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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References 31A Appendix 331. Is the geometrical lemma tight for Kitaev's original construction?2. Can the circuit-to-Hamiltonian construction be improved with regard to universal adiabatic computation?3. Can we modify the Feynman-Kitaev construction-by introducing clock transitions with varying weights, or by adding branching or loops as analysed in the context of unitary labeled graphs-in order to improve on the known Ω(T −3 ) bound on the UNSAT penalty?In the next section, we motivate each of these questions and formally state our results.Accepted in Quantum 2018-08-03, click title to verify 1 The term "UNSAT" derives from the related QUNSAT ψ = e∈E ψ| he |ψ /|E| quantity that was previously defined and analyzed using the detectability lemma [18]. We use the term UNSAT penalty to emphasize that it is the energy difference between accepting and non-accepting computations.

We consider Hamiltonian simulation using the first order Lie-Trotter product formula under the assumption that the initial state has a high overlap with an energy eigenstate, or a collection of eigenstates in a narrow energy band. This assumption is motivated by quantum phase estimation (QPE) and digital adiabatic simulation (DAS). Treating the effective Hamiltonian that generates the Trotterized time evolution using rigorous perturbative methods, we show that the Trotter step size needed to estimate an energy eigenvalue within precision using QPE can be improved in scaling from to 1/2 for a large class of systems (including any Hamiltonian which can be decomposed as a sum of local terms or commuting layers that each have real-valued matrix elements). For DAS we improve the asymptotic scaling of the Trotter error with the total number of gates M from O(M −1 ) to O(M −2 ), and for any fixed circuit depth we calculate an approximately optimal step size that balances the error contributions from Trotterization and the adiabatic approximation. These results partially generalize to diabatic processes, which remain in a narrow energy band separated from the rest of the spectrum by a gap, thereby contributing to the explanation of the observed similarities between the quantum approximate optimization algorithm and diabatic quantum annealing at small system sizes. Our analysis depends on the perturbation of eigenvectors as well as eigenvalues, and on quantifying the error using state fidelity (instead of the matrix norm of the difference of unitaries which is sensitive to an overall global phase).

Quantum fluctuations driven by non-stoquastic Hamiltonians have been conjectured to be an important and perhaps essential missing ingredient for achieving a quantum advantage with adiabatic optimization. We introduce a transformation that maps every non-stoquastic adiabatic path ending in a classical Hamiltonian to a corresponding stoquastic adiabatic path by appropriately adjusting the phase of each matrix entry in the computational basis. We compare the spectral gaps of these adiabatic paths and find both theoretically and numerically that the paths based on non-stoquastic Hamiltonians have generically smaller spectral gaps between the ground and first excited states, suggesting they are less useful than stoquastic Hamiltonians for quantum adiabatic optimization. These results apply to any adiabatic algorithm which interpolates to a final Hamiltonian that is diagonal in the computational basis.

Perturbed Hamming weight problems serve as examples of optimization instances for which the adiabatic algorithm provably out performs classical simulated annealing. In this work we study the efficiency of the adiabatic algorithm for solving the "the Hamming weight with a spike" problem by using several methods to compute the scaling of the spectral gap at the critical point, which apply for various ranges of the height and width of the barrier. Our main result is a rigorous polynomial lower bound on the minimum spectral gap for the adiabatic evolution when the bit-symmetric cost function has a thin but polynomially high barrier. This is accomplished by the use of a variational argument with an improved ansatz for the ground state, along with a comparison to the spectrum of the system when no spike term is present. We also give a more detailed treatment of the spin coherent path-integral instanton method which was used by Farhi, Goldstone, and Gutmann in arXiv:quant-ph/0201031, and consider its applicability for estimating the gap for different scalings of barrier height and width. We adapt the discrete WKB method for an abruptly changing potential, and apply it to the construction of approximate wave functions which can be used to estimate the gap. Finally, the improved ansatz for the ground state leads to a method for predicting the location of avoided crossings in the excited states of the energy spectrum of the thin spike Hamiltonian, and we use a recursion relation to determine the ordering of some of these avoided crossings, which may be a useful step towards understanding the diabatic cascade phenomenon which occurs in spike Hamiltonians.

We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute. Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist?We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of [[N, k, d, ε]] approximate QLDPC codes that encode k = Ω(N) logical qubits into N physical qubits with distance d = Ω(N) and approximation infidelity ε = O(1/ polylog(N)). The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in O(polylog N) projectors. We prove the existence of an efficient encoding map and show that the spectral gap of the code Hamiltonian scales as Ω(N −3.09 ). We also show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth.Our family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes (Nirkhe, et al., ICALP 2018) to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes (Brown and Fawzi, ISIT 2013). Then, in order to obtain a code with sparse checks and strong detection of local errors, we use a spacetime circuit Hamiltonian construction in order to take advantage of the parallelism of the Brown-Fawzi circuits.The analysis of the spectral gap of the code Hamiltonian is the main technical contribution of this work. We show that for any depth D quantum circuit on n qubits there is an associated spacetime circuit-to-Hamiltonian construction with spectral gap Ω(n −3.09 D −2 log −6 (n)). To lower bound this gap we use a Markov chain decomposition method to divide the state space of partially completed circuit configurations into overlapping subsets corresponding to uniform circuit segments of depth log n, which are based on bitonic sorting circuits. We use the combinatorial properties of these circuit configurations to show rapid mixing between the subsets, and within the subsets we develop a novel isomorphism between the local update Markov chain on bitonic circuit configurations and the edge-flip Markov chain on equal-area dyadic tilings, whose mixing time was recently shown to be polynomial (Cannon, Levin, and Stauffer, RANDOM 2017). Previous lower bounds on the spectral gap of spacetime circuit Hamiltonians have all been based on a connection to exactly solvable quantum spin chains and applied only to ...