††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Overcoming leakage in scalable quantum error correction

Kevin C. Miao  Google Quantum AI, Santa Barbara, CA 93117, USA  Matt McEwen
Google Quantum AI, Santa Barbara, CA 93117, USA  Department of Physics,
University of California, Santa Barbara, CA 93106, USA  Juan Atalaya  Dvir
Kafri  Leonid P. Pryadko  Andreas Bengtsson  Alex Opremcak  Kevin J. Satzinger
Zijun Chen  Paul V. Klimov  Chris Quintana  Google Quantum AI, Santa Barbara,
CA 93117, USA  Rajeev Acharya  Kyle Anderson  Markus Ansmann  Frank Arute
Kunal Arya  Abraham Asfaw  Google Quantum AI, Santa Barbara, CA 93117, USA
Joseph C. Bardin  Google Quantum AI, Santa Barbara, CA 93117, USA  Department
of Electrical and Computer Engineering, University of Massachusetts, Amherst,
MA 01003, USA  Alexandre Bourassa  Jenna Bovaird  Leon Brill  Bob B. Buckley
David A. Buell  Tim Burger  Brian Burkett  Nicholas Bushnell  Juan Campero
Ben Chiaro  Roberto Collins  Paul Conner  Alexander L. Crook  Ben Curtin
Dripto M. Debroy  Sean Demura  Andrew Dunsworth  Catherine Erickson  Reza
Fatemi  Vinicius S. Ferreira  Leslie Flores Burgos  Ebrahim Forati  Austin G.
Fowler  Brooks Foxen  Gonzalo Garcia  William Giang  Craig Gidney  Marissa
Giustina  Raja Gosula  Alejandro Grajales Dau  Jonathan A. Gross  Google
Quantum AI, Santa Barbara, CA 93117, USA  Michael C. Hamilton  Google Quantum
AI, Santa Barbara, CA 93117, USA  Department of Electrical and Computer
Engineering, Auburn University, Auburn, AL 36849, USA  Sean D. Harrington
Paula Heu  Jeremy Hilton  Markus R. Hoffmann  Sabrina Hong  Trent Huang
Ashley Huff  Justin Iveland  Evan Jeffrey  Zhang Jiang  Cody Jones  Julian
Kelly  Seon Kim  Fedor Kostritsa  John Mark Kreikebaum  David Landhuis  Pavel
Laptev  Lily Laws  Kenny Lee  Brian J. Lester  Alexander T. Lill  Wayne Liu
Aditya Locharla  Erik Lucero  Steven Martin  Anthony Megrant  Xiao Mi  Shirin
Montazeri  Alexis Morvan  Ofer Naaman  Matthew Neeley  Charles Neill  Ani
Nersisyan  Michael Newman  Jiun How Ng  Anthony Nguyen  Murray Nguyen  Rebecca
Potter  Charles Rocque  Pedram Roushan  Kannan Sankaragomathi  Christopher
Schuster  Michael J. Shearn  Aaron Shorter  Noah Shutty  Vladimir Shvarts
Jindra Skruzny  W. Clarke Smith  George Sterling  Marco Szalay  Douglas Thor
Alfredo Torres  Theodore White  Bryan W. K. Woo  Z. Jamie Yao  Ping Yeh
Juhwan Yoo  Grayson Young  Adam Zalcman  Ningfeng Zhu  Nicholas Zobrist
Google Quantum AI, Santa Barbara, CA 93117, USA  Hartmut Neven  Vadim
Smelyanskiy  Google Quantum AI, Venice, CA 90291, USA  Andre Petukhov  Google
Quantum AI, Santa Barbara, CA 93117, USA  Alexander N. Korotkov  Google
Quantum AI, Santa Barbara, CA 93117, USA  Department of Electrical and
Computer Engineering, University of California, Riverside, CA 92521, USA
Daniel Sank  Yu Chen  Google Quantum AI, Santa Barbara, CA 93117, USA

(August 27, 2024)

###### Abstract

Leakage of quantum information out of computational states into higher energy
states represents a major challenge in the pursuit of quantum error correction
(QEC). In a QEC circuit, leakage builds over time and spreads through multi-
qubit interactions. This leads to correlated errors that degrade the
exponential suppression of logical error with scale, challenging the
feasibility of QEC as a path towards fault-tolerant quantum computation. Here,
we demonstrate the execution of a distance-3 surface code and distance-21 bit-
flip code on a Sycamore quantum processor where leakage is removed from all
qubits in each cycle. This shortens the lifetime of leakage and curtails its
ability to spread and induce correlated errors. We report a ten-fold reduction
in steady-state leakage population on the data qubits encoding the logical
state and an average leakage population of less than $1\times 10^{-3}$
throughout the entire device. The leakage removal process itself efficiently
returns leakage population back to the computational basis, and adding it to a
code circuit prevents leakage from inducing correlated error across cycles,
restoring a fundamental assumption of QEC. With this demonstration that
leakage can be contained, we resolve a key challenge for practical QEC at
scale.

