Articulación-Fraseo

Due to intermediate population of other Rydberg states, the ideal unitary after the sequence in FIG.7.2a will not equal the CZ gate that was discussed in the beginning of this chapter. Instead, the ideal evolution is given by

ˆ

U_CZ,~φ= diag(eiφ00_{, e}iφ01_{, e}iφ10_{, e}iφ11_{), (7.8)}

where φij ≡ φij,ij is a shorthand notation for phases on the diagonal elements. To turn the CZ-like gate in equation (7.8) into an entangling CNOT-like gate we follow the procedure that turns a common CZ into a CNOT gate: Applying a Hadamard gate on the target qubit before and after the CZ operation recovers a CNOT. Similarly for our case, we find that, instead of a Hadamard gate, a general π/2 rotation

ˆ R(~h) = √1 2  eih00 _eih01 eih10 _eih11  (7.9) with phases ~h = (h00, h01, h10, h11) can be used to turn, up to relative phases, the CZ-like gate (7.8) into a CNOT. If the corresponding entangling phase φent = φ00− φ01− φ10+ φ11 equals exactly π, the transformation

 ˆ 1⊗ ˆR(π, ˜φ,_{− ˜}φ, 0)UˆCZ,φ  ˆ 1⊗ ˆR(0, 0, 0, π) (7.10)

75 7.3. GATE ANALYSIS 30 40 50 60 70 80 90 100 110 10−5 10−4 10−3 10−2 10−1 100 Gate time tg (ns) B ell st a te in fi d elit y 1 − FB S1 – DRAG (d/r) S1 – DRAG (d) S1 – DRAG S1 – Gaussian S1 – Square S2 – DRAG (d/r)

FIG. 7.5 – Unitary Bell state infidelity as a measure for entanglement gener- ated by the pulse sequence in FIG.7.2using square pulses, Gaussians, DRAG, detuned (d) DRAG controls and detuned DRAG controls with rescaled (r) amplitude for setting S1, as well as optimal DRAG controls in S2. Detuning DRAG pulses on the target atom accounts for wrong phases and combines less leakage with high degrees of entanglement. The necessary detuning Λ decreases proportionally to 1/τ2

t with a value of Λ/2π = 124.07 MHz at τt= 25 ns.

with ˜φ = φ10−φ11produces a maximally entangling CNOT-like gate. In order to quantify the degree of entanglement of our pulse sequence, we pick (|00i + |10i)/√2 as an initial state. Ideally, under evolution (7.10) this yields, up to local phases, the maximally entangled Bell state_|Φ+i = (|00i + |11i)/

√ 2. To quantify the performance we evaluate the state overlap fidelity (4.2) between two density matrices ρ and ρt according to

F2 =  TrQ q_√ ρρid√ρ 2 . (4.2)

Here, we partially trace over the computational subspace Q, spanned by the computational basis states_{{|00i , |01i , |10i , |11i}, to disregard irrelevant} information about non-computational states.

For the specific target state ρt=|Φ+ihΦ+| we refer to the fidelity as Bell state fidelity FB. The corresponding results are depicted in FIG.7.5, whereby we assume that the π/2 gates on the qubit subspace are perfect. We observe that Gaussian controls seem to achieve better results than a naive DRAG

CHAPTER 7. ENTANGLING RYDBERG ATOMS 76

control. However, the main reason for DRAG pulses to perform poorly at a first glance is wrong phases. In the first publication of DRAG, a real-time control of the carrier frequency ωd accounts for such phase errors [40].

However, it is also possible to employ a constant detuning Λ from res- onance, i.e. ωd = ωr − ωq + Λ [195], with the benefit of less experimental effort being required. We find that detuning only the 2π rotation of the target atom is sufficient to achieve low enough errors. As a consequence of off-resonant driving, rotation errors will be induced which can be compen- sated by slightly rescaling the pulse amplitudes (up to at most 3% for the fastest gates). The difference between the solid black and the dotted red lines in FIG. 7.5 demonstrates that the combination of constant detuning and a rescaled amplitude indeed account for both errors, yielding at least two orders of magnitude improvement over Gaussian waveforms. As one would expect from previous results [40], the optimal detuning is proportional to the Rabi frequency squared, yielding approximately a 1/τ2

t power law. In order to obtain the optimal detuning Λ as a function of gate time we ran a single-parameter optimization using the Nelder-Mead algorithm described in section4.2.1. The functional dependence of the optimal detuning is illustrated in FIG.7.6. We find that we are able to produce Bell states with a fidelity of 0.9999 for tg ∼ 50 ns using detuned DRAG pulses with amplitude correction.

An alternate approach to account for phase issues is by waiting an ap- propriate time tgap between the pulses [89], or to track phases in software and correct for them afterwards. The former approach will noticeably prolong the gate times compared to our approach. Overall, detuned DRAG pulses yield an improvement of more than two orders of magnitude compared to conventional shapes. Furthermore, the necessary gate times are less than 10−7 of the few second coherence times that have been demonstrated with neutral atom qubits [86], substantiating that Rydberg gates are a promising approach for scalable quantum computing.