Letter reSeArCH
show no oscillations regardless of the input state. The completely ran-
dom initial state, ρRR, shows no oscillations because the equally prob-
able ∣⟩↑↓ and ∣⟩↓↑ oscillations cancel each other out. As a result, we
observe near-flat outputs with evenly distributed populations. The
deviations from the expected measurement probabilities for the differ-
ent input states are attributed to initialization and readout errors.
On the basis of the four different initial-state measurements, we cal-
culate the truth table for the SWAP, SWAP and SWAP^2 (identity)
gates after removing initialization and readout errors (see Fig. 4g and
Supplementary Information section V). The truth table gives the input
and output of the gate for the four different spin basis states,
{∣⟩∣⟩∣⟩∣⟩↓↓,,,}↑↓ ↓↑ ↑↑. To quantitatively analyse the gate performance
we calculate the logical basis fidelity, Fzz (the gate fidelity in the z basis
of both qubits), using the experimental and theoretical truth
table matrices^26 E = (Eij) and T = (Tij). The logical basis fidelities
for the SWAP, SWAP and SWAP^2 gates are Fzz,S=±90% 3%,
Fzz,S = 79 % ± 3% and Fzz,S 2 =±69%2%, respectively. These fidelities
provide an upper bound on the overall two-qubit gate fidelity^26 , F ≤ Fzz,
which we estimate based on theoretical calculations to be
FS≈.86 7%±.02%. Using theoretical process tomography calcula-
tions we estimate that σε < 10 μeV should be sufficient to achieve a
SWAP fidelity greater than 99%, which we believe is achievableDetuning,H (mV)Magnetic eld (T)–0.2 –0.1 00 .1 0.2ProbabilityDetuning, H (mV)Step 1234LRLoad
|S〉5 msH = 02,41,53(1,3) (2,3)(1,2) (2,2)H < 0H > 0
5LRc
f
ab
d e
5 ms721,63,54(1,3) (2,3)(1,2) (2,2)Load
↓Load
randomStep 1234567–3 –2 –100.50.10.20.30.40.6–4 0JH 0 mVH 0 mV–5ΔEZ/h = 200 MHz–8–40–6–22EZ = ΔEZ^2 + J(H,tc)^2Read
|S〉, |T〉Read
↓,↑↑↑
↑↓
↓↑
↓↓Read
↓,↑tc/h = 4.3 ± 0.4 GHzFig. 2 | Electrostatic control over the electron-exchange interaction.
a, b, Spin funnel measurement protocol. The voltage pulse (rise time of
about 10 μs) along the detuning axis (dashed red line) is followed by a
readout phase that allows us to distinguish between the singlet and triplet
states^18. c, Spin funnel measurement. The anticrossing point between
S(1,3) and T− is mapped as a function of detuning ε and magnetic field.
By fitting the shape of the spin funnel to an equation of the form
EEZZ=Δ^2 +Jt(,ε c)^2 we obtain the difference in Zeeman energy between
two electrons, ΔEZ/h = 0.2 ± 0.1 GHz, and the tunnel coupling,
tc/h = 4.3 ± 0.4 GHz. d, e, Two-spin correlation measurement protocol. A
random spin is loaded onto L while R is deterministically prepared as spin-
down. This preparation step is then followed by a 5-ms exchange pulse
(rise time of about 10 μs) at a given detuning voltage ε, which is varied in
this experiment. Finally, the spins on both quantum dots are measured.
f, Measured two-spin probabilities of the left and right quantum dots. As
the detuning pulse approaches ε = 0, P↓↑ and P↑↓ become equal owing to
the increasing exchange interaction between the S(1,3) and T 0 states. The
∣⟩↓↓ and ∣⟩↑↑ probabilities remain constant, as the T+ and T− states are
not affected by the exchange interaction because Zeeman splitting is much
larger than the exchange energy for all ε ≲ 0 mV (B = 2.5 T). The error bars
represent uncertainty of one standard deviation in the measured values.
10 15 20 25
Time, W (ns)54 3 2 1 0 0T 2 SWAP ≈ 4 nsT 2 SWAP ≈ 5 nsT 2 SWAP ≈ 6 nsT 2 SWAP ≈ 14 nsΩ/2π ≈ 184 MHzΩ/2π ≈ 580 MHzΩ/2π ≈ 130 MHzΩ/2π ≈ 300 MHzT–T 0S(1,3)
S(2,2)H = 0 T+~tc hΩ^ ≈^ JDetuning, H
dca b
2,7163,54(1,3) (2,3)(1,2) (2,2)LR
WStep 1234567EnergyJProbability,P
↑↓H 0 mV5630ΔEZEZ =JeB
2 πLoad
randomLoad
↓Read
↓,↑
Read
↓,↑↓↓↓↑↑↓↑↑Fig. 3 | Exchange-driven coherent spin–spin oscillations. a, b, Pulse
sequence employed to observe the coherent exchange oscillations. The
sequence is similar to that shown in Fig. 2d, e, but the detuning position is
fixed at various values (shown by the circles in c) and the duration of the
exchange pulse is varied up to 30 ns with a rise time of about 100 ps.
c, Energy diagram showing the evolution of singlet and triplet levels as a
function of detuning. d, Measured probabilities of the ∣⟩↑↓ spin state; error
bars (1 s.d.) are smaller than the marker size. The four datasets were taken
at different detuning voltages, marked with blue circles in c (offset by 1.5
for clarity). The T 2 SWAP times were determined from fits of the data (solid
lines) to the results of the charge noise model (see Supplementary
Information section III). We note the deviations from theory around 14 ns
(Ω/2π ≈ 184 MHz) and 20 ns (Ω/2π ≈ 130 MHz), which can be explained
by charge noise coupling to the SET charge sensor shifting the optimal
readout position.18 JULY 2019 | VOL 571 | NAtUre | 373