Quantum Error-Detection at Low Energies

Abstract

Motivated by the close relationship between quantum error-correction, topological order, the holographic AdS/CFT duality, and tensor networks, we initiate the study of approximate quantum error-detecting codes in matrix product states (MPS). We first show that using open-boundary MPS to define boundary to bulk encoding maps yields at most constant distance error-detecting codes. These are degenerate ground spaces of gapped local Hamiltonians. To get around this no-go result, we consider excited states, i.e., we use the excitation ansatz to construct encoding maps: these yield error-detecting codes with distance $\Omega (n^{1-\nu })$ for any $\nu \in (0,1)$ and $\Omega (\log n)$ encoded qubits. This shows that gapped systems contain $-$ within isolated energy bands $-$ error-detecting codes spanned by momentum eigenstates. We also consider the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks. We show that it contains $-$ within its low-energy eigenspace $-$ an error-detecting code with the same parameter scaling. All these codes detect arbitrary $d$-local (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic error-detecting features.

arXiv comment79 pages, 17 figures. Version 2: added references

References in corpus (53)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Good Quantum Error-Correcting Codes Exist
• Efficient classical simulation of slightly entangled quantum computations
• Periodic table for topological insulators and superconductors
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Efficient simulation of one-dimensional quantum many-body systems
• Topological quantum memory
• String-net condensation: A physical mechanism for topological phases
• Quantum Error Correction for Quantum Memories
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• A class of quantum many-body states that can be efficiently simulated
• Topological phases of fermions in one dimension
• Entanglement Renormalization and Holography
• Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
• Bulk Locality and Quantum Error Correction in AdS/CFT
• Classifying quantum phases using Matrix Product States and PEPS
• Spectral Gap and Exponential Decay of Correlations
• Holographic duality from random tensor networks
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Complete classification of 1D gapped quantum phases in interacting spin systems
• A fault-tolerant one-way quantum computer
• The Ryu-Takayanagi Formula from Quantum Error Correction
• Topological quantum order: stability under local perturbations
• PEPS as ground states: degeneracy and topology
• Fermionic Projected Entangled Pair States
• Solving Gapped Hamiltonians Locally
• Post-Matrix Product State Methods: To tangent space and beyond
• On Quantum Fidelities and Channel Capacities
• Entanglement renormalization and topological order
• Exponential Decay of Correlations Implies Area Law
• Topological Order at Non-zero Temperature
• Anyons and matrix product operator algebras
• Thresholds for topological codes in the presence of loss
• Explicit tensor network representation for the ground states of string-net models
• An area law for entanglement from exponential decay of correlations
• Holographic Renyi Entropy from Quantum Error Correction
• Faster Methods for Contracting Infinite 2D Tensor Networks
• General conditions for approximate quantum error correction and near-optimal recovery channels
• Elementary excitations in gapped quantum spin systems
• Exact entanglement renormalization for string-net models
• Permutation-invariant quantum codes
• Geometry of Matrix Product States: metric, parallel transport and curvature
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Preparing projected entangled pair states on a quantum computer
• Numerical Contraction of the Tensor Network generated by the Algebraic Bethe Ansatz
• The Algebraic Bethe Ansatz and Tensor Networks
• Towards holography via quantum source-channel codes
• Entanglement renormalization, quantum error correction, and bulk causality
• Tensor-product representations for string-net condensed states

Cited by in corpus (7)

• Continuous symmetries and approximate quantum error correction
• New perspectives on covariant quantum error correction
• Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes
• Covariant Quantum Error-Correcting Codes with Metrological Entanglement Advantage
• Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians
• Classes of topological qubits from low-dimensional quantum spin systems
• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry