We construct an explicit and solvable toy model for the AdS/CFT correspondence in the form of an approximate quantum error correction code with a non-trivial center 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 in the form of 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.

%B Journal of High Energy Physics %V 2021 %8 5/14/2021 %G eng %U https://arxiv.org/abs/2010.05960 %R https://doi.org/10.1007/JHEP05(2021)127 %0 Journal Article %J Physical Review B %D 2021 %T Conformal field theories are magical %A Christopher David White %A ChunJun Cao %A Brian Swingle %X"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 Z3 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.

%B Physical Review B %V 103 %P 075145 %8 2/25/2021 %G eng %U https://arxiv.org/abs/2007.01303 %N 7 %R https://journals.aps.org/prb/pdf/10.1103/PhysRevB.103.075145 %0 Journal Article %D 2021 %T From Quantum Codes to Gravity: A Journey of Gravitizing Quantum Mechanics %A ChunJun Cao %XIn this note, I review a recent approach to quantum gravity that "gravitizes" quantum mechanics by emerging geometry and gravity from complex quantum states. Drawing further insights from tensor network toy models in AdS/CFT, I propose that approximate quantum error correction codes, when re-adapted into the aforementioned framework, also has promise in emerging gravity in near-flat geometries.

%8 11/30/2021 %G eng %U https://arxiv.org/abs/2112.00199 %0 Journal Article %D 2021 %T Hyper-Invariant MERA: Approximate Holographic Error Correction Codes with Power-Law Correlations %A ChunJun Cao %A Jason Pollack %A Yixu Wang %XWe consider a class of holographic tensor networks that are efficiently contractible variational ansatze, manifestly (approximate) quantum error correction codes, and can support power-law correlation functions. In the case when the network consists of a single type of tensor that also acts as an erasure correction code, we show that it cannot be both locally contractible and sustain power-law correlation functions. Motivated by this no-go theorem, and the desirability of local contractibility for an efficient variational ansatz, we provide guidelines for constructing networks consisting of multiple types of tensors that can support power-law correlation. We also provide an explicit construction of one such network, which approximates the holographic HaPPY pentagon code in the limit where variational parameters are taken to be small.

%8 3/15/2021 %G eng %U https://arxiv.org/abs/2103.08631 %0 Journal Article %D 2021 %T Magic State Distillation from Entangled States %A Ning Bao %A ChunJun Cao %A Vincent Paul Su %XMagic can be distributed non-locally in many-body entangled states, such as the low energy states of condensed matter systems. Using the Bravyi-Kitaev magic state distillation protocol, we find that non-local magic is distillable and can improve the distillation outcome. We analyze a few explicit examples and show that spin squeezing can be used to convert non-distillable states into distillable ones.

Our analysis also suggests that the conventional product input states assumed by magic distillation protocols are extremely atypical among general states with distillable magic. It further justifies the need for studying a diverse range of entangled inputs that yield magic states with high probability.

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation. More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion. Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network. The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as non-stabilizer codes over qubits and qudits. For a contractible tensor network, the sequence of contractions also constructs a decoding/encoding circuit. To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct non-trivial codes. These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a 3d stabilizer code, and other stabilizer codes with interesting properties. Surprisingly, we find that the surface code is equivalent to the 2d Bacon-Shor code after "dualizing" its tensor network encoding map.

%8 9/16/2021 %G eng %U https://arxiv.org/abs/2109.08158 %0 Journal Article %J Journal of High Energy Physics %D 2020 %T Building Bulk Geometry from the Tensor Radon Transform %A ChunJun Cao %A Xiao-Liang Q %A Brian Swingle %A Eugene Tang %XUsing the tensor Radon transform and related numerical methods, we study how bulk geometries can be explicitly reconstructed from boundary entanglement entropies in the specific case of AdS3/CFT2. We find that, given the boundary entanglement entropies of a 2d CFT, this framework provides a quantitative measure that detects whether the bulk dual is geometric in the perturbative (near AdS) limit. In the case where a well-defined bulk geometry exists, we explicitly reconstruct the unique bulk metric tensor once a gauge choice is made. We then examine the emergent bulk geometries for static and dynamical scenarios in holography and in many-body systems. Apart from the physics results, our work demonstrates that numerical methods are feasible and effective in the study of bulk reconstruction in AdS/CFT.

%B Journal of High Energy Physics %V 2020 %P 1-50 %8 12/4/2020 %G eng %U https://arxiv.org/abs/2007.00004 %N 12 %0 Journal Article %J Classical and Quantum Gravity %D 2020 %T More of the Bulk from Extremal Area Variations %A Ning Bao %A ChunJun Cao %A Sebastian Fischetti %A Jason Pollack %A Yibo Zhong %XIt was shown recently, building on work of Alexakis, Balehowksy, and Nachman that the geometry of (some portion of) a manifold with boundary is uniquely fixed by the areas of a foliation of two-dimensional disk-shaped surfaces anchored to the boundary. In the context of AdS/CFT, this implies that (a portion of) a four-dimensional bulk geometry can be fixed uniquely from the entanglement entropies of disk-shaped boundary regions, subject to several constraints. In this Note, we loosen some of these constraints, in particular allowing for the bulk foliation of extremal surfaces to be local and removing the constraint of disk topology; these generalizations ensure uniqueness of more of the deep bulk geometry by allowing for e.g. surfaces anchored on disconnected asymptotic boundaries, or HRT surfaces past a phase transition. We also explore in more depth the generality of the local foliation requirement, showing that even in a highly dynamical geometry like AdS-Vaidya it is satisfied.

%B Classical and Quantum Gravity %V 38 %P 047001 %8 12/24/2020 %G eng %U https://arxiv.org/abs/2009.07850 %N 4 %R https://iopscience.iop.org/article/10.1088/1361-6382/abcfd0/pdf %0 Journal Article %D 2019 %T How Low Can Vacuum Energy Go When Your Fields Are Finite-Dimensional? %A ChunJun Cao %A Aidan Chatwin-Davies %A Ashmeet Singh %XAccording to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this essay, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein-Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

%8 05/11/2019 %G eng %U https://arxiv.org/abs/1905.11199 %0 Journal Article %D 2019 %T Towards Bulk Metric Reconstruction from Extremal Area Variations %A Ning Bao %A ChunJun Cao %A Sebastian Fischetti %A Cynthia Keeler %XThe Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi formulae suggest that bulk geometry emerges from the entanglement structure of the boundary theory. Using these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to show that in four bulk dimensions, the entanglement entropies of boundary regions of disk topology uniquely fix the bulk metric in any region foliated by the corresponding HRT surfaces. More generally, for a bulk of any dimension , knowledge of the (variations of the) areas of two-dimensional boundary-anchored extremal surfaces of disk topology uniquely fixes the bulk metric wherever these surfaces reach. This result is covariant and not reliant on any symmetry assumptions; its applicability thus includes regions of strong dynamical gravity such as the early-time interior of black holes formed from collapse. While we only show uniqueness of the metric, the approach we present provides a clear path towards an\textit {explicit} spacetime metric reconstruction.

%8 04/09/19 %G eng %U https://arxiv.org/abs/1904.04834