of 26
nature physics
https://doi.org/10.1038/s41567-024-02408-0
Artic�e
Deterministic generation of
multidimensional photonic cluster states
with a single quantum emitter
In the format provided by the
authors and unedited
Supplementary Information for:
“Deterministic Generation of Multidimensional Photonic Cluster States with a Single
Quantum Emitter”
CONTENTS
I. Measurement Setup
1
A. Device fabrication
1
B. Measurement Setup
1
II. Device Characterization
3
A. Slow-light Waveguide
3
B. Qubits
5
C. Purcell Filter
6
III. Flux Control for Shaped Photon Emission and
Qubit-Photon CZ Gate
8
A. Distortion Pre-Compensation of Square Flux
Pulses
8
B. Photonic Pulse Shaping
9
IV. Radiation Field Quantum State Tomography
11
A. Measurement of
a
out
(
t
)
and
a
out
(
t
)
a
out
(
t
)
11
B. Absolute Power Calibration
12
C. Measuring Expectation Values of Radiation
Field Moments
13
D. MLE
14
E. Quantum State Tomography Results
15
V. Process Tomography of the Time-Delayed
Feedback Operation
18
A. QPT Experiment Design
18
B. MLE
19
C. Generalized state-preparation superoperator 20
D. Readout Error Correction
20
E. State Fidelity Confidence Intervals
21
VI. Sources of Infidelity and Future Improvements
21
A. Infidelity Analysis
21
B. Scaling the Size of the Cluster State
22
I. MEASUREMENT SETUP
A. Device fabrication
The device used in this work is fabricated on a 2 cm
×
1
cm high-resistivity (10 kΩ) silicon substrate. Fabrication
details follow closely those of references [1, 2]. A resist
layer is deposited on the silicon substrate, and the ground
plane, waveguides, resonator and qubit capacitors are
patterned on this resist using electron-beam lithography.
This is followed by deposition of 120 nm of aluminum
(Al) at a rate of 1 nm/s using an electron-beam evapo-
rator. The device is left in N-methyl-2-pyrrolidinone at
80
C for 2.5 hours to perform liftoff of the resist layer.
The junctions are fabricated by double-angle evapora-
tion of a suspended Dolan bridge patterned using electron
beam lithography. A 60 nm Al evaporation at 1 nm/s
constitutes the first junction layer, and a 120 nm Al evap-
oration constitutes the second, between which there is an
intervening 20 minute oxidation period when 10 mBar of
oxygen gas is admitted to the chamber of the electron
beam evaporator to grow the insulating aluminum oxide
layer of the junction. After deposition of the two junction
layers, electron beam lithography is again used to pat-
tern a ‘bandage’ layer comprising regions of aluminum
that galvanically connect the bottom and top layers of
the junctions. Before deposition of the Al bandages, the
device is Argon milled for 5 minutes to remove surface
oxides and facilitate galvanic connection of the top and
bottom junction layers to the bandage.
Air bridges connecting the ground plane on either
side of our coplanar waveguides are patterned using a
grayscale electron-beam lithography technique. After de-
velopment of the airbridges, resist is made to reflow by
placing the device on a hot plate at 105
C, after which
140 nm of Al is deposited at 1 nm/s on the developed re-
sist. Prior to the Al deposition, the device is Argon milled
for 5 minutes to remove surface oxide of the ground plane
to facilitate galvanic connection between it and the feet
of the air bridges.
B. Measurement Setup
A schematic of the fridge wiring and our room-
temperature analog signal processing electronics is shown
in Fig. S1. Measurements are performed in a 3He/4He
dry dilution refrigerator. The sample is wirebonded to a
CPW printed circuit board (PCB) with coaxial connec-
tors, and is housed inside a copper box that is mounted
to the MXC plate of the fridge with
T
MXC
= 7 mK.
A coil is placed on top of the copper box for static flux
tuning of the qubits, and the sample is enclosed in two
layers of magnetic shielding to suppress effects of stray
magnetic fields. See refs [1–3] for more details on device
fabrication.
Attenuators are placed at several temperature stages of
the fridge to provide thermalization of the coaxial input
lines and to reduce thermal microwave noise at the in-
put to the sample. Our gigahertz microwave lines (XY
E
,
XY
M
, Readout IN, SLWG IN, TWPA Pump) have sig-
nificantly more attenuation than our fast flux lines (Z
E
,
Z
M
) for reasons explained in ref. [4]. In addition, fast flux
2
SLWG
IN
20 dB
300 K
50 K plate
4 K plate
XY
E, M
Z
E, M
Readout
IN
HEMT
OUT
Cold plate
TWPA
Pump
10 dB
20 dB
20 dB
20 dB
20 dB
20 dB
20 dB
TWPA
20 dB
20 dB
20 dB
6 dB
MXC plate
Coil
80
kHz
Device
PFWG
SLWG
+ –
Attenuator
DC block
Circulator
16 dB
dir. coupler
LP Filter
(0.85, 8 GHz)
Eccosorb
50 Ω
terminator
XY
E
OUT
Q
I
LO1
ADC
PHOTON
ADC
RO
LP Filter
160 MHz
Readout
IN
Q
I
LO2
BP Filter
600-900 MHz
TWPA Pump
Notch Filter
IQ Mixer
Room Temp.
