General error estimate for adiabatic quantum computing

Abstract

Most investigations devoted to the conditions for adiabatic quantum computing are based on the first-order correction Ψ ground (t)|Ḣ(t)|Ψ excited (t) /∆E 2 (t) ≪ 1. However, it is demonstrated that this first-order correction does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections as well and shows that the computational error can be made exponentially small -which facilitates significantly shorter evolution times than the above first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time T of order of the inverse minimum energy gap ∆Emin is sufficient and necessary, i.e., T = O(∆E −1 min ). For some examples, these analytical investigations are confirmed by numerical simulations.

Preservation of positivity by dynamical coarse graining

“…(24) one can deduce that (unlike in the pure dephasing case) the Lamb-shift term will contribute, since (although diagonal) it is not proportional to the identity matrix anymore. Likewise, we also observe here a decoupled evolution of diagonal ρ τ S00 (t) = γ 01,01 γ 01,01 + γ 10, 10 1 − e −(γ01,01+γ10,10)t +e −(γ01,01+γ10,10)t ρ τ S00 (0)…”

“…[9,10]. If a|ċ ≈ 0, one obtains with the adiabatic approximation u ad ab (t) = δ ab e iγa(t) which implies for the time evolution operator in the adiabatic limit …”

“…Consequently, the maximum transformation rate (where the final excitations are acceptably small) corresponds to the computational complexity of the adiabatic algorithm. For closed systems, it is related to the spectral properties of the time-dependent system Hamiltonian [9,10]. For a reservoir at sufficiently low temperatures, this scheme is thought to be robust against decoherence [6] and might even be aided by it [11].…”

Fast adiabatic qubit gates using onlyσzcontrol

“…The general problem of achieving fast adiabatic performance is of great interest to the physics community [6][7][8][9], with applications in coherent manipulation, precision measurements, and quantum computing. Finding an improved control methodology therefore has potential for applications in a variety of quantum systems.…”

Necessary condition for the quantum adiabatic approximation

“…It is important to remark that some familiar adiabatic approximations are known to be insufficient [9] and sometimes unnecessary [10,11]. The growing interest in the adiabatic approximation has spurred work on corresponding rigorous conditions [10,12].…”