Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA, also known as adiabatic optimization) finds the minimum of some cost functions exponentially more quickly than Simulated Annealing (SA), which is arguably a classical analogue of QA.One such cost function is the simple "Hamming weight with a spike" function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While this problem can be solved by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [16] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [28] to include barriers with width n ζ and height n α for ζ + α ≤ 1/2. This advantage could be explained in terms of quantum-mechanical "tunneling."Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling.Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general. * Caltech

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 groundspaces. More recently, in the context of the AdS/CFT 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 work we establish new connections between quantum chaos and translationinvariance 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 exhibiting the Eigenstate Thermalization Hypothesis (ETH) have eigenstates forming approximate quantum errorcorrecting codes. Then we show that AQECC can be obtained probabilistically from translationinvariant 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 non-stabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models.

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