ABSTRACT
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 models form quantum codes. In this Letter we establish new connections between quantum chaos and translation invariance in many-body spin systems, on one hand, and approximate quantum error correcting codes (AQECC), on the other hand. We first observe that quantum chaotic systems obeying the eigenstate thermalization hypothesis have eigenstates forming approximate quantum error-correcting codes. Then we show that AQECC can be obtained probabilistically from translation-invariant energy eigenstates of every translation-invariant spin chain, including integrable models. Applying this result to 1D classical systems, we describe a method for using local symmetries to construct parent Hamiltonians that embed these codes into the low-energy subspace of gapless 1D quantum spin chains. As explicit examples we obtain local AQECC in the ground space of the 1D ferromagnetic Heisenberg model and the Motzkin spin chain model with periodic boundary conditions, thereby yielding nonstabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models.
ABSTRACT
BACKGROUND: Gaucher disease is a rare hereditary glycolipid storage disease. One of the rare complications is neurodeficits due to vertebral involvement. CASE PRESENTATION: An 18-year-old female patient presented to the outpatient clinic with cauda equina syndrome due to sacral involvement of type 1 GD. Bilateral laminectomy via posterior approach without posterior stabilization was performed. CONCLUSION: Maximum excision of the mass avoiding destabilization of the spinal column can provide long-term vertebral stability and improvement in neurodeficits.
Subject(s)
Cauda Equina Syndrome/etiology , Gaucher Disease/complications , Adolescent , Cauda Equina Syndrome/diagnostic imaging , Cauda Equina Syndrome/surgery , Decompression, Surgical , Enzyme Replacement Therapy/methods , Female , Gaucher Disease/surgery , Gaucher Disease/therapy , Humans , Laminectomy/methods , Magnetic Resonance Imaging , Neurosurgical Procedures/methods , Treatment OutcomeABSTRACT
Blood specimens were obtained from 281 inhabitants of an Eti-Turk village with a population of about 500. Starch gel (pH 8.6) and agar gel (pH 6.45) electrophoresis were performed in 279 of the specimens. Hb S was present in 105 partially interrelated persons (37.36%), three of whom had sickle-cell anaemia. Hb E was detected in 5 persons (1.79%), one of whom was a double heterozygote for Hb S and Hb E. One Hb S+alpha-thalassaemia and 7 Hb S with elevated Hb A'2 combinations were found. The beta-thalassaemia gene prevalence was 0.0377. Hb A2 was found in 4 persons (1.42%), and Hb F was slightly increased in 37 (22.3%) persons with a normal haemoglobin picture. Erythrocyte G-6-PD deficiency was 10% among males.
