of 21
Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons
H. Levine ,
1
A. Haim,
1,2
J. S. C. Hung,
1
N. Alidoust,
1
M. Kalaee,
1
L. DeLorenzo,
1
E. A. Wollack ,
1
P. Arrangoiz-Arriola,
1
A. Khalajhedayati ,
1
R. Sanil,
1
H. Moradinejad,
1
Y. Vaknin,
1,3
A. Kubica ,
1
D. Hover,
1
S. Aghaeimeibodi ,
1
J. A. Alcid,
1
C. Baek,
1
J. Barnett,
1
K. Bawdekar,
1
P. Bienias,
1
H. A. Carson,
1
C. Chen,
1
L. Chen ,
1
H. Chinkezian,
1
E. M. Chisholm,
1
A. Clifford,
1
R. Cosmic,
1
N. Crisosto,
1
A. M. Dalzell,
1
E. Davis,
1
J. M. D
Ewart,
1
S. Diez,
1
N. D
Souza,
1
P. T. Dumitrescu ,
1
E. Elkhouly,
1
M. T. Fang,
1
Y. Fang ,
1
S. Flammia,
1
M. J. Fling,
1
G. Garcia,
1
M. K. Gharzai,
1
A. V. Gorshkov ,
1
M. J. Gray,
1
S. Grimberg,
1
A. L. Grimsmo ,
1,4
C. T. Hann,
1
Y. He,
1
S. Heidel ,
1
,
S. Howell,
1
M. Hunt,
1
J. Iverson,
1
I. Jarrige,
1
L. Jiang,
1,5
W. M. Jones ,
1
R. Karabalin,
1
P. J. Karalekas ,
1
A. J. Keller ,
1
D. Lasi,
1
M. Lee ,
1
V. Ly,
1
G. MacCabe,
1
N. Mahuli,
1
G. Marcaud ,
1
M. H. Matheny,
1
S. McArdle,
1
G. McCabe ,
1
G. Merton,
1
C. Miles,
1
A. Milsted,
1
A. Mishra ,
1
L. Moncelsi ,
1
M. Naghiloo,
1
K. Noh,
1
E. Oblepias,
1
G. Ortuno,
1
J. C. Owens,
1
J. Pagdilao,
1
A. Panduro,
1
J.-P. Paquette,
1
R. N. Patel,
1
G. Peairs,
1
D. J. Perello,
1
E. C. Peterson,
1
S. Ponte,
1
H. Putterman,
1
G. Refael,
1,6,7
P. Reinhold,
1
R. Resnick,
1
O. A. Reyna,
1
R. Rodriguez,
1
J. Rose,
1
A. H. Rubin ,
1
,
M. Runyan,
1
C. A. Ryan ,
1
A. Sahmoud,
1
T. Scaffidi,
1
B. Shah,
1
S. Siavoshi,
1
P. Sivarajah,
1
T. Skogland,
1
C.-J. Su,
1
L. J. Swenson,
1
J. Sylvia,
1
S. M. Teo,
1
A. Tomada,
1
G. Torlai,
1
M. Wistrom,
1
,
K. Zhang,
1
I. Zuk ,
1,3
A. A. Clerk,
1,5
F. G. S. L. Brandão,
1,6
A. Retzker,
1,3
and O. Painter
1,6,8
,*
1
AWS Center for Quantum Computing, Pasadena, California 91125, USA
2
Institute of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel
3
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel
4
Centre for Engineered Quantum Systems, School of Physics, The University of Sydney,
Sydney, New South Wales 2006, Australia
5
Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
6
Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA
7
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
8
Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute,
California Institute of Technology, Pasadena, California 91125, USA
(Received 28 September 2023; revised 27 December 2023; accepted 16 January 2024; published 20 March 2024)
Quantum error correction with erasure qubits promises significant advantages over standard error
correction due to favorable thresholds for erasure errors. To realize this advantage in practice requires a qubit
for which nearly all errors are such erasure errors, and the ability to check for erasure errors without dephasing
the qubit. We demonstrate that a
dual-rail qubit
consisting of a pair of resonantly coupled transmons can
form a highly coherent erasure qubit, where transmon
T
1
errors are converted into erasure errors and residual
dephasing isstrongly suppressed, leadingtomillisecond-scalecoherencewithinthe qubit subspace. Weshow
that single-qubit gates are limited primarily by erasure errors, with erasure probability
p
erasure
¼
2
.
19
ð
2
Þ
×
10
3
per gate while the residual errors are
40
times lower. We further demonstrate midcircuit detection of
erasure errors while introducing
<
0
.
