02137nas a2200169 4500008004100000245004200041210004200083260001300125520166200138100002101800700001701821700002401838700002101862700002201883700002501905856003701930 2021 eng d00aBehavior of Analog Quantum Algorithms0 aBehavior of Analog Quantum Algorithms c7/2/20213 a

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/2107.0121801652nas a2200133 4500008004100000245005900041210005900100260001500159520124500174100001901419700002201438700002101460856003701481 2021 eng d00aLefschetz Thimble Quantum Monte Carlo for Spin Systems0 aLefschetz Thimble Quantum Monte Carlo for Spin Systems c10/20/20213 a

Monte Carlo simulations are often useful tools for modeling quantum systems, but in some cases they suffer from a sign problem, which manifests as an oscillating phase attached to the probabilities being sampled. This sign problem generally leads to an exponential slow down in the time taken by a Monte Carlo algorithm to reach any given level of accuracy, and it has been shown that completely solving the sign problem for an arbitrary quantum system is NP-hard. However, a variety of techniques exist for mitigating the sign problem in specific cases; in particular, the technique of deforming the Monte Carlo simulation's plane of integration onto Lefschetz thimbles (that is, complex hypersurfaces of stationary phase) has seen success for many problems of interest in the context of quantum field theories. We extend this methodology to discrete spin systems by utilizing spin coherent state path integrals to re-express the spin system's partition function in terms of continuous variables. This translation to continuous variables introduces additional challenges into the Lefschetz thimble method, which we address. We show that these techniques do indeed work to lessen the sign problem on some simple spin systems.

1 aMooney, T., C.1 aBringewatt, Jacob1 aBrady, Lucas, T. uhttps://arxiv.org/abs/2110.1069901409nas a2200157 4500008004100000245006100041210006100102260001400163520091900177100002101096700002901117700002101146700002201167700002501189856003701214 2020 eng d00aOptimal Protocols in Quantum Annealing and QAOA Problems0 aOptimal Protocols in Quantum Annealing and QAOA Problems c3/19/20203 a

Quantum 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