right]\Big|_{\alpha=1}$$

Two novel Monte Carlo schemes based on Metropolis-Hastings sampling are introduced for estimating the mutual $2$-SRE, which overcome inefficiencies arising from local update proposals. Additionally, the statevector simulation method is improved to $O(8^{N/2})$ time and $O(2^N)$ memory, demonstrated for systems with up to 24 qubits.

3. Applications: Quantum Phase Transitions and Circuit Randomness

Magic capacity serves as a sensitive marker for transitions in quantum many-body systems and random circuit ensembles.

• Transverse-field Ising model (TFIM): The mutual SRE, constructed on top of magic capacity, robustly signals the quantum critical point, independent of local basis choice. Whereas the bare SRE can be strongly basis-dependent, mutual SRE and capacity are basis-independent at criticality.
• Heisenberg XXX model: Scaling of $C_M$ with system size $N$ jumps from linear (volume law, $C_M \propto N$) to quadratic ($C_M \propto N^2$) in highly atypical ground states as the anisotropy is tuned.
• Clifford+$T$ circuit randomness: Magic capacity distinguishes typical states (where $C_M$ is $O(1)$ in the thermodynamic limit) from atypical states (with $C_M \propto N$), and delineates the threshold $z=N_T/N$ at which these circuits approach Haar randomness.

4. Monte Carlo Sampling and Statevector Simulation Techniques

The efficiency of magic capacity computation is underpinned by two algorithmic innovations:

• Metropolis-Hastings with Optimal Proposal: Utilizing the perfect sampling distribution $p(P)$ as a proposal yields high acceptance rates for states with nearly flat spectra, since the relevant Kullback-Leibler divergence $D_{KL}(p(P) || \Pi(P))$ is small when $M_1 - M_2$ is small.
• Hybrid Schrödinger–Feynman Algorithm: Sampling the first $N$ bits without explicit state storage, then using calculated marginals for the rest, reduces both memory and time complexity, facilitating simulation for system sizes beyond conventional methods.

These advances enable tractable estimation of magic capacity even in nontrivial many-body systems.

5. Criticality, Anti-Flatness, and Sampling Complexity

Magic capacity assesses how far the Pauli spectrum is from flatness (the anti-flatness), quantifying the complexity required to estimate the SRE in practice. States with large $C_M$ (high anti-flatness) are more complex to sample, consistent with increased non-stabilizerness.

• Critical points: Mutual SRE and $C_M$ yield consistent, basis-independent extremal values at phase transitions, as observed at the TFIM quantum critical point ($h\sim 1$).
• Sampling Cost: The number of required samples scales inversely with $C_M$, making it essential for benchmarking numerical methods and for practical simulation of quantum many-body systems.

6. Broader Implications and Future Directions

The improved computational tractability of magic capacity advances several research frontiers:

• Quantum complexity characterization: Enables high-resolution studies of non-stabilizerness as a resource alongside entanglement, bridging quantum information theory and many-body physics.
• Quantum algorithm and error correction design: Quantitative understanding of magic capacity informs the necessary non-Clifford resources, guiding the development of more efficient and accurate simulation and fault-tolerant circuit architectures.
• Universal quantum simulation: Provides a diagnostic to distinguish quantum phases, transitions, and randomness, relevant to both quantum simulation and the verification of quantum devices.

Efficient mutual SRE and magic capacity thus form a foundational toolkit for probing and harnessing the complexity and computational power of quantum states and circuits, especially in regimes where classical resources become insufficient. The advancements reported in (Tarabunga et al., 9 Apr 2025) set the stage for both large-scale theoretical studies and the practical deployment of magic-based quantum technologies.