1%
dephasing error per check. Finally, we show that the suppression of
transmon noise allows this dual-rail qubit to preserve high coherence over a broad tunable operating range,
offering an improved capacity to avoid frequency collisions. This work establishes transmon-based dual-rail
qubits as an attractive building block for hardware-efficient quantum error correction.
DOI:
10.1103/PhysRevX.14.011051
Subject Areas: Quantum Physics, Quantum Information
*
Corresponding author: ojp@amazon.com
Present address: OpenAI, San Francisco, California, USA.
Present address: Department of Physics and Electrical and Computer Engineering, University of California, Davis, California 95616,
USA.
§
Present address: Department of Physics and Astronomy, University of California, Irvine, California 92697, USA.
Present address: Google, 1600 Amphitheatre Parkway, Mountain View, California 94043, USA.
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International
license. Further
distribution of this work must maintain attribution to the author(s) and the published article
s title, journal citation, and DOI.
PHYSICAL REVIEW X
14,
011051 (2024)
Featured in Physics
2160-3308
=
24
=
14(1)
=
011051(21)
011051-1
Published by the American Physical Society
I. INTRODUCTION
The path toward useful quantum error correction requires
qubits which have physical error rates well below the error
correction threshold. While many physical qubit platforms
have reached error rates below the surface code threshold
[1
5]
, a major frontier challenge is to continue driving
down error rates to reduce the overhead necessary to protect
logical qubits. In parallel with this effort, the growing field
of hardware-efficient error correction seeks to leverage or
engineer the noise properties of physical qubits to more
efficiently correct errors, effectively raising thresholds and
reducing overhead
[6
10]
. Recent examples include noise-
bias engineering and robust qubit encoding in supercon-
ducting bosonic modes
[11,12]
.
The recently developed
erasure qubit
offers a compelling
path toward more relaxed error correction requirements
[13
16]
. While most physical qubits exhibit undetectable
errors that occur within the qubit subspace, erasure qubits
are those in which the dominant error is leakage out of the
computational subspace, and for which these leakage errors
can be detected in real time. These
erasure errors
can be
corrected more efficiently, leading to higher thresholds and
reduced logical error rates
[13
15,17
19]
. To leverage this
advantage, erasurequbits must demonstrate (1) a largeerasure
noise bias,whichistheratio oferasureerrorstoresidual errors
within the qubit subspace, and (2) the ability to detect erasure
errors without introducing additional errors within the
subspace. While erasure qubit implementations have been
proposed in several architectures including neutral atoms
[13,19]
,ions
[20,21]
, and superconducting qubits
[14,16]
,
these ideas have only recently been implemented experimen-
tally using neutral atoms, demonstrating that a large fraction
of gate errors can be detected midcircuit and converted into
erasures
[22]
, as well as the excision of state preparation and
Rydberg decay errors detected as erasures
[23]
.
In this paper, we demonstrate that a superconducting
dual-rail qubit
formed by two resonantly coupled trans-
mons meets both requirements for serving as an erasure
qubit. We show that the dual-rail qubit converts transmon
T
1
errors into detectable erasure errors while passively
suppressing residual transmon dephasing, enabling a large
erasure noise bias while idling and during single-qubit
gates. We then demonstrate high-fidelity midcircuit erasure
detection using an ancilla qubit, and show that this
detection minimally degrades coherence within the dual-
rail subspace. Finally, we demonstrate that the dual-rail
qubit
s passive noise suppression allows it to maintain high
coherence while remaining broadly tunable, offering an
extra knob to dodge frequency collisions and improving the
prospects for qubit yield at scale.
II. DUAL-RAIL QUBIT
The dual-rail qubit is defined in the single-excitation
manifold of a pair of superconducting modes
[14,16,24
28]
.
We use a pair of resonantly coupled transmons, as first
demonstrated in Ref.
[28]
, where the logical subspace
consists of the hybridized symmetric and antisymmetric
states
j
0
L
i¼ð
1
=
ffiffiffi
2
p
Þðj
01
i
j
10
and
j
1
L
i¼ð
1
=
ffiffiffi
2
p
Þ
ðj
01
iþj
10
. This encoding allows
T
1
decay of the under-
lying transmons, which constitutes the fundamental limit to
transmon coherence, to be converted to erasure errors in the
form of leakage to the
j
00
i
state [Figs.
1(a)
1(c)
].
Crucially, in addition to converting
T
1
errors into erasure
errors, the resonant coupling between
j
01
i
and
j
10
i
with
strength
g
acts as a passive decoupling mechanism which
strongly suppresses the impact of low-frequency noise on
the underlying transmons
[14,28]
, analogous to continuous
dynamical decoupling in driven systems
[29
31]
. This is
illustrated by the dual-rail energy gap
E
DR
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
2
g
Þ
2
þ
δ
2
p
,
where
δ
is the frequency difference between the two
transmons which inherits frequency noise from each trans-
mon. When
g
δ
, frequency noise is suppressed accord-
ing to
E
DR
2
g
þ
δ
2
=
4
g
, allowing the dual-rail energy gap
to be orders of magnitude more stable than the underlying
transmons (see Appendix
M
for a more detailed analysis).
