Holographic tensor network models and quantum error correction: A topical review
HHolographic tensor network models and quantum error correction:A topical review
Alexander Jahn and Jens EisertDahlem Center for Complex Quantum Systems,Freie Universit¨at Berlin, 14195 Berlin, Germany
Abstract
Recent progress in studies of holographic dualities, originally motivated by insights from string the-ory, has led to a confluence with concepts and techniques from quantum information theory. A partic-ularly successful approach has involved capturing holographic properties by means of tensor networkswhich not only give rise to physically meaningful correlations of holographic boundary states, but alsoreproduce and refine features of quantum error correction in holography. This topical review provides anoverview over recent successful realizations of such models. It does so by building on an introductionof the theoretical foundations of AdS/CFT and necessary quantum information concepts, many of whichhave themselves developed into independent, rapidly evolving research fields.
Contents a r X i v : . [ qu a n t - ph ] F e b Introduction
This topical review covers the intersection of two fields of physics that have recently been identified as be-ing closely related to each other. One of these two fields is high-energy physics, in form of the holographicprinciple as realized by the anti-de Sitter/conformal field theory correspondence (AdS/CFT), a striking du-ality conjecture that in itself brings together notions of gravity and of conformal field theories. The otherresearch field is that of quantum information, which has turned out to offer surprising new insights intothe properties of holographic dualities. This applies specifically to notions of quantum error correction that have arisen in the context of quantum computing, connecting a topic of immense practical relevance toone developed from pure theory. An increasing number of connections between holography and quantuminformation is being unearthed as research within a steadily growing research community progresses, bridg-ing the extensive theoretical foundation underlying research on AdS/CFT and quantum error correction. Inthese endeavours, one particularly useful and concrete tool that captures key aspects of holography is that oftensor networks. Originally born out of condensed matter and mathematical physics, they have taken centerstage in many questions at the heart of quantum information science due to their inherent relation to theconcept of quantum entanglement, but they have also found wide-ranging applications that include numeri-cal analysis, machine learning and probabilistic modelling. In this review, we describe how tensor networkshave developed from a practical tool for computing properties of low-dimensional quantum systems to anindispensable component of holographic models. However, the full range of such models would already ex-tend the bounds of a topical review due to the vast amount of research work produced in the last few years.For this reason, we focus here on those tensor networks exploring a particular salient quantum informationfeature of AdS/CFT, that of quantum error correction. While the subfield of holographic quantum errorcorrection extends significantly beyond tensor network methods, these offer a particularly transparent andpractically computable framework in which to explore such questions.Before exploring the state of current research in this field, Sec. 2 begins with a detailed introduction intothe foundations of the field both on the high-energy and quantum information side. We provide a lightningreview of string theory and the specific setup of the AdS/CFT correspondence which served as the startingpoint of much of the vast amount of research on holography in the past twenty years. Then we introducethe two topics of quantum information theory essential for this review: Tensor networks and quantum errorcorrection, for both of which we sketch their original development and basic principles. In Sec. 3, wethen describe how quantum information and holography became connected, starting with the discoveryof holographic entanglement entropy and the role of entanglement in modern notions of quantum gravity.The inclusion of tensor networks and quantum error-correcting codes into the scope of holography is thenlaid out, laying the groundwork for the core topic of this review. In Sec. 2 and 3 we end each subsectionwith a brief referral to more detailed introductory work to each topic where such is available. Bringingall the previously introduced topics together, Sec. 4 then discusses holographic tensor network models ofquantum error correction. A large focus of this section is the class of holographic toy models known as
HaPPY codes which reproduce many of the features of continuum models of quantum error correctionin AdS/CFT. Subsequently, we introduce the language of
Majorana dimer states that makes it possible tocompute directly many of the boundary properties of the codes, which we discuss in detail. We then describeways in which the original HaPPY proposal can be extended to produce more general holographic models,and what different approaches exist for capturing holographic quantum error correction in tensor networks.Sec. 5 then closes with an outlook on the future of the field as well as acknowledgements.
Foundations
Before exploring the concrete duality proposals relevant for this topical review, we begin with the foundationupon which they all rest, the holographic principle . Its origins can be traced back to the 1970s, when physi-cists began to consider black holes from a new vantage point: Based on the work of Stephen W. Hawkingand Jacob D. Bekenstein, it was realized that black holes are thermodynamical objects with a well-definedtemperature and entropy [1, 2]. For a black hole with a given mass M , these are given by T H = (cid:126) c πk B GM , S BH = 4 GM (cid:126) c = c A hor (cid:126) G , (1)commonly called the
Hawking temperature and
Bekenstein-Hawking entropy of a black hole, respectively.When expressed in terms of the area A hor of the event horizon, the latter equation, more succinctly writtenin natural units as S BH = A hor / G , contains a surprising insight: Rather than growing with its volume,as a conventional thermodynamical system, a black hole’s entropy grows with its surface area! This hasled to the suggestion that the information of a black hole’s microstates are holographically encoded on itshorizon. The “resolution” of this encoding is on the order of the Planck scale, as we can see by writingthe denominator of (1) in terms of the Planck length l P as (cid:126) G/c = 4 l P . From this observation, LeonardSusskind and Gerardus ’t Hooft declared that a consistent theory of quantum gravity would have to obeya holographic principle : The dynamics of gravity in -dimensional spacetime in such a theory wouldhave be reducible to an effective -dimensional description [3, 4]. While the entropy scaling in termsof area rather than volume appeared in gravitational settings other than black holes [5], the holographicprinciple was fundamentally vague: It neither specified which theory of quantum gravity would producesuch a holographic mapping between systems in different dimensions, nor how this mapping would beimplemented.For this reason, the conjecture of the AdS/CFT correspondence , a specific holographic duality betweena d +1 -dimensional gravitational theory and a d -dimensional quantum field theory by Juan M. Maldacenain 1997 [6] was met with a tremendous amount of research activity. It fundamentally changed the field ofstring theory, from which it was derived, and had repercussions in a wide range of research areas beyondthe high-energy theory community. The basic setup of AdS/CFT, along with the necessary string-theoreticconcepts it has been built upon, will be reviewed in the next sections. The development of quantum field theory in the second half of the 20th century led to a consistent and pre-cise description of high-energy processes occurring in nature. With the
Standard Model of particle physics,quantum field theory unified electromagnetic, weak and strong interactions into one formulation. Quantumfield theory, however, is an inherently effective theory. The fields of the Standard Model require renor-malization : Their naive formulation leads to diverging physical quantities, requiring the introduction of aregulating energy or length scale (similar to the lattice scale in solid state models) and leading to physicalobservables such as coupling “constants” depending on the energy at which the system is probed. This im-plies that as higher and higher energies are considered, the behavior of the theory changes; a quantum fieldtheory valid at lower energies may need to be replaced by a more complicated one at higher energies, e.g.,by introducing new intermediate particles. While the Standard Model with its finite parameters describes the During the remainder of this review, natural units with (cid:126) = c = 1 will be used. Newton’s constant G will be kept explicit, asit acts as a useful scale in AdS/CFT. gravitons ) can be easily con-structed at low energies. However, at high energies the process of renormalization requires the introductionof increasingly many parameters to cancel out divergences, making the theory useless for actual predictions.This suggests that a naive field quantization of gravity is only an effective theory for a more fundamentaltheory of quantum gravity appearing at exceedingly high energies.One candidate for such a theory is given by string theory. Rather than the fundamental point-like parti-cles appearing in quantum field theory, this approach proposes the quantization of one-dimensional objectscalled strings. Similar to how the trajectories of point particles correspond to worldlines in spacetime, astring traces out a two-dimensional worldsheet X µ ( τ, σ ) parametrized by two coordinates τ and σ . Notethat the D -vector X µ can describe a point in a target spacetime of arbitrary dimension D > . Extendingthe action of a point particle in special relativity to a two-dimensional object, one arrives at the Nambu-Goto action S NG = − π √ l s (cid:90) d τ d σ (cid:115)(cid:18) ∂X µ ∂τ ∂X µ ∂σ (cid:19) − (cid:18) ∂X µ ∂τ ∂X µ ∂τ (cid:19) (cid:18) ∂X µ ∂σ ∂X µ ∂σ (cid:19) , (2)where X µ = η µν X ν (with the Minkowski metric η µν ) and the constant l s is known as the string length (often replaced by a coefficient α (cid:48) = l s ).Solutions to the action (2) can be either open or closed strings; in the former case, this means that thestring endpoints need to be associated with Dirichlet or Neumann boundary conditions. It was later realizedthat in the first case, the dynamics of the endpoints are related to higher-dimensional objects known as Dirichlet-branes or D-branes for short.The spectrum of possible excitations on strings can be identified with particles of mass M . For thepurely bosonic action (2), the vacuum state of both open and closed strings leads to unphysical tachyons with M < . The first excited states, however, become massless if the target space dimension is chosen as D = 26 . These states can be identified with gauge bosons for open strings and gravitons for closed strings.In addition, closed string excitations contain scalar dilatons and an antisymmetric tensor field.Interactions in string theory are considerably more constrained than in regular quantum field theory,where coupling constants are usually free parameters to be determined by experiment. In contrast, inter-actions of strings follow directly from geometrical considerations: For example, by pinching together twopoints of a closed string it is split into two new ones. Similarly, open strings can turn into closed onesthrough the joining of endpoints. The effective string coupling g s is determined by the vacuum expectationvalue of the dilaton field.While twenty-six dimensions could be reduced to our familiar four by compactification of the remainingdimensions to small scales, leading to new effective lower-dimensional fields, the problem of a tachyonicground state is not easily circumvented. However, after extending the bosonic action to a supersymmet-ric one containing both bosonic and fermionic degrees of freedom (see Summary 1), the tachyonic statescan be removed through the GSO projection [8]. This projection removes states of even fermionic par-ity, including the unphysical vacuum. In the case of supersymmetric string theory, the critical number ofdimensions necessary to produce massless states is reduced to ten. Different projections led to different ten-dimensional superstring theories. For closed superstrings, due to different possible choices of worldsheet(anti-)periodicity of the left- and right-moving modes, two consistent models known as type IIA and typeIIB superstring theory emerge. In addition, by separately placing bosonic and supersymmetric modes in Named after Yoichiro Nambu and Tetsuo Goto, though no formal publication of theirs introduces it. In practice, a reformulationin terms of the equivalent
Polyakov action [7] is more conveniently used. ummary 1: Supersymmetry Classifying quantum field theories by their symmetries has driven much of the development of theStandard Model over the late 20th century. While bosons and fermions in the Standard Model areintimitely related by gauge symmetries, it is possible to extend its field content to allow for a directsymmetry between bosonic and fermionic fields, called supersymmetry . Following Noether’s theo-rem, supersymmetry implies the existence of supercharges Q . These act as operators on fields thatchange their spin by / , turning bosonic fields into fermionic ones and vice-versa. The number N of possible distinct supercharges is subject to physical constraints. Models of extended supersym-metry with N > preserve chiral symmetry and are thus incompatible with the Standard Model.In dimensions, a theory with spins s ≤ can have N = 4 at most. A particular exampleof such a theory is N = 4 super-Yang-Mills theory, which also possesses conformal symmetry.Supersymmetries with N > imply fields with spin s > .the left- and right-moving sector, another consistent solution known as heterotic string theory is recovered. Finally, another possibility is given by type I superstring theory, which contains both open and closed unori-ented strings. This web of consistent string theories was later found to be connected by dualities that mapfrom one theory to another. Furthermore, it was speculated that these theories might be related to a uniqueeleven-dimensional theory called
M theory [9].For the purposes of the AdS/CFT correspondence, we are mostly interested in type IIB superstring the-ory. In the low-energy limit, only the lowest string excitions are relevant, leading to an effective quantumfield theory known as type IIB supergravity . In addition to the graviton, this theory contains a number of ad-ditional fields that preserve supersymmetry. Interestingly, this theory allows for non-perturbative (solitonic)objects known as D-branes [10] that fill out some of the ten dimensions. Beyond containing endpoints ofopen strings, as mentioned earlier, D-branes can themselves carry masses and charges and perturb the metricaround them. These two perspectives on D-branes are essential for the construction that led to AdS/CFT.For more information on string theory, refer to one of the several textbooks on the subject available both atthe introductory (undergraduate) [11, 12] and advanced level [13, 14].
