Fast quantum annealing for the infinite-range Ising model...

Fast quantum annealing for the infinite-range Ising modelby mean-field counterdiabatic driving

Takuya HatomuraDepartment of Physics, The University of Tokyo

hatomura@spin.phys.s.u-tokyo.ac.jp

Abstract Construct an approximated counter-diabatic Hamiltonian for the infinite-range Ising

model by using the mean-field approximation Mean-field counter-diabatic Hamiltonian becomes local Effectiveness is demonstrated through numerical simulations of quantum annealing Compare with the variational method for local counter-diabatic driving

IntroductionThe concept of “Shortcuts to adiabaticity [1]” was proposed, which mimics adiabatic

dynamics in finite time. In the counter-diabatic driving approach, an auxiliary Hamiltonian is introduced to cancel out diabatic changes and adiabatic dynamics of an original Hamiltonian is realized. This auxiliary Hamiltonian is called the counter-diabaticHamiltonian.

In order to construct the counter-diabatic Hamiltonian, the eigenstates of the original Hamiltonian are required. This requirement makes it difficult to construct the counter-diabatic Hamiltonian for quantum many-body systems. It is also problematic that the counter-diabatic Hamiltonian for quantum many-body systems is non-local. This non-locality makes it difficult to realize in experiments.

MethodShortcuts to adiabaticity by counter-diabatic driving [1].―Consider a time-dependent Hamiltonian and its adiabatic dynamics

Where is an arbitrary real and we take the dynamical phase

Find a Hamiltonian which mimics above adiabatic dynamics

Counter-diabatic Hamiltonian

Variational approach for local counter-diabatic driving [2].―Consider the Schrödinger equation

Introduce the unitary transformation which diagonalizes the Hamiltonian

⇒ Diabatic changes take place due to the gauge potential⇒ Hamiltonian which mimics adiabatic dynamics

We call the adiabatic gauge potential(equivalent to the counter-diabatic Hamiltonian)

“The problem to find an approximated gauge potential under certain constraints is nothing but the problem to find the minimum of the Hilbert-Schmidt norm of the following function with a trial gauge potential [2]”

(If there is no constraint, then the trial gauge potential should be equal to the adiabatic gauge potential)

Shortcut to adiabaticity in the infinite-range Ising model by mean-field counter-diabaticdriving [3].―Infinite-range Ising model

Mean-field Hamiltonian

Mean-field counter-diabatic Hamiltonian

Suppose that the mean-field is given by

ResultsQuantum annealing processes.―Polynomial schedule going through the vicinity of the critical point smoothly

where and

Magnetization dynamics(a) without any assist (b) with the mean-field CD driving

Fidelity to adiabatic dynamics(a) system-size dependence (b) operation-time dependence

Comparing mean-field counter-diabatic driving with local counter-diabatic driving by the variational approach.―Suppose a trial gauge potential to be local

Function

Minimizing the Hilbert-Schmidt norm of with respect to and obtain

⇒ This approach cannot take time-dependence of interactions into account properly

Summary Introduced “shortcuts to adiabaticity by mean-field counter-diabatic driving” and

applied to the infinite-range Ising model Found the mean-field counter-diabatic Hamiltonian for the infinite-range Ising model

consists of only local operators Demonstrated that mean-field counter-diabatic driving can mimic quasi-adiabatic

dynamics via quantum annealing processes Confirmed that our approach takes higher order contributions into account than the

variational approach for local counter-diabatic driving

TH, arXiv:1705.03168.

Reference[1] (As a review article) Erik Torrontegui, et al., “Shortcuts to Adiabaticity”, Adv. At. Mol. Opt. Phys. 62, 117 (2013). [2] Dries Sels and Anatoli Polkovnikov, “Minimizing irreversible losses in quantum systems by local counterdiabatic driving”, PNAS 114, E3909 (2017). [3] Takuya Hatomura, “Shortcuts to adiabaticity in the infinite-range Ising model by mean-field counter-diabatic driving”, arXiv:1705.03168.

Exact CD driving Local CD driving MF CD driving

Adiabaticity ◎ △ ○

Applicability △ ◎ △

Locality × ◎ ◎

Critical point

Quantum annealing schedule