This noise suppression enables a large erasure noise bias, in
which dephasing errors within the subspace are suppressed
and the dominant error mechanism is erasure errors due to
the underlying transmon
T
1
.
Our experiments utilize a superconducting quantum
circuit with three tunable transmons, two of which (Q1
and Q2) form the dual-rail qubit and are parked on
resonance with one another, while the third (Q3) is used
as an ancilla qubit (Fig.
1
). Experimental sequences
involving dual-rail erasure qubits are illustrated in Fig.
1(d)
,
with calibration routines presented in Appendices
D
and
E
.
Each sequence begins with a microwave pulse on Q1 to
initialize the logical
j
1
L
i
. Single-qubit gates within the
dual-rail subspace are performed by flux modulation of Q2
at the frequency
2
g
¼
2
π
×
180
MHz, with the phase of the
flux modulation defining the axis of the drive field. At the
end of the circuit, the dual-rail qubit is measured by
adiabatically separating the two transmons with a flux
pulse on Q2 such that the two logical states
j
0
L
i
;
j
1
L
i
map
to the pair states
j
01
i
and
j
10
i
, respectively, whereupon the
two transmons are jointly read out
[28]
. Dual-rail qubit
operation is illustrated in Fig.
1(f)
by a spin-echo experi-
ment, where the final readout shows both coherent oscil-
lations within the
j
0
L
i
;
j
1
L
i
subspace as well as leakage to
j
00
i
due to transmon
T
1
decay.
During the experimental sequence, periodic erasure
checks may be performed to identify if the system decayed
into
j
00
i
. This is done by leveraging a dispersive shift on
the ancilla qubit which depends on whether the dual rail is
in
j
00
i
or if it remains in the logical subspace [Fig.
1(e)
].
The erasure check consists of a conditional
π
pulse on the
ancilla which is resonant only if the dual rail is in
j
00
i
,
followed by ancilla state readout. We note that a similar
procedure using a directly coupled readout resonator could
H. LEVINE
et al.
PHYS. REV. X
14,
011051 (2024)
011051-2
also be used for erasure detection, but would require
more carefully engineered symmetric coupling strengths
to avoid dephasing as discussed in Ref.
[14]
. Such erasure
checks are critical tools when using erasure qubits for
quantum error correction, and as discussed below, are also
necessary for characterizing the coherence properties of the
dual-rail qubit.
III. COHERENCE WITHIN THE
DUAL-RAIL SUBSPACE
A key metric for erasure qubits is the erasure noise bias,
which should be
1
in order to benefit most significantly
from erasure conversion
[13]
. This should hold for errors
while idling, requiring that the coherence within the logical
subspace is much longer than the erasure lifetime, as well
as during gates. To probe error rates within the subspace,
we perform coherence and gate benchmarking experiments
and postselect on shots where the system stays in the dual-
rail subspace. We note that postselection is used here only
to characterize the nonerasure error rates and would not be
needed in a concatenated surface code architecture as
described in Ref.
[14]
.
We postselect against leakage by relying on both the
final readout of the dual-rail pair as well as a sequence
of erasure checks performed during the measurement
[Fig.
2(a)
]. These midcircuit erasure checks are important
to catch events in which the system decays into
j
00
i
during
the circuit but then heats back into the dual-rail subspace;
without such checks, this decay and reheating process
limits the dephasing time that can be measured using only
the final readout. Erasure checks are evenly spaced during
the coherence measurement and successfully mitigate the
contributions of decay and reheating events as long as they
are repeated with a spacing shorter than the transmon
T
1
times (see further discussion in Appendix
J
).
We find that while the two transmons Q1 and Q2
individually have
T
CPMG
2
at the microsecond and tens of
microseconds scales, the
T
CPMG
2
within the dual-rail sub-
space (postselected against erasure errors) is
>
500
μ
s,
ranging from
543
ð
23
Þ
μ
s for one echo pulse to 1.25(8) ms
for 256 echo pulses [Figs.
2(b)
and
2(c)
]. The dual-rail
T
1
fits to an extrapolated
T
1
¼
906
ð
15
Þ
μ
s [Fig.
2(d)
]. These
millisecond-scale coherence times are far longer than the
timescale for decay out of the subspace into
j
00
i
, which
occurs at a characteristic erasure lifetime
T
erasure
30
μ
s
[Fig.