The original AdS/CFT setup [6] is based on type IIB superstring theory in D =10 spacetime dimensions.This theory supports non-perturbative D-brane solutions. We consider a stack of N parallel D3-branes fillingout three of the nine spatial dimensions. This setup has the following parameters: The string coupling g s ,the string length l s , the distance d between the branes and their number N .As has been mentioned in the previous section, D-branes serve as endpoints of open strings and carryfields. We consider the low-energy limit of vanishing string length l s → , i.e., we “zoom out” to scaleswhere all excitations (of order /l s ) beyond the ground state become negligible. To keep the mass of stringmodes between different branes constant, U = d/l s is fixed (so that d → , as well). Considering the openstrings as small perturbations (in N g s ) of the branes, one finds an effective U ( N ) Yang-Mills theory withcoupling constant g YM = 2 πg s in the low-energy limit. Specifically, this theory has N = 4 supercharges(half of the original type IIB theory, broken by the D-branes) and is known as N = 4 super-Yang-Mills(SYM) theory. The limit l s → also removes interactions between the open strings on the branes and the Due to different possible symmetries of this construction, there are actually two heterotic string theories: SO (32) and E × E . EFT : At small coupling
N g s , openstrings between N D-branes form an effective U ( N ) gauge theory in the low-energy limit, with decoupledclosed strings described by IIB supergravity in a flat R , background. R IGHT : At large
N g s , the D-branes deform the spacetime background filled with closed strings. At low energies, strings near to and faraway from the D-branes decouple; both are approximated by IIB supergravity, with the metric described byAdS × S and flat R , , respectively.closed strings in the type IIB background, so that the U ( N ) theory decouples. The remaining closed stringscan be treated by the low-energy limit of type IIB strings, given by type IIB supergravity.Alternatively, we can look at the branes as a massive perturbation of the background of the closed stringsin the type IIB string theory. The metric around N D3-branes is given by [15]d s = d x (cid:112) πN g s ( l s /r ) + (cid:112) πN g s ( l s /r ) (cid:0) d r + r d Ω (cid:1) , (3)where x are the spacetime coordinates along the branes and r is the radial coordinate away from the D-branes. At large r , this metric simply describes flat 10-dimensional space. However, at small r we canrewrite it as d s = r α d x + α r d r + α d Ω , (4)where we defined α = (4 πN g s ) / l s . The metric (4) describes -dimensional anti-de Sitter (AdS)spacetime in the Poincar´e coordinates ( x, r ) in addition to the angular coordinates Ω of the 5-sphere S .This combination is denoted as AdS × S . The AdS radius α is also the radius of the 5-sphere in this setup.AdS spacetime has constant negative curvature of the same magnitude as the positive curvature of S . Asexplained in Summary 2, the branes at r = 0 form a horizon that is infinitely spatially separated from theremaining spacetime.We again consider the low-energy limit of this setup: As l s → , the closed strings at both large andsmall r are described by type IIB supergravity while decoupling from one another. At large r , the spacetimebackground is flat and we find the same supergravity theory as in the previous setup where we consideredthe open string dynamics between branes. However, at small r we find a theory of supergravity on an AdSbackground, rather than the U ( N ) theory resulting from the previous analysis using open strings. Assumingthat both descriptions of the D-brane setup are equally valid across the whole range of couplings N g s , itappears that both theories should be equivalent, as well. This leads to the following duality: N = 4 SU ( N ) SYM theory on R , ≡ Type IIB superstring theory on AdS × S . ummary 2: Anti-de Sitter spacetime A particularly symmetric class of D -dimensional spacetimes are those with constant scalar cur-vature R at all points. For vanishing R , we find flat Minkowski spacetime R D − , . The cases R > and R < are known as de Sitter (dS) and anti-de Sitter (AdS) spacetimes, respectively.The latter case can be expressed in different metrics; most commonly used are global coordinates ( τ, ρ, Ω , . . . , Ω D − ) withd s = − (cid:18) ρ α (cid:19) d τ + α α + ρ d ρ + ρ d Ω D − , (5)and Poincar´e coordinates ( t, r, x , . . . , x D − ) with d s = r α (cid:0) − d t + d (cid:126)x (cid:1) + α r d r . (6)The AdS radius α determines the scalar curvature R = − D ( D − /α . Characteristic of AdSspacetime is a horizon at spatial infinity ( ρ → ∞ or r → ) that no timelike geodesics can reach.The metric at this horizon is given by flat D − -dimensional Minkowski spacetime R D − , . AdSspacetime has SO ( D − , symmetry, the same symmetry as a D − -dimensional conformal fieldtheory (CFT), an important cornerstone of the AdS/CFT correspondence.Note that we changed the gauge group from U ( N ) to SU ( N ) , as a set of U (1) modes on the boundary isnon-dynamical [16]. The coupling constants are related via g YM = 2 πg s and N g YM = ( α/l s ) . This leadsto a remarkable property. The effective coupling constant in the SYM theory is given by λ = N g YM = α l s . (7)As we are working in the l s (cid:28) α limit, λ is large and the SYM theory is thus strongly coupled. However, ifwe are also taking the N → ∞ limit, the string coupling g s = λ/ (2 πN ) is weak and the type IIB superstringtheory can be studied perturbatively. This duality between supersymmetric gauge theory and supergravity(or gauge/gravity duality, for short) is thus often called a strong/weak duality. The name
AdS/CFT corre-spondence comes from a particular property of the SU ( N ) SYM theory: It possesses conformal invariance(see Summary 3) and is thus belongs to the class of conformal field theories (CFTs).The range of applicability of the AdS/CFT correspondence appears to be much larger than the specificexample just given: Rather than a relationship between supergravity in -dimensional AdS and a -dimensional CFT ( N = 4 SYM), similar AdS D +1 /CFT D dualities can be constructed for different D . This is consistent with the symmetries of both theories: The spacetime symmetries of AdS D +1 are givenby SO ( D, , as it can be embedded onto a hyperbola in flat R D, spacetime. This exactly matches thespacetime and conformal symmetries of a CFT D , which taken together also form SO ( D, . The obser-vation that the algebra of AdS D +1 symmetry generators turns into the D -dimensional conformal algrebra The large N limit at fixed λ is usually called the ’t Hooft limit , after an earlier observation that Yang-Mills theory in this limithas a perturbation series similar to that of a quantized string [17]. Other examples for D = 1 , , and were already proposed in Maldacena’s original work [6] using different D-brane setupsand compactifications. ummary 3: Conformal field theory As stated by the Coleman-Mandula theorem [18], it is generally not possible to combine internalsymmetries of quantum fields with spacetime symmetries in any nontrivial way. One exception to thistheorem is supersymmetry, based on a graded Lie algebra beyond the scope of Coleman-Mandula,which directly relates bosonic and fermionic fields (see Summary 1). [19] As the theorem is basedon the properties of the S -matrix describing scattering between asymptotic particles, it also breaksdown in theories without a length scale. This includes scale-invariant and conformally invariantmodels. Conformal transformations g µν ( x ) → Ω ( x ) g µν ( x ) (with positive Ω ( x ) ) preserve localangles but not lengths. This greatly restricts the physical properties of a conformal field theory (CFT).Correlations generally depend polynomially on distances, with the form of two- and three-pointfunctions fixed by symmetry. Extending D -dimensional Poincar´e symmetries (translations, rotationsand Lorentz transformations) with conformal symmetry leads to the conformal group SO ( D, .While higher-dimensional CFTs are hard to study analytically, in dimensions many examples(such as the critical Ising model [20]) are exactly solvable. Further requiring supersymmetry leadsto superconformal theories, a class that also includes N = 4 super-Yang-Mills (see Summary 1).at the asymptotic boundary of AdS D +1 spacetime was already observed in the D = 2 case long beforethe AdS/CFT correspondence [21]. Supersymmetry on the field theory side corresponds to the additionalcompact dimensions of matching symmetry on the gravity side: For the AdS /CFT case, the N = 4 su-persymmetry corresponds to an SU (4) symmetry that again matches the SO (6) ∼ SO (4) symmetry of S . This relationship between supersymmetry and additional compact dimensions suggests the existenceof some non-supersymmetric duality between gravity in AdS D +1 and CFT D . Similarly, one may speculateabout the validity of AdS/CFT at small ’t Hooft coupling λ , where a weakly coupled CFT in the aboveconstruction appears to be related to a strongly interacting — i.e., non-perturbative — theory of quantumgravity. While examples in both directions have been constructed, the general validity of AdS/CFT remainsunknown. This is intimitely tied to the problem that in a strong/weak duality, one of the two sides of theduality will always be hard to treat analytically. For this reason, a fundamental motivation for the focusof this review is the construction of simpler models than can be more directly studied with analytical andnumerical tools.Beyond being a duality between theories, AdS/CFT can also be formulated more concretely as a dictio-nary between degrees of freedom on both sides. First, note that the flat background spacetime of the gaugetheory side is associated with the location of the D-brane stack, which lies at the asymptotic boundary r = 0 of the AdS spacetime (4). As explained in Summary 2, this is a natural identification, as AdS D +1 spacetimeindeed has a flat R D − , horizon at spatial infinity. It is customary to refer to the AdS D +1 spacetime as the bulk and to the asymptotic R D − , as the boundary . The relation between the two is visualized in Fig. 2. Thebulk/boundary mapping effected by AdS/CFT generally relates fields φ in the AdS background to operators O in the boundary conformal field theory. The dynamics on both sides are equivalent; in the language ofpartition functions, Z bulk [ φ ] = Z boundary [ O ] . (8)This relationship has first been proposed by Edward Witten [22]. Concretely, consider a CFT operator O with scaling dimension ∆ , i.e., with two-point correlations (cid:104)O ( x ) O ( y ) (cid:105) CFT ∝ | x − y | , (9)8igure 2: Bulk/boundary relation in the AdS/CFT correspondence: The bulk AdS D +1 spacetime (shadedcylinder) has a flat asymptotic boundary at spatial infinity. Each time-slice t = const is a hyperbolic spacewith negative curvature.between two boundary points x and y . Assume that the AdS/CFT dictionary relates O to a dual field φ withboundary values φ . The bulk configuration of φ is determined as a boundary value problem from a given φ , so that the bulk action can