Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2112.05099 · Physical review. D/Physical review. D.

Renormalization group and approximate error correction

Keiichiro Furuya, Nima Lashkari, Mudassir Moosa

Published Nov 10, 2022• 10 citations
arXivHTMLPDF

Abstract

We show that in renormalization group (RG) flow the low-energy states form a code subspace that is approximately protected against the local short-distance errors. To demonstrate how general this connection is, we consider three examples: the classical Ising model in one dimension, free relativistic scalar quantum field theory (QFT) in two spacetime dimensions, and holographic field theories as examples of strongly coupled systems. As a concrete example of real-space RG in QFT, we consider the continuous multiscale renormalization ansatz for massive free fields and show that the low-energy coherent states are approximately protected from the quantum errors caused by the high-energy localized coherent operators. In holographic RG flows, we study the phase transition in the entanglement wedge of a single region and argue that one needs to define the price and the distance of the code with respect to the reconstructable wedge.

DOI
10.1103/physrevd.106.105007

Related papers

Aspects of the renormalization group in three-dimensional quantum field theory

Benjamin R. Safdi

The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.

•Sep 25, 2014•0 citations

Entanglement Wedge Reconstruction without Holographic Quantum Error Correction

Seiji Terashima

Bulk reconstruction is a central problem in AdS/CFT, and entanglement wedge reconstruction is its subregion version. We argue that this subregion statement should be separated from the stronger holographic quantum error correction interpretation, in which one region-independent logical bulk operator has code-preserving representatives in several boundary regions. A simple locality argument shows that such a common reconstruction must commute with the code-preserving local algebras in the complementary regions. This is the mechanism realized in HaPPY-type codes: the erased regions are blind to a protected logical algebra. An ordinary finite $N$ holographic CFT does not have such a protected invisible sector for supergravity fields. Its low-energy local observables, in particular, suitably smeared stress tensors, detect the physical support and gravitational dressing of ordinary bulk operators, up to possible center or superselection data. Thus, there is no such holographic quantum error correction and the $N=\infty$ agreement of global and subregion HKLL formulae is a free-theory statement. What remains is entanglement wedge reconstruction without holographic quantum error correction, or subregion complementarity: each boundary region has its own code-preserving low-energy algebra and its own region-adapted bulk interpretation, rather than a shared logical operator.

hep-thCross-listedarXiv:2607.08684•Jul 10, 2026

Real-space renormalization, error correction and conditional expectations

Keiichiro Furuya, Nima Lashkari, Shoy Ouseph

We show that the real-space renormalization group (RG), as a map from the observable algebra to the subalgebra of long-distance observables, is an error correction code, best described by a conditional expectation. It is comprised of a coarse-graining step followed by an isometric embedding. The coarse-graining is the error map and the long-distance observables are the correctable operators. We show that if there is a state that is preserved under renormalization the coarse-graining step is the Petz dual of the isometric embedding (the Petz map). We demonstrate that a set of states are preserved under this map if and only if their pairwise relative entropies do not change when we restrict to the long-distance observables. We study the operator algebra quantum error correction in the GNS Hilbert space which applies to any quantum system including the local algebra of quantum field theory. We show that the recovery map is an isometric embedding of the correctable subalgebra. Similar to the RG, the composition of the error map followed by the recovery map forms a conditional expectation (a projection in the GNS Hilbert space). In gauge/gravity dualities, the bulk relative entropy of holographic states is the same as their boundary relative entropies which implies that the holographic map is an error correction code, and hence a conditional expectation. It follows that the boundary to the bulk map is a Petz map.

•Dec 30, 2020•1 citation

Real-space RG, error correction and Petz map

Keiichiro Furuya, Nima Lashkari, Shoy Ouseph

Abstract There are two parts to this work: first, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large N and a large gap in the spectrum of operators in the emergence of complementary recovery. Second, we study the operator algebra exact quantum error correction for any von Neumann algebra. We show that similar to the finite dimensional case, for any error map in between von Neumann algebras the Petz dual of the error map is a recovery map if the inclusion of the correctable subalgebra of operators has finite index.

arXiv:2012.14001•Journal of High Energy Physics•Jan 1, 2022•19 citations

Entanglement Renormalization for Quantum Fields in Real Space

Jutho Haegeman, Tobias J. Osborne, Henri Verschelde +1 more

We show how to construct renormalization group (RG) flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wave functions arising from this RG flow are translation invariant and exhibits an entropy-area law. We illustrate the construction for a free nonrelativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.

arXiv:1102.5524•Physical Review Letters•Mar 5, 2013•241 citations

Multiscale Entanglement Renormalization Ansatz: Causality and Error Correction

Domenico Pomarico

Computational complexity reduction is at the basis of a new formulation of many-body quantum states according to tensor network ansatz, originally framed in one-dimensional lattices. In order to include long-range entanglement characterizing phase transitions, the multiscale entanglement renormalization ansatz (MERA) defines a sequence of coarse-grained lattices, obtained by targeting the map of a scale-invariant system into an identical coarse-grained one. The quantum circuit associated with this hierarchical structure includes the definition of causal relations and unitary extensions, leading to the definition of ground subspaces as stabilizer codes. The emerging error correcting codes are referred to logical indices located at the highest hierarchical level and to physical indices yielded by redundancy, framed in the AdS-CFT correspondence as holographic quantum codes with bulk and boundary indices, respectively. In a use-case scenario based on errors consisting of spin erasure, the correction is implemented as the reconstruction of a bulk local operator.