2(b)
] set by the underlying transmon
T
1
values
15
35
μ
s (Table
I
). The ratio of these timescales defines
the erasure noise bias for idling errors,
T
CPMG
2
=T
erasure
20
, confirming that the dominant error on the dual-rail qubit
while idling is erasure errors.
Several effects can limit the coherence within the dual-
rail subspace at the millisecond level and are discussed in
Appendix
N
. The most dominant contribution is thermal
Johnson-Nyquist noise on the flux lines, which is expected
FIG. 1. Dual-rail erasure qubit encoding. (a) Our circuit has three tunable transmons: Q1 and Q2 form the dual-rail qubit, and Q3 is an
ancilla qubit used for erasure detection (insets: tuning ranges, with primary operating point marked). (b) Microwave spectroscopy of Q1
as Q2 is tuned into resonance, exhibiting an avoided crossing with gap
2
g=
2
π
¼
180
MHz. (c) The dual-rail subspace is the symmetric
and antisymmetric combination of
j
01
i
and
j
10
i
. Decay of the underlying transmons leads to detectable leakage to
j
00
i
, constituting an
erasure error. (d) Circuits begin with a 40 ns microwave pulse to initialize
j
1
L
i
, followed by single-qubit gates enacted by flux
modulation of Q2, and finally the qubits are adiabatically separated and jointly read out. (e) We perform spectroscopy on the ancilla after
initializing the dual-rail pair in
j
00
i
(blue),
j
1
L
i
(orange), or
j
0
L
i
(green, trace not shown). Erasure checks are performed by applying a
microwave pulse on the ancilla which is resonant only if the dual-rail pair is in
j
00
i
(blue vertical line). (f) Spin-echo experiment on the
dual-rail qubit where the phase of the final
π
=
2
is stepped, showing both coherent fringes within the subspace and also decay out of the
subspace with a lifetime of
30
μ
s. Error bars denote 68% confidence intervals unless otherwise specified.
DEMONSTRATING A LONG-COHERENCE DUAL-RAIL ERASURE
...
PHYS. REV. X
14,
011051 (2024)
011051-3
to limit the
T
1
within the dual-rail subspace to
1
ms. We
therefore hypothesize that this is the limiting contribution
for our measured
T
1
; this effect can be mitigated with
further cryogenic attenuation of the fast-flux line and will
be studied in future work. Other effects, including residual
sensitivity to transmon frequency noise, photon fluctua-
tions in readout resonators, and leakage into the two-photon
manifold, can all limit dual-rail coherence at the level of
several milliseconds. While the measured
T
1
is attributed to
Johnson noise, the
T
2
may be limited by a combination of
these effects and requires further investigation.
We find that the dual-rail
T

2
phase coherence without
dynamical decoupling is limited by a separate effect: slow,
seconds-scale telegraph noise on the dual-rail frequency
(Fig.
13
). We correlate this frequency instability with
similar telegraph noise on one of the underlying transmons
(Q1) and attribute this to a nearby toggling two-level
system (TLS) defect at 4.98 GHz (Appendix
K
). This
behavior is consistent with a dispersive coupling model
discussed in Appendix
N3
and may offer a new probe for
TLS behavior. Nonetheless, the switching time is suffi-
ciently slow that this noise is effectively mitigated with
dynamical decoupling.
To validate that long dual-rail coherence times with
dynamical decoupling translate to high-fidelity gates,
we characterize single-qubit X90 gates on the dual-rail
qubit, enacted by 48 ns flux-modulation pulses on Q2
(Appendix
E
). We perform Clifford randomized bench-
marking (Fig.
3
), in parallel with frequent erasure checks,
and measure an erasure error probability of
2
.
19
ð
2
Þ
×
10
3
per X90, consistent with the duration of the gate relative to
FIG. 3. Single-qubit randomized benchmarking. (a) The ran-
domized benchmarking circuit uses random Cliffords, each
implemented as two X90 pulses with appropriate phase shifts
[33]
. In parallel, we perform 11 erasure checks evenly spaced
throughout the circuit, regardless of depth. The X90 pulse is a
Gaussian-shaped 48 ns flux pulse on Q2 along with a 0.067 rad
phase correction. An example timing diagram is presented in
Fig.
9
. (b) We plot the postselection probability (green) as a
function of circuit depth and fit to an exponential decay to 0 to
extract the erasure error per Clifford of
4
.
38
ð
3
Þ
×
10
3
. The
postselected survival probability within the dual-rail subspace
(blue), averaged over 200 random circuits for each depth, is fit to
an exponential decay with offset
1
=
2
and gives a residual error
rate of
1
.