be expressed purely in terms of φ . The bulk action then follows from asimple coupling between O and φ as Z bulk [ φ ] ≡ Z bulk [ φ ] = (cid:28) exp (cid:90) d D x φ O (cid:29) CFT . (10)The boundary operator O k thus acts as a source term for the boundary bulk field φ k . Conversely, forexpectation values on the boundary, φ k acts as a source term for the operator O k . Remarkably, (10) leadsto a direct relationship between the mass m of a massive bulk field and the scaling dimension ∆ of its dualoperator, given by [22] ∆ = 12 (cid:16) D + (cid:112) D + 4 m (cid:17) . (11)AdS/CFT thus implies a concrete relationship between asymptotic bulk fields and boundary operators in aconformal field theory. While the AdS/CFT correspondence is still a conjecture, many specific examples of the AdS/CFT dic-tionary with applications from high-energy to condensed matter physics have been found, with its impact onquantum information theory being a particular focus of this review. With now more than fifteen thousandcitations, Juan Maldacena’s original work has led to a vast amount of research whose end is nowhere insight. Beyond technical work, a number of introductory texts to AdS/CFT have been written, from formal As the fields φ are technically divergent at the boundary, one generally defines φ = lim r → ( r ∆ φ ) , where r is the radial AdScoordinate from (4) and ∆ the scaling dimension of its dual field. The dependence of the bulk fields φ on the boundary fields φ can be written in a diagrammatic expansion known as Witten diagrams . Note that the operator/state correspondence allows each CFT state to be characterized by a single, local operator, as scaleinvariance allows us to effectively project the path integral evolution of any state onto a point. Specifying CFT states and operatorsis thus equivalent.
Hilbert spaces of physical systems are generally huge in their dimension. While most problems in classicalmechanics can be reduced to a small parameter space that is approachable with efficient analytical andnumerical techniques, the state spaces of quantum mechanics rarely offer such a relief. Beyond perturbativemethods that can describe problems close to one of the few analytically solvable, usually non-interacting ones, only approximate numerical techniques are available. To see how the size of Hilbert spaces becomesa fundamental problem in this approach, consider a simple system of N quantum mechanical degrees offreedom each corresponding to an M -level system (e.g., spins for M = 2 ). To describe a single purequantum state ψ in this system, we use a basis representation | ψ (cid:105) = M (cid:88) k ,k ,...,k N =1 T k ,k ,...,k N | k , k , . . . , k N (cid:105) , (12)where each basis state can be expressed as a direct product of local state vectors | k , k , . . . , k N (cid:105) ≡ | k (cid:105) ⊗ | k (cid:105) ⊗ · · · ⊗ | k N (cid:105) . (13)The state (1) is thus expressed by the M N amplitudes T k ,...,k N ∈ C . We can view T as a complex-valued rank N tensor. The dimension of each index, often called the bond dimension χ , is given by χ = M . A fundamental problem of any numerical method to tackle a quantum-mechanical problem — e.g.,finding the ground state of a Hamiltonian — is that describing a quantum state and optimizing over itscomponents takes an exponential amount of memory. A spin chain of only fifty sites already requires ≈ . · complex numbers to store, which in the C++ type complex
EFT :The full network acting on an initial coarse-grained state vector | ψ (cid:105) . R IGHT : Identities of the isometriesand disentanglers for contractions of each tensor with its Hermitian conjugate over two legs.a MERA ansatz [45]. Much of the mindset of MERA also carries over to hyperinvariant tensor networks [46, 47]. The usefulness of tensor networks in understanding and modeling properties of AdS/CFT is themain motivation of a large part of the work presented in this topical review, extending beyond the tensornetwork approaches presented in this introduction. Reviews for a broader introduction to tensor networkmethods include Refs. [48, 49, 50, 51].
Information storage and transmission is susceptible to errors. Even in a purely classical system — digital oranalog — is affected by corruption of the physical medium carrying the information. In case of informationstorage, this includes corruption of bits on hard drives or SSDs, while in the case of transmission, noise inthe conducting material can affect the signal from which the data is later read.As it is impossible to preclude errors completely, error correction becomes necessary. This means thatinformation is encoded such that recovery of the logical data is still possible after small errors have occurred.The simplest way of achieving such resilience is by simply storing or transmitting multiple copies of theoriginal data in what is called a repetition code . For example, one may transfer the bit sequences and in place of the logical bits and . If one of the bits becomes corrupted, the remaining two still allowthe reconstruction of the original logical bit (e.g., from ).Classical codes are often categorized by the notation [ n, k, d ] , which denotes an encoding of k logicalbits in n physical ones, with a Hamming distance d . The latter is the minimal number of physical single-biterrors required to map one logical state (sequence of bits) to another. If we think of all possible physicalbitstrings for a given code block as nodes in a graph, and of single-bit errors as edges connecting them, d becomes the minimal graph distance between the bitstrings corresponding to logical states, showcasing thenotion of code distance. In the given notation, and n -fold repetition code for a single logical bit is denoted asan [ n, , n ] code, as n physical bits need to be flipped in order to change the bitstring to and vice-versa.Classical codes in practical use are much more complicated that simple repetition codes, but rely onthe same concept of spreading out the information of logical bits over larger bitstrings. For example, thepopular class of Reed-Solomon codes interprets k logical values as coefficients in a polynomial functionwhose result is mapped onto n > k physical ones [52], leading to an [ n, k, n − k + 1] code.The appearance of errors and methods for their correction are fundamentally different for quantumsystems. When interacting with an environment, isolated quantum systems exhibit decoherence , i.e., the14reakdown of quantum superposition and in turn, entanglement. As entanglement is a necessary resourcefor any quantum computation, its breakdown must be avoided if computational power beyond classicallimits is desired.Methods of quantum error correction are thus required to store and manipulate quantum informationwith a certain resilience to coupling with an environment. The most useful approach in classical errorcorrection, the duplication of information, is impossible for quantum systems due to the no-cloning theorem :No unitary operator, and thus no physical time evolution, can duplicate an arbitrary quantum state [54, 55].Quantum error correction thus requires other approaches. The most popular and relevant for this review isthe use of stabilizer codes , first introduced by Daniel Gottesman in his PhD thesis [56], extending earlierapproaches to the problem by Peter Shor and Andrew Steane [57, 58]. The idea of stabilizer codes is toencode quantum information in ground states of Hamiltonians H S = − m (cid:88) i =1 S i (24)that are given by the sum of orthogonal operators S i , called the generators of the stabilizer S = { S , S , . . . , S m } . (25)The generators are chosen to commute with one another and act as “parity checks” on different parts of theHilbert space, i.e., have eigenvalues ± . The space of ground states of H S , given an n -qubit system, is thus n − m -dimensional and contains all states that are in the +1 -eigenspace of each generator. For qubits, it isconvenient to choose stabilizer generators that are tensor products of the Pauli operators σ x , σ y , σ z and theidentity , as well as using them as a basis set for operators that represent local errors. This ensures thatany product of such errors either commutes or anti-commutes with each generator. The errors thus flip theeigenvalue of one or more of the generators, leading to a measured pattern or syndrome from which the typeof error can be deduced and reversed.Stabilizer codes are generally denoted as [[ n, k, d ]] codes, in a generalization of the notation for classicalcodes introduced above. Here n and k again denote the number of physical and logical sites, respectively,usually qubits. The code distance d , however, has a slightly more nuanced meaning than the classicalHamming distance. Consider, for example, a single logical qubit encoded in a basis of states ¯0 and ¯1 (readas “logical zero” and “logical one”): | ¯ ψ (cid:105) = α | ¯0 (cid:105) + β | ¯1 (cid:105) , (26)where | α | + | β | = 1 . The quantum analogon of classical bit flip errors is a basis flip ¯0 ↔ ¯1 , expressed byan operator O b that interchanges the basis as O b | ¯ ψ (cid:105) = β | ¯0 (cid:105) + α | ¯1 (cid:105) . (27)Clearly this operator fulfills the condition O b = , which followed directly from our expression of errors interms of products of Pauli operators. However, there exists another type of error which fulfills this conditionas well; these phase flip errors, expressed by an operator O p act on a logical qubit basis as O p | ¯ ψ (cid:105) = α | ¯0 (cid:105) − β | ¯1 (cid:105) . (28)Note that this type of error maps the ¯0 basis state onto itself, but adds a phase e − i π to ¯1 . This implies thatwhen calculating the error distance d , we have to count the minimal number of fundamental error operations Though necessary, more entanglement does not automatically make a quantum system more useful for computations [53]. ummary 4: The 5-qubit code The [[5 , , quantum error correcting code [60, 61] is built from the stabilizers S = { σ x σ z σ z σ x , σ x σ z σ z σ x , σ x σ x σ z σ z , σ z σ x σ x σ z } . (30)Note that all generators are cyclic permutations of one another, and that multiplying all of themyields a fifth generator σ z σ z σ x σ x that is precisely the missing permutation. This code is optimal ina variety of ways: It saturates the quantum Hamming bound [62] as well as the quantum Singletonbound [56], which follows from conditions on reconstructability after erasures [63] and is given by n ≥ d −