Quantum error correction (QEC) promises to exponentially suppress uncorrelated
errors in quantum computing devices, bridging the gap between achievable
physical error rates and the low logical error rates required for useful
quantum algorithms [1, 2, 3]. The surface code is a promising candidate for
experimental implementations of QEC, where a repetitive stabilizer circuit
protects a logical qubit state.

Superconducting transmon qubits [4, 5] represent a leading platform for
implementing surface code QEC, with recent demonstrations of architectures
compatible with QEC and capable of scaling [6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17]. However, the transmon is only weakly nonlinear, with transitions
between successive states closely spaced in frequency. Transitions from the
qubit _computational_ states to higher energy _leakage_ states are therefore
difficult to avoid. These leakage states can be significantly populated by
single-qubit gates [18, 19], entangling gates [20, 21, 22, 23, 24], and
measurement [25, 26].

Leakage is particularly dangerous in the context of QEC [27, 28, 29, 30]. A
key underlying assumption of QEC is that the physical errors to be suppressed
are sufficiently uncorrelated in both space and time. Contrary to this
requirement, a qubit in a leakage state can induce errors on multiple
neighboring qubits, even causing them to leak as well [29]. The correlated
spread of errors through the device represents a major problem for
experimental QEC. Identifying and post-selecting out leakage events has
permitted cutting-edge experiments on the surface code [16, 15], and partial
leakage removal has been integrated into surface code circuits [14, 17].
However, all these experiments displayed a characteristic rise in the number
of detected errors as the code progressed, indicative of accumulating leakage
population in the device. A demonstration of leakage removal from all qubits
in a surface code circuit has not yet been reported. Further, stabilizing the
leakage populations such that error rates do not grow over time is a
requirement for scalable QEC, and this remains an important open challenge.

Here, we study and remove the effects of leakage in a surface code circuit on
an array of transmon qubits. First, we detail the dynamics of leakage in the
QEC circuit and the spread of errors through space and time. We quantify the
effect of leaked qubits undergoing multi-qubit interactions, which is the
primary vehicle for spatial propagation of leakage. Second, we demonstrate the
effective removal of leakage from all qubits involved in the surface code
circuit. We show residual leakage populations averaged over all qubits are
suppressed to below $1\times 10^{-3}$, and do not grow as the code is extended
in time. Finally, we show that removing leakage improves logical performance.
Using a distance-21 bit-flip code with leakage removal, injected leakage
impacts logical performance equivalently to injected Pauli errors. This
confirms that leakage removal is effective in suppressing the correlated
nature of leakage-induced errors. Then, using a distance-3 surface code, we
show that leakage removal both decreases the rate of logical errors and
prevents the code performance from declining over time, proving that QEC can
be stable when carried out over many cycles. We extrapolate this behavior to
larger code distances operating well below threshold, where we find that
injected leakage impacts logical error rates in the same fashion as
uncorrelated Pauli errors. In summary, leakage removal overcomes an important
obstacle to growing QEC to algorithmically relevant scales.

##  1 Characterizing the spread of leakage

Figure 1:  Leakage in a structured QEC circuit.  a) The energy potential of a
transmon qubit, illustrating the computational energy levels
$\left|0\right\rangle$ and $\left|1\right\rangle$ (blue) and the leakage
levels $\left|2\right\rangle$ and higher (red). b) The circuit for surface
code QEC, showing a square grid comprised of measure qubits (light blue
circle) and data qubits (orange squares). The cycle consists of four layers of
entangling gates, along with intervening single qubit rotations, followed by
the measurement (M) and reset (R). The reset operation here is shown across
all qubits; it may be implemented as single qubit operations on the measure
qubit, or include entangling operations with various neighboring data qubits.
c) The time decay (main, blue) and spatial spread (inset) of leakage in a
distance-3 surface code following the injection of $\left|2\right\rangle$ on
the central data qubit. Each cycle takes approximately 1 $\mu$s, and leakage
population is measured at the end of each cycle. The expected decay of
$\left|2\right\rangle$ from $T_{1}$ relaxation on the leaked qubit alone is
indicated (dashed red). Excess leakage population is defined as the
subtraction of leakage population in the absence of injection from the leakage
population in the presence of injection.