01
ð
1
Þ
×
10
4
per Clifford. Individual outcomes for
random circuits are plotted with transparency. We symmetrize
readout errors by alternating measurements in which the ideal
outcome is
j
0
L
i
or
j
1
L
i
. The X90 gate errors are half those of the
Clifford gate, with erasure error
2
.
19
ð
2
Þ
×
10
3
and residual error
5
.
06
ð
6
Þ
×
10
5
. Similar error rates are computed when postse-
lecting only on midcircuit erasure checks (Appendix
F
).
FIG. 2. Millisecond-scale coherence within the dual-rail sub-
space. (a) We measure
T
2
coherence with a CPMG (Carr-Purcell-
Meiboom-Gill) sequence
[32]
consisting of
N
π
pulses with a
total evolution time of
N
τ
. In parallel, we perform a sequence of
N
e
¼
11
erasure checks, evenly spaced by
τ
e
¼
N
τ
=
ð
N
e
1
Þ
.
Coherence data are postselected against leakage using both the
final dual-rail readout and also the midcircuit erasure checks.
(b) Example experiment with
N
¼
64
π
pulses, where the final
π
=
2
pulse phase is stepped to produce fringes. The postselection
probability decays with time according to a
29
.
3
ð
4
Þ
μ
s erasure
lifetime, while the postselected shots are fit to an exponential
decay with extrapolated
T
2
¼
849
ð
51
Þ
μ
s. 330 000 shots were
taken per point, but only a small fraction of these survive
postselection. (c) CPMG measurements on each underlying
transmon, if parked alone at the operating point, as well as on
the dual-rail qubit. Error bars are standard deviation of many
individual measurements. (d)
T
1
within the dual-rail subspace.
We measure the probability of ending in
j
1
L
i
, postselected
against erasure errors, after initializing in either
j
1
L
i
(blue) or
j
0
L
i
(orange). The difference between these traces is fit to an
exponential decay with fixed offset 0, giving
T
1
¼
906
ð
15
Þ
μ
s.
The 24 hour time trace of data comprising these dual-rail
T
1
,
T
2
plots is shown in Appendix
L
.
H. LEVINE
et al.
PHYS. REV. X
14,
011051 (2024)
011051-4
the erasure lifetime. Postselected against erasures, we find a
low residual error rate of
5
.
06
ð
6
Þ
×
10
5
per X90, averaged
over a 24 hour period, with the remaining error likely
limited by a combination of residual decoherence within
the logical subspace and coherent calibration errors. These
error rates constitute an erasure noise bias of 43(1), and
show that single-qubit gates preserve the large noise bias
achieved while idling.
IV. MIDCIRCUIT ERASURE DETECTION
USING AN ANCILLA QUBIT
We next turn to the characterization of midcircuit erasure
detection. Our erasure check relies on a dispersive shift
(
2
π
×
1
.
6
MHz) induced on the ancilla qubit when the
dual-rail qubit is in the logical subspace (
j
0
L
i
;
j
1
L
i
) relative
to when it is in
j
00
i
[Fig.
1(e)
; precise shifts given in
Appendix
G
]
[34]
. The erasure check consists of a 540 ns
square microwave pulse on the ancilla qubit whose length
is chosen to excite the ancilla only if the dual rail is in
j
00
i
and not if the dual rail is in its logical subspace [Fig.
1(e)
].
After the microwave pulse, we perform a 340 ns readout of
the ancilla transmon (Appendix
B
).
There are three metrics to evaluate the performance of
erasure detection, motivated by the proposed use of erasure
checks in an error-correcting code
[13,14]
. First,
false-
positive
errors are those in which no error occurred but an
erasure is falsely flagged. Second,
false-negative
errors are
those in which an erasure error occurs but is not correctly
flagged. Finally,
erasure-check-induced dephasing
is the
dephasing on the dual-rail qubit induced by checking for
an erasure error when no such error had occurred. To
establish target performance levels for these metrics, we
consider a surface code protocol in which an erasure check
is performed alongside each 2-qubit gate to prevent the
diffusion of errors. Erasure check errors will add to the
effective error rates of the gates, and in instances where
the erasure check is slower than the gates, an additional
error will also be introduced due to the time associated
with the erasure check. For concreteness, we benchmark
against 1% erasure error per gate and 0.1% Pauli error per
gate; this pair of error rates is comfortably below the
surface code threshold
[14]
.
False positives are characterized by initializing the dual
rail in
j
1
L
i
, performing an erasure check, and then reading
out the dual-rail pair [Fig.
4(a)
]. We postselect on the final
pairwise readout showing the correct state was initialized,
and measure the probability that the erasure check
correctly indicated no error. We find a false-positive rate
of 0.58(1)% for
j
1
L
i
, limited by accidental excitation of
the ancilla when the dual rail has not decayed. We find a
higher false-positive rate of
0
.
