Delft University of Technology
High-Fidelity Controlled- Z Gate with Maximal Intermediate Leakage Operating at the
Speed Limit in a Superconducting Quantum Processor
Negîrneac, V.; Ali, H.; Muthusubramanian, N.; Battistel, F.; Sagastizabal, R.; Moreira, M. S.; Marques, J. F.;
Vlothuizen, W. J.; Beekman, M.; Zachariadis, C.
DOI
10.1103/PhysRevLett.126.220502
Publication date
2021
Document Version
Final published version
Published in
Physical Review Letters
Citation (APA)
Negîrneac, V., Ali, H., Muthusubramanian, N., Battistel, F., Sagastizabal, R., Moreira, M. S., Marques, J. F.,
Vlothuizen, W. J., Beekman, M., Zachariadis, C., Haider, N., Bruno, A., & Dicarlo, L. (2021). High-Fidelity
Controlled- Z Gate with Maximal Intermediate Leakage Operating at the Speed Limit in a Superconducting
Quantum Processor.
Physical Review Letters
,
126
(22), [220502].
https://doi.org/10.1103/PhysRevLett.126.220502
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High-Fidelity Controlled-Z Gate with Maximal Intermediate Leakage Opera ting at the
Speed Limit in a Superconducting Quantum Processor
V. Negîrneac ,
1,2,*
H. Ali ,
1,3,*
N. Muthusubramanian ,
1,3
F. Battistel ,
1
R. Sagastizabal,
1,3
M. S. Moreira ,
1,3
J. F. Marques ,
1,3
W. J. Vlothuizen ,
1,4
M. Beekman ,
1,4
C. Zachariadis ,
1,3
N. Haider,
1,4
A. Bruno ,
1,3
and L. DiCarlo
1,3
1
QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
2
Instituto Superior T´ecnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal
3
Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
4
Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 96864, 2509 JG The Hague, Netherlands
(Received 3 September 2020; accepted 6 April 2021; published 4 June 2021)
Simple tuneup of fast two-qubit gates is essential for the scaling of quantum processors. We introduce
the sudden variant (SNZ) of the net zero scheme realizing controlled-Z (CZ) gates by flux control of
transmon frequency. SNZ CZ gates realized in a multitransmon processor operate at the speed limit of
transverse coupling between computational and noncomputational states by maximizing intermediate
leakage. Beyond speed, the key advantage of SNZ is tuneup simplicity, owing to the regular structure of
conditional phase and leakage as a function of two control parameters. SNZ is compatible with scalable
schemes for quantum error correction and adaptable to generalized conditional-phase gates useful in
intermediate-scale applications.
DOI: 10.1103/PhysRevLett.126.220502
Superconducting quantum processors have recently
reached important milestones [1], notably the demonstra-
tion of quantum supremacy on a 53-transmon processor [2].
On the path to quantum error correction (QEC) and fault
tolerance [3], recent experiments have used repetitive parity
measurements to stabilize two-qubit entanglement [4,5]
and to perform surface-code quantum error detection in a
7-transmon processor [6]. These developments have relied
on two-qubit controlled-phase (CPhase) gates realized by
dynamical flux control of transmon frequency, harnessing
the transverse coupling J
2
between a computational state
j11i and a noncomputational state such as j02i [7,8].
Compared to other implementations, e.g., cross resonance
using microwave-frequency pulses [9] and parametric
radio-frequency pulsing [10], baseband flux pulses achieve
the fastest controlled-Z (CZ) gates (a special case of
CPhase), operating near the speed limit t
lim
¼ π=J
2
[11].
Over the last decade, baseband flux pulsing for two-qubit
gating has evolved in an effort to increase gate fidelity and
to reduce leakage and residual ZZ coupling. In particular,
leakage became a main focus for its negative impact on
QEC, adding complexity to error-decoder design [12]
and requiring hardware and operational overhead to seep
[13–17]. To reduce leakage from linear-dynamical distor-
tion in flux-control lines and limited time resolution in
arbitrary waveform generators (AWGs), unipolar square
pulses [8,18] have been superseded by softened counter-
parts [19,20] based on fast-adiabatic theory [21].In
parallel, coupling strengths have reduced to J
2
=2π ∼
10–20 MHz to mitigate residual ZZ coupling, which
affects single-qubit gates and idling at bias points, and
produces crosstalk from spectator qubits [22]. Many groups
are actively developing tunable coupling schemes to sup-
press residual coupling without incurring slowdown
[23–27].