Leakage states (Figure 1a) are particularly problematic in structured QEC
circuits because they are long-lived and spread through the device, inducing
correlated errors in both space and time. The surface code circuit displayed
in Figure 1b shows a single cycle, which consists of a number of moments. A
moment is a grouping of gates operated concurrently in time. Four such moments
correspond to CZ gates used to measure the surface code stabilizers. When a
qubit in the circuit leaks, subsequent gates involving that qubit produce
additional errors.

Figure 1c illustrates the dynamics of leakage in a distance-3 surface code
circuit. At the cycle labeled $0$, we inject a full
$\left|1\right\rangle\rightarrow\left|2\right\rangle$ rotation on the central
data qubit, producing an expected near-50% $\left|2\right\rangle$ population.
It takes many surface code cycles before this injected leakage population
decays sufficiently, with an exponential decay constant around $4.4$ cycles.
However, this decay is somewhat faster than the expected decay from $T_{1}$
relaxation of $\left|2\right\rangle$ alone. The insets show that the leakage
population does not stay on the injected qubit, but is also transported to
neighboring qubits as the circuit progresses. At the small code distance being
considered, this transport is enough to affect every qubit involved in the
circuit.

Without any attempt to remove it, a single leakage event persists for many
rounds and spreads a significant distance through the device, affecting many
measurements and inducing many error detection events. The number of
uncorrelated errors required to produce the same effect is the _decomposed
weight_ of the leakage event [27]. This high weight of leakage events when
decomposed into uncorrelated errors makes them especially problematic for QEC.

Figure 2:  Leakage transport and phase errors in CZ gates.  a) The eigenenergy
ladder for a pair of qubits satisfying the resonance condition for a diabatic
CZ gate, where the qubits are detuned by their common nonlinearity
$\left|\eta\right|$. We denote the two-qubit states $\left|HL\right\rangle$
with the higher (lower) energy qubit first (second). In addition to the
intended resonance
($\left|20\right\rangle\leftrightarrow\left|11\right\rangle$), higher levels
also satisfy a resonance condition, either directly
($\left|31\right\rangle\leftrightarrow\left|22\right\rangle$) or mediated by a
two-photon process
($\left|30\right\rangle\leftrightarrow\left|12\right\rangle$). b) The relative
population transport (net change in state populations) $\Delta P_{t}$ for the
diabatic CZ gate, including the first two leakage levels. The rotation in
$\left|20\right\rangle\leftrightarrow\left|11\right\rangle$ has been
calibrated to $2\pi$. Highlighted are the off-diagonal elements due to the
couplings between higher levels, with average relative population transport
$\overline{|\Delta P_{t}|}$ shown below. c) The two circuits used to measure
the relative population transport shown in (b). We subtract the population
transport $P_{t}$ in the baseline experiment without a CZ gate (right) from
the experiment with a CZ gate (left). d) The circuit for the modified Ramsey
experiment shown in (e) with an interleaved CZ gate to a neighboring qubit at
a higher frequency, followed by tomography on the lower frequency qubit. e)
The measured phase shift $\phi$ during the modified Ramsey experiment with the
neighboring qubit prepared in $\left|0\right\rangle$, $\left|1\right\rangle$,
or $\left|2\right\rangle$, shown in an ECDF over 20 qubit pairs, with the mean
value indicated by the dashed line. The CZ gate should produce a phase shift
of $\phi=0$ for an input $\left|0\right\rangle$, and a shift of $\phi=\pi$ for
an input $\left|1\right\rangle$. A spurious phase shift near $\phi\approx
0.65\pi$ is produced when the higher-energy qubit is prepared in
$\left|2\right\rangle$.