1) + k , (31)for an arbitrary [[ n, k, d ]] code. The two local eigenstates ¯0 and ¯1 of the 5-qubit code can be distin-guished by the total parity σ ⊗ z , which thus acts as the logical parity operator ¯ σ z . Similarly, ¯ σ x = σ ⊗ x and ¯ σ y = σ ⊗ y act as the remaining logical Pauli operators. Conveniently, a Jordan-Wigner transfor-mation maps ¯0 and ¯1 to fermionic states that are Gaussian, i.e., can be expressed as ground states ofa Hamiltonian that is only quadratic in fermionic operators.(local Pauli operators) that not only map logical basis states to other eigenstates of the stabilizer Hamiltonian,but also include errors than produce basis-dependent phases. Note that in a classical system, an operation ofarbitrary complexity that maps each bit string to itself produces no effective error, a simplification that nolonger applies for code states in a quantum superposition. As in classical codes, to increase d one generallyneeds to increase n , the number of physical sites, as well. This is quantified by the quantum Hammingbound [59], which can be derived from the following argument: The full n -qubit Hilbert space can contain n orthogonal states, k of which are logical states. If the code distance is d , then (cid:98) d − / (cid:99) errors can becorrected, i.e., lead to distinct orthogonal states. There are n possible local errors, one for each physicalqubit and Pauli operator, and m (cid:0) nm (cid:1) possibilities of applying exactly m non-trivial errors. As all errorsapplied to the logical states must be distinct and contained in the physical Hilbert space, we arrive thefollowing bound (cid:98) d − (cid:99) (cid:88) m =0 m (cid:18) nm (cid:19) ≤ n k (29)for an [[ n, k, d ]] code. For a single logical qubit, the quantum Hamming bound leads to the requirement of n ≥ physical sites. Indeed, a [[5 , , code that can correct an arbitrary Pauli-type error on one logicalqubit exists, often simply called the “5-qubit code”. This code, which will be highly relevant troughout thiswork, is explained in more detail in Summary 4.A widely used class of stabilizer codes are Calderbank-Shor-Steane (CSS) codes [64, 58], built froma combination of two classical codes. Each is mapped onto stabilizer generators containing, up to localidentities, only σ x or only σ z operators, respectively, which makes it easier to realize such codes in prac-tice; indeed, they were the first quantum codes to be realized experimentally [65]. Recently, topologicalcodes [66] like the surface code [67] and the color code [68] have enjoyed large popularity for potentialquantum error correction in large systems of qubits (see, e.g., Ref. [69]). In addition, an experimental real-ization of the 9-qubit Bacon-Shor code , a subsystem code that extends stabilizer codes by a notion of gauge Note that we can detect d − errors, but may not be able to identify the original logical state. In recent years, viewing the AdS/CFT correspondence through the lens of quantum information theory hasled to surprising connections between both fields. These “holographic” descriptions of quantum informationconcepts deepen our understanding of holography itself, but also offer potential approaches to problems thatwere not originally thought to be associated with AdS/CFT.
Probably the first connection between AdS/CFT and quantum information was introduced by Shinsei Ryuand Tadashi Takayanagi, when they considered the following question: What is the dual AdS d +1 bulkdescription of the entanglement entropy S A of a boundary subsystem A in a holographic CFT d ? The answeris provided by a startling generalization of the black hole entropy formula (1) which relates the black holeentropy to its horizon area. It turns out that the bulk quantity dual to S A is the area of a d − -dimensionalminimal surface γ A homologous to A , i.e., with ∂A = ∂γ A (see Fig. 5, left). This is quantified by the Ryu-Takayanagi (RT) formula [76] S A = | γ A | G , (32)where | γ A | is the area of γ A and G is the gravitational constant in the d +1 -dimensional bulk spacetime. Notethat this formula has no dependence on the actual holographic model, e.g., the degree of supersymmetry andthe bulk structure. Given an excited CFT with a dual bulk geometry other than “pure”, undeformed AdS, theshape and area of γ A change, reflecting entanglement produced or destroyed by the excitation. For example,consider a thermal CFT, whose bulk dual is given by an AdS black hole geometry (see Fig. 5, right). Thehorizon deforms the minimal surface towards the AdS boundary, increasing its area and thus reproducing thethermal entanglement associated with a finite-temperature CFT [76]. If we start growing the subsystem A until it encompasses the entire boundary, the minimal surface starts wrapping around the black hole horizon,as this horizon is itself extremal. In that limit, (32) becomes the Bekenstein-Hawking formula (1) for theclassical entropy S , showing the intimate connection between both formulae.Strictly speaking, the definition of γ A is only properly coordinate-independent if we restrict the boundarysystem to the time-slice of a static spacetime geometry. For more general space-like boundary regions A ,we have to consider a space-like bulk surface γ A that is extremal : It is minimal with regard to space-likevariations and maximal with regard to time-like ones. This generalized form of the RT formula is oftencalled the Hubeny-Rangamani-Takayanagi (HRT) formula [77].While originally a conjecture within AdS/CFT, the RT formula was proven first for -dimensionalCFTs [78, 79] and shortly afterwards for the more general case [80]. It was soon understood that (32) onlyholds in the AdS/CFT limit of large G (classical bulk gravity) and N → ∞ , where N is the rank of the gauge17igure 5: L EFT : Minimal surface γ A homologous to a boundary region A in an AdS time-slice (blue-shadedthroat region). R IGHT : Deformed minimal surface ˜ γ A in an AdS geometry with black hole horizon H .group SU ( N ) of the boundary CFT, and that quantum corrections lead to additional terms constant in G and N that can be interpreted as entanglement between bulk regions [81]. For an introduction to holographicentanglement entropy, the extended version of Shinsei Ryu and Tadashi Takayanagi’s original work [82] isa good starting point. There also exists a book on the topic [83], a shortened version of which is availableonline [83]. A particularly fascinating feature of AdS/CFT is that it relates a theory with (quantum) gravity to onewithout it. This has led to the suggestion that gravity, whose failure at consistent quantization on arbitrarilysmall scales was one of the driving motivations behind the development of string theory, is indeed not afundamental force at all, but rather holographically emergent from quantum degrees of freedom. Moreconcretely, Mark Van Raamsdonk suggested that the connectivity between regions of spacetime could be aconsequence of entanglement between them [84]: First, he interpreted an earlier setup relating a maximallyentangled AdS black hole spacetime to two copies of a CFT [85] as an example of a non-entangled systembeing spatially separated. An entangled system, on the other hand, would be characterized by a nonvanishing mutual information I ( A : B ) = S A + S B − S A ∪ B , (33)between two regions A and B , serving as an upper bound to two-point correlation functions between bothregions [86]. In a holographic theory, where such correlators are expected to decay exponentially withgeodesic distance through the bulk, an increase in entanglement would thus imply a closer spatial bulkdistance.While this conjectured connection between entanglement and the emergence of gravity remains far frombeing understood, many similar ideas have appeared throughout the decade since its proposal. For example,a similar behavior has been found in the holographic description of quantities other than entanglemententropy. This includes the entanglement of purification [88], an entanglement measure for mixed statesrelated holographically to the minimal cross-section of the throat-like RT surface of a two-component region[87, 89] (see Fig. 6 for details), again relating spatial connectivity and entanglement. Another example isthe suggestion that the state complexity of a quantum state, quantifying the number of local operations orquantum gates required to construct it from a reference state [90], may possess a holographic dual either interms of a bulk volume [91] (“complexity equals volume”) or an action evaluated on as specific bulk region[92] (“complexity equals action”). The validity of either proposal is still hotly debated.18igure 6: Holographic prescription of the entanglement of purification E P for a disjoint boundary region A ∪ B : The minimal (Ryu-Takayanagi) surface γ AB connects both boundary regions when A and B areclose together. The area of the minimal entanglement wedge cross-section Σ AB is conjectured to be relatedto the entanglement of purification via E P = | Σ min AB | / G N [87].As most of these proposals are motivated around conceptual issues of quantum gravity, it may be sur-prising that a holographic description of gravity may have a bearing on actual qubit experiments: It wasproposed that certain experimental setups for quantum teleportation , the transmission of quantum states viaentanglement, may be effectively described by a holographic construction involving traversable wormholes[93]. Though no such experiments have yet been conducted, this emphasizes the potential of bridging thefields of quantum computation and fundamental physics. Many of the early ideas of connecting gravity andentanglement are surveyed in Mark Van Raamsdonk’s lecture notes on the same topic [94]. The form of the Ryu-Takayanagi formel (32) bears a striking resemblance to the entanglement entropybound (22) in generic tensor networks, both involving a minimal surface through a geometry than extendsthe direct geometry of the boundary state. This leads to a straightforward question: Can the time-slice of anAdS spacetime, which we consider in the RT formula, be expressed as a tensor network? The first proposalin this direction was made by Brian Swingle, who suggested the MERA tensor network for this particularinterpretation [43]. This identification is tempting, because the MERA implements entanglement renormal-ization between discretized quantum systems at different scales. Similarly, we expect a timeslice in the AdSmetric (6) at fixed radius r from the AdS boundary to describe an increasingly fine-grained system as r is decreased. Furthermore, the gapless states produced by the MERA resemble those expected in the con-formal boundary theories of AdS/CFT, though being ground states of much simpler critical Hamiltoniansthat usually feature neither supersymmetry nor non-Abelian gauge symmetries. While the MERA producesboundary states with conformal symmetries, its tensor network geometry does not exactly match an AdStime-slice — the hyperbolic Poincar´e disk — and leads to inconsistencies when treated as such [95]. Al-ternatively, the MERA geometry was interpreted both as a time-like surface in positively curved de Sitter(dS) spacetime [45] and as a path integral discretization of an AdS light-like surface [96]. In either case,using the MERA is a discrete realization of AdS/CFT requires abandoning the simple time-slice picture inwhich the RT formula was derived, leading to a setting whose relationship to holography is not yet fullyunderstood. 