Dynamics•Sep 18, 2023•6 citations
We show that in renormalization group (RG) flow the low-energy states form a code subspace that is approximately protected against the local short-distance errors. To demonstrate how general this connection is, we consider three examples: the classical Ising model in one dimension, free relativistic scalar quantum field theory (QFT) in two spacetime dimensions, and holographic field theories as examples of strongly coupled systems. As a concrete example of real-space RG in QFT, we consider the continuous multiscale renormalization ansatz for massive free fields and show that the low-energy coherent states are approximately protected from the quantum errors caused by the high-energy localized coherent operators. In holographic RG flows, we study the phase transition in the entanglement wedge of a single region and argue that one needs to define the price and the distance of the code with respect to the reconstructable wedge.
The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.
Bulk reconstruction is a central problem in AdS/CFT, and entanglement wedge reconstruction is its subregion version. We argue that this subregion statement should be separated from the stronger holographic quantum error correction interpretation, in which one region-independent logical bulk operator has code-preserving representatives in several boundary regions. A simple locality argument shows that such a common reconstruction must commute with the code-preserving local algebras in the complementary regions. This is the mechanism realized in HaPPY-type codes: the erased regions are blind to a protected logical algebra. An ordinary finite NNN holographic CFT does not have such a protected invisible sector for supergravity fields. Its low-energy local observables, in particular, suitably smeared stress tensors, detect the physical support and gravitational dressing of ordinary bulk operators, up to possible center or superselection data. Thus, there is no such holographic quantum error correction and the N=∞N=\inftyN=∞ agreement of global and subregion HKLL formulae is a free-theory statement. What remains is entanglement wedge reconstruction without holographic quantum error correction, or subregion complementarity: each boundary region has its own code-preserving low-energy algebra and its own region-adapted bulk interpretation, rather than a shared logical operator.
We show that the real-space renormalization group (RG), as a map from the observable algebra to the subalgebra of long-distance observables, is an error correction code, best described by a conditional expectation. It is comprised of a coarse-graining step followed by an isometric embedding. The coarse-graining is the error map and the long-distance observables are the correctable operators. We show that if there is a state that is preserved under renormalization the coarse-graining step is the Petz dual of the isometric embedding (the Petz map). We demonstrate that a set of states are preserved under this map if and only if their pairwise relative entropies do not change when we restrict to the long-distance observables. We study the operator algebra quantum error correction in the GNS Hilbert space which applies to any quantum system including the local algebra of quantum field theory. We show that the recovery map is an isometric embedding of the correctable subalgebra. Similar to the RG, the composition of the error map followed by the recovery map forms a conditional expectation (a projection in the GNS Hilbert space). In gauge/gravity dualities, the bulk relative entropy of holographic states is the same as their boundary relative entropies which implies that the holographic map is an error correction code, and hence a conditional expectation. It follows that the boundary to the bulk map is a Petz map.
Abstract There are two parts to this work: first, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large N and a large gap in the spectrum of operators in the emergence of complementary recovery. Second, we study the operator algebra exact quantum error correction for any von Neumann algebra. We show that similar to the finite dimensional case, for any error map in between von Neumann algebras the Petz dual of the error map is a recovery map if the inclusion of the correctable subalgebra of operators has finite index.
We show how to construct renormalization group (RG) flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wave functions arising from this RG flow are translation invariant and exhibits an entropy-area law. We illustrate the construction for a free nonrelativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.
Computational complexity reduction is at the basis of a new formulation of many-body quantum states according to tensor network ansatz, originally framed in one-dimensional lattices. In order to include long-range entanglement characterizing phase transitions, the multiscale entanglement renormalization ansatz (MERA) defines a sequence of coarse-grained lattices, obtained by targeting the map of a scale-invariant system into an identical coarse-grained one. The quantum circuit associated with this hierarchical structure includes the definition of causal relations and unitary extensions, leading to the definition of ground subspaces as stabilizer codes. The emerging error correcting codes are referred to logical indices located at the highest hierarchical level and to physical indices yielded by redundancy, framed in the AdS-CFT correspondence as holographic quantum codes with bulk and boundary indices, respectively. In a use-case scenario based on errors consisting of spin erasure, the correction is implemented as the reconstruction of a bulk local operator.