[Paper Review] Holographic codes

José I. Latorre, Germán Sierra|arXiv (Cornell University)|Feb 23, 2015

Advanced Memory and Neural Computing17 citations

TL;DR

This paper introduces the H-code, a holographic quantum code constructed from a four-qutrit state with absolute maximal entanglement, enabling a perfect mapping from boundary to bulk degrees of freedom via a neutralization rule. The resulting state exhibits topological order, is stabilized by a highly non-local Hamiltonian, and features a topological entanglement entropy of -1, demonstrating a novel class of topologically ordered quantum memories with exponentially large Hamming distances between bulk states.

ABSTRACT

There exists a remarkable four-qutrit state that carries absolute maximal entanglement in all its partitions. Employing this state, we construct a tensor network that delivers a holographic many body state, the H-code, where the physical properties of the boundary determine those of the bulk. This H-code is made of an even superposition of states whose relative Hamming distances are exponentially large with the size of the boundary. This property makes H-codes natural states for a quantum memory. H-codes exist on tori of definite sizes and get classified in three different sectors characterized by the sum of their qutrits on cycles wrapped through the boundaries of the system. We construct a parent Hamiltonian for the H-code which is highly non local and finally we compute the topological entanglement entropy of the H-code.

Motivation & Objective

• To construct a holographic quantum code where boundary and bulk degrees of freedom are strictly related via a balanced superposition.
• To realize a many-body quantum state with absolute maximal entanglement in all partitions, enabling robust quantum memory.
• To define a tensor network based on a four-qutrit maximally entangled state that enforces a neutralization rule (sum to zero mod 3) across triangles.
• To derive a parent Hamiltonian for the H-code that is highly non-local and stabilizes the ground state with topological order.
• To compute the topological entanglement entropy and demonstrate its invariance under local deformations.

Proposed method

• The H-code is constructed using a tensor network of triangular simplices on a 2D triangular lattice, with ancillary qutrits enforcing a neutralization rule: s₁ + s₂ + s₃ ≡ 0 (mod 3) for each up-pointing triangle.
• The underlying four-qutrit state |ψ⟩ = ∑_{s,i} |s⟩_phys |i, i+s, i+2s⟩_ancillae ensures absolute maximal entanglement and acts as a building block for the network.
• The network enforces that physical indices in each row are determined by the sum of indices in the row above, via a rule equivalent to s₃ = -s₁ - s₂ (mod 3).
• The construction is invariant under different choices of holographic direction, ensuring consistency across the lattice.
• A parent Hamiltonian H = H_Z + H_X is derived, where H_Z enforces the neutralization rule and H_X breaks degeneracy via non-local operators with exponential support.
• The Hamiltonian H_X is a highly non-local operator involving products of X and X⁻¹ over all up triangles, with support growing as 6^k, ensuring the H-code is the unique ground state in a given topological sector.

Experimental results

Research questions

• RQ1 Can a holographic quantum code be constructed such that every boundary state uniquely determines a bulk state via a unitary mapping?
• RQ2 What are the entanglement and topological properties of a many-body state built from absolute maximally entangled four-qutrit states?
• RQ3 How can a non-local Hamiltonian be designed to stabilize a topological quantum code with maximal entanglement and large Hamming distances?
• RQ4 What is the topological entanglement entropy of such a code, and does it remain invariant under local changes in the subsystem partition?
• RQ5 Can the H-code be classified into distinct topological sectors, and are these sectors distinguishable via bulk observables?

Key findings

• The H-code is a balanced superposition of boundary and bulk product states, with the bulk state uniquely determined by the boundary via a neutralization rule.
• The topological entanglement entropy of the H-code is exactly -1, computed via the formula S_top = S_ABC - S_AB - S_AC - S_BC + S_A + S_B + S_C.
• The reduced density matrix of any bulk region A with m < 3^k qutrits is maximally mixed: ρ_A = (1/3^m) I_{3^m}, indicating complete disorder in small subsystems.
• The parent Hamiltonian H = H_Z + H_X stabilizes the H-code as its unique ground state in a given topological sector, with H_X being highly non-local and supporting 6^k operators.
• The H-code exhibits exponentially large Hamming distances between bulk states, making it a natural candidate for quantum memory due to maximal distinguishability.
• The three distinct H-code states on a torus (labeled by S) are topologically indistinguishable via bulk observables, confirming topological order.