19n principle, the indices of any tensor network can be seperated into two sets between which it acts as alinear map on states. Clearly, labeling these two sets “bulk” and “boundary” and expectating the map to showany holographic features is pointless for most setups. Invoking the RT formula, we at minimum desire anansatz that gives the correct entanglement entropy scaling for minimal cuts through the network. Choosinga tensor network whose geometry discretizes the Poincar´e disk is not generally sufficient; we also need theentanglement entropy bound (22) to saturate for any choice of subsystem. The right choice of tensors is thuscrucial. Surprisingly, choosing random tensors already reproduces many of the expectated properties, suchas polynomially decaying correlation functions, as long as the bond dimension is large [97, 98]. Rather thana mapping between individual states, such a construction considers averages of random bulk configurations,leading to a bulk partition that is in fact equivalent ot the classical Ising model. Unfortunately, many of theobserved holographic results break down at finite bond dimension, and even in the infite limit, the R´enyientropies S ( n ) A = 11 − n log(tr ρ nA ) (34)do not reproduce the expected behavior for CFTs. Which other choices of tensors are possible? Conditions toconstrain suitable tensors, as we will see in the next section, are found by considering quantum informationquantities beyond simple entanglement measures. While a broad introduction to tensor network holographyremains to be written, the initial proposal by Brian Swingle [43] contains many of the key ideas that variousimplementations over the past decade have been based on. As a duality between bulk and boundary, AdS/CFT contains a complicated mapping of quantum informationbetween both sides of the duality. Considering subregions of bulk or boundary, the question arises if theinformation encoded in local bulk regions is contained in a local region on the boundary and vice-versa.In general, this does not appear to be the case. Reconstructing a bulk field φ ( x ) at a point x generallyrequires information about a boundary region that increases in size as x is moved further into the bulk; inother words, information about fields close to the boundary can be recovered from a small boundary region,while information deep in the bulk is “smeared out” over the boundary [99]. More precisely, given a regionon the boundary there exists a causal wedge in the bulk spacetime whose content can be reconstructed[100, 101, 102]. As different boundary regions correspond to sometimes overlapping wedges in the bulk,local bulk information can in fact be reconstructed on different boundary regions [103]. This leads to aconundrum: Consider, as shown in Fig. 7 (left), two causal wedges W A and W B containing a point x . If wecan reconstruct the bulk field φ ( x ) in both of the corresponding boundary regions A and B , does this implythat its information is encoded in A ∩ B ? This conclusion cannot be correct, as x can be chosen so that itis not contained within the causal wedge W A ∩ B and hence the information in A ∩ B must be insufficient toreconstruct φ ( x ) .The only resolution to this problem — other than assuming that reconstructed operators are all trivial,acting as an identity — is to conclude that φ ( x ) can be represented as different equivalent operators ondifferent boundary regions, an insight transparently captured in work by Ahmed Almheiri, Xi Dong, andDaniel Harlow [104]. Its information of φ ( x ) is thus stored redundantly, as removing parts of the boundaryrequired for the reconstruction along one causal wedge does not prevent its recovery via another; in otherwords, bulk information is stored on the boundary in the manner of a quantum error-correcting code. Bulkquantum information stored at a point in the center is protected against the erasure of large parts of theboundary, as the causal wedge of the erased region usually does not penetrate far enough into the bulk toaffect the center. In contrast, bulk information at a point near the boundary is completely lost by an erasure20f a small region whose causal wedge contains it. Properties of quantum error correction, in the form of a subregion duality between boundary and bulk subsystems, thus appear to be a generic feature of AdS/CFTand can be shown to hold even in deformed holographic theories [105].The behavior of holographic codes can be reproduced in an exceedingly simple discrete toy model firstproposed in Ref. [104] and refined in Ref. [106]: This model relies on the [107] which encodesa logical qutrit (a three-level quantum state) in three physics ones. It contains the three logical basis states | ¯0 (cid:105) = | (cid:105) + | (cid:105) + | (cid:105)√ , (35) | ¯1 (cid:105) = | (cid:105) + | (cid:105) + | (cid:105)√ , (36) | ¯2 (cid:105) = | (cid:105) + | (cid:105) + | (cid:105)√ . (37)This code enables quantum secret sharing : To reconstruct a logical state vector (some superposition of thelogical basis states), one requires access to the state on any two of the physical qutrits from which it can berecovered via a suitable unitary transformation. Conversely, the reduced density matrix of any single physi-cal qutrit is maximally mixed, and no information about the logical state can be recovered from it at all. Theholographic interpretation of this code now comes about if we identify the logical qutrit with the bulk andthe three physical ones with the boundary: We interpret the need for two physical qutrits for reconstructionof the logical one as the requirement that only large boundary regions have a causal wedge large enough torecover information far in the bulk. This model reproduces a number of quantum error correction propertiesexpected from the earlier AdS/CFT discussion: First, a bulk local operator (an operator acting on the log-ical state space) commutes with all local boundary operators (acting on the physical qutrits). This followsdirectly from the erasure property of the 3-qutrit code: If any single physical qutrit can be erased withoutaffecting the logical state then such single-site operators must commute with the logical ones. Second, thereexists a subregion duality between any two physical qutrits on the boundary and the bulk information (thelogical state): Any logical operator can be represented as an operator acting on only two boundary qutrits.Third, the entanglement entropies of boundary subregions have a holographic interpretation as well: For amixed logical state encoded in the density matrix ¯ ρ , the 1- and 2-site entanglement entropies are S = log 3 , S = log 3 + S ( ¯ ρ ) . (38)We can identify the log 3 contribution as an “area term” that corresponds to a minimal cut through thegeometry over a single bond of dimension χ = 3 . For boundary regions large enough to reconstruct thebulk logical state, we further find a “bulk entropy term” that depends on mixing within the logical state. Incontinuum AdS/CFT, such terms appear as higher-order corrections to the Ryu-Takayanagi formula [81] andappear to be deeply related to properties of quantum error correction [106]. This 3-qutrit model can evenbe interpreted as containing a notion of a black hole: For any state in the code subspace we can reconstructthe logical qutrit and thus the full “bulk geometry”, but any state outside of the code subspace makes itinaccessible, as if hidden behind a black hole horizon.Clearly a model of three qutrits does not really describe the bulk geometry of an AdS space-time, but theunderlying ideas were the starting point of the more elaborate construction presented in Ref. [108] whichwill we explore in great detail in the next section.The connection between quantum error correction and holography has only recently been establishedand is the subject of much ongoing research. While a formal review does not yet exist, a short introductionby Beni Yoshida, which also introduces the tensor network realization of a holographic code presented inthe next section, is available on Caltech’s Quantum Frontiers blog [109].21igure 7: L
EFT : Causal wedges W A and W B in an AdS timeslice (Poincar´e disk), corresponding to tworegions in which a bulk field φ ( x ) can be reconstructed as two operators O A and O B with support onboundary regions A and B . The causal wedge W A ∩ B of the intersection of A and B is insufficient. R IGHT :The discrete form of causal wedges in the hyperbolic pentagon code. The logical state on the markedpentagon can be reconstructed from either of the two boundary state vectors | ψ A (cid:105) and | ψ B (cid:105) . In its continuum formulation with infinitely many degrees of freedom, the code picture of AdS/CFT is dif-ficult to treat with the language usually used in the context of quantum information. Can we instead builda discrete toy model, based on simple quantum error-correcting codes that are already familiar to us? Aclass of such models has indeed been constructed and are now known as
HaPPY codes [108], an acronymcontaining the authors’ initials. Before defining the entire class of codes, first consider a particularly simpleinstance of this code built on the previously mentioned five-qubit code: This hyperbolic pentagon code [108]has the geometry of a regular hyperbolic discretization built from pentagons, each pentagon being identifiedwith one [[5 , , code and each pentagon edge with a physical qubit (or pairs thereof for each edge sharedby two neighboring pentagons). Representing the code as a six-leg tensor that maps between the logicaland physical qubits, the five “physical legs” are then contracted following the adjacency of edges in the dis-cretized geometry. In this construction, visualized in Fig. 7 (right), the reconstruction along different causalwedges discussed in the previous section follows directly from the [[5 , , code’s properties: Starting froma set of uncontracted physical sites on the (asymptotic) boundary, we can reconstruct the logical states onthe near-boundary pentagons from just three physical sites. Recovering the physical state on the remainingedges, we then use these as inputs on the next layer of pentagons, reconstructing their logical states in turn.Through this procedure we gradually recover the logical states in the bulk from a boundary region, buildingup a discretized causal wedge until we can no longer find three physical sites around the same pentagon,i.e., until the boundary of the wedge is no longer concave. This process, known as the greedy algorithm ,can be applied to any given boundary region, the state of which is determined by all logical states within thewedge. Conversely, the logical state on a single pentagon affects all physical boundary states in subsystemswhose wedges include it. The hyperbolic pentagon code thus gives a concrete mapping between bulk andboundary states with the quantum error-correcting features of AdS/CFT.22he hyperbolic pentagon code is only a special case of a large class of tensor networks with similarproperties: Their crucial ingredient is that of perfect tensors, which act as isometries between any bipartitionof its indices as long as the number of output indices is at least as large as the number of input indices. Forexample, a perfect five-index tensor T i,j,k,l,m would fulfill constraints such as (cid:88) i,k,m T (cid:63)i,j,k,l,m T i,n,k,o,m ∝ δ j,n δ l,o . (39)Perfect tensors can be identified with absolutely maximally entangled states [110] by means of exploitingan equivalence with pure multi-partite quantum states, and in turn give rise to instances of quantum error-correcting codes. Tensor networks