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

Transcription

1 with Ultracold Trapped Atoms at the Quantum Speed Limit Michael Goerz May 31, 2011

2 with Ultracold Trapped Atoms Prologue:

3 with Ultracold Trapped Atoms Classical Computing: 4-Bit Full Adder Inside the CPU: Bits: 0 ˆ= low voltage 1 ˆ= high voltage from: Calculations: logical functions of bits mapped to electronic gates Gates combine to more complex gates Gates can be decomposed into NAND-gates

4 with Ultracold Trapped Atoms A Single Qubit Definition of a Single Qubit with Ψ 1q = α α 1 1 α α 1 2 = 1 Vector Representation ( 1 0 = 0) ( 0 1 = 1) ( ) α0 Ψ 1q = α 1

5 with Ultracold Trapped Atoms Two Qubits Definition of a Two-Qubit System with Ψ 2q = α α α α In general, Ψ 2q can be entangled, i.e. it cannot be written as a product state ( ) ( ) α (1) α (1) 1 1 α (2) α (2) 1 1 Vector Representation α = = = = 0 0 Ψ 1q = α 10 α α 11

6 with Ultracold Trapped Atoms One and Two Qubit Gates 1 Qubit Gate: Hadamard 2 Qubit Gate: CNOT Ψ 1q,i H Ψ 1q,t Ψ 2q,i CNOT Ψ 2q,t Ψ 1q,i H Ψ 1q,t Ψ 2q,i Ψ 2q,t ( ) Ψ q,i = Ψ 1q,t Ψ 2q,i = Ψ 2q,t

7 with Ultracold Trapped Atoms Quantum Circuits Inside the CPU: Bits: 0 ˆ= low voltage 1 ˆ= high voltage from: Calculations: logical functions of bits mapped to electronic gates Gates combine to more complex gates Gates can be decomposed into NAND-gates

8 with Ultracold Trapped Atoms Quantum Circuits : Qubits: Eigenstates 0, 1 Superposition states Ψ = α α 1 1 from: Calculations: logical functions of bits mapped to electronic gates Gates combine to more complex gates Gates can be decomposed into NAND-gates

9 with Ultracold Trapped Atoms Quantum Circuits : Qubits: Eigenstates 0, 1 Superposition states Ψ = α α 1 1 Calculations: unitary transformations of qubits mapped to quantum gates from:

10 with Ultracold Trapped Atoms Quantum Circuits H H H H H H H H H H H H : Qubits: Eigenstates 0, 1 Superposition states Ψ = α α 1 1 X Z Z X Z Z X Z Z Z X X Z X X Z X X X Z X X Z Z Figure Quantum circuit for measuring the generators of the Steane code, to give the error syndrome. Th top six qubits are the ancilla used for the measurement, and the bottom seven are the code qubits. from: Nielsen, Chuang: Quantum Information and Calculations: unitary transformations of qubits mapped to quantum gates Gates combine to more complex gates Gates can be decomposed into single-qubit and CNOT

11 with Ultracold Trapped Atoms Quantum Circuits H H H H H H H H H H H H X Z Z X Z Z X Z Z Z X X Z X X Z X X X Z X X Z Z Figure Quantum circuit for measuring the generators of the Steane code, to give the error syndrome. Th top six qubits are the ancilla used for the measurement, and the bottom seven are the code qubits. from: Nielsen, Chuang: Quantum Information and A few explicit points: Universal Gate Theorem: only single-qubit gates and (two-qubit) CNOT. Restrictions on quantum circuit due to unitarity Power of quantum computing: Quantum Parallelism But: complex wavefunctions cannot be measured Clever algorithms like Shor-algorithm for prime decompositions General problem: Decoherence

12 with Ultracold Trapped Atoms with Ultracold Trapped Atoms Implement a Controlled Phasegate on Calcium Atoms

13 with Ultracold Trapped Atoms The Controlled Phasegate 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