8%
for the logical state
L
i
[measured independently in Fig.
4(c)
], which we
take as an effective average rate that does not distinguish
between assignment errors and any mechanism which
may induce an erasure error during the check. In a surface
code context, this type of error would
inject
extra
erasure errors into the system, and should thus be
compared to the 1% erasure error target.
False negatives are characterized by initializing the dual
rail in
j
00
i
, performing an erasure check, and then post-
selecting on a final readout which shows
j
00
i
[Fig.
4(b)
].
We find a false-negative rate of 1.54(1)%, limited by
T
1
decay of the ancilla during readout. In a surface code,
the quantity of interest is the probability that an erasure
event occurred and was not properly flagged in detection,
thus persisting and possibly propagating to undetectable
Pauli errors. The probability of such an event is the
chance of an erasure error during the erasure check,
FIG. 4. Characterizing midcircuit erasure detection. (a),(b) We
characterize false-positive and false-negative errors by initializing
a target state, either
j
1
L
i
or
j
00
i
, performing an erasure check, and
then analyzing its results postselected on a final readout con-
firming the correct initialization. We ensure initialization of the
ancilla in
j
0
i
by postselecting on an additional prereadout before
the sequence begins (not shown). (c) We characterize erasure-
check-induced dephasing by inserting a variable number of
checks into a spin-echo experiment with fixed total evolution
time
T
¼
114
μ
s. The
N
erasure checks are evenly spaced with
separation time
τ
e
¼
T=N
. We vary the phase of the final
π
=
2
pulse to measure the remaining degree of coherence, postselected
on all erasure checks indicating no erasure. Coherence appears to
grow with a small number of checks due to properly catching and
postselecting against leakage into
j
00
i
which reheats into the
subspace. This effect saturates, after which we observe little
dephasing per check at the level of
<
0
.
1%
(see Appendix
I
). The
postselection probability (right-hand subplot) decreases with
increasing erasure checks, first due to discarding shots with
leakage, then saturating at a rate of
0
.
8%
per check which we
interpret as a combination of false-positive assignment errors and
check-induced erasure events. We note that a positive erasure
detection event can with low probability, 0.0293(3)%, reexcite
the dual-rail qubit (see Appendix
B
).
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T
check
=T
erasure
2
.
9%
, multiplied by the false negative rate.
The probability of a missed erasure is then 0.04%,
comfortably below the 0.1% target Pauli level.
Finally, we measure the dephasing induced by each
erasure check by performing a spin-echo measurement on
the dual rail and inserting a variable number of erasure
checks [Fig.
4(c)
]. The dual-rail coherence appears to
improve and then plateau when inserting a small number
of erasure checks, as these checks correctly eliminate shots
in which the system decayed to
j
00
i
and then heated back
into the subspace. As more checks are inserted, remaining
phase coherence degrades only minimally, from which we
compute a conservative upper bound of
<
0
.
1%
error per
erasure check (Appendix
I
). Such dephasing errors inject
undetectable Pauli errors in each check, and thus should be
compared to the target 0.1% Pauli error level.
While the fidelities for these three metrics are below
their respective surface co
de thresholds, the current
erasure check time (880 ns) is slower than 2-qubit gate
protocols in this architecture of
200
ns
[14,28]
,and
would contribute a larger erasure error of
T
check
=T
erasure
2
.
9%
per gate. Faster erasure checks can be achieved
through larger dispersive couplings and optimized ancilla
readout
[35]
, or by using a single symmetrically coupled
readout resonator
[14]
. Additionally, ancilla reset and
reinitialization of the dual rail after an erasure event are
still needed, and we expect implementations can be
closely adapted from standard transmon reset and leakage
reduction protocols
[36
38]
.
V. ROBUST OPERATION AT FLEXIBLE
OPERATING POINTS
The dual-rail qubit enables a large erasure noise bias as
well as high-fidelity erasure detection. Importantly, these
features are not fine-tuned based on special operating
frequencies, but instead can be preserved at a broad range
of operating points. This robustness offers a significant
advantage over standard architectures in which tunable
transmons are expected to operate close to their flux-
insensitive sweet spots, but may be unable to due to
frequency collisions, parasitic modes, or TLSs. While
operating tunable transmons away from the sweet spot is
possible, extensive dynamical decoupling is needed to
recover
T
2
performance, and this decoupling may compli-
cate single-qubit or multiqubit gates
[39]
. In the dual-rail
qubit, however, this noise suppression is achieved pas-
sively, and carries over into the operation of single-qubit
gates, erasure detection, and proposed implementations of
multiqubit gates
[14]
.
We test this robustness by parking the two dual-rail
transmons at several operating points over a 350 MHz band
from 4.75 to 5.1 GHz which is mutually accessible by each.
