Quantum speedup at zero temperature via coherent catalysis

Abstract

It is known that secondary non-stoquastic drivers may offer speed-ups or catalysis in some models of adiabatic quantum computation accompanying the more typical transverse field driver. Their combined intent is to raze potential barriers to zero during adiabatic evolution from a false vacuum to a true minimum; first order phase transitions are softened into second order transitions. We move beyond mean-field analysis to a fully quantum model of a spin ensemble undergoing adiabatic evolution in which the spins are mapped to a variable mass particle in a continuous one-dimensional potential. We demonstrate the necessary criteria for enhanced mobility or 'speed-up' across potential barriers is actually a quantum form of the Rayleigh criterion. Quantum catalysis is exhibited in models where previously thought not possible, when barriers cannot be eliminated. For the 3-spin model with secondary anti-ferromagnetic driver, catalysed time complexity scales between linear and quadratically with the number of qubits. As a corollary, we identify a useful resonance criterion for quantum phase transition that differs from the classical one, but converges on it, in the thermodynamic limit.

Role of nonstoquastic catalysts in quantum adiabatic optimization

“…Our results are consistent with the conclusions of Ref. [34], whereby the exponential advantage of the non-stoquastic catalyst over the stoquastic catalyst is only maintained if the relative strength of the non-stoquastic catalyst to the problem Hamiltonian grows with system size. For a fixed non-stoquastic catalyst strength, the scaling returns to an exponential scaling but it is milder than the stoquastic case, as shown in Fig.…”

“…The case of p = 3 is different because the mean-field potential always exhibits a discontinuous jump in its global minimum [30]. A careful treatment of this model shows that it still exhibits the exponential advantage for the non-stoquastic catalyst [34], but it requires an optimized λ that grows with system size unlike the examples we studied here. We show this in Appendix E.…”

“…We therefore observe a similar incremental change to the ground state wavefunction in the subspace of the evolution as in the p-spin model. It is also useful to consider the semiclassical potential derived in the Villain representation [25,34,[37][38][39][40]:…”

“…A general mechanism for this enhancement was provided in Ref. [34] in terms of a quantum Rayleigh limit, whereby the ground state profile coalesces from bimodal to unimodal. This beyond mean-field treatment also demonstrated the enhancement for the p = 3 case.…”

Symmetry‐Protected Quantum Adiabatic Evolution in Spontaneous Symmetry‐Breaking Transitions

“…It is possible to verify our theory in that platform. Besides, our findings also can be applied for the constrained quantum annealing and speedup the adiabatic process in quantum computation . Our study will not only deepen the understandings of quantum adiabatic evolution and SSB transitions, but also provide promising applications ranging from quantum state engineering to topological Thouless pumping.…”