Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains.

  title={Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains.},
  author={Fernando G. S. L. Brand{\~a}o and Elizabeth Crosson and Mehmet Burak Şahinoğlu and John Bowen},
  journal={Physical review letters},
  volume={123 11},
Quantum error correction was invented to allow for fault-tolerant quantum computation. Systems with topological order turned out to give a natural physical realization of quantum error correcting codes (QECC) in their ground spaces. More recently, in the context of the anti-de Sitter/conformal field theory correspondence, it has been argued that eigenstates of CFTs with a holographic dual should also form QECCs. These two examples raise the question of how generally eigenstates of many-body… 

Constructing linear distance quantum codes in the ground space of local translation-invariant spin chains

This work constructs explicit quantum error correcting codes with linear distance that reside in the ground space of a new semi-exactly solvable local quantum spin chain and constructs a 2-local Hamiltonian, whose quantum error correction properties the authors analytically prove.

Near-Optimal Covariant Quantum Error-Correcting Codes from Random Unitaries with Symmetries

The results not only indicate (potentially efficient) randomized constructions of optimal U (1)- and SU ( d )-covariant codes, but also reveal fundamental properties of random symmetric unitaries, which yield important solvable models of complex quantum systems that have attracted great recent interest in quantum gravity and condensed matter physics.

Continuous Symmetries and Approximate Quantum Error Correction

The approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and the five-rotor code can be stacked to form a covariant holographic code.

Quantum variational learning for quantum error-correcting codes

VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit, sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.

Asymptotic reversibility of thermal operations for interacting quantum spin systems via generalized quantum Stein’s lemma

For quantum spin systems in any spatial dimension with a local, translation-invariant Hamiltonian, we prove that asymptotic state convertibility from a quantum state to another one by a

Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians

A new 2-local frustration free quantum spin chain Hamiltonian whose ground space the authors analytically characterize completely is introduced and it is demonstrated that the ground space contains explicit quantum codes with linear distance.

New perspectives on covariant quantum error correction

New and powerful lower bounds on the infidelity of covariant quantum error correction are proved, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds.

Quantum error-detection at low energies

This work considers the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks, and shows that it contains — within its low-energy eigenspace — an error-detecting code with the same parameter scaling.

Eigenstate thermalization hypothesis and approximate quantum error correction

This paper explores the properties of ETH as an error correcting code and shows that there exists an explicit universal recovery channel for the code, and discusses a generalization that all chaotic theories contain error correcting codes.

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

This family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes.



Theory of quantum error-correcting codes

A general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions is developed and necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction are obtained.

Towards Holography via Quantum Source-Channel Codes.

It is argued that quantum source-channel codes are of independent interest beyond holography and given guarantees on its erasure decoding performance from calculations of an entropic quantity called conditional mutual information, this gives rise to the first concrete interpretation of a bona fide conformal field theory as a quantum error correcting code.

Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].

The Ryu–Takayanagi Formula from Quantum Error Correction

I argue that a version of the quantum-corrected Ryu–Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the

Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems

We consider the problem of determining the energy distribution of quantum states that satisfy exponential decay of correlation and product states, with respect to a quantum local Hamiltonian on a

Adiabatic quantum state generation and statistical zero knowledge

The ASG approach to quantum algorithms provides intriguing links between quantum computation and many different areas: the analysis of spectral gaps and groundstates of Hamiltonians in physics, rapidly mixing Markov chains, statistical zero knowledge, and quantum random walks.

Power law violation of the area law in quantum spin chains

Entanglement between two quantum systems is a non-classical correlation between them. Entanglement is a feature of quantum mechanics which does not appear classically, and it serves as a resource for

Chaos and quantum thermalization.

  • Srednicki
  • Physics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1994
It is shown that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey Berry's conjecture, and argued that these results constitute a sound foundation for quantum statistical mechanics.

Randomizing Quantum States: Constructions and Applications

It is shown that there exists a set of roughly d’log d unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d2 operators required to randomize perfectly.

Bulk locality and quantum error correction in AdS/CFT

A bstractWe point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the