We park each transmon individually at each of these
operating points to first characterize their individual proper-
ties, and find that Q2 has fairly uniform coherence over this
range since it is already far from its sweet spot, while Q1
s
coherence degrades as it is tuned away from the sweet spot
(Fig.
5
). Across this full range, except for a narrow band at
4.96 GHz impacted by a TLS (discussed in Appendix
O
),
the dual-rail qubit preserves
T
echo
2
coherence of hundreds
of microseconds, from 366(20) to
550
ð
30
Þ
μ
s. While more
investigations are needed to fully understand coherence
limitations, these measurements highlight the resilience of
the dual-rail qubit to flux noise which normally precludes
operating at highly flux-sensitive points. Additional adjust-
ments of the qubit parameters, such as tuning ranges and
coupling strengths, may improve performance and robust-
ness further.
FIG. 5. Robust operation away from sweet spot. (a) The dual-
rail qubit can be operated anywhere the two transmons can be
brought onto resonance, forming an engineered
tunable
sweet
spot. (b) At a range of operating points, we measure
T
echo
2
of each
underlying transmon individually as well as of the dual-rail qubit.
With the exception of a narrow region around 4.96 GHz, the dual-
rail qubit maintains a coherence of several hundred microseconds
over most of this range [
366
ð
20
Þ
550
ð
30
Þ
μ
s], with the modest
degradation likely due to increased effects of Q1
s flux noise
away from its sweet spot (Appendix
N
). These coherence times
are over an order of magnitude larger than the higher-coherence
transmon (Q1) and more than 2 orders of magnitude larger than
Q2. The dip in dual-rail
T
2
at 4.96 GHz is likely explained by a
TLS coupled to Q2 which is near resonant with the upper hybrid
mode at this operating point; a similar reduction is present on Q2
directly at an offset operating point, as visible in supplementary
datasets presented in Appendix
O
. Error bars are standard
deviations of many fit results (individual dots) for Q1 and Q2,
and are fit uncertainties for the dual-rail qubit.
H. LEVINE
et al.
PHYS. REV. X
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VI. DISCUSSION AND OUTLOOK
These results show that the dual-rail qubit is a prom-
ising, highly coherent building block for erasure-based
error correction architectur
es. Coherence within the dual-
rail subspace, even for transmons which are individually
noisy, is approaching the millisecond regime, while the
erasure lifetime is set by the transmon
T
1
. Subsequent
studies may complete the toolbox for operating an erasure
surface code, including developing 2-qubit gates between
dual-rail pairs
[28]
, reinitializing dual-rail qubits after
erasure errors, and exploring novel erasure detection
mechanisms
[14]
.
While the dual-rail transmon architecture is more com-
plex than standard transmon architectures, an optimized
approach can be taken which uses a single resonator for
both readout and erasure detection
[14]
. In this setup, a
dual-rail qubit involves only slightly more space on the
processor and only one additional control line compared to
standard transmons (Appendix
P
). For this modest increase
in complexity, the conversion of
T
1
decay to erasure errors
enables leveraging of higher thresholds and effectively
larger code distances as compared with Pauli errors,
offering a positive prospect for accelerating the path to
near-term below-threshold operation of error-corrected
processors and for long-term logical qubit performance.
Additionally, the broad effort toward scaling transmon
systems and improving transmon coherence will translate
naturally into the dual-rail architecture; for example,
improvement in transmon
T
1
will result in lower erasure
error rates.
The passive noise suppression also opens opportunities
to build dual-rail qubits out of other components which
have long
T
1
but shorter
T
2
coherence times. Candidate
components would include transmonlike circuits which
operate at lower values of
E
J
=E
C
, phase qubits, or
potentially fluxonium. Such physical qubits, which in
some regimes are undesirable when used on their own
due to short coherence, may form highly robust logical
qubits in a dual-rail architecture with improved erasure
lifetime while still maintaining strong phase coherence
within the dual-rail subspace.
Finally, in addition to prospects for enhancing logical
qubit performance, we note that the ability of erasure qubits
to postselect against errors may already offer significant
advantages for some tasks including quantum simulation
[23,40]
, sampling
[41]
, and nonverifiable algorithms
[40,42,43]
. For these applications, postselecting on the
lack of erasure errors enables more accurate reconstruction
of the probability distributions than would be achieved by
conventional quantum processors even with lower overall
error rates
[23]
.
Note added.
Recently, we became aware of related work
on the dual-rail encoding in microwave cavities
[44]
.
ACKNOWLEDGMENTS
We thank the technical support from across the AWS
Center for Quantum Computing, including the teams
involved with theory, design, fabrication, device packaging,
cryogenics, signals, software, procurement, and lab infra-
structure. We also thank Simone Severini, Bill Vass, and
AWS for supporting the quantum computing program.