built from such tensors allow for a variant of the greedy algorithm tobe applied and thus lead to Ryu-Takayanagi-like entanglement scaling. While the property to be a perfecttensor makes perfect sense from the perspective of quantum error correction and renders the analysis ofthe resulting holographic code very transparent, from a physical perspective this is a rather strong property.In fact, random tensors drawn from a suitable probability measure are with high probability close to suchperfect tensors [97] so that much of the analysis of Ref. [108] carries over to the case of random tensornetworks .We can think of the resulting codes as being akin to an omnidirectional MERA: The MERA tensor con-straints (compare Fig. 4) reduce the computation of local observables to a problem of evaluating a localizedpart of the tensor network, as most of the contractions simply reduce to identities. Similarly, an operatorapplied to a HaPPY code boundary can only affect the result of contractions within the wedge obtained fromthe boundary by application of the greedy algorithm. Unlike the MERA, HaPPY codes have no inherentdirectionality, as a regular hyperbolic tilings has the same geometrical structure around any given tensor.These codes can be defined on any such tiling; they are generally labeled by the Schl¨afli symbol { n, k } de-noting an n -gon tiling where k n -gons meet at each vertex (or equivalently, whose dual tiling that replacesvertices and n -gon centers yields a k -gon tiling). By considering the angles of such tilings one finds thathyperbolic tilings require f ( n, k ) = nk − n + k ) to be positive (if f ( n, k ) is zero the tiling is flat, andthe integers satisfying f ( n, k ) < characterize regular polyhedra). The standard hyperbolic pentagon codeis thus also refered to as a { , } holographic code. Any [[ n, m, d ]] code can in principle be embedded intoan n -gon tiling, with those whose encoding isometry can be representated as a perfect tensors inheriting theHaPPY properties.Both the choice of a hyperbolic bulk and its discretization by a regular tilings determine the geometricalfeatures of the boundary, which can be seen to have a fractal structure in two different ways: First, thequestion of which boundary regions are necessary for reconstructing information at a point x in the bulkleads to the insight that not the entirety of the boundary whose entanglement wedge contains x is needed;in fact, in the continuum an infinite number of subregions of decreasing size can be “punched out” of theboundary while still being reconstructable from the properties of the code. This means that from an operatoralgebra perspective, only operators on a fractal subset of this boundary are required to reconstruct the bulk, aproperty called uberholography [111]. In the regular tiling discretization, this ability to remove pieces of theboundary with impunity is greatly restricted, but replaced by another notion of fractal geometry: Cutting offthe tiling after a finite number of inflation steps (discussed in more detail below) leads to a boundary whoseboundary geometry is inherently quasiregular [112], meaning it has self-similar geometric structures re-sembling those of a fractal. More generally, it has been proposed that tensor networks on regular hyperbolicgeometries naturally encode the symmetry transformations of quasiregular conformal field theory (qCFT),a theory with discretely broken conformality, possessing properties distinct from continuum CFTs [113].23 .2 Majorana dimer codes What kind of physical boundary states do holographic quantum error-correcting codes lead to? In thecase of the hyperbolic pentagon code and its generalizations, this is not a question that can be studiedin a straightforward manner: Tensor contraction of large-scale version of this code is, unlike the MERA,computationally inefficient. However, we can use the code properties of this model to study its correlationand entanglement structure in an almost completely analytical manner.In the case of entanglement, these properties follow directly from the application of the greedy algorithm[108]: For boundary regions A and their complement A C that both reduce to the same minimal cut γ A through application of the greedy algorithm on either region, the property of maximal entanglement of eachindividual tensor enforces maximal entanglement entropy across the cut. That is, the entanglement followsa discrete Ryu-Takayanagi formula S A = | γ A | log χ , (40)where | γ A | is the number of edges composing γ A and χ is the constant bond dimension of each perfecttensor bond, e.g. χ = 2 for the hyperbolic pentagon code (as the [[5 , , code is a spin code). Whilethis reduction of A and A C to the same discrete bulk geodesic γ A is possible for most regions A (evendisjoint ones), there exist pathological cases where a residual bulk region separates γ A from γ A C , affectingthe resulting entanglement entropy.Beyond entanglement, one can consider two-point correlation functions, which appear to be simpleat first. In the spin picture of the hyperbolic pentagon code, for example, consider a two-point function (cid:104) X j X k (cid:105) between Pauli X k operators on two boundary sites j and k . If the operators act on edges of separatepentagons, then the effect of each operator corresponds to a correctable error, as the logical bulk state oneach pentagon can still be reconstructed from its four other physical sites. This remains true even if bothoperators act on the same pentagon, as we can reconstruct the logical state from merely three physical sitesas well. We thus conclude that all two-point functions between single Pauli operators must vanish. Doesthis hold for more complicated two-point functions as well? Fig. 8 illustrates that this is not the case. Bycarefully choosing pairs of neighboring sites, we can produce entire bulk regions stretching between twoboundary regions on which the logical state that can no longer be reconstructed. This implies that two-pointfunctions of such boundary operators with two-site support are generally nonzero.Fig. 8 also suggests than these non-zero correlations are associated with endpoints of discrete geodesicsof the tiling, an intuition that can be clarified by formulating the pentagon code not in the language of spins,but that of Majorana fermions . We already mentioned in Summary 4 that the code states ¯0 and ¯1 of the5-qubit code simplify in a fermionic setting; explicitly, an operator transformation of the form γ k − = σ z ⊗ ( k − ⊗ σ x ⊗ ⊗ ( N − k − , (41) γ k = σ z ⊗ ( k − ⊗ σ y ⊗ ⊗ ( N − k − , (42)that maps Pauli operators on N = 5 spins to N Majorana operators γ k that obey anticommutation relations { γ j , γ k } = 2 δ j,k . The four stabilizers S k of the [[5 , , code (and its fifth permutation S ) then take theform S = σ x σ z σ z σ x = − i γ γ , (43) S = σ x σ z σ z σ x = − i γ γ , (44) S = σ x σ x σ z σ z = − i P tot γ γ , (45) S = σ z σ x σ x σ z = − i P tot γ γ , (46) S = σ z σ z σ x σ x = − i P tot γ γ . (47)24igure 8: L EFT : A non-reconstructable bulk region (red-shaded region) created by the insertion of fouroperators (red dots). The greedy algorithm will be unable to reach this region from any starting region.R
IGHT : By pulling the four operators to the boundary along a discrete geodesics, the non-reconstructableregion divides the tensor network in two. Expectation values of the four operators can be nonzero.Here P tot is the total parity operator expressed as σ ⊗ z in spin and − i γ γ . . . γ in Majorana operators.We immediately notice that up to parity, these stabilizers are products of only two Majorana operators: Thisimplies that the ground state space of the stabilizer Hamiltonian H = − (cid:80) k S k is spanned by two parityeigenstates that are Gaussian , i.e., correspond to a non-interacting fermionic model. This immensely simpli-fies working with these states: Gaussian states are entirely characterized by their two-point correlations, asthe lack of interaction terms in the Hamiltonian of which they are ground states implies that all higher-ordercorrelations are computable from Wick’s theorem! For example, for any fermionic Gaussian state vector | ψ (cid:105) the four-point Majorana correlation function is given by (cid:104) ψ | γ i γ j γ k γ l | ψ (cid:105) = (cid:104) ψ | γ i γ j | ψ (cid:105) (cid:104) ψ | γ k γ l | ψ (cid:105) − (cid:104) ψ | γ i γ k | ψ (cid:105) (cid:104) ψ | γ j γ l | ψ (cid:105) + (cid:104) ψ | γ i γ l | ψ (cid:105) (cid:104) ψ | γ j γ k | ψ (cid:105) . (48)Note that minus sign appearing at any odd permutation of the operators due to their anticommutation re-lations. Gaussianity allows us not only to describe quantum states using quadratic instead of exponentialmemory but also to contract tensor networks of Gaussian tensors (i.e., tensors describing Gaussian states)efficiently [114]. For the HaPPY code, this implies that if we fix the bulk legs on each pentagon to a basisstate input — either ¯0 or ¯1 locally — then the resulting tensor network is efficiently computable. While thiscan be performed with methods suitable for general Gaussian tensors [115], the HaPPY code tensors haveadditional structure than simplifies their contraction to simple graphical rules. Let us begin with a singletensor describing a logical basis state ¯0 or ¯1 on one pentagon, i.e., on ten Majorana modes. We can visualize25oth state vectors with the diagrams | ¯0 (cid:105) = , | ¯1 (cid:105) = . (49)Here each arrow from Majorana mode j to k , called a Majorana dimer [116, 117], denotes a stabilizer term i γ j γ k . The states are uniquely defined by these diagrams up to a complex phase, as the Hamiltoniansadmits no ground state degeneracy. Alternatively, we can associate with each dimer an operator γ j + i γ k that annihilates the state; intuitively, we can thus picture each dimer as a delocalized fermion shared betweensites j and k . Note that the orientation of a dimer matters, as the Majorana operators do not commute. Inthe visualization, dimer arrows from a mode j to k are shaded in blue if j < k and in orange if j > k . As γ j + i γ k ∝ γ k − i γ j , we can think of each “inverted dimer” as a fermionic hole. Each dimer can thus beassociated with a dimer parity p j,k for a dimer from j to k , given by p j,k = (cid:40) +1 if j < k − if j > k . (50)The name “parity” comes from the observation that the total parity P tot (the eigenvalue of P tot ) of a Majoranadimer state is simply [117] P tot = ( − N c (cid:89) ( j,k ) p j,k , (51)where the product runs over all dimers ( j, k ) in the state and n c is the number of crossings of dimers inthe diagram (which can only be changed by an even number when deforming it). The Majorana dimerpicture becomes particularly useful when performing contractions: For example, contracting two tensorscorresponding to two adjacent pentagons encoding a ¯0 and ¯1 state, respectively, leads to the new state = . (52)Here the two dashed lines symbolizing contraction between the two pentagon edges (i.e., two indices) areintentionally suggestive of the result of the contraction of Majorana dimer state: Two pairs of dimers oneither side of the contraction are paired up into two new dimers