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Locality, code stability, and bulk dynamics: a three-way trade-off for emergent-gravity dictionaries

Zongmin Liu

Published Aug 5, 2026• 0 citations
Publisher

Abstract

A conserved charge cannot move a logical operator faster than the operator's own nonlocality permits. We make this exact and parameter-free. Let $A$ be a logical observable of a quantum code reconstructible with error $\epsilon$ on a region $S$, and let a Hermitian charge $\hat B$ supported on a region disjoint from $S$ induce the logical Hamiltonian $H$. Then $\|[H,A]\|\le\Delta_{\hat B}\,\epsilon$, where $\Delta_{\hat B}$ is the spectral spread of the charge: nothing else about the code enters, the constant is attained, and the exact case says a logical operator with an exact representative on any one charge-free region is dynamically inert. For a $U(1)$ charge on $|T|$ qubits this is an accuracy floor $\epsilon\ge\|[H,A]\|/2|T|$, the localized-charge counterpart of the $1/n$ suppression in transversal covariant codes. A complementary identity, $[H,A]=[G,H]+(E^{\dagger}R-R^{\dagger}E)$, resolves the obstruction into logical bandwidth and code leakage $\eta$, and we show $\eta^{2}=\max_{\psi}[\Var_{V\psi}(\hat B)-\Var_{\psi}(H)]$. We then compute $\epsilon$ itself. For $A$ an involution exchanging two logical states with charge amplitude distributions $a,b$, the smallest reconstruction error achievable by \emph{any} code realising $(a,b)$ is an explicit convex program in those two distributions, attained by an explicit code; for two charge levels its value is exactly the purified distance $\sqrt{1-F^{2}}$ between them. Reconstruction locality on a charge-free region is charge-distribution distinguishability. This collapses certification of $\epsilon$ from a semidefinite program of size $2^{2|S|}$ to two measured charge histograms, and it closes the achievable region in closed form: at two levels it is disconnected precisely for $\eta<\sqrt3/2$. Applied to emergent gravity, where the Hamiltonian is a boundary charge, the exact case is the sharp form of Marolf's theorem.

DOI
10.5281/zenodo.21842632

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A conserved charge cannot move a logical operator faster than the operator's own nonlocality permits. We make this exact and parameter-free. Let AAA be a logical observable of a quantum code reconstructible with error ϵ\epsilonϵ on a region SSS, and let a Hermitian charge B^\hat BB^ supported on a region disjoint from SSS induce the logical Hamiltonian HHH. Then ∥[H,A]∥≤ΔB^ ϵ\|[H,A]\|\le\Delta_{\hat B}\,\epsilon∥[H,A]∥≤ΔB^​ϵ, where ΔB^\Delta_{\hat B}ΔB^​ is the spectral spread of the charge: nothing else about the code enters, the constant is attained, and the exact case says a logical operator with an exact representative on any one charge-free region is dynamically inert. For a U(1)U(1)U(1) charge on ∣T∣|T|∣T∣ qubits this is an accuracy floor ϵ≥∥[H,A]∥/2∣T∣\epsilon\ge\|[H,A]\|/2|T|ϵ≥∥[H,A]∥/2∣T∣, the localized-charge counterpart of the 1/n1/n1/n suppression in transversal covariant codes. A complementary identity, [H,A]=[G,H]+(E†R−R†E)[H,A]=[G,H]+(E^{\dagger}R-R^{\dagger}E)[H,A]=[G,H]+(E†R−R†E), resolves the obstruction into logical bandwidth and code leakage η\etaη, and we show η2=max⁡ψ[\VarVψ(B^)−\Varψ(H)]\eta^{2}=\max_{\psi}[\Var_{V\psi}(\hat B)-\Var_{\psi}(H)]η2=maxψ​[\VarVψ​(B^)−\Varψ​(H)]. We then compute ϵ\epsilonϵ itself. For AAA an involution exchanging two logical states with charge amplitude distributions a,ba,ba,b, the smallest reconstruction error achievable by \emph{any} code realising (a,b)(a,b)(a,b) is an explicit convex program in those two distributions, attained by an explicit code; for two charge levels its value is exactly the purified distance 1−F2\sqrt{1-F^{2}}1−F2​ between them. Reconstruction locality on a charge-free region is charge-distribution distinguishability. This collapses certification of ϵ\epsilonϵ from a semidefinite program of size 22∣S∣2^{2|S|}22∣S∣ to two measured charge histograms, and it closes the achievable region in closed form: at two levels it is disconnected precisely for η<3/2\eta<\sqrt3/2η<3​/2. Applied to emergent gravity, where the Hamiltonian is a boundary charge, the exact case is the sharp form of Marolf's theorem.
Bulk reconstruction is a central problem in AdS/CFT, and entanglement wedge reconstruction is its subregion version. We argue that this subregion statement should be separated from the stronger holographic quantum error correction interpretation, in which one region-independent logical bulk operator has code-preserving representatives in several boundary regions. A simple locality argument shows that such a common reconstruction must commute with the code-preserving local algebras in the complementary regions. This is the mechanism realized in HaPPY-type codes: the erased regions are blind to a protected logical algebra. An ordinary finite NNN holographic CFT does not have such a protected invisible sector for supergravity fields. Its low-energy local observables, in particular, suitably smeared stress tensors, detect the physical support and gravitational dressing of ordinary bulk operators, up to possible center or superselection data. Thus, there is no such holographic quantum error correction and the N=∞N=\inftyN=∞ agreement of global and subregion HKLL formulae is a free-theory statement. What remains is entanglement wedge reconstruction without holographic quantum error correction, or subregion complementarity: each boundary region has its own code-preserving low-energy algebra and its own region-adapted bulk interpretation, rather than a shared logical operator.
We explicitly construct a class of holographic quantum error correction codes with non-trivial centers in the code subalgebra. Specifically, we use the Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer code with gauge-fixing), which we call the holographic hybrid code. This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the sub-region duality. We then construct approximate versions of the holographic hybrid codes by “skewing” the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography. These approximate hybrid codes are not necessarily stabilizer codes, but they can be expressed as the superposition of holographic tensor networks that are stabilizer codes. For such constructions, different logical states, representing different bulk matter content, can “back-react” on the emergent geometry, resembling a key feature of gravity. The locality of the bulk degrees of freedom becomes subspace-dependent and approximate. Such subspace-dependence is manifest from the point of view of the “entanglement wedge” and bulk operator reconstruction from the boundary. Exact complementary error correction breaks down for certain bipartition of the boundary degrees of freedom; however, a limited, state-dependent form is preserved for particular subspaces. We also construct an example where the connected two-point correlation functions can have a power-law decay. Coupled with known constraints from holography, a weakly back-reacting bulk also forces these skewed tensor network models to the “large N limit” where they are built by concatenating a large N number of copies.
We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four σσσ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-σσσ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply Pr⁡L(fail)≤C∣ΩL∣e−cL\Pr_L(\mathrm{fail})\le C|Ω_L|e^{-cL}PrL​(fail)≤C∣ΩL​∣e−cL below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.
Abstract We explicitly construct a class of holographic quantum error correction codes with non-trivial centers in the code subalgebra. Specifically, we use the Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer code with gauge-fixing), which we call the holographic hybrid code. This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the sub-region duality. We then construct approximate versions of the holographic hybrid codes by “skewing” the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography. These approximate hybrid codes are not necessarily stabilizer codes, but they can be expressed as the superposition of holographic tensor networks that are stabilizer codes. For such constructions, different logical states, representing different bulk matter content, can “back-react” on the emergent geometry, resembling a key feature of gravity. The locality of the bulk degrees of freedom becomes subspace-dependent and approximate. Such subspace-dependence is manifest from the point of view of the “entanglement wedge” and bulk operator reconstruction from the boundary. Exact complementary error correction breaks down for certain bipartition of the boundary degrees of freedom; however, a limited, state-dependent form is preserved for particular subspaces. We also construct an example where the connected two-point correlation functions can have a power-law decay. Coupled with known constraints from holography, a weakly back-reacting bulk also forces these skewed tensor network models to the “large N limit” where they are built by concatenating a large N number of copies.
We develop a novel holographic framework to study dynamical phases in random quantum circuits with a global symmetry GGG. Viewing the circuit as a tensor network, we decompose it into two parts: a symmetric layer, which defines an emergent gauge wavefunction in one higher dimension, and a non-symmetric layer, composed of random multiplicity tensors. For G = ZNG\,{=}\,\mathbb{Z}_NG=ZN​ symmetric circuits consisting of local unitary gates interspersed with local symmetric noise channels, averaging over the non-symmetric layer yields a dynamically generated noisy ZN\mathbb{Z}_{N}ZN​ surface code. This allows us to interpret ZN\mathbb{Z}_{N}ZN​ symmetric circuits in the volume-law phase as quantum error-correcting codes with a distinguished set of logical spin states that inherit the topological protection of the bulk code. By establishing equality of bulk and boundary coherent information, we show that quantum information encoded in these logical states is maximally protected against symmetric noise up to a finite threshold. We further study weakly monitored ZN\mathbb{Z}_{N}ZN​ symmetric circuits which exhibit a charge-sharpening transition. We show that the point at which the observer gains classical information about the global charge coincides with the point at which measurements destroy the underlying quantum information encoded in the bulk surface code. This also allows for a natural interpretation of the sharpening transition as a confinement transition in the gauge theory. For N ≤ 4N\,{\leq}\,4N≤4, weak measurements drive a single transition from a charge-fuzzy phase with exponential sharpening time t#∼eLt_{\#}\sim e^{L}t#​∼eL to a charge-sharp phase with t#∼O(1)t_{\#}\sim \mathcal{O}(1)t#​∼O(1). On the other hand, for N>4N>4N>4, the circuit can enter an intermediate phase with a linear sharpening time t#∼O(L)t_{\#}\sim \mathcal{O}(L)t#​∼O(L). In this regime, the bulk gauge theory realizes a Coulomb phase with emergent gapless photons.
Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the error-correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems nor the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.