Amplier
a
b
FIG. S1.
Measurement Setup a
Schematic of the mea-
surement wiring inside the dilution refrigerator. See Supple-
mentary text for further details (“dir.” is shorthand for “di-
rectional”, “LP” is shorthand for “Low Pass”, and “BP” is
shorthand for “Band Pass”.). See Fig. 1 for electrical connec-
tions at the sample.
b
Simplified diagram of measurement
wiring outside the dilution refrigerator.
lines are filtered by an 850 MHz low-pass filter below the
MXC plate, which suppresses thermal noise photons at
higher frequencies while still maintaining short rise and
fall times of square flux pulses, as well as allowing trans-
mission of AC flux drives. The tuning coil is differentially
biased by two DC input lines, with 80 kHz low-pass filters
at the 4K stage to further suppress noise photons. Fur-
thermore, Gigahertz microwave input lines are filtered
by an 8GHz lowpass filter and all microwave lines have
an Eccosorb filter, in order to ensure strong suppression
of thermal noise photons at very high frequencies. Note
also that all 50-Ω terminations are thermalized to the
MXC plate in order to suppress thermal noise from their
resistive elements.
Output signals from the Purcell filter waveguide
(PFWG) and slow-light waveguide (SLWG) device lines
are merged to a single amplifier chain in the following
manner. Their corresponding coaxial lines are connected
to a circulator as shown in Fig. S1a, such that signals
exiting the SLWG continue directly to the output chain,
while signals exiting the Purcell filter are first routed to
the SLWG device line and subsequently reflect off of the
finite-bandwidth structure, thus finally routing them to
the output chain. Note that input signals to the SLWG
IN line undergo similar routing in order to arrive at the
device.
Our amplifier chain at the “OUT” line consists of
a quantum-limited traveling-wave parametric amplifier
(TWPA) [5] as the initial amplification stage, followed
by a Low Noise Factory LNF-LNC4
8C high mobility
electron transistor (HEMT) amplifier mounted at the 4K
plate. For operation of the TWPA, a microwave pump
signal from Rohde & Schwarz SMB100A is added to the
amplifier via the coupled port of a 16 dB directional cou-
pler, with its isolated port terminated in 50-Ω. We in-
clude two isolators between the directional coupler and
the sample in order to shield the sample from the strong
TWPA pump.
Outside the fridge, we further amplify output signals
with amplification that is suitable for the dynamic range
of our ADC. We note that we use a Micro Lambda
Wireless MLBFR-0212 tunable notch filter to reject the
TWPA pump signal in order to prevent saturation of the
following room temperature electronics. Additionally, we
use IF amplifiers (0-1GHz bandwidth) for downconverted
signals due to IQ mixer saturation power limits.
Due to their different frequencies, we route SLWG and
PFWG signals to different downconversion stages via a
2-way power splitter, followed by a circulator at each
branch to prevent crosstalk between the two branches.
The “PHOTON” branch is connected to a IQ mixer for
downconversion of
4.8 GHz photonic signals, which
are then measured by an Alazartech ATS9371 digitizer
(ADC PHOTON); measurement of both photonic signal
quadratures
I
(
t
) and
Q
(
t
) comprise the heterodyne mea-
surement of time-dependent photon signals alluded to in
Supplementary Section IV. Meanwhile, the other branch
of the power splitter is also connected to an IQ mixer for
downconversion of
7.5GHz readout signals, which are
then measured by a Keysight M3102 digitizer (ADC RO).
We note that downconversion mixers share LO signals
(generated by Rohde & Schwarz SMB100A microwave
signal generators) with their upconversion counterparts
(where a Zurich HDAWG is used), in order to ensure
phase drift/jitter of LO’s during upconversion are can-
celled out during downconversion. And crucially, we
place additional filters before measurement at the ADC
in order to suppress noise outside of the IF measurement
band of interest. This not only allows for better utiliza-
tion of the ADC dynamic range, but also rejects noise at
irrelevant Nyquist bands that “fold” over to the band-
width of measured signals; we note that this effectively
3
improved the
n
noise
of our photon measurement chain by
almost a factor of 2 (see Supplementary Section IV for
more details).
II. DEVICE CHARACTERIZATION
A. Slow-light Waveguide
As discussed in the 2D cluster state generation protocol
proposed in ref [6], one of the dimensions of the resultant
cluster state is limited by the number of photons that can
be held in the delay line simultaneously, necessitating a
delay line with a sufficiently large round trip time
τ
d
. In
this work, we realize such a delay line via implementation
of a slow-light waveguide (SLWG), which provides large
group delay for time-delayed feedback. In addition, the
SLWG also provides spectral constriction of propagating
modes to a passband with a finite bandwidth, where the
photonic density of states (DOS) sharply decreases at the
bandedges and is negligible outside the passband, thus
enabling selective emission of the
Q
E
’s
|
f
⟩ −→ |
e
transi-
tion, as discussed in the main text. The SLWG is physi-
cally realized as a periodic array of capacitively coupled
lumped-element superconducting microwave resonators,
with low resonator loss and negligible resonator frequency
disorder, as was demonstrated in our prior work [2]. It
can be shown that such a design allows for large group
delay per resonator
1
2
J
, where
J
is the photon hopping
rate between adjacent resonators, as well as strong emis-
sion of transmon qubits only at qubit frequencies within
the SLWG passband.