A main limitation to the fidelity of flux-based CPhase
gates is dephasing from flux noise, as one qubit is displaced
0.5–1 GHz below its flux-symmetry point (i.e., sweet spot
[28]) to reach the j11i–j02i resonance. To address this
limitation, Ref. [29] introduced a bipolar variant [termed
net zero (NZ)] of the fast-adiabatic scheme, which provides
a built-in echo reducing the impact of low-frequency flux
noise. The double use of the transverse interaction also
reduces leakage by destructive interference, as understood
by analogy to a Mach-Zehnder interferometer (MZI).
Finally, the zero-average characteristic avoids the buildup
of long-timescale distortions in the flux-control lines,
significantly improving gate repeatability. NZ pulsing
has been successfully used in several recent experiments
[4,6,30], elevating the state of the art for CZ gate fidelity to
99.72 0.35% [1]. However, NZ suffers from complicated
tuneup, owing to the complex dependence of conditional
phase and leakage on fast-adiabatic pulse parameters. This
limits the use of NZ for two-qubit gating as processors
grow in qubit count.
In this Letter, we introduce the sudden variant (SNZ) of
the NZ scheme implementing CZ, which offers two
advantages while preserving the built-in echo, destructive
leakage interference, and repeatability characteristic of
conventional NZ (CNZ). First, SNZ operates at the speed
limit of transverse coupling by maximizing intermediate
leakage to the non-computational state. The second and
PHYSICAL REVIEW LETTERS 126, 220502 (2021)
0031-9007=21=126(22)=220502(6) 220502-1 © 2021 American Physical Society
main advantage is greatly simplified tuneup: the landscapes
of conditional phase and leakage as a function of two pulse
parameters have regular structure and interrelation, easily
understood by exact analogy to the MZI. We realize SNZ
CZ gates among four pairs of nearest neighbors in a
seven-transmon processor and characterize their perfor-
mance using two-qubit interleaved randomized benchmark-
ing (2QIRB) with modifications to quantify leakage
[29,31,32]. The highest performance achieved from one
2QIRB characterization has 99.93 0.24% fidelity and
0.10 0.02% leakage. SNZ CZ gates are fully compatible
with scalable approaches to QEC [33]. The generalization
of SNZ to arbitrary CPhase gates is straightforward
and useful for optimization [34], quantum simulation
[35], and other noisy intermediate-scale quantum (NISQ)
applications [36].
A flux pulse to the j11i–j02i interaction implements the
unitary
U
CPhase
¼
0
B
B
B
B
B
B
B
@
10 0 0 0
0 e
iϕ
01
00 0
00e
iϕ
10
00
00 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − 4L
1
p
e
iϕ
11
ffiffiffiffiffiffiffiffi
4L
1
p
e
iϕ
02;11
00 0
ffiffiffiffiffiffiffiffi
4L
1
p
e
iϕ
11;02
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − 4L
1
p
e
iϕ
02
1
C
C
C
C
C
C
C
A
in the fj00i; j01i; j10i; j11i; j02ig subspace, neglecting
decoherence and residual interaction between far off-
resonant levels. Here, ϕ
01
and ϕ
10
are the single-qubit
phases, ϕ
11
¼ ϕ
01
þ ϕ
10
þ ϕ
2Q
, where ϕ
2Q
is the condi-
tional phase, and L
1
is the leakage, The ideal CZ
gate simultaneously achieves ϕ
01
¼ ϕ
10
¼ 0ð mod 2πÞ,
ϕ
2Q
¼ πð mod 2πÞ (phase condition PC), and L
1
¼ 0
(leakage condition LC), with arbitrary ϕ
02
.
The SNZ CZ gate is realized with two square half pulses
with equal and opposite amplitude A and duration t
p
=2
each. To understand its action, consider first the ideal
scenario with perfectly square half pulses (infinite band-
width), infinite time resolution, t
p
¼ t
lim
, and A ¼ 1
(corresponding to j11i and j02i on resonance). The unitary
action of each complete half pulse (rising edge, steady
level, and falling edge combined) implements one of two
beam splitters in the MZI analogy: BS1 fully transmits j11i
to −ij02i (producing maximal intermediate leakage), and
BS2 fully transmits −ij02i to −j11i, yielding an ideal CZ
gate. SNZ adds an idling period t
ϕ
between the half pulses
to perfect the analogy to the MZI, allowing accrual of
relative phase ϕ between j02i and j11i in between the beam
splitters.