14 with Ultracold Trapped Atoms Calcium Term Scheme Qubit Encoding 10 5 cm d 6p 6p 5d 6s 4p s 5p 4d 4d 4p 5p 3.5 5s 5s p 4s 4s 1.5 4p s2 1 S 1 P 1 D 3 S 3 P 3 D

15 with Ultracold Trapped Atoms Calcium Term Scheme Qubit Encoding 10 5 cm d 6p 6p 5d 6s 4p s 5p 4d 4d 4p 5p 3.5 5s 5s aux 4s 4s 1.5 ω L = 23652cm S 1 P 1 D 3 S 3 P 3 D

16 with Ultracold Trapped Atoms Two-Qubit Gates on Trapped Neutral Atoms Calcium: 1 P 1 a 1 P 3 1 ω L = cm -1 1 S 0 0 d x 1 x 2 Low-Lying states in Alkaline-Earth atoms or Rydberg states Atoms in optical lattice or optical tweezers

17 with Ultracold Trapped Atoms 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?

18 with Ultracold Trapped Atoms The Objective Problem QC with atomic collisions: adiabaticity slow. Strong interaction fast gates? only if ignoring motion. Quantum Speed limit Outline 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? 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 arxiv:

19 with Ultracold Trapped Atoms Two-Qubit-Hamiltonian, Optimization with Krotov

20 with Ultracold Trapped Atoms System Hamiltonian integrate out COM d x 1 x 2 R 0 = d

21 with Ultracold Trapped Atoms System Hamiltonian integrate out COM d x 1 x 2 R 0 = d ) Ĥ = (Ĥ1q 1 1q + 1 1q Ĥ 1q 1 R + 1 1q 1 1q Ĥ trap + Ĥ int = ik ik [ˆT + ˆV trap(r) + ˆV ik ] BO (R) + Ê ik + i,k +ɛ(t) [ ] ik jk + ki kj ˆµij i j,k

22 with Ultracold Trapped Atoms System Hamiltonian integrate out COM d x 1 x 2 R 0 = d cm -1 aa cm -1 a1 1a cm cm cm -1 0a a cm -1 00

23 with Ultracold Trapped Atoms The Logical Subspace Full System Hamiltonian Ĥ = (Ĥ1q 1 1q + 1 1q Ĥ 1q ) 1 R + 1 1q 1 1q Ĥ trap + Ĥ int Dimension of Ĥ: 3 3 N R Dimension of Ô: 4 How does that work...?

24 with Ultracold Trapped Atoms The Logical Subspace Full System Hamiltonian Ĥ = (Ĥ1q 1 1q + 1 1q Ĥ 1q ) 1 R + 1 1q 1 1q Ĥ trap + Ĥ int Dimension of Ĥ: 3 3 N R Dimension of Ô: 4 How does that work...? 4 initial states: ijϕ 0 = ij ϕ 0, i, j = 0, 1 with ϕ 0 (R) the vibrational ground state of the harmonic trap. After pulse: projection onto logical subspace There should be no population left in the auxiliary electronic states The vibrational state after the pulse should again be ϕ 0(R) (up to a phase factor)

25 with Ultracold Trapped Atoms The Logical Subspace Full System Hamiltonian Ĥ = (Ĥ1q 1 1q + 1 1q Ĥ 1q ) 1 R + 1 1q 1 1q Ĥ trap + Ĥ int Dimension of Ĥ: 3 3 N R Dimension of Ô: 4 How does that work...? 4 initial states: ijϕ 0 = ij ϕ 0, i, j = 0, 1 with ϕ 0 (R) the vibrational ground state of the harmonic trap. After pulse: projection onto logical subspace There should be no population left in the auxiliary electronic states The vibrational state after the pulse should again be ϕ 0(R) (up to a phase factor) General concept! Having a logical subspace in a large Hilbert space of the physical system is quite common in implementations of quantum computation.