The precise dynamics of leakage depends primarily on the details of the
entangling gate used in the circuit. Here, we focus on the diabatic CZ gate
used in the Sycamore architecture [31, 14, 17]. This gate involves biasing
qubits to satisfy the resonance conditions indicated in Figure 2a, and tuning
the interaction strength to achieve a rotation of $2\pi$ in
$\left|11\right\rangle\leftrightarrow\left|20\right\rangle$. We maintain the
convention that the higher-energy qubit state is listed first in two-qubit
states $\left|HL\right\rangle$. This resonance condition also aligns other
resonances that involve leakage states. In particular, the
$\left|30\right\rangle\leftrightarrow\left|12\right\rangle$ resonance enables
a two-photon process which allows $\left|2\right\rangle$ on the lower-energy
qubit to move to $\left|3\right\rangle$ on the higher-energy qubit. Similarly,
the $\left|31\right\rangle\leftrightarrow\left|22\right\rangle$ resonance
enables $\left|3\right\rangle$ on the higher-energy qubit to cause the lower-
energy qubit to leak to $\left|2\right\rangle$, while the higher-energy qubit
remains leaked in $\left|2\right\rangle$. These so-called _leakage transport_
processes are what allow leakage to spread, even in a single QEC cycle.

The amount of leakage transport a gate produces is not normally calibrated,
and so depends on the chosen gate length and effective coupling between
levels. Figure 2b shows how a calibrated CZ gate affects populations, as
measured by the circuits shown in Figure 2c. In this device, we find around
18% of the population of $\left|30\right\rangle$ is transported to
$\left|12\right\rangle$ and vice versa. The transport population is around 61%
for $\left|31\right\rangle\leftrightarrow\left|22\right\rangle$. We can also
see the first indications of expected higher resonances such as
$\left|42\right\rangle\leftrightarrow\left|33\right\rangle$. Data for each
individual experiment and further characterisation of the readout can be found
in Supplementary Information Section S1.

Even in the absence of leakage transport, we find that leakage induces
additional errors in the CZ gate. When the higher-energy qubit is in
$\left|2\right\rangle$ and the lower-energy qubit is in the computational
basis, leakage transport is not possible but a significant phase error is
imparted on the non-leaked qubit. When a CZ gate is applied as in Figure 2d
with the higher-energy qubit in $\left|0\right\rangle$, we expect to see no
phase shift $\phi=0$ on the lower-energy qubit. With the higher-energy qubit
prepared in $\left|1\right\rangle$, we expect to see a phase shift $\phi=\pi$,
indicating a well-calibrated CZ gate. Figure 2e shows the relative phase for
20 pairs of qubits. When computational states $\left|0\right\rangle$ and
$\left|1\right\rangle$ are prepared we see tight groupings around the expected
phase shifts $\phi=0$ and $\phi=\pi$, respectively. However, when a leakage
state is prepared on the higher-energy qubit, we see a phase shift near
$\phi\approx 0.65\pi$. This represents a significant computational error on
the non-leaked qubit, and is a significant source of errors to be detected and
corrected as leakage spreads.

These results illuminate the dangers of leakage: A single leakage event on any
qubit will expose many CZ gates to a leaked input state before it decays
sufficiently. Each of these interactions has a significant probability to
introduce new computational errors, move the leakage to another qubit, or
induce additional leakage on previously non-leaked qubits. In QEC circuits,
these effects are damaging enough that they must be included in simulations to
achieve good agreement with experimental performance [17]. Accordingly, we are
motivated to remove leakage in the code circuit so as to suppress these
effects.

##  2 Suppressing leakage populations during a QEC circuit

Having better understood the dangers of leakage in QEC circuits, we turn to
removing it. An unconditional reset gate can remove all energy from a qubit,
including when it starts in a leakage state, and can be applied to the measure
qubits at the end of each cycle [32, 33, 34, 35, 36]. However, our study of
leakage transport motivates the need to remove leakage from the data qubits as
well. Leaving the computational state intact is incompatible with
unconditional reset and requires a more delicate _leakage removal_ operation.

Three broad approaches for leakage removal have been proposed: swap-type [27,
37], where the roles of measure and data qubits are exchanged at a regular
interval by the use of additional operations; feedback-type [29, 30] where the
leakage is identified classically from measurement patterns and feedback is
applied to return the qubit to the computational subspace; and direct-type
[38] where an operation is used to remove leakage from a qubit without
disturbing the computational states. In light of our findings on leakage
transport, swap-type strategies become more difficult to justify; only half
the qubits are reset in each cycle, and so leakage may still move between
qubits and thereby spread through time. Similarly, the conditional nature of
feedback-type approaches prevents them from fully solving the leakage problem
– leakage states cause several errors before they are noticed and corrected.
Hence, we pursue a direct removal approach.

