Robust Diabatic Grover Search by Landau–Zener–Stückelberg Oscillations

Abstract

Quantum computation by the adiabatic theorem requires a slowly varying Hamiltonian with respect to the spectral gap. We show that the Landau-Zener-Stückelberg oscillation phenomenon, that naturally occurs in quantum two level systems under non-adiabatic periodic drive, can be exploited to find the ground state of an N dimensional Grover Hamiltonian. The total runtime of this method is O( √ 2 n ) which is equal to the computational time of the Grover algorithm in the quantum circuit model. An additional periodic drive can suppress a large subset of Hamiltonian control errors using coherent destruction of tunneling, providing superior performance compared to standard algorithms.Adiabatic Quantum Computation (AQC) [1, 2] is a computational model, motivated by the physical phenomenon described by the adiabatic theorem, which states that if a system is prepared in the ground state of an initial Hamiltonian, and the Hamiltonian slowly varies in time, then it is guaranteed that the evolution will be adiabatic -meaning that the system will remain close to its instantaneous ground state throughout [3,4]. By encoding a solution for a computational problem in the ground state of the finally applied Hamiltonian, one can exploit this phenomenon to produce the aforementioned ground state, and thus produce a solution to the problem. The maximal rate of change allowed for such evolution usually scales with the inverse square of the energy gap between the ground state and the first excited state [1].The Grover problem [5], also known as The Unstructured Search Problem is one of the few problems solvable by a native adiabatic algorithm, which achieves the same performance as the best possible algorithm in the circuit model [6] (for other native algorithms see [7] and the partially adiabatic [8]). The input to the problem is an n qubit Hamiltonian, which can only be used as a black box, i.e., can be switched on or off [9]where I N is the N × N identity matrix with N = 2 n , and the problem is to find the unknown string y. The problem is comparable to finding the ground state of a known multiple-qubit Hamiltonian; the ground state might be computationally hard to find and therefore can be considered "computationally unknown" [10]. An adiabatic algorithm for the search problem was suggested by [1]. The system is initialized to a symmetric superposition of states denoted |u = |+ · · · + , and then evolves by the time-dependent Hamiltonian H G (s(t)) =(1 − s(t)) · (I N − |u u|)where the control function s(t) : [t i , t f ] → [0, 1] is initialized to 0 and increases monotonically with time to 1. The minimal gap for n qubit systems is ∆ = √ 2 −n . Evolving with a linear s(t) requires O(2 n ) time, while a specially tailored control function, whose rate matches the instantaneous spectral gap, generates the ground state of H p in the optimal time, O( √ 2 n ) [11,12]. In this work, we introduce a diabatic algorithm for the Grover problem, denoted algorithm A, whose performance matches both the optimized adiabatic and the circuit model algor...

Quantum Control via Landau-Zener-Stückelberg-Majorana Transitions

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Reservoir-engineering shortcuts to adiabaticity

“…The dynamics here predicted can be verified in several setups, some prominent examples are a single trapped ion [29], a single trapped atom in a resonator [9,30], and a superconducting qubit in circuit QED [6]. The QND type of Hamiltonian here discussed can be realised by means of the protocol of Ref.…”

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“…Strategies for implementing relatively fast and efficient adiabatic transformations are being actively investigated. These include the application of optimal control theory [4][5][6] and the active use of projective measurements [7]. The ultimate goal is to identify a general concept which can allow one to arbitrarily reduce the error of protocols based on quantum adiabatic dynamics.…”

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