Finally, we thank Jeff Thompson, Manuel Endres, and
David Schuster for helpful discussions and feedback on
the manuscript.
APPENDIX A: SUPERCONDUCTING DEVICE
The superconducting quantum circuit is shown in Fig.
6
,
and device parameters are summarized in Table
I
. Each of
the three transmons has a dedicated readout resonator, all of
which are coupled to a single transmission line. Q1 and Q3
have additional Purcell filters on their resonators. Transmon
capacitive couplings are realized via wedge couplers
[45]
.
Our device is fabricated using electron-beam lithography
(EBL) to pattern components onto a 100 nm aluminum
ground plane. The ground plane is connected across
coplanar waveguides through crossovers, realized as alu-
minum air bridges. The Josephson junctions are aluminum
based, fabricated using EBL followed by double-angle
evaporation. The junction electrodes are shorted to the base
layer metallization using Al-based bandages.
TABLE I. Summary of device parameters. Transmon frequen-
cies and coherence times, coupling strengths
g
ij
, and readout
resonator(RO)frequencies
ω
RO
and
κ
,
χ
values, reported at the
qubit idling points. Coherence values are the median of
25
measurements, with the listed uncertainty being the standard
deviation of those measurements (* except
T

2
for Q3, which is
from a single measurement). All coherence fits are to expo-
nential decays which properly capture the qualitative behavior
but neglect some small systematic deviations from exponential
behavior such as beating in Ramsey
T

2
fringes. The coupling
g
12
is directly measured, while the couplings
g
13
and
g
23
are
calculated based on the measured
ancilla dispersive shifts in
Appendix
G
.
Property
Q1
Q2
Q3 (ancilla)
ω
min
=
2
π
(GHz)
3.1
3.3
2.5
ω
max
=
2
π
(GHz)
5.1
6.1
3.95
ω
idle
=
2
π
(GHz)
5.1
5.1
3.74
η
=
2
π
(MHz)
193
204
196
T
1
(
μ
s)
36(12)
14(4)
38(5)
T

2
(
μ
s)
31(14)
1.29(6)
4.4(2)*
g
12
=
2
π
(MHz)
90.1
90.1
g
13
=
2
π
(MHz)
8.4
8.4
g
23
=
2
π
(MHz)
81.7
81.7
ω
RO
=
2
π
(GHz)
7.749
7.511
7.341
χ
RO
=
2
π
(MHz)
3.73(7)
0.32(1)
2.53(3)
κ
RO
=
2
π
(MHz)
9.3(4)
0.87(2)
6.7(2)
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The device is wire bonded to a printed circuit board
and cooled to
10
mK at the base of a Bluefors dilution
refrigerator. Microwave and baseband signals are delivered
from custom control hardware based on a Xilinx RFSoC,
with the signal chains shown in Fig.
7
.
APPENDIX B: TRANSMON READOUT
Fast transmon readout, particularly for the ancilla qubit,
is critical for erasure qubit experiments. We use a traveling
wave parametric amplifier (TWPA)
[46]
from MIT Lincoln
Laboratory to amplify readout signals and Purcell filters to
facilitate fast readout without degrading qubit lifetimes.
The ancilla readout consists of a 200 ns microwave tone
on the transmission line, while we integrate the readout
signal against a linear matched filter which extends for
an additional 140 ns after the drive tone ends
[47]
.We
assign thresholds in the IQ plane to preferentially minimize
false-positive rates (typically reaching
0
.
1%
) at the cost
of slightly increasing false-negative rates, which are mainly
limited by the few-percent probability of
T
1
decay during
readout. This choice of threshold is helpful to minimize
false-positive rates during erasure detection.
The dual-rail transmons Q1 and Q2 are read out at the
end of the circuit over
1
μ
s. During this period, Q2 is flux
pulsed up to its maximum frequency at 6.1 GHz, while Q1
remains at 5.1 GHz. For the data in Fig.
5
in which we tune
the operating point of the dual-rail qubit below 5.1 GHz,
the final readout always occurs while Q2 is flux pulsed
back to its maximum frequency.
We observe that for some operating points, reading out
the ancilla can stimulate transitions on the dual-rail qubit.
We attribute these to measurement-induced state transitions
[48,49]
. In particular, with the ancilla idling at 3.74 GHz,
we find that if the ancilla is excited, there is a small,
0.0293(3)%, chance that its readout will excite the dual-rail
FIG. 6. Optical image of superconducting device. False-color overlays indicate the functional purpose of each component needed to
define the transmon qubits, control them with microwave and flux lines, and perform readout.
H. LEVINE
et al.
PHYS. REV. X
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