26 with Ultracold Trapped Atoms Optimal Control Generally: we have some knobs that we can turn to influence the dynamics of a system, and we want find the optimal way to turn them to reach a desired outcome. E.g. Curling: the goal: bring the stone as close as possible to the target at time T Static control : speed, direction, and spin of thrown rock Dynamic control (at every point in time): sweeping where to sweep how hard to sweep take into account physical constraints: boundaries of the playing field, sweeping speed and strength of players

27 with Ultracold Trapped Atoms Optimal Control Generally: we have some knobs that we can turn to influence the dynamics of a system, and we want find the optimal way to turn them to reach a desired outcome. In Quantum Mechanics: Drive a quantum state from an initial to a target state (or unitary transformation) System dynamics given by Hamiltonian Control: some parameter in the Hamiltonian; in our case: amplitude of laser pulse over time. Take into account constraints, e.g. finite pulse amplitude iterative optimization algorithms

28 with Ultracold Trapped Atoms Optimizing the Laser Pulse Target Functional J = 1 [tr (Ô Û )] N Re }{{} F T + 0 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, (2003) 11 Ô ɛ (1) ɛ (0) Ô Ô Ô 00 t 0 t T

29 with Ultracold Trapped Atoms The Krotov Algorithm ɛ 1 ɛ 2 ɛ nt 2 ɛ nt 1 Ψ bw (t 0) Ψ i Ψ bw (t) Ψ fw (t) Ψ bw (t) Ψ fw (t) Ψ bw (t) ˆµ ˆµ.. ˆµ. ˆµ = τ Ψ fw (t) Ψ t Ψ t Ψ fw (T ) fw (T ɛ 1 ɛ 2 ɛ nt 2 ɛ nt 1 Propagate target state backward with guess pulse Calculate pulse update Propagate forward with updated pulse

30 with Ultracold Trapped Atoms Measures of Merit Fidelity F and cost functional J are not very informative. Control over the Motional Degree of Freedom F 00 = 00ϕ 0 Û(T, 0; ɛ opt 2 ) 00ϕ 0 Does 00 return to it s initial vibrational eigenstate? Gate Phases ( ) φ 00 = arg 00ϕ 0 Û(T, 0; ɛ opt ) 00ϕ 0 ) What is the phase change relative to the initial state?

31 with Ultracold Trapped Atoms Cartan Decomposition Local Two-Qubit Gate X (( ) ) ( ( )) = X Distinguish local two-qubit gate from non-local gate like CNOT, that cannot be decomposed this way! (cf. product states vs entangled states) Cartan Decomposition Zhang et al. PRA 67, (2003) Û = ˆk 1 ˆk 2 ˆk 1, ˆk 2 : local operations; Â: purely non-local operation Only  has entangling power Cartan decomposition defines equivalence class of two-qubit gates ( Locally equivalent )

32 with Ultracold Trapped Atoms Measures of Merit Fidelity F and cost functional J are not very informative. Control over the Motional Degree of Freedom F 00 = 00ϕ 0 Û(T, 0; ɛ opt 2 ) 00ϕ 0 Does 00 return to it s initial vibrational eigenstate? Gate Phases ( ) φ 00 = arg 00ϕ 0 Û(T, 0; ɛ opt ) 00ϕ 0 ) What is the phase change relative to the initial state? True Two-Qubit Phase Cartan Decomposition leads to χ = φ 00 φ 01 φ 10 + φ 11 Concurrence (Entanglement) C = sin χ 2

33 with Ultracold Trapped Atoms For which gate durations can we reach a high-fidelity CPHASE?

34 with Ultracold Trapped Atoms Parameters of the Optimization Short internuclear distance sufficient interaction d = 5 nm Peak intensity ɛ 0 ɛ 0 to induce 1 Rabi cycle 0 T Pulse duration between T 1 rad int = 1.23 ps and T v = 800 ps 1 T 1 rad int 0a 0 π T v 00 E d 00 R

35 with Ultracold Trapped Atoms Optimization Success over Pulse Duration (F, χ,f 00 ) pulse duration T (ps) F χ F For small T, vibrational purity is lost with increasing two-qubit phase High two-qubit phase and high vibrational only for long pulse durations