whose dimer parity is the product of theoriginal two dimer parities (up to crossing terms appearing in self-contractions) [117]. Equipped with thisgraphical method of contracting code basis states of the [[5 , , code, we can immediately evaluate theboundary states of the entire HaPPY code for fixed local bulk input. For a ¯0 input on every pentagon, the26esult can be visualized in a single picture: → (53)As we can see, for the full HaPPY code of infinitely many pentagons, the dimers meet up in pairs atthe (asymptotic) boundary of the hyperbolic disk. Furthermore, these pairs of dimers trace the discretegeodesics of the { , } hyperbolic tiling, which shows that the dimers produce the non-zero correlationalong these geodesics that we deduced in Fig. 8 from the code properties. This statement can now be quan-tified: Fermionic two-point correlations G j,k = i2 (cid:104) γ j γ k − γ k γ j (cid:105) are simply zero if no dimer connectsboundary site j and k , and − p j,k otherwise. The average falloff of correlations between two large boundaryregions is thus given by a histogram of the number of dimers plotted over the boundary distance d betweenthe two sites that they connect. Such a histogram shows that correlations fall off with ∝ /d [115], just asone would expect in a critical spin model (such as the c = Ising CFT) under a Jordan-Wigner transforma-tion to fermions. Note that due to the contraction rules of Majorana dimers, logical basis state input otherthan a global ¯0 input only changes the signs of these correlations but not their structure. It is also possible toshow that for arbitrary logical input, i.e., superpositions of ¯0 and ¯1 logical states, the resulting non-Gaussianboundary state still has two-point correlation functions that follow the dimer pattern [118].The Majorana dimer picture also explains the HaPPY code’s entanglement structure without recourse tothe greedy algorithm: As each dimer carries an entanglement entropy of log 2 , computing the entanglemententropy of a compact boundary region A amounts to counting dimers between A and its complementregion A C . From this the geodesic structure of dimers immediately implies a discrete version of the Ryu-Tayanagi formula: A discrete cut γ A through the bulk with the same endpoints as A and which is of minimallength, i.e., discretizes a bulk geodesic, can only cut through such a dimer once. This is because no twoshortest paths through a network (nor two continuous geodesics) can meet at more than one point. Thisleads to the formula S A = | γ A | log 2 , (54)where | γ A | is the number of edges in γ A . The only possible loophole to this argument requires a geometrywhere geodesics are not unique; these cases correspond precisely to the residual bulk regions unreachableby the greedy algorithm. Such a scenario is shown in Fig. 9: For a pathological boundary region A , thegreedy algorithm gets stuck at two adjacent pentagons which cannot be passed, as each only presents twoedges which are insufficient to reconstruct the state. The tensor network thus cannot be a full isometrybetween A and its complement region A C . The Majorana dimer picture (Fig. 9, right) resolves this issue:For a basis-state input, the entanglement entropy acquires a correction from the dimers passing through γ A twice, leading to S A = ( | γ A | −
1) log 2 . For a general input that corresponds to a superposition of dimer If define entanglement entropy in the spin picture, as is relevant for discussing the original HaPPY code, non-compact regionsdo not preserve operator locality after a transformation to fermions. esidual bulk region (wedge picture) A A C Residual bulk region (dimer picture)
A A C Figure 9: L
EFT : A residual bulk region formed when the greedy algorithm for a boundary region A and itscomplement A C do not converge to the same discrete bulk geodesic γ A . R IGHT : Even though moving alonggeodesics themselves, certain dimers (shaded red) passing through a residual bulk region can pass through γ A twice, reducing the entanglement entropy of region A .states, this correction depends on the exact logical input on the two pentagons in the residual bulk region[117].The discrete Ryu-Takayanagi-like behavior of entanglement entropy S A implies that, up to the smallcorrections produced by residual bulk regions, the HaPPY code’s boundary states follow the logarithmicscaling (23) expected of ground states of conformal field theories on a circle. But what is the effective centralcharge c of these states, i.e., the exact coefficient of entanglement scaling? Perhaps surprisingly, its valuedepends on the geometrical construction of the HaPPY code’s tiling. As it consists of infinitely many tiles,properly defining a HaPPY boundary state requires a cutoff procedure or equivalently, a renormalizationgroup (RG) step that iteratively extends the tiling. Such a step is not unique: Depending on the choiceof inflation rule chosen to grow the tiling (two of which are shown in Fig. 10), the asymptotic boundaryis approached differently. This leads to different effective RG steps of the boundary states. Again, underthe assumption of basis state input the Majorana dimer picture helps quantify these statements: Expressedin terms of dimers, each inflation step acts as a fermionic analogue of an RG step of the strong-disorderrenormalization group (SDRG) [119], leading to an apriodic correlation and entanglement structure that canbe exactly solved [118]. For the two inflation methods shown in Fig. 10, this leads to two different analyticalvalues of the central charge, c { , } v = 9 log 2log (cid:0) √ (cid:1) ≈ . , c { , } e = 6 ln 2ln √ ≈ . , (55)where the subscripts denote either vertex or edge inflation. Extensions to tilings other than the { , } oneare also possible [118]. 28 , } vertex inflation { , } edge inflation Figure 10: Inflation methods for the { , } tiling, with layers in the inflation color-coded. L EFT : Vertexinflation, where each inflation step adds polygons vertex-adjacent to the previous layer, forming a closedband. R
IGHT : Edge inflation, where each inflation step adds polygons edge-adjacent to the previous layer.
The conditions for a holographic quantum error-correcting code following the HaPPY proposal is quiteconstrained, as the perfect tensor property allows only for few solutions. For example, the nonexistenceof absolutely maximally entangled states on four spins [120, 121] forbids a HaPPY code embedded into a { , k } tiling, at least for a bond dimension χ = 2 (i.e., for a spin model). Fortunately, it was shown thatmany properties of HaPPY codes are preserved in a tensor network model with block-perfect tensors [122].While a perfect tensor corresponds to an isometry for any bipartition of indices, a block-perfect one is onlyisometric for bipartitions into adjacent sets of indices, i.e., if neither A nor A C are disjoint regions. Thisweaker constraint allows one to build holographic codes based on the more widely studied Calderbank-Shor-Steane (CSS) codes [122], a special and particularly important class of quantum error correcting codes thatare constructed from pairs of classical error correcting codes that share a number of desirable properties. Thespecific construction made use of is based on the . For any such holographic quantumerror correcting code, the question of a suitable decoding appears — as for any quantum error correctingcode — so that of a classical algorithm that based on the syndrome arising from local measurements wouldassign the likely error that has actually occurred. This line of thought of exploring decoders specificallyfor holographic quantum error correcting codes has been explored in Ref. [123], by suggesting an integeroptimization decoder.Block-perfect generalizations of the [[5 , , -based holographic pentagon code can also be constructedin the Majorana dimer setting. To ensure that connected subregions are maximally entangled, basis states ofsuch an n -gon code have to consist of dimers connecting modes i and ( i + n ) mod 2 n on opposite ends.Ensuring that not only the basis states but also their superposition have the same entanglement structureleads to conditions on the dimer parities that can be fulfilled only if n = 4 i + 1 , i ∈ Z [117]. The first29on-HaPPY example of this construction is a “nonagon code” with basis state vectors | ¯0 (cid:105) = , | ¯1 (cid:105) = . (56)Unfortunately, this code does not become more resilient to errors at larger n ; for example, the operator X Z n +12 X n (where X i and Z i are Pauli σ x and σ z operators acting on the i th site) has different eigenvalueson the basis states and therefore corresponds to a phase flip error of weight , regardless of n . While suchcodes are thus not useful in practical applications, they can be used as a tool to study holographic codes: Forexample, it can be shown that the effect of residual bulk regions becomes negligible in generalized Majoranadimer codes as n becomes large [118].The generalization to more general types of tensors as such, however, is by no means the only line ofthought to generalize the HaPPY prescription. One can also introduce degrees of freedom on the edges ofan associated tensor network, connected to further copies of the HaPPY code by an appropriate isometry.This mindset leads to a generalization of HaPPY holographic quantum error-correcting code to provide toymodels for bulk gauge fields or linearized gravitons [124].Rather than studying static properties of holographic codes, one may also wonder if they admit dynam-ics . Such a line of research was first carried out [125] for the particularly symmetric limit of holograpiccodes on { n, k } tilings were k → ∞ , the so-called ideal regular tilings . The boundary symmetries of suchtilings are encapsulated by Thompson’s group T which bears some similarities to the conformal group. Thecorresponding bulk symmetries are then given by the Ptolemy group P t describing a form of discretized dif-feomorphisms, generated by
Pachner moves than re-arrange edges within the bulk, breaking regularity. Theresulting bulk/boundary dynamics differs from continuum notion of time evolution in some ways, specifi-cally in that there is no well-defined Hamiltonian. The Pachner moves also act highly non-locally, as eachedge in an ideal tiling stretches all the way to the asymptotic boundary. This makes it difficult to definenotions of locality in the IR, which would be required for defining a discrete analogue of a particle in thebulk. Similarly, the tree tensor network structure of such models makes it difficult to produce boundarystates with the entanglement structure of physical CFTs, though such geometries do appear in p-adic mod-els of AdS/CFT [126]. However, further studies along these lines in more general geometries may lead to adiscrete bulk/boundary dictionary of holographic codes.Another approach to boundary dynamics may be provided by the SDRG picture that arises from de-scribing the HaPPY code in terms of Majorana dimers: Most traditional SDRG models, whose RG stepsis formulated in terms of spin singlets rather than dimers, result from a strong disorder limit of certainHamiltonians with only nearest-neighbor coupling. For example, consider the
