Counterdiabaticity and the quantum approximate optimization algorithm

Abstract

The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the p→∞ limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of p large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter "error" terms to approximate adiabatic gauge potentials which suppress diabatic excitations arising from finite ramp speed. In our construction, these "error" terms are helpful, not detrimental, to QAOA. Using this matching to link QAOA with quantum adiabatic algorithms (QAA), we show that the approximation ratio converges to one at least as 1−C(p)∼1/pμ. We show that transfer of parameters between graphs, and interpolating angles for p+1 given p are both natural byproducts of CD-QAOA matching. Optimization of CD-QAOA angles is equivalent to optimizing a continuous adiabatic schedule. Finally, we show that, using a property of variational adiabatic gauge potentials, QAOA is at least counterdiabatic, not just adiabatic, and has better performance than finite time adiabatic evolution. We demonstrate the method on three examples: a 2 level system, an Ising chain, and the MaxCut problem.

Avoiding barren plateaus via transferability of smooth solutions in Hamiltonian Variational Ansatz

“…On a theoretical side, it might be interesting to prove analytically the transferability and landscape properties of smooth solutions found via INTERP. A possible link between this class of solutions and adiabaticity might be investigated [39,41,58]. Finally, our scheme could be directly tested with near-term technology on real quantum devices, beyond the size limits of classical computation.…”

“…This ansatz state can be regarded as a generalization of the Quantum Approximate Optimization Algorithm (QAOA) [7], originally devised for classical combinatorial optimization problems. Remarkably, by means of appropriate iterative schemes for constructing the layer parameters γ m,j , it is often possible to efficiently single-out optimal or nearly-optimal variational parameters that are smooth functions [35][36][37][38][39][40][41] of the layer index m.…”

Augmenting QAOA Ansatz with Multiparameter Problem-Independent Layer

“…The performance of QAOA relies on the choice of these parameters. The problem of identifying optimal QAOA parameters has generated a large amount of theoretical [34]- [37] and computational [38]- [44] work, including such methods as reinforcement learning [45], [46] and reusing preoptimized QAOA parameters from related problem instances [47]- [50]) to significantly reduce the computational cost of parameter optimization.…”

Meta-Learning Digitized-Counterdiabatic Quantum Optimization

“…In particular, a study solving a class of optimiza-tion problems called "digitized counterdiabatic quantum optimization" has reported a polynomial enhancement over the current methods, in which CD interaction serves as a nonstoquastic catalyst [32]. Among the algorithms using CD protocols in QAOA [33][34][35][36], the digitized-counterdiabatic quantum approximate optimization algorithm (DC-QAOA) [34] has shown drastic improvement over standard QAOA. In DC-QAOA, appropriate terms are supplemented to the circuit ansatz corresponding to the adiabatic gauge potentials [20] to reach the ground state more efficiently.…”