Optimal Controlled Phasegates for Trapped Neutral Atoms at the Quantum Speed Limit

Transcription

1 at the Quantum Speed Limit Michael Goerz Tommaso Calarco Christiane P. Koch Universität Kassel Universität Ulm DPG Spring Meeting Dresden March 6, 2

2 Universal Quantum Computing Controlled Phasegate e iχ Ô(χ) = CPHASE(χ) = Controlled-Not X X CNOT = H X O(π) X H CPHASE(π) equivalent to CNOT Universal Quantum Computing CPHASE is used in Quantum Fourier Transform

3 Two-Qubit Gates on Trapped Neutral Atoms Calcium: P a P 3 ω L = cm - S d x x 2 Low-Lying states in Alkaline-Earth atoms or Rydberg states Atoms in optical lattice or optical tweezers

4 The Objective Problem QC with atomic collisions: adiabaticity slow. Strong interaction fast gates? only if ignoring motion. Quantum Speed limit QSL: What is the maximum speed at which a quantum system can evolve? What limits on the gate duration can we find through optimization? How do gate durations depend on the interaction strength?

5 The Objective Problem QC with atomic collisions: adiabaticity slow. Strong interaction fast gates? only if ignoring motion. Quantum Speed limit QSL: What is the maximum speed at which a quantum system can evolve? What limits on the gate duration can we find through optimization? How do gate durations depend on the interaction strength? Approach Describe the system including the motional degree of freedom. Optimize for varying times / interaction strengths: I Two Calcium atoms at fixed distance (fixed interaction): vary T II For fixed T, two atoms with artificial dipole-dipole interaction V (R) = C 3 /R 3 : vary C 3

6 Two-Qubit-Hamiltonian, Optimization with Krotov

7 System Hamiltonian integrate out COM d x x 2 R = d

8 System Hamiltonian integrate out COM d x x 2 R = d cm - aa cm - a a cm cm cm - a a. cm -

9 Optimizing the Laser Pulse Target Functional J = [tr (Ô Û )] N Re }{{} F T + α S(t) ɛ2 (t) dt; Ô = CPHASE Û = e iĥ(ɛ(t))t ɛ Krotov: pulse update ɛ minimizing J ɛ Im Ψ bw ˆµ Ψ fw Palao, Kosloff, PRA 68, 6238 (23) Ô ɛ () ɛ () Ô Ô Ô t t T

10 Measures of Merit Fidelity F and cost functional J are not very informative. Control over the Motional Degree of Freedom F = (R) Û(T, ; 2 ɛopt ) (R) Does return to it s initial vibrational eigenstate? Gate Phases ( ) φ = arg (R) Û(T, ; ɛopt ) (R) What is the phase change relative to the initial state? True Two-Qubit Phase Cartan Decomposition leads to χ = φ φ φ + φ Concurrence (Entanglement) C = sin χ 2

11 For which gate durations can we reach a high-fidelity CPHASE?

12 Parameters of the Optimization Short internuclear distance sufficient interaction d = 5 nm Peak intensity ɛ ɛ to induce Rabi cycle T Pulse duration between T rad int =.23 ps and T v = 8 ps T rad int a π T v E d R

13 Optimization Success over Pulse Duration two-qubit phase vibrational purity fidelity Optimization time T [ps] For small T, vibrational purity is lost with increasing two-qubit phase High two-qubit phase and high vibrational only for long pulse durations

14 System Dynamics for 8 ps Pulse population time t (ps) ( 6 V/m) [τ] - - [τ ] F =.997 [τ] - - [τ ] amplitude (arb. units) frequency ω ( 4 cm - ) [τ] - - [τ ] [τ] - - [τ ] τ = (R) Û(T, ; ɛopt ) (R)

15 Two Atoms at Long Distance under Strong Dipole-Dipole Interaction Can we avoid vibration with very short pulses, but very strong interaction?

16 Parameters of the Optimization Fixed short pulse duration T = ps, T =.5 ps Realistic lattice spacing with strong interaction C 3 R 3 d = 2 nm Vary C 3 : C 3 = 6 Action over ps for Calcium at d = 5 nm, scaled to d = 2 nm Increase by three orders of magnitude Action over 8 ps for Calcium at d = 5 nm, scaled to d = 2 nm d a C 3 = 6. C 3 = 9

17 Optimization Success over Dipole Interaction Strength two-qubit phase vibrational purity fidelity e+6 e+7 e+8 e+9 e+ interaction strength C 3 [atomic units] Increasing two-qubit-phase with increasing interaction strength For small T, vibrational purity is lost with increasing two-qubit phase

18 Conclusions

19 Conclusions two-qubit phase.2 vibrational purity fidelity Optimization time T [ps] e+6 e+7 e+8 e+9 e+ interaction strength C 3 [atomic units] Long gate duration can reach arbitrarily high fidelities. For short gate durations, the two-qubit phase is at the expense of the vibrational purity. If T < QSL, not all measures of merit can be fulfilled. Time scale for a successful gate is determined by max (T int, T vib ).

20 Acknowledgements AG Koch Christiane Koch Daniel Reich Mamadou Ndong Ruzin Ağanoğlu Giulia Gualdi Anton Haase Martin Berglund Funding Financial support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged (Grant No. KO232)