Conformal field theories are magical

TL;DR

The paper investigates mana, a magic monotone that quantifies non-Clifford content, in the ground state of the 1D $q=3$ Potts model and shows that mana peaks at the critical point $ heta_c=rac{}$, reflecting long-range correlations and an infrared, scale-spanning magic. A MERA-based tensor-counting framework explains the origin and distribution of mana across scales, predicting finite-subsystem and two-point mana behavior that largely matches DMRG/MPS results, and revealing that the critical CFT is magical with nonlocal gate structure. A mean-field analysis for general $q$ indicates mana remains extensive but submaximal for $q>3$, while the multiscale nature of magic has implications for quantum simulation of field theories and constrains holographic tensor-network models of AdS-CFT. Overall, mana serves as a circuit- and information-structure-based diagnostic for ground-state preparation difficulty and for understanding how quantum information features encode the IR physics of conformal field theories.

Abstract

"Magic" is the degree to which a state cannot be approximated by Clifford gates. We study mana, a measure of magic, in the ground state of the $\mathbb Z_3$ Potts model, and argue that it is a broadly useful diagnostic for many-body physics. In particular we find that the $q = 3$ ground state has large mana at the model's critical point, and that this mana resides in the system's correlations. We explain the form of the mana by a simple tensor-counting calculation based on a MERA representation of the state. Because mana is present at all length scales, we conclude that the conformal field theory describing the 3-state Potts model critical point is magical. These results control the difficulty of preparing the Potts ground state on an error-corrected quantum computer, and constrain tensor network models of AdS-CFT.

Figures (14)

• Figure 1: Ground state mana density for $N = 6$-site systems (black) and $l$-site subsystems of an $N = 128$-site system (orange/red). For $N = 128$ the mana density is peaked at the phase transition and appears to take the form $m \propto |\theta - \theta_c|$. Comparison with the $N = 6$ mana density shows that the subsystem mana is a good estimator of the whole-system mana density; the peak of the $N = 6$ mana density is rounded and shifted away from the true critical point due to finite-size effects. The light-grey region shows where the correlation length is $2\xi > 7$, where finite-subsystem effects become important and our results substantially underestimate the mana density.
• Figure 2: Two-point mana: connected component of mana density $m_{\mathrm{cc}}(A,B)$ (Eq. \ref{['eq:mcc-def']}) as a function of the distance $\delta x$ between subsystems $A$ and $B$ for $A,B$ each one site (top) or two sites (bottom). We take $A$ at site $32$ and $B$ right of $A$. The black curve is mana density at the critical $\theta = \pi/4$; the two red lines are two different crude power-law fits. For $A,B$ each one site the mana density is $m_{\textrm{cc}}(A,B) \lesssim 10^{-14}$ (numerically zero) for $\delta x > 60$, because $\rho_{AB}$ enters $\textnormal{STAB}$ as it approaches $\rho_{A} \otimes \rho_B$. For $A,B$ each two sites and $\delta x \gg 1$ we find good agreement with the predicted exponent $\delta x^{-4/15}$, though we cannot rule out other small exponents.
• Figure 3: MERA representation of a critical ground state. In this work, we read a MERA from top to bottom as a quantum circuits preparing the state in question. The blue squares are unitary "disentanglers"; the green triangles are "isometries" that embed a small Hilbert space in a larger Hilbert space. The isometries can always be re-written as unitary gates by expanding the (smaller) input Hilbert space. This is guaranteed by the Stinespring dilation theorem. The network structure results in (1) scale invariance, (2) entanglement logarithmic in subsystem size, (3) power-law correlation functions, and (4) concrete predictions for subsystem and two-point mana.
• Figure 4: Causal domains of an $l$-site subsystem of a MERA state. Left, the "past domain of dependence": the set of gates that influence only the subsystem in question, and no other degrees of freedom. Right, the "past causal cone": the set of all gates that influence the subsystem in question, whether or not they also influence other gates.
• Figure 5: Iteratively constructing larger regions with larger past domains of dependence (cf Fig. \ref{['fig:mera:causal-domains']} left, Sec. \ref{['ss:mera:finite']}). Interpret the MERA as a top-to-bottom specification for a circuit that---after each set of blue square disentanglers---produces a valid ground state for the CFT. Start with a single blue disentangler; this is layer $k = 0$, and the past domain of dependence of two sites. Add two green isometries and then three blue disentanglers; this is layer $k = 2$, and the past domain of dependence of two sites. Proceeding in this way, at layer $2k-1$ we add $n_\triangle^{(2k-1)} = 2^{k+1}-2$ green isometries, and at layer $2k$ we add $n_\square^{(2k)} = 2^{k+1} - 1$ blue disentanglers, at which point we have constructed the past domain of dependence for $\ell^{(2k)} = 2^{2k+2} - 2$ sites.