Quantum Computational Resources and Conformal Field Theory: Unifying Spins, Bosons, and Fermions

Abstract: Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, can capture aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer Rényi entropy has revealed universal features of nonstabilizerness that are qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic Rényi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. This allows us to reveal common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary phase transition through bulk-induced boundary renormalization-group flows. As a concrete demonstration, we present a detailed CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters $K=1/3$ and $K=3$. We perform numerical calculations that confirm our field-theoretical predictions. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.

• The paper introduces magic Rényi entropy to unify quantification of quantum magic across spins, bosons, and fermions.
• It employs a convolution protocol with replica path integrals and boundary CFT to derive universal scaling laws and critical behaviors.
• Numerical validation via exact diagonalization confirms analytical predictions, revealing boundary RG flows and magic phase transitions.

Unified Characterization of Quantum Computational Resources via Conformal Field Theory

Introduction and Motivation

The quantification of quantum computational resources such as entanglement and magic has proven pivotal for both quantum many-body theory and the resource theory underpinning quantum computation. While entanglement is universally recognized as a fundamental resource across spins, bosons, and fermions, the "magic" resource—quantifying the state's deviation from efficiently simulable classes (stabilizer, Gaussian)—has, until now, lacked an encompassing, field-theoretical framework valid across all physical settings.

This paper presents a comprehensive theoretical construction that unifies the quantification of nonstabilizerness (spins/qudits) and non-Gaussianity (bosons/fermions) via a universal measure—magic Rényi entropy (MRE)—and describes its field-theoretical underpinning through conformal field theory (CFT). The authors introduce this measure, analytically derive several universal scaling results via boundary CFT, and corroborate these results through numerically exact diagonalization.

The Magic Rényi Entropy: Unified Construction and Properties

The central conceptual advance is the introduction of the magic Rényi entropy (MRE), $M_n$, constructed through a quantum convolution protocol: for a pure state $\rho$, $M_n$ is defined as the Rényi entropy of order $n$ of the reduced state resulting from an $n$-replica unitary convolution followed by tracing out $n-1$ replicas.

Figure 1: Schematic of quantum convolution—after the replica-mixing unitary, only resourceful (nonfree) quantum states generate inter-replica correlations detectable via the MRE.

For spins with local Hilbert-space dimension $d$, $M_n$ recovers the stabilizer Rényi entropy (SRE)—a well-studied measure of magic. In the continuous-variable case (bosons, fermions), $M_n$ is defined such that it precisely quantifies non-Gaussianity. This construction satisfies faithfulness, invariance under free unitaries (Clifford or Gaussian), and additivity for pure states. Under certain conditions, monotonicity under free operations is also established.

The convolution's defining unitary can be made explicit in all settings, with the Helmert (or beam-splitter for $n=2$) transformation for bosons/fermions, ensuring the measure is both computationally tractable and physically meaningful across platforms.

Field-Theoretical Formulation: Replica Path Integrals and Boundary Conditions

A central technical achievement is the path-integral representation of $\rho$0 for many-body thermal or ground states, recasting the convolution operation within replicated Liouville-space path integrals. The key observation is that the convolution and purity calculation map to a path-integral with a defect line (the convolution unitary) and a nontrivial boundary condition determined by partial traces and replica mixing.

Figure 2: Folded Liouville path-integral representations of the partition function and state purity, emphasizing the boundary sewing structure induced by the MES.

Figure 3: Schematic for the MRE path-integral, showing equivalence of the rotated boundary (upper, convolution as a boundary operation) and rotated bulk (lower, convolution as bulk deformation with unrotated boundary) perspectives.

Boundary CFT techniques reveal that the universal, size-independent contribution to MRE in critical one-dimensional systems is determined by the Affleck-Ludwig $\rho$1 factor—a measure of boundary entropy reflecting universal data of the infrared fixed-point boundary condition induced by the convolution. Two equivalent perspectives are analyzed:

• Rotated-boundary picture: The convolution unitary acts as a defect at the edge, changing the boundary state's symmetry and thus altering $\rho$2.
• Rotated-bulk picture: The bulk Hamiltonian is conjugated by the convolution, generating inter-replica couplings while preserving a simple boundary sewing structure.