In the following sections, we present and compare three leakage removal
strategies. First, No Reset forgoes any operations at the end of the cycle,
representing the best case for a simple Pauli error model, but the worst case
for leakage. Second, MLR applies multi-level reset (MLR) gates [35] on measure
qubits immediately after measurement at the end of every cycle. This adds
additional error to the cycle due to additional data qubit idle time while the
gate is performed, but has been previously shown to remove leakage population
and improve logical performance compared with the baseline No Reset strategy
[35]. Finally, in DQLR we perform a multi-level reset on the measure qubits
followed by data qubit leakage removal (DQLR), consisting of a two-qubit
interaction to transport leakage from data to measure qubits and a reset gate
on measure qubits. Additional details on the DQLR process and constituent
operations are included in Supplementary Information Section S2.

Figure 3:  Leakage population during surface code execution.  a) Average
leakage populations for data qubits (squares) and measure qubits (circles)
measured at the end of each surface code cycle with No reset (red), MLR
(green), and DQLR (blue). b) Top: The surface code circuit shown for a pair of
neighboring measure and data qubits. Each surface code cycle is highlighted
(rounded rectangles). Bottom: A single surface code cycle showing each moment
in the cycle. c) Leakage populations after each moment in the cycle for MLR
(green) and DQLR (blue) leakage removal strategies, averaged over data qubits
(squares) and measure qubits (circles) and over cycles 25–30.

To compare the leakage dynamics for the three strategies, we implement a
distance-3 surface code on a Sycamore processor. We measure the evolution of
leakage population as the surface code progresses by truncating the circuit in
time and performing a measurement that can resolve $\left|2\right\rangle$ on
all qubits [35]. In Figure 3a, we perform this truncation at the end of each
surface code cycle (top of Figure 3b). Using No Reset, we observe a gradual
rise in leakage populations over all qubits, reaching nearly 5% average
leakage population for data qubits and nearly 3% for measure qubits over 30
cycles. We note that, even after 30 cycles, leakage populations have not
stabilized and continue to grow. Using MLR reduces average measure qubit
leakage populations to about $3\times 10^{-4}$, but average data qubit
populations still rise to over 1.5%. Using DQLR suppresses average leakage
populations to around $10^{-3}$ for data qubits and less than $10^{-4}$ for
measure qubits. Most importantly, DQLR maintains these levels throughout the
full 30 cycles.

We can use the same technique to study the dynamics of leakage within a
surface code cycle, by truncating the circuit at each moment midway through a
cycle (bottom of Figure 3b). Figure 3c shows the leakage population measured
after each moment in the cycle, averaged over cycles 25–30 where the leakage
populations have stabilized. We neglect the No Reset strategy here, as leakage
populations do not stabilize. With MLR, the average leakage population on the
data qubits saturates to a stable value around 1.5%, consistent with Figure
3a. However, the average measure qubit leakage population starts each cycle at
a very low value near $2\times 10^{-4}$, grows over the course of the cycle as
operations produce leakage, and is then reduced back to its initial low value
by the reset procedure. This lets us estimate that the operations produce
around $5\times 10^{-3}$ leakage in each cycle. With DQLR, we see that leakage
populations for both measure and data qubits grow over the course of the
cycle, and are removed by the reset procedure. The data qubits start each
cycle with around $1\times 10^{-3}$ leakage population, again increasing to
around $5\times 10^{-3}$ immediately following measurement, before it is
removed. The measure qubits attain even lower leakage populations compared to
MLR.

These results demonstrate that our DQLR procedure successfully suppresses
steady-state leakage populations to previously unachievable levels and
stabilizes those levels over the course of a long QEC circuit. The removal
strategy also contains the leakage dynamics to a single cycle. However, the
residual ability for leakage to spread and induce correlated errors within a
single round [28] should be the subject of further study.

##  3 Effect on QEC logical performance

Having achieved low leakage populations in both data qubits and measure qubits
with our DQLR procedure, we turn to evaluating logical performance. We
consider two codes providing complementary information: a distance-21 bit-flip
code and a distance-3 surface code. Our physical qubit error rates place the
surface code close to threshold, whereas the bit-flip code is well below
threshold [14, 17]. The vastly lower logical error rates for the bit-flip code
give us finer resolution on the effect of leakage within the code. In
contrast, the surface code is a more challenging circuit for calibration and
operation, and is sensitive to both bit-flip and phase-flip errors, providing
an environment where more potentially adverse effects from reset can be
detected and measured.