36 with Ultracold Trapped Atoms System Dynamics for 800 ps Pulse population time t (ps) 1 (10 5 V/m) [τ00] [τ00] 1 F = [τ01] [τ01] amplitude (arb. units) frequency ω (10 4 cm -1 ) [τ0] [τ0] [τ1] [τ1] 00ϕ 0 FIG. 9: Phase dynamics induced by the optimized pulse (T = 800ps) τ 00 in = the complex 00ϕ 0 plane Û(T for, the 0; two-qubit ɛ opt ) single-qubit states, analogously to Fig. 6. excited state that sets the speed limit for two atoms resonantly Optimal excited Controlled to an Phasegates interacting for Trapped state, we Neutral vary the Atoms in- FIG. 10: (c vibrational fi state, F00 fo state for tw converged t d =200nm. fidelity imp ing the C3 c whether pi can be ach strong, i.e. large. Bas local phase

37 with Ultracold Trapped Atoms The Reduced Optimization Scheme full reduced target 00 e i(φ+φ T ) e iφ T e iφ T e iφ T e i(φ+φ T ) 00 0 e iφ T /2 0 gate phases φ 00 φ 10 = φ 01 φ 11 = φ 00 = φ 0 + φ 1 = 2φ 1 non-local phase χ = φ 00 φ 01 φ 10 + φ 11 χ = φ 00 2φ 0

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

39 with Ultracold Trapped Atoms Parameters of the Optimization Fixed short pulse duration T = 1 ps, T = 0.5 ps Realistic lattice spacing with strong interaction C 3 R 3 d = 200 nm Vary C 3 : C 3 = Action over 1 ps for Calcium at d = 5 nm, scaled to d = 200 nm Increase by three orders of magnitude Action over 800 ps for Calcium at d = 5 nm, scaled to d = 200 nm d 0a C 3 = C 3 =

40 with Ultracold Trapped Atoms Optimization Success over Dipole Interaction Strength F (1 ps) χ (1 ps) F 00 (1 ps) (0.5 ps) (0.5 ps) (0.5 ps) interaction strength C 3 (atomic units) FIG. Increasing 10: (color two-qubit-phase online) Fidelity with increasing F, non-local interaction phase strength χ and vibrational For smallfidelity, T, vibrational i.e. projection purity is lost onto with the increasing vibrational two-qubit target phase state, F 00 for increasing interaction strength C 3 in the excited

41 with Ultracold Trapped Atoms Conclusions

42 with Ultracold Trapped Atoms Conclusions [τ0] (F, χ,f 00 ) pulse duration T (ps) F χ F [τ1] 0 0 FIG. 10: (color online) Fidelity F, non-local phase χ and Long gate duration can reach arbitrarily vibrational highfidelity, fidelities. i.e. projection onto the vibrational target state, F00 for increasing interaction strength C3 in the excited For short gate durations, the two-qubit statephase for two is different at the gateexpense times T. All ofoptimizations the vibrational have -1-1 converged to F < The interatomic distance is -1 purity d =200nm. [τ 0] [τ If T < QSL, not all measures 1] of merit can be fulfilled F (1 ps) χ (1 ps) F00 (1 ps) (0.5 ps) (0.5 ps) (0.5 ps) interaction strength C3 (atomic units) FIG. 9: Phase Time dynamics scale induced forby a the successful optimized pulse gate (T = is determined fidelity implementations by max (Twith int optimal, T vib ). control by increasing 800ps) in the complex plane for the two-qubit single-qubit states, analogously to Fig. 6. the C 3 coefficient. In particular, we pose the question whether picosecond and sub-picosecond gate durations can be achieved given that the interaction is sufficiently strong, i.e. given that the C 3 coefficient is sufficiently excited state that sets the speed limit for two atoms resonantly large. Based on the results of Sec. III A where a non- excited to an interacting state, we vary the in- local Optimal phase of Controlled π was achieved Phasegates within for Trapped 50 ps, Neutral we estimate Atoms

43 with Ultracold Trapped Atoms Thanks!