Behavior of Analog Quantum Algorithms

Abstract

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.

Using copies to improve precision in continuous-time quantum computing

“…Practical demonstrations of the precision improvement could be done on hardware with a suitable native graph -results for spin chains indicate that minor embedding can reduce rather than increase precision, which could lead to inconclusive results. Furthermore, it is an open question whether the techniques proposed here could be applied to gate-model machines through QAOA (quantum approximate optimisation algorithm), given the recently highlighted connections between QAOA and quantum annealing [65]. Whether these techniques can be extended beyond classical optimization problems and classical repetition codes to fully universal quantum error correcting codes, e.g., for quantum simulators, where quantum Hamiltonians are evolved continuously in time, is also an interesting open question.…”

Quantum Alternating Operator Ansatz (QAOA) Phase Diagrams and Applications for Quantum Chemistry

“…A more detailed comparison of the methods would require finding optimal adiabatic and optimal QAOA schedules. In the context of optimization there is a close connection between optimal QAOA parameters and smooth adiabatic schedules [10,28,35,36], showing conditions under which optimal schedules exist composed of "bang-bang" and "annealing" regimes, as well as the connection to counteradiabatic effects for suppressing excitations [37]. Cases where there are relatively few free parameters may be especially amenable to generalizations of our phase diagram approach.…”

Counterdiabaticity and the quantum approximate optimization algorithm

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