The SLWG is implemented by periodically placing
N
= 50 unit cells across the device as seen in Fig. 1d,
where a unit cell consists of a lumped-element resonator
realized with tightly meandered lines providing the ma-
jority of the inductance, wider rectangular features pro-
viding the majority of the capacitance, and with capac-
itive coupling between adjacent resonators achieved via
their long capacitive wings, as shown in Fig. 1c. At
the output side of the SLWG, the Bloch impedance of
the SLWG is matched to its output 50 Ω CPW via a “ta-
per section” comprised of two lumped element resonators,
where their coupling capacitances towards the output are
gradually increased, and their capacitances to ground are
correspondingly gradually decreased to compensate for
resonance frequency changes. Crucially, in order to pre-
vent distortion of
Q
E
photon emission, at the terminated
side of the single-ended SLWG a capacitance to ground
via a long capacitive wing is placed at the left of the first
unit cell resonator (Fig. 1c), thus maintaining the res-
onance frequency of the first resonator to be the same
as the frequency of the other resonators, which ensures
monotonic emission from
Q
E
(as observed in separate
modeling).
The corresponding circuit model of the SLWG waveg-
uide coupled to
Q
E
and
Q
M
is depicted in Fig. S2b. In
the regime of
C
g
C
0
, the dispersion of the SLWG is
approximately,
ω
k
=
ω
p
+ 2
J
cos (
k
)
(1)
where
ω
0
= 1
/
L
0
C
0
is the resonance frequency of unit
cell resonators,
J
=
ω
0
C
g
2
C
0
,
ω
p
=
ω
0
2
J
is the center
frequency of the passband, and the passband width is 4
J
.
To mitigate the deleterious effects in the time-domain
shape of emitted photons emerging from the higher-order
dispersion [7], a sufficiently large
J
is required. On the
other hand, our requirement for large group delay
τ
d
=
N
J
necessitates a sufficiently small
J
. In order to balance the
conflicting requirements of large delay and manageable
dispersion, we chose
J
= 33
.
5 MHz as a target parameter
that corresponds to the round-trip delay of
τ
d
= 237 ns.
We thus aimed for the following target circuit param-
eters:
L
0
= 3.1 nH,
C
0
= 353 fF,
C
g
= 5.05 fF,
C
1
=
347 fF,
C
1
g
= 8.6 fF,
C
2
= 267 fF, and
C
2
g
= 87 fF,
yielding
J/
2
π
= 33.5 MHz,
ω
p
/
2
π
= 4.744 GHz, and the
requisite impedance matching at the boundary. As seen
in Fig. S2a, for the taper section the increasing coupling
capacitances are implemented as longer capacitive wings
or interdigitated capacitors, and adjustments to the res-
onance frequencies are achieved by both shortening the
length of the meandered lines and modifying the head ca-
pacitances. In addition, the coupling capacitance of
Q
E
and
Q
M
to their respective unit cells, as depicted in Fig.
1c and Fig. S2a, were designed to be 2
.
41 fF and 5
.
37
fF, respectively. This yields the qubit-unit cell coupling
g
uc
= 38
.
5 MHz of
Q
E
and
g
M
uc
= 85
.
6 MHz of
Q
M
via
the following relation:
g
uc
=
C
E
qg
2
q
(
C
0
+ 2
C
g
)(
C
E
Σ
+
C
E
qg
)
ω
p
(2)
where
g
M
uc
is obtained by a similar calculation. As dis-
cussed in the next subsection of the Supplementary, these
small coupling capacitances lead to large emission rates
due to the slow-light nature of the SLWG, where a small
group velocity
v
g
=
∂ω
∂k
is commensurate with a large
density of states
1
/
|
v
g
|
, which enhances emission rates.
[8, 9]
In order to characterize the SLWG, we investigated
the transmittance of the SLWG boundary for an itiner-
ant pulse by sending coherent gaussian pulses of variable
carrier frequency through the SLWG IN line and mea-
suring their outgoing intensity at ADC PHOTON after
they pass through the device. The measurement result,
comprising distinct features separated in time that cor-
respond to different reflection events, is shown in S2c.
First, when the pulses arrive at the SLWG boundary,
due to the finite reflectance of the taper section, a frac-
tion of the incident pulse is reflected (and thus does not
enter the SLWG) and is measured as the first bright fea-
ture in Fig. S2c. Next, the transmitted fraction of the
pulse propagates through the SLWG, completes a round-
trip, and arrives at the SLWG boundary again. While
a small fraction of the pulse is again reflected due to fi-
nite reflectance, most of the energy transmits through