Fibonacci XXZ model withHamiltonian H = (cid:88) i J i ( X i X i +1 + Y i Y i +1 + ∆ Z i Z i +1 ) , (57)where the X i , Y i , Z i are Pauli spin operators acting on the i th site and the J i are coupling terms that varyalong the sites according to an aperiodic Fibonacci sequence. This model is non-Gaussian for ∆ (cid:54) = 0 andgenerally difficult to solve for generic J i . However, in the case of strong disorder, i.e., when the value of30 lternating square/hexagon tiling { , } HaPPY code with black hole
Figure 11: Tensor network generalizations of HaPPY codes, with logical states represented as red dots.L
EFT : An alternating hyperbolic tiling of squares and hexagons used in Ref. [131] with logical degreesof freedom encoded in a Bacon-Shor code on the squares. R
IGHT : The black hole geometry from Refs.[108, 132], where a central tensor is removed, leading to additional logical “horizon” degrees of freedom onthe remaining open edges.the couplings changes significantly with the aperiodic sequence, its ground state is approximately given bya configuration of singlets than can be recursively computed with the SDRG approach [127, 128]. Giventhat the boundary states of the HaPPY code can be produced in a similar SDRG process relying on fermionsrather than spins [117], it is thus plausible to speculate that the resulting boundary states are also the ap-proximate ground state of a Hamiltonian with only local couplings, according to which one could definelocal time dynamics. Note that the HaPPY boundary states are of course ground states of a non-local Hamil-tonian: For basis-state input, this is simply the free Hamiltonian coupling the endpoints of each Majoranadimer configuration.The question of dynamics of holographic codes can also be approached from the stabilizer picture.Inflation of the hyperbolic tiling embedding the HaPPY code can be associated with a mapping betweenisometries, each additional inflation layer adding both bulk and boundary degrees of freedom, that allows foran explicit construction of the resulting stabilizers at each layer [129]. The form of the resulting stabilizersimplies that most long-range correlations (in the spin picture) vanish, which is equivalent to the sparseness ofcorrelations we already saw in the Majorana dimer picture. However, the stabilizer picture can be used makestatements about finite-temperature dependence of the resulting boundary model. This inflation process canalso be equivalently described in terms of C (cid:63) algebras [130], from which boundary entanglement is morereadily computable. While the HaPPY model of a holographic code is already quite versatile and captures a number of holo-graphic properties, it is not the only possible way to construct such a code. Two plausible directions for31onstructing more general codes is to consider tensors describing more general encoding isometries and toconsider tilings that are more complicated than simple regular { n, k } ones. One recent approach combiningboth directions is presented in Ref. [131], where the tensors are chosen to represent a Bacon-Shor code which generalizes quantum error-correcting codes by including gauge degrees of freedom. This code isthen embedded into an alternating hyperbolic tiling composed of squares and hexagons where the logicalqubits are encoded only on the squares while the hexagons contain perfect tensors without bulk degreesof freedom. The resulting setup is visualized in Fig. 11 (left). This construction inherits some of the is-sues of the original HaPPY model (such as residual bulk regions) but allows for a deformation to a skewedcode with only approximate error-correcting properties whose effect on entanglement wedges resembles agravitational back-reaction of a massive bulk deformation.Another approach to constructing holographic codes is to consider them as a mapping between localHamiltonians, an idea developed in Ref. [132]: Such a mapping is indeed possible for a discretized hyper-bolic bulk of three or more spatial dimensions (compared to two in the HaPPY model) in an approximatemanner using perturbation gadgets , a tool from Hamiltonian simulation theory. These higher-dimensionalmodel require a generalization of the regular tilings to a tesselation with polytopes that follow the symme-tries of a Coxeter group. Intriguingly, these models preserve locality both on the boundary and in the bulk,which makes it possible to consider a generic form of time evolution that reproduces certain aspects of blackhole formation. These constructions also reaffirm an idea present in the original HaPPY paper [108] anddeveloped in more detail in Ref. [133]: That low-energy bulk excitations should be describable by changeswithin the logical code space while high-energy ones (e.g. black holes, shown in Fig. 11, right) explicitlybreak the code space and modify the bulk geometry into which the holographic code is embedded.Previous tensor network models of holography can also be included into the framework of quantumerror-correcting codes: Given the strong resemblance of MERA with discrete instances of hyperbolic ge-ometries as featuring in the AdS/CFT correspondence [43], it comes as no surprise that the connectionbetween quantum error correction and MERA has been explored. Indeed, MERA serves as an example tosolidify the idea of creating quantum error-correcting codes arising from an encoding map from the bulktheory to the boundary theory. Ref. [134] explores the connection between quantum error correction andMERA, in fact guided by a bold motivation: Here, it is not only attempted to realize some specific holo-graphic quantum error correction code. Instead, the point is made that if there is a unitary equivalence ofconformal field theory and a quantum theory of gravity in AdS space-time, one should be able to explainholographic codes as emerging directly from properties of the underlying CFT.That said, in order to establish such a connection in the framework of Ref. [134], notions of quantumerror correction have to be slightly weakened to an approximate quantum error correction . An erasure ofa given region is correctable — appropriately modified in an approximate version thereof — if and only ifthat region does not contains any logical information, and hence, if and only if that region is uncorrelatedwith the purifying space for all the code words. Refining this insight, a notion of local correctability [135]extends this to a connection between local correctability and the degree to which different subsystems areseparated in correlations. In Ref. [134], these notions are applied to and put into the context of MERA codes.Specifically, assume that A is a simply connected region “shielded” by a region B such that AB ≡ A ∪ B contains all sites within a distance x from A , and that the MERA contains sufficiently layers s so that | AB | < s . If we further denote with C the complement of AB , then it is shown that there exists a recoverymap R reconstructing the region AB from B under the bound (cid:107)R ( ρ BCR ) − ρ ABCR (cid:107) ≤ c (cid:18) | A | c (cid:19) ν/ , (58)for all purified code states ρ ABCR (involving a further purifying system R ), where c > is a constant and32 > is the scaling dimension of the isometries of the MERA. (cid:107) . (cid:107) here is the trace norm meaningfullyquantifying the statistical distinguishability of quantum states. This notion of recoverability provides abroader basis to the understanding that low energies of the critical systems should have a certain errorcorrection properties [136] without demanding tensors to obey such strict bounds as perfect tensors. Theseresults apply broadly to critical ground states within the extent that they can be approximated by the MERA. In this topical review, we have laid out some developments of a growing field of research at the interfaceof high energy physics and quantum information theory that aims at fleshing out aspects of holography ina particularly transparent picture. The two main ingredients in this endeavour are on the one hand tensornetworks that capture the natural underlying entanglement structure of quantum states. On the other hand,these are notions of quantum error corrections, concepts originally having arisen in the context of quantumcomputing in the presence of noise, but actually being closely intertwined with notions of holography.We have explored here the roots in string theory and high energy physics, made the connection to tensornetworks, and moved on to explain the connection to quantum error correction. We have discussed in greatdetail holographic quantum error correction in tensor networks and toy models of holography in a stabilizerpicture as well as their fermionic representation in terms of coupled Majorana modes, but also the variousefforts made to generalize these approaches in the search of more comprehensive tensor network models ofholographic quantum error correction.And yet, many exciting questions remain open, and much of what has been said here can be seen as aninvitation to pursue these steps. The study of the dynamics of holographic models — only hinted upon above— is just beginning to unfold [125, 129, 132]. Just as importantly, questions of meaningful continuum limitsof tensor networks relating discrete and continuous models of holography are being extensively pursued[137, 138, 139, 140], with the particularly interesting potential of describing regimes of interacting quantumfields [141]. Notions of circuit and state complexity [90] have also taken center stage in recent discussionsof holography [142, 143, 144, 145, 146], not the least due to the bold “complexity equals volume” [91] and“complexity equals action” [92] conjectures due to Leonard Susskind, Douglas Stanford, Adam R. Brown,Brian Swingle and others mentioned above. The precise connection of complexity to tensor network modelsof holography is yet to be fully established but may be related to path integral approaches to holography[147, 148]. Holographic tensor network models can also be used to construct practical stabilizer codes withnatural decoders [149], showing that insights from these models can become useful in an applied setting.It is the hope that the present article, beyond merely giving an overview over current developments of anexciting field, will serve as a source of inspiration for further endeavours exploring the intricate connectionsbetween notions of holography, quantum error correction, and tensor networks.
Acknowledgements
We would like to warmly thank numerous colleagues for stimulating discussions on the topics addressedin this review. A list of those colleagues includes but is by no means limited to A. Altland, B. Czech, G.Evenbly, M. Gluza, L. F. Hackl, D. Harlow, M. Heller, R. C. Myers, F. Pastawski, J. Prior, S. Singh, M.Steinberg, T. Takayanagi, G. Vidal, M. Walter, C. Wille, H. Wilming, X.-L. Qi, and Z. Zimboras. This workhas been supported by the DFG (CRC 183 and EI 519/15-1) and the FQXi. This review includes parts ofthe doctoral thesis submitted by AJ at the Free University of Berlin.33 eferences [1] J. D. Bekenstein, “Black holes and entropy,”
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