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Fast quantum annealing for the infinite-range Ising modelby mean-field counterdiabatic driving
Takuya HatomuraDepartment of Physics, The University of Tokyo
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