Figure 4:  Bit-flip code logical performance and dependence on injected
errors.  a) Logical error probability for a distance-21 bit-flip code run for
60 cycles, under the effect of either injected leakage (dark circles) or
injected Pauli errors (light squares). Three leakage removal strategies, No
reset (red, unfilled), MLR (green, semi-filled), and DQLR (blue, filled), are
considered. Lines are fits to experimental data using a power law with an
offset. Below: Highlights of fits to experimental data (solid) and numerical
simulations (dashed) for the MLR and DQLR strategies. b) Circuits for the bit-
flip code, showing the error injection locations for both leakage (left) and
Pauli errors (right).

Figure 4a shows the logical error probability of a distance-21 bit-flip code
carried out to 60 cycles while introducing both leakage and Pauli errors. We
inject leakage population $P_{L}$ into all qubits by applying a
$\left|1\right\rangle\leftrightarrow\left|2\right\rangle$ rotation on each
qubit immediately after the first Hadamard gate layer (Figure 4b, left), where
the rotation angle $\theta_{L}$ is

$\displaystyle\theta_{L}$
$\displaystyle=2\sin^{-1}\left(\sqrt{2P_{L}}\right).$

We compare $P_{L}$ to injected Pauli error “population” $P_{P}$, which is
produced by $X$ and $Z$ rotations on the data and measure qubits (Figure 4b,
right), respectively, taking advantage of the classical nature of the bit-flip
code. The Pauli error rotation angle $\theta_{P}$ is

$\displaystyle\theta_{P}$ $\displaystyle=2\sin^{-1}\left(\sqrt{P_{P}}\right),$

where the missing factor of 2 relative to the definition of leakage population
accounts for Pauli rotations always affecting the qubit state in the
computational basis, whereas leakage injection only applies to qubit
population in $\left|1\right\rangle$. We fit the experimental data and
numerical simulations to an offset power law as a guide, as detailed in
Section S5 of the Supplementary Information.

With No Reset, even small amounts of injected leakage population less than 1%
cause the logical error probability to rise above 40%. This is in contrast
with correctable Pauli errors, which can be introduced to around 5% population
before similar logical error probabilities are encountered. With MLR, the
logical error probability is drastically lowered without injection, consistent
with prior measurements in bit-flip codes [35]. Still, the logical error
probability rises much more rapidly when injecting leakage compared to
injecting Pauli errors. We attribute this to unmitigated leakage accumulation
on the data qubits, which leads to high decomposed weight of uncorrelated
errors and ultimately logical errors. When we prevent this leakage buildup
with DQLR, we observe a much smaller difference between the code’s response to
injected leakage compared to injected Pauli errors. This is strong evidence
that the DQLR operation has successfully reduced the decomposition weight of a
leakage event to near 1. In this situation, leakage has around the same
influence on logical performance as an equivalent amount of Pauli error, and
has been prevented from effectively spreading and inducing correlated errors.

We also note the good agreement between data and numerical simulation for
injected leakage and Pauli errors, quantifying our understanding of the
effects of leakage in the code with both MLR and DQLR strategies. In both
cases, we note that we slightly underestimate the logical error induced by
injected leakage, illustrating the difficulties of fully capturing the effect
of correlated errors even with DQLR preventing substantial spread across
cycles, and emphasising the importance of future work on leakage dynamics
inside a single cycle. Nonetheless, the close correspondence of the Pauli
simulation to the injected leakage experimental data for DQLR helps justify
future Pauli simulations as useful estimates of final code performance when
leakage is removed each cycle.

Figure 5:  Surface code logical performance and dependence on injected errors.
a) The detection probability averaged for the weight-4 stabilizers in a
distance-3 surface code under the three leakage removal strategies studied in
this work. b) Logical error probability for a distance-3 surface code run for
15 cycles, under varying injected leakage population and the three different
leakage removal strategies studied in this work. The inset shows that the
circuit has an included layer where leakage is injected by performing a
$\left|1\right\rangle\leftrightarrow\left|2\right\rangle$ rotation. c)
Dependence of projected surface code error budget $1/\Lambda_{5/7}$ (the
inverse of the exponential error suppression factor between a distance-5 and
distance-7 surface code) under injected leakage population compared between
MLR (green) and DQLR (blue). Solid lines are fits to a ratio of offset power
laws, while the dotted light blue line is a linear fit of the data using DQLR.