The key advantage of SNZ over CNZ is the straightfor-
ward procedure to simultaneously meet PC and LC. To
appreciate this, consider the landscapes of ϕ
2Q
and L
1
as a
function of A and t
ϕ
[Figs. 1(c) and 1(d)] in this ideal
scenario. The landscapes have a clear structure and link to
each other. The L
1
landscape shows a vertical leakage
valley at A ¼ 1 arising from perfect transmission at each
beam splitter (LC1), and also two vertical valleys arising
from perfect reflection (LC2). Leakage interference gives
rise to additional diagonal valleys (LC3). Crucially, juxta-
posing the ϕ
2Q
¼ 180° contour shows that PC is met
periodically, at the crossing of LC1 and LC3 valleys,
where Δ
max
02
t
ϕ
¼ 0ð mod 2πÞ (Δ
max
02
is the detuning
between j02i and j11i at the bias point). This regular
leakage landscape therefore provides useful crosshairs for
simultaneously achieving PC and LC. We note that ϕ
2Q
ðt
ϕ
Þ
changes monotonically along the LC1 valley, allowing for
CPhase gates with arbitrary ϕ
2Q
. We leave this generali-
zation for future work.
There are practical reasons to include t
ϕ
in experiment:
any flux-pulse distortion remaining from the first half pulse
(e.g., due to finite pulse decay time) will break the
symmetry between BS1 and BS2. Because of the time
resolution t
s
of the AWG used for flux control, ϕ can only
increment in steps of −Δ
max
02
t
s
. Typically Δ
max
02
=2π ¼
0.5–1 GHz and t
s
∼ 1 ns, so the number of intermediate
sampling points only provides coarse control. For fine
control, we propose to use the amplitude B of the first and
last sampling points during t
ϕ
[37].
(a)
(c) (d)
(b)
FIG. 1. Numerical simulation of an ideal SNZ pulse (infinite
bandwidth and time resolution) using parameters for pair
Q
L
-Q
M2
(see Table I). (a) Schematic of the ideal SNZ flux
pulse, with t
p
¼ t
lim
and variable A and t
ϕ
. The amplitude A is
normalized to the j11i–j02i resonance. Inset: MZI analogy for
A ¼ 1. (b) Transition frequency from j00i to levels jiji in the
two-excitation manifold as a function of instantaneo us pulse
amplitude. (c),(d) Landscapes of conditional phase ϕ
2Q
(b) and
leakage L
1
(c) as a function of A and t
ϕ
.
PHYSICAL REVIEW LETTERS 126, 220502 (2021)
220502-2
We now turn to the experimental realization of SNZ CZ
gates between nearest-neighbor pairs among four transmons.
High- and low-frequency transmons (Q
H
and Q
L
,respec-
tively) connect to two mid-frequency transmons (Q
M1
and
Q
M2
) using bus resonators dedicated to each pair [con-
nectivity diagram shown in Fig. 4(a) inset]. Each transmon
has a flux-control line for two-qubit gating, a microwave-
drive line for single-qubit gating, and dedicated readout
resonators [4,38] (see Supplemental Material [37] for
details). Each transmon is statically flux biased at its sweet
spot to counter residual offsets. Flux pulsing is performed
using a Zurich Instruments HDAWG-8 ðt
s
¼ 1=2.4 nsÞ.
Following prior work [29,44], we compensate the band-
width-limiting effect of attenuation in the flux-control
coaxial line (skin ef fect) and cryogenic reflectiv e and
absorptive low-pass filters using real-time digital filters in
the AWG. In this way, we produce on-chip flux waveforms
with rise time t
rise
on par with that of the AWG (0.5 ns).
We exemplify the tuneup of SNZ using pair Q
L
-Q
M2
(Fig. 2). We first identify t
lim
for the j11i–j02i interaction
and amplitude A bringing the two levels on resonance. Both
are extracted from the characteristic chevron pattern of
j2i-population P
j2i
in Q
M2
as a function of the amplitude
and duration of a unipolar square flux pulse acting on j11i
[Fig. 2(a)]. The symmetry axis corresponds to A ¼ 1. The
difference in consecutive pulse durations achieving P
j2i
maxima along this axis gives an accurate estimate of t
lim
unaffected by initial transients. We set t
p
≡ 2nt
s
, where n is
the number of sampling points achieving the first P
j2i
maximum. Using the measured positive difference t
p
− t
lim
and numerical simulation (data not shown), we estimate
t
rise
≈ 0.5 ns. Next, we use standard conditional-oscillation
experiments [29] to measure the landscapes of ϕ
2Q
and
leakage estimate
˜
L
1
for SNZ pulses over amplitude ranges
A ∈ ½0.9; 1.1 and B ∈ ½0;A, keeping t
ϕ
≳ 3t
rise
.As
expected, the landscape of
˜
L
1
[Fig. 2(c)] reveals a vertical
valley at A ¼ 1 and a diagonal valley. Juxtaposing the
ϕ
2Q
¼ 180° contour from Fig. 2(b), we observe the
matching of PC at the crossing of these valleys, in excellent
agreement with a numerical two-qutrit simulation
[Fig. 2(d)].
Experimentally, due to the discreteness of t
s
,itis
unlikely to precisely match t
p
=2 to the half-pulse duration
that truly maximizes P
j2i
. To understand the consequences,
we examine the ϕ
2Q
and
˜
L
1
landscapes for SNZ pulses
upon intentionally changing t
p
by 6t
s
(Fig. 3). While the
PC contour remains roughly unchanged in both cases, there
are significant effects on
˜
L
1
. In both cases, we observe that
˜
L
1
lifts at the prior crossing of LC1 and LC3 valleys where
(a) (b)
(c) (d)
FIG. 2. Calibration of the SNZ pulse for pair Q
L
-Q
M2
and
comparison to simulation. (a) j2i-state population of Q
M2
as a
function of the amplitude and duration of a unipolar square pulse
making j11i interact with j02i. (b),(c) Landscapes of conditional
phase ϕ
2Q
and leakage estimate
˜
L
1
as a function of SNZ pulse
amplitudes A and B, with t
ϕ
¼ 1.67 ns. The juxtaposed ϕ
2Q
¼
180° contour runs along the opposite diagonal compared to
Figs. 1(b), 1(c) because increasing B (which decreases Δ
02
)
changes ϕ in the opposite direction from t
ϕ
. Data points marked
with dots are measured with extra averaging for examination in
Fig. 3. (d) Numerical simulation of leakage L
1
landscape and
ϕ
2Q
¼ 180° contour with parameters and flux-pulse distortions
from experiment. All landscapes (also in Fig. 3) are sampled
using an adaptive algorithm based on [39].
(a)
(c)
(b)
FIG. 3. (a),(b) Landscapes of the leakage estimate
˜
L
1
for
intentionally short and long SNZ half pulses on Q
M2
. (c) Ex-
tracted
˜
L
1
along the ϕ
2Q
¼ 180° contours from (a),(b), and
Fig. 2( c).
PHYSICAL REVIEW LETTERS 126, 220502 (2021)
220502-3
ϕ
2Q
¼ 180°. For too-short pulses [Fig. 3(a)], there remain
two valleys of minimal
˜
L
1
, but these are now curved and do
not cross ϕ
2Q
¼ 180°. For too-long pulses [Fig. 3(b)], there
are also two curved valleys. Crucially, these cross the
ϕ
2Q
¼ 180° contour, and it remains possible to achieve PC
and minimize leakage at two ðA; BÞ settings. Extracting
˜
L
1
along the ϕ
2Q
¼ 180° contours [Fig. 3(c)] confirms that
too-long pulses can achieve the same minimal
˜
L
1
as when
using the nominal t
p
. The impossibility to achieve minimal
leakage at ϕ
2Q
¼ 180° for too-short pulses manifests the
speed limit set by J
2
. In turn, the demonstrated possibility
to do so for too-long pulses (even overshooting by several
sampling points) proves the viability of the SNZ pulse in
practice.
W ith these insights, we proceed to tune the remaining
SNZ CZ gates following similar procedures. We use final
weak bipolar pulses of total duration t
1Q
¼ 10 ns to null the
single-qubit phases in the frame of microwave drives. Since
our codeword-based control electronics has a 20 ns timing
grid, and 40 ns <t
total
¼ t
p
þ t
ϕ
þ t
1Q
< 60 ns for all
pairs, we allocate 60 ns to ev ery CZ gate. Some pair-specific
details must be noted. Owing to the frequency overlap of
Q
M1
and Q
M2
, implementing CZ between Q
H
and Q
M1
ðQ
M2
Þ requires a bipolar parking flux pulse on Q
M2
ðQ
M1
Þ
during the SNZ pulse on Q
H
[6,33]. For most pairs, we
employ the j11i–j02i interaction, which requires the small-
est flux amplitude (reducing the impact of dephasing from
flux noise) and does not require crossing any other inter-
action. However, for Q
L
-Q
M1
, we cannot reliably use this
interaction as there is a flickering two-level system (TLS)
overlapping with the j0i–j1i transition in Q
M1
at this
amplitude [37]. For this pair, we therefore employ the
j11i–j20i interaction. Here, SNZ offers a side benefit: it
crosses the Q
M1
-TLS, j11i–j02i,andj01i–j10i resonances
as suddenly as possible, minimizing population exchange.
Table I summarizes the timing parameters and perfor-
mance attained for the four SNZ CZ gates. The CZ gate
fidelity F and leakage L
1
are extracted using a 2QIRB
protocol [29,32]. For each pair, we report the average and
standard deviation of both based on at least 10 repetitions of
the protocol spanning more than 8 h [37]. Several obser-
vations can be drawn. First, CZ gates involving Q
H
perform
better on average than those involving Q
L
. This is likely
due to the shorter t
lim
and correspondingly longer time
60 ns −t
p
spent near the sweetspot. Additionally, the
frequency downshifting required of Q
H
to interact with
the midfrequency transmons is roughly half that required of
the latter to interact with Q
L
. This reduces the impact of
dephasing from flux noise during the pulse. Not surpris-
ingly, performance is worst for Q
L
− Q
M1
. Here, the pulse
must downshift Q
M1
the most to reach the distant j11i–j20i
interaction, increasing dephasing from flux noise. Also,
there may be residual exchange at the crossed resonances.
Overall, there is significant temporal variation in perfor-
mance as gleaned by repeated 2QIRB characterizations. We
believe this reflects the underlying variability of qubit
relaxation and dephasing times and flux offsets, which,
however, were not tracked simultaneously. In addition to
having the best average performance, pair Q
M2
-Q
H
dis-
plays the maximum F of 99.93 0.24% (Fig. 4) extracted
from a single 2QIRB characterization. To the best of our
knowledge, this is the highest CZ fidelity extracted from
one 2QIRB characterization in a multitransmon processor.
To understand the dominant sources of infidelity ε ¼
1 − F and leakage, we run numerical simulations [29], for
both SNZ and CNZ, with experimental input parameters for
pair Q
M2
-Q
H
. We dissect an error budget versus various
models finding similar contributions for both gates (see
Supplemental Material [37]). Nevertheless, the results
suggest that SNZ slightly outperforms CNZ, likely due
to a shorter time spent away from the sweetspot during the
fixed 60 ns allocated for both variants. This confirms that
the temporary full transfer from j11i to j02i does not
compromise the gate fidelity.
In summary, we have proposed and realized high-fidelity
CZ gates using the sudden version of the net zero bipolar
fluxing scheme. SNZ CZ gates operate ever closer to the
TABLE I. Summary of SNZ CZ pulse parameters and achieved performance for the four transmon pairs. Single-qubit phase
corrections are included in t
total
. Gate fidelities and leakage are obtained from 2QIRB keeping the other two qubits in j0i. Statistics
(average and standard deviation) are taken from repeated 2QIRB runs (see Supplemental Material [37] for technical details). The
maximum F and minimum L
1
quoted are not necessarily from the same run.
Parameter Q
M1
-Q
H
Q
M2
-Q
H
Q
L
-Q
M1
Q
L
-Q
M2
t
lim
(ns) 31.0 27.6 38.4 33.8
t
p
, t
ϕ
(ns) 32.50,2.92 29.10,3.75 40.83,1.25 35.83,1.67
t
total
(ns) 45.42 42.91 52.08 47.50
Interaction j11i–j02ij11i–j02ij11i–j20ij11i–j02i
Parked qubit Q
M2
Q
M1
Avg. F (%) 98.89 0.35 99.54 0 . 27 93.72 2.10 97.14 0.72
Avg. L
1
(%) 0.13 0.02 0.18 0. 04 0.78 0.32 0.63 0.11
Max. F (%) 99.77 0.23 99.93 0 . 24 99.15 1.20 98.56 0.70
Min. L
1
(%) 0.07 0.04 0.10 0.02 0.04 0.08 0.41 0.10
PHYSICAL REVIEW LETTERS 126, 220502 (2021)
220502-4
![TABLE I. Summary of SNZ CZ pulse parameters and achieved performance for the four transmon pairs. Single-qubit phase corrections are included in ttotal. Gate fidelities and leakage are obtained from 2QIRB keeping the other two qubits in j0i. Statistics (average and standard deviation) are taken from repeated 2QIRB runs (see Supplemental Material [37] for technical details). The maximum F and minimum L1 quoted are not necessarily from the same run.](/figures/table-i-summary-of-snz-cz-pulse-parameters-and-achieved-107dav8b.webp)
