Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the quantum approximate optimization algorithm (QAOA). These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters. In this work, we explore connections between these analog algorithms, as well as limits in which they become approximations of the optimal procedure.Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state that effect the coherent error cancellation of diabatic transitions. Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern. Additionally, the ratios of the QAOA bangs are determined by the smooth, non-oscillatory part of the optimal procedure. We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure. With these arguments, we conclude that different analog algorithms can emulate the optimal protocol under different limits and approximations. Finally, we present a new algorithm for better approximating the optimal protocol using the analytic and numeric insights from the rest of the paper. In practice, numerically, we find that this algorithm outperforms standard QAOA and naive quantum annealing procedures.

1 aBrady, Lucas, T.1 aKocia, Lucas1 aBienias, Przemyslaw1 aBapat, Aniruddha1 aKharkov, Yaroslav1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2107.0121801944nas a2200181 4500008004100000245006800041210006700109260001400176520137900190100002201569700001801591700001801609700002401627700002801651700002101679700002501700856003701725 2021 eng d00aDiscovering hydrodynamic equations of many-body quantum systems0 aDiscovering hydrodynamic equations of manybody quantum systems c11/3/20213 aSimulating and predicting dynamics of quantum many-body systems is extremely challenging, even for state-of-the-art computational methods, due to the spread of entanglement across the system. However, in the long-wavelength limit, quantum systems often admit a simplified description, which involves a small set of physical observables and requires only a few parameters such as sound velocity or viscosity. Unveiling the relationship between these hydrodynamic equations and the underlying microscopic theory usually requires a great effort by condensed matter theorists. In the present paper, we develop a new machine-learning framework for automated discovery of effective equations from a limited set of available data, thus bypassing complicated analytical derivations. The data can be generated from numerical simulations or come from experimental quantum simulator platforms. Using integrable models, where direct comparisons can be made, we reproduce previously known hydrodynamic equations, strikingly discover novel equations and provide their derivation whenever possible. We discover new hydrodynamic equations describing dynamics of interacting systems, for which the derivation remains an outstanding challenge. Our approach provides a new interpretable method to study properties of quantum materials and quantum simulators in non-perturbative regimes.

1 aKharkov, Yaroslav1 aShtanko, Oles1 aSeif, Alireza1 aBienias, Przemyslaw1 aVan Regemortel, Mathias1 aHafezi, Mohammad1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2111.0238501409nas a2200157 4500008004100000245006100041210006100102260001400163520091900177100002101096700002901117700002101146700002201167700002501189856003701214 2020 eng d00aOptimal Protocols in Quantum Annealing and QAOA Problems0 aOptimal Protocols in Quantum Annealing and QAOA Problems c3/19/20203 aQuantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) are two special cases of the following control problem: apply a combination of two Hamiltonians to minimize the energy of a quantum state. Which is more effective has remained unclear. Here we apply the framework of optimal control theory to show that generically, given a fixed amount of time, the optimal procedure has the pulsed (or "bang-bang") structure of QAOA at the beginning and end but can have a smooth annealing structure in between. This is in contrast to previous works which have suggested that bang-bang (i.e., QAOA) protocols are ideal. Through simulations of various transverse field Ising models, we demonstrate that bang-anneal-bang protocols are more common. The general features identified here provide guideposts for the nascent experimental implementations of quantum optimization algorithms.

1 aBrady, Lucas, T.1 aBaldwin, Christopher, L.1 aBapat, Aniruddha1 aKharkov, Yaroslav1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2003.08952