Approximate Quantum Codes From Long Wormholes
arXiv:2310.07770 · doi:10.22331/q-2024-08-14-1439
Abstract
We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, , of fermions goes to infinity. For SYK, the distance scales as , and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. , while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size . We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity.
arXiv comment81 pages, 22 figures, revised and substantially expanded version to be published in Quantum, main results are unchanged and new results added
References in corpus (65)
- Fault-tolerant quantum computation by anyons
- Unpaired Majorana fermions in quantum wires
- Holographic Derivation of Entanglement Entropy from AdS/CFT
- Chaos and Quantum Thermalization
- Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
- Comments on the Sachdev-Ye-Kitaev model
- Improved Simulation of Stabilizer Circuits
- Fermionic quantum computation
- A Theory of Quantum Error-Correcting Codes
- Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime
- Quantum corrections to holographic entanglement entropy
- Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
- Bulk Locality and Quantum Error Correction in AdS/CFT
- Stabilizer Codes and Quantum Error Correction
- The Spectrum in the Sachdev-Ye-Kitaev Model
- Entanglement entropy in free quantum field theory
- The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual
- Direct estimations of linear and non-linear functionals of a quantum state
- Universal Quantum Estimator
- Measuring entanglement growth in quench dynamics of bosons in an optical lattice
- Fermionic Localization of the Schwarzian Theory
- Quantum Low-Density Parity-Check Codes
- Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
- Approximate quantum error correction can lead to better codes
- AdS holography and the SYK model
- Eternal traversable wormhole
- Power-law out of time order correlation functions in the SYK model
- An introduction to the SYK model
- Some Results on Mutual Information of Disjoint Regions in Higher Dimensions
- Majorana Fermion Codes
- Quantum Corrections to Holographic Mutual Information
- Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains
- Looking at supersymmetric black holes for a very long time
- Quantum Gravity in the Lab: Teleportation by Size and Traversable Wormholes, Part II
- Local Hamiltonians Whose Ground States are Hard to Approximate
- NLTS Hamiltonians from good quantum codes
- Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames
- Pedagogical introduction to SYK model and 2D Dilaton Gravity
- Holography for people with no time
- Error Correction of Quantum Reference Frame Information
- New perspectives on covariant quantum error correction
- Low-rank Sachdev-Ye-Kitaev models
- Dirac Fast Scramblers
- Near-optimal covariant quantum error-correcting codes from random unitaries with symmetries
- Traversable wormholes in four dimensions
- Majorana dimers and holographic quantum error-correcting codes
- Quenched vs Annealed: Glassiness from SK to SYK
- A Sparse Model of Quantum Holography
- Limits on the storage of quantum information in a volume of space
- Approximate Quantum Error Correction Revisited: Introducing the Alpha-bit
- Short random circuits define good quantum error correcting codes
- Holographic measurement and quantum teleportation in the SYK thermofield double
- Quantum Error Correction for Complex and Majorana Fermion Qubits
- SYK wormhole formation in real time
- Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
- Quasi-exact quantum computation
- Eigenstate Thermalization Hypothesis and Approximate Quantum Error Correction
- Good Quantum LDPC Codes with Linear Time Decoders
- NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians
- Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes
- An efficient decoder for a linear distance quantum LDPC code
- Decoding quantum Tanner codes
- Approaching the Quantum Singleton Bound with Approximate Error Correction
- Large Matrix Quantum Mechanics as a Quantum Memory
- Fermionic Hamiltonians without trivial low-energy states