Both approaches yield identical universal results, though their applicability and computational convenience differ depending on the context.

Analytical Results: Tomonaga-Luttinger Liquid and Universal Scaling

A concrete demonstration of the field-theoretical formalism is provided for the spinless fermion chain at half-filling, whose low-energy description is given by a Tomonaga-Luttinger liquid (TLL). The key parameter is the Luttinger $\rho$3, with $\rho$4 corresponding to the free-fermion, Gaussian fixed point.

The universal constant $\rho$5 (size-independent contribution to $\rho$6) is analytically obtained:

• In the vicinity of $\rho$7, $\rho$8 with $\rho$9. This result encapsulates the continuous renormalization of non-Gaussianity, with even dependence on $M_n$0 reflecting the $M_n$1 duality symmetry.
• Beyond a critical interaction ($M_n$2, i.e., $M_n$3 or $M_n$4), a boundary RG flow is induced, triggering a boundary phase transition. The boundary CFT analysis shows that in this case, $M_n$5 decreases for increasing system size, in accordance with the $M_n$6 theorem for boundary entropy monotonicity.

  Figure 4: (a) Unified structure of magic across spins, bosons, and fermions. (b) Top: Universal $M_n$7 scaling for the TLL as a function of $M_n$8. Dashed curve: analytical result. Bottom: Boundary RG flow in $M_n$9-space, indicating critical points at $n$0.

Numerical Validation and Finite-Size Scaling

Extensive exact diagonalization confirms these field-theoretical results:

• Numerical fits of $n$1 to scaling forms robustly extract $n$2 for various system sizes and interaction strengths.
• Data plotted as a function of $n$3 exhibit collapse under the $n$4 duality, confirming analytical predictions.

  Figure 5: Universal constant $n$5 extracted for the TLL as a function of $n$6, showing nonmonotonicity and agreement with field-theoretical predictions.

  

  Figure 6: Collapse of $n$7 data for $n$8 vs. $n$9, illustrating the duality $n$0.

A finite-size scaling analysis based on RG considerations yields the precise boundary phase transition point (numerically located at $n$1), in excellent agreement with the theoretical value $n$2.

Figure 7: Finite-size scaling crossing analysis used to locate the boundary phase transition at $n$3. Inset: zoom near the transition point.

Broader Implications, Open Questions, and Theoretical Outlook

This unified approach provides a rigorous, universal framework for many-body quantum computational resources. Notably:

• The MRE, as constructed, is proven faithful and additive, and satisfies monotonicity under key classes of free operations (including adaptive Gaussian/Stabilizer protocols for certain constructions).
• The field-theoretical framework, especially the connection to $n$4-factors and boundary RG flows, enables analytic computation of magic across diverse microscopic models, including systems with complex interactions, disorder, or topological features.
• The formalism subsumes earlier results for the SRE and advances understanding of non-Gaussianity’s universal structure in interacting bosonic and fermionic systems, including phenomena such as magic phase transitions and chaos indicators.
• The framework is readily extendable to the analysis of critical dynamics, measurement-induced transitions, hybrid systems, and potentially quantum gravity contexts (SYK models, holography).

Several open questions and challenges remain:

• Full operational meaning of the MRE for bosons and fermions at general $n$5; extension of monotonicity and robustness results to more general adaptive protocols.
• Development of efficient computational algorithms (e.g., tensor network or Monte Carlo methods) for MRE in large-scale systems—this is critical for numerically probing magic phase diagrams in higher dimensions or in the presence of disorder.
• Exploration of the interplay between magic and other complexity measures, and further elucidation of universal relationships between magic, entanglement, and chaos.

Conclusion

This work establishes a rigorous, unifying framework for quantum computational resources in many-body systems, bridging spin, boson, and fermion settings using the magic Rényi entropy and boundary conformal field theory. The field-theoretical insights allow for the analytical and numerical study of universal properties of "magic," including how many-body interactions can drive boundary RG flows and phase transitions. These results strengthen the connection between quantum information theory and low-energy field theories, positioning the MRE as a fundamental tool for the analysis and classification of quantum resources in condensed matter, quantum computation, and beyond.