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 Ω(n 1−ν ) for any ν ∈ (0, 1) and Ω(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 errordetecting features.

Continuous Symmetries and Approximate Quantum Error Correction

“…Approximate quantum error-correcting codes also arise naturally in many-body quantum systems [8,9]. We anticipate that constraints on correlation functions of many-body quantum states can be derived from the covariance properties of the corresponding codes.…”

“…Likewise, quantum error-correcting codes often have approximate or exact symmetries with important implications. In the case of a time-translationinvariant many-body system, for example, certain energy subspaces are known to form approximate quantum errorcorrecting codes [8,9], which are preserved under time evolution. Limits to sensitivity in quantum metrology are related to the degree of asymmetry of probe states, a notion formalized in the resource theory of asymmetry and reference frames [10,11].…”

New perspectives on covariant quantum error correction

“…In general, a quantum code is covariant with respect to a logical Hamiltonian H L and a physical Hamiltonian H S if any symmetry transformation e −iH L θ is encoded into a symmetry transformation e −iH S θ in the physical system. Besides important implications to fault-tolerant quantum computation, covariant QEC is also closely connected to many other topics in quantum information and physics, such as quantum reference frames and quantum clocks [9,11,12], symmetries in the AdS/CFT correspondence [10,[12][13][14][15][16][17][18] and approximate QEC in condensed matter physics [19]. Although covariant codes cannot be perfectly local-error-correcting, they can still approximately correct errors with the infidelity depending on the number of subsystems, the dimension of each subsystem, etc.…”

Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes

“…The edge code behaves similarly with holographic code, while the bulk code can support some topological logical gates from the SPT order [29,30]. Although the exact distance might be a small constant due to the short-range entanglement of ground states [16,31], the quasi distance could be larger. With a proper local error model, we also estimate the quasi-thresholds, which depend on the weak code distances and they are essentially the same for the three types of VBS codes.…”

“…It was observed that [7] approximation can be considered as a quantum resource due to non-orthogonality of quantum states. Some approximate QEC (AQEC) codes were studied in many settings decades ago [7][8][9][10], and recently AQEC is attracting wide attentions partly due to the connection with holography and quantum gravity [11][12][13][14][15][16][17][18]. It is thus a question whether in general AQEC codes can be used for FTQ computation.…”