Figure 5a shows the average detection probabilities corresponding to the
weight-4 stabilizers in the distance-3 surface code. Detection probabilities
are the fraction of the total number of experiments where an error was
detected on a given stabilizer. With No Reset, the buildup of leakage
population produces more errors as the code progresses, creating a rising
pattern of detection probability. With MLR, a large portion of this rise is
mitigated, but the detection probability still rises by 2.5% over the course
of the first 15 cycles. With DQLR the detection probability immediately
stabilizes to around 18% and remains steady throughout the code duration. We
attribute this to the recurrent removal of leakage on all qubits preventing
growth in leakage populations and resulting correlated errors over time. This
resolves a key concern in state of the art QEC work [16, 15, 17] where
detection probabilities were found to rise even with partial leakage removal
or post-selection. These results confirm the relationship between rising
detection probability and rising leakage populations and demonstrate the
resolution of this effect.

In Figure 5b, we evaluate the three leakage removal strategies by measuring
the logical error probability of a distance-3 surface code after 15 cycles. At
0% injected leakage the circuit corresponds to the standard code circuit with
an additional idle where the injection is otherwise inserted. Over the range
of injected leakage population values, No Reset exhibits the worst logical
performance, followed by MLR, with DQLR having the lowest logical error
probability. This confirms that DQLR improves logical errors by suppressing
correlated errors from leakage, despite the additional cycle time and errors
introduced by the DQLR operations. Further, No Reset and MLR degrade in
logical performance faster with more injected leakage when compared to DQLR.

In order to study surface code performance in a regime further below
threshold, we turn to numerical simulations of distance-5 and distance-7
surface codes. To consider scaling performance, we use the exponential error
suppression factor $\Lambda_{5/7}$, defined as
$\Lambda_{5/7}=\varepsilon_{5}/\varepsilon_{7}$, where $\varepsilon_{5}$ and
$\varepsilon_{7}$ are the logical error rates for a distance-5 and distance-7
surface code, respectively. In Figure 5c, we investigate $\Lambda_{5/7}$ for a
hypothetical device with lower component errors than what is currently
realizable (see Supplementary Information Section S6 for details). In
particular, we set intrinsic leakage rates to zero and vary the probability of
leakage injection. With no leakage in the system, $\Lambda_{5/7}\approx 7.2$,
independent of leakage removal strategy. However, when injecting up to
$4\times 10^{-3}$ leakage population per round (comparable to intrinsic
leakage rates in current devices), the surface code error budget
$1/\Lambda_{5/7}$ [17] rises rapidly and nonlinearly for MLR. In contrast,
with DQLR, leakage increases $1/\Lambda_{5/7}$ much more slowly and with a
near-linear dependence on injected leakage population, characteristic of an
uncorrelated error source [14, 17]. With this ability to maintain effective
error suppression in the presence of leakage, DQLR successfully mitigates the
dangers of correlated leakage-induced errors to scalable QEC.

##  4 Summary and Outlook

We have demonstrated the effective removal of leakage from all qubits involved
in a surface code QEC circuit. Moreover, we have shown that when leakage is
removed on all qubits, correlated leakage-induced errors are suppressed. At
the same time, the logical performance of the code improves outright and
stabilizes in time. We confirm the conjecture that growth in logical errors is
attributable to leakage, and we do not uncover other major sources of logical
error that grow as the code continues in time.

With these findings, we unequivocally resolve the longstanding concern that
qubits with weak nonlinearity cannot successfully implement QEC at long times
due to correlated leakage-induced errors. As such, we confirm that large
arrays of transmon qubits are a viable and promising architecture for QEC at
scale.

## Declarations

### Acknowledgments

The authors would like to acknowledge and thank Rami Barends for his early
work and guidance on the subject of data qubit leakage removal.

### Competing interests

The authors declare no competing interests.

### Availability of data and materials

The data necessary to reproduce the findings for this study are available upon
reasonable request, or at 10.5281/zenodo.7302032.

### Author contributions

K.C.M. and M.M. developed leakage removal operations, designed and performed
the experiments, analyzed the data and prepared the manuscript. J.A. provided
theoretical models and analysis. D.K. and L.P. provided simulation tools.
A.B., A.O., K.J.S., Z.C., P.V.K., C.Q. provided and maintained key components
of the experimental setup. D.S. and Y.C. provided technical guidance and
helped prepare the manuscript. All authors contributed to the fabrication
process, experimental set-up and operation, and revision of the manuscript.

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Supplementary information for _Overcoming leakage in scalable quantum error
correction_

Supplementary information for _Overcoming leakage in scalable quantum error
correction_

##  S1 Effects of leakage on diabatic CZ gate

During the diabatic CZ gate, additional levels are placed on resonance and
contribute to the leakage transport phenomenon depicted in the main text
Figure 2. The resonance
$\left|30\right\rangle\leftrightarrow\left|12\right\rangle$ allows a two-
photon transition mediated by $\left|21\right\rangle$, which is detuned by
around the nonlinearity $\eta$. If $g$ is the induced coupling between
$\left|11\right\rangle$ and $\left|20\right\rangle$, then the effective
couplings are:

$\displaystyle g_{\left|30\right\rangle\leftrightarrow\left|21\right\rangle}$
$\displaystyle=\sqrt{3}g,$ $\displaystyle
g_{\left|21\right\rangle\leftrightarrow\left|12\right\rangle}$
$\displaystyle=2g,$ $\displaystyle
g_{\mathrm{eff}}=g_{\left|30\right\rangle\leftrightarrow\left|12\right\rangle}$
$\displaystyle=-g_{\left|21\right\rangle\leftrightarrow\left|12\right\rangle}\times
g_{\left|30\right\rangle\leftrightarrow\left|21\right\rangle}/\eta.$

Then, for a CZ gate where $g$ is maintained for time $t$, the population
transport $P_{t}$ for
$\left|30\right\rangle\leftrightarrow\left|12\right\rangle$ can be estimated
as

$\displaystyle P_{t}=\sin^{2}\left(g_{\mathrm{eff}}t\right).$

Figure S1:  Measuring leakage transport.  a) Readout clouds for the pair of
qubits used in the leakage transport experiment, showing good
distinguishability between all of the lowest four qubit energy states. b) Raw
measured population transport matrices for the two transport experiments. In
both matrices, we can see the effect of $T_{1}$ decay during measurement,
which is enhanced for the higher levels. Subtracting “Baseline” from “With CZ
gate” produces the matrix shown in Figure 2b of the main text. c) Individual
relative population transport values $\Delta P_{t}$ for relevant leakage
transport mechanisms
$\left|30\right\rangle\leftrightarrow\left|12\right\rangle$ and
$\left|31\right\rangle\leftrightarrow\left|22\right\rangle$. The average of
the norm of these values produce the population transport values at the bottom
of Figure 2b of the main text.

To measure the leakage transport in the diabatic CZ gate, we calibrate a
readout pulse capable of distinguishing all of the four lowest qubit energy
levels, as shown in Figure S1a. When we encounter $\left|4\right\rangle$
during measurement using this readout pulse, we assign and identify it as
$\left|3\right\rangle$. The baseline experiment consists of preparing a given
two-qubit state using microwave drives and then performing simultaneous
readout of the qubits. The “Baseline” matrix of Figure S1b shows the results
of this experiment, illustrating that performing readout simultaneously does
not impact the high distinguishability between all two-qubit states. We can
also see that the majority of the error in this simultaneous readout is due to
$T_{1}$ decay during the readout process. These decay channels reduce the
populations on the main diagonal by a few percent, and become more prominent
for higher levels. The “With CZ gate” matrix of Figure S1b shows the same
experiment with a CZ gate inserted between state preparation and measurement.
We then see new off-diagonal processes corresponding to leakage transport. We
subtract the “Baseline” matrix from the “With CZ gate” matrix to produce the
matrix shown in Figure 2b of the main text. The leakage transport processes
that are most relevant to the spread of leakage in our system correspond to
the $\left|30\right\rangle\leftrightarrow\left|12\right\rangle$ and
$\left|31\right\rangle\leftrightarrow\left|22\right\rangle$ rotations. In
Figure S1c, we numerically show the individual relative population transport
values $\Delta P_{t}$ for these two processes. We take the mean of the
absolute value of the relative population transport values to determine the
values at the bottom of Figure 2b of the main text.

Figure S2:  Measuring leakage phases.  a) Raw data for the leakage phase
experiment on a single pair of qubits. Each of $\left|0\right\rangle$,
$\left|1\right\rangle$, and $\left|2\right\rangle$ is prepared on the higher-
energy qubit, and then a Ramsey experiment with an interleaved CZ gate is
performed on the lower-energy qubit. The solid lines are sinusoidal fits and
the dashed lines indicate the extracted phase shifts $\phi$