of 2
Supplementary material for “Al transmon qubits on silicon-on-insulator for
quantum device integration”
Andrew J. Keller,
1, 2
Paul B. Dieterle,
1, 2
Michael Fang,
1, 2
Brett Berger,
1, 2
Johannes M. Fink,
1, 2, 3
and Oskar
Painter
1, 2
1)
Kavli Nanoscience Institute and Thomas J. Watson Laboratory of Applied Physics,
California Institute of Technology, Pasadena, CA 91125 USA
2)
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena,
CA 91125 USA
3)
Institute for Science and Technology Austria, 3400 Klosterneuburg, Austria
(Dated: 24 June 2017)
I. MEASUREMENT SETUP
A Tektronix AWG5014C arbitrary waveform generator
(AWG) generates shaped in-phase (I) and quadrature (Q)
pulses at IF = 100 MHz for both qubit readout and XY
drive. Each output of the AWG passes through its own
home-made dissipative Gaussian filter with 320 MHz cut-
off. The waveforms are each amplified with a home-made
differential amplifier and passed to the I and Q ports of
IQ mixers (Marki IQ-0307MXP for the XY drive, IQ-
0409MXP for readout). Carrier tones are supplied by
CW microwave sources (Rohde & Schwarz SMB100A)
to the local oscillator (LO) ports of the mixers. As a
result, the readout and XY pulses are single-sideband-
upconverted to microwave frequencies. We attenuate
and filter these signals at several temperature stages of a
cryogen-free dilution refrigerator.
Flux biasing is provided by a programmable DC source
(Yokogawa GS200) which is filtered at 4 K (Therma-uD-
25G from Aivon Oy, Helsinki, Finland) and again at the
mixing chamber plate with a reflective microwave filter
(Minicircuits).
The DC and AC signals reach the device, which is
mounted on a gold-plated PCB inside a copper box inside
two concentric magnetic shields (Magnetic Shields Ltd.,
Staplehurst, UK) consisting of 1.5 mm thick Cryophy ma-
terial heat-treated to MSL1154-HTC specification. The
inner shield is 51mm ID by 168mm high and the outer
shield is 67mm ID by 185mm high. The copper box and
magnetic shields are mounted to a copper coldfinger at-
tached to the mixing chamber plate. A shield on the
mixing chamber is painted in an infrared-absorbing car-
bon/silica/epoxy mixture to minimize quasiparticle gen-
eration in the aluminum.
1,2
The output is protected from room-temperature noise
by two circulators (Raditek RADC-4-8-Cryo-0.02-4K-
S23-1WR-b) and at 4 K, a HEMT (Low Noise Factory
LNF-LNC4
8C) amplifies by 42 dB, with 68 dB further
amplification at room temperature. We used two room-
temperature power amplifiers, a Miteq AFS42-00101200-
22-10P-42 with 50 dB of gain and a home-made amplifier
with 18 dB of gain designed by the Martinis group. The
readout signal is then downconverted and the resulting
I and Q are simultaneously digitized using a 1GS/s 2-
channel PCIe digitizer (AlazarTech ATS9870). In soft-
ware, the I and Q are mixed with 100 MHz tones to yield
a single point in the I-Q plane for a single readout pulse.
The semirigid coaxial cable in our fridge is stainless-
stainless .085” above the 4 K plate and NbTi-NbTi .085”
below.
II. AL-ON-SI QUBIT
To demonstrate the compatibility of our SOI qubit pro-
cess with Si qubit processes, we fabricated and measured
an aluminum qubit on silicon by omitting the first and
last step of the SOI process depicted in Fig. 1(a). We use
float zone (FZ) grown, 525
μ
m thickness,
>
10 kΩ-cm
resistivity silicon and, for a device designed similarly to
that presented in the manuscript, measured parameters:
f
q
=
ω
q
/
2
π
= 4
.
962 GHz,
η/
2
π
=
260 MHz,
ω
r
/
2
π
=
6
.
868 GHz,
χ/
2
π
= 1
.
2 MHz,
E
J
/h
= 13
.
1 GHz,
g/
2
π
= 135 MHz,
Q
i
= 5
.
8
×
10
3
, and
Q
e
= 12
.
9
×
10
3
.
We note that
Q
i
is more then two-orders of magnitude
smaller than expected from previous resonator-only tests
we have performed on Si. Evidence of frequency jitter
in the read-out resonator of this sample was observed,
which may explain an under-estimate of the
Q
i
from the
swept frequency measurement used here.
Indeed, characterization of the Al-on-Si qubit (Fig. S1)
at the flux-insensitive point yields
T
1
= 27
μ
s and
T
2
= 6.6
μ
s, suggesting that the intrinsic loss is much
lower than the resonator measurement suggests. Using
the same estimate discussed in the manuscript, we es-
timate the Purcell-limited
T
1
= 18
.
5
μ
s. However, a
more conservative estimate which assumes we are in-
deed under-estimating
Q
i
due to frequency jitter takes
Q
Q
e
, and yields a Purcell-limited
T
1
of 57
μ
s. We
also performed randomized benchmarking on the Si qubit
(Fig. S2) using 40 random Clifford sequences and mea-
suring two gates (
X
π/
2
and
X
π
). We find the average
Clifford group gate fidelity
̄
f
(
C
) = 0
.
9952(5).
III. RANDOMIZED BENCHMARKING
In Clifford group randomized benchmarking proto-
cols,
3–5
a qubit initialized in its ground state has 2
N
gates performed on it—
N
random gates from the Clif-
2
12
(b)
(a)
(c)
Normalized
excited state
population
Duration
(μs)
Delay (μs)
0
0.0
0.5
1.0
1
0
0.0
0.5
Normalized excited
state population
Log of normalized
excited state population
Drive f (GHz)
τ
Delay (μs)
-3
-2
-1
0
60
0
5.01
4.91
τ
τ
τ
Read
τ
Read
Xπ/2 Xπ/2
τ
Read
FIG. S1. Al-on-Si qubit characterization. (a) Excited state
population (normalized to the unit interval) as a function of
XY drive frequency and pulse duration
τ
. (b) Natural log of
the excited state population as a function of waiting time
τ
yields
T
1
= 27
μ
s (points are data, blue trace is fit). Here
we used a 45 ns X
π
pulse. (c) Ramsey oscillations obtained
with a 30 ns X
π/
2
pulse yield
T
2
= 6.6
μ
s (points are data,
blue trace is fit). In (a–c) we use a rectangle-windowed 500 ns
readout pulse.
N
0
20
40
60
80
100
excited state population
(log
10
scale, arb. oset)
f( X
π/2
) = 0.996(2)
f( C) = 0.9952(5)
f( X
π
) = 0.9961(6)
FIG. S2. Gate fidelity (with an arbitrary offset given by the
readout fidelity) as a function of
N
using 30 ns pulses. Error
bars are 1 standard error in the measurements averaged over
40 random Clifford sequences. Uncertainties in the gate fideli-
ties represent 1 standard deviation of
f
due to the statistical
uncertainty of the parameter
p
in the exponential fit described
in the randomized benchmarking section of this supplement.
ford group (labeled
C
i
) interleaved with
N
of the same
Clifford gate (e.g.,
X
π
). After application of all 2
N
gates,
we perform the Clifford gate that puts the qubit in its
excited state (labeled
C
|
1
) and read out the qubit state.
The probability of being in the excited state as a function
of
N
, is then compared against the same procedure, but
without
N
of the same interleaved gate. This procedure
yields two exponentials of the form
c
1
+
c
2
p
N
, the latter
with depolarizing parameter ̄
p
and the former with depo-
larizing parameter
p
G
. These two parameters are related
to the average Clifford group gate fidelity,
̄
f
(
C
), and gate
fidelity of interest,
f
(
G
), through
̄
f
(
C
) = 1
1
̄
p
2
and
f
(
G
) = 1
1
p
G
/
̄
p
2
.
3
For the data presented in Fig. 4 and
Fig. S2, we performed 20000 repetitions of each random
Clifford group sequence.
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