Computing with Spin: Cohomology and Mathematical Applications of Holographic Satisfiability

We extend the holographic test of entanglement cohomology from $d=2$ to higher-dimensional conformal field theories. For a $d$ -dimensional CFT with a spherical entangling region $A$ , the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdS $_{d+1}$ , and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for $d=3$ and $d=4$ and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients $κ_d Δ^2 / C_T$ , where $κ_3 = π^2/12$ and $κ_4 = 3π^3/16$ . The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the $d=2$ case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}We extend the holographic test of entanglement cohomology from $d=2$ to higher-dimensional conformal field theories. For a $d$-dimensional CFT with a spherical entangling region $A$, the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdS$_{d+1}$, and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for $d=3$ and $d=4$ and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients $κ_d Δ^2 / C_T$, where $κ_3 = π^2/12$ and $κ_4 = 3π^3/16$. The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the $d=2$ case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}

arXiv:2610.25969Oct 19, 20260 citations

Quantum demonstration of a bio-molecular solution of the satisfiability problem on spin-based ensemble

Ren Ting-Ting, Feng Mang, Weng-Long Chang +1 more

DNA computation (DNAC) has been proposed to solve the satisfiability (SAT) problem due to operations in parallel on extremely large numbers of strands. This paper attempts to treat the DNA-based bio-molecular solution for the SAT problem from the quantum mechanical perspective with a purpose to explore the relationship between DNAC and quantum computation (QC). To achieve this goal, it first builds up the correspondence of operations between QC and DNAC. Then it gives an example for the case of two variables and three clauses for details of this theory. It also demonstrates a three-qubit experiment for solving the simplest SAT problem with a single variable on a liquid-state nuclear magnetic resonance ensemble to verify this theory. Some discussions are made for the potential application and for further exploration of the present work.DNA computation (DNAC) has been proposed to solve the satisfiability (SAT) problem due to operations in parallel on extremely large numbers of strands. This paper attempts to treat the DNA-based bio-molecular solution for the SAT problem from the quantum mechanical perspective with a purpose to explore the relationship between DNAC and quantum computation (QC). To achieve this goal, it first builds up the correspondence of operations between QC and DNAC. Then it gives an example for the case of two variables and three clauses for details of this theory. It also demonstrates a three-qubit experiment for solving the simplest SAT problem with a single variable on a liquid-state nuclear magnetic resonance ensemble to verify this theory. Some discussions are made for the potential application and for further exploration of the present work.

Chinese Physics BDec 1, 20090 citations

A Formalism for Quantum Computing and a Satisfiability Problem

César Bautista-Ramos, M. Castro-Cardona, Alfred Zehe

Elsevier eBooksJan 1, 20050 citations

Quantum mathematics: holonomic computing algorithms and their applications. Pt. 2

Anatoliy K. Prykarpatsky, Ufuk Taneri, Denis Blackmore

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w KrakowieJan 1, 20040 citations

Bases Collapse in Holographic Algorithms

Jin‐Yi Cai, Pinyan Lu

Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2.Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2.

IEEE Conference on Computational Complexity (CCC)Jun 1, 200721 citations

We extend the holographic test of entanglement cohomology from

d=2

to higher-dimensional conformal field theories. For a

d

-dimensional CFT with a spherical entangling region

A

, the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdS

_{d+1}

, and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for

d=3

and

d=4

and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients

κ_d Δ^2 / C_T

, where

κ_3 = π^2/12

and

κ_4 = 3π^3/16

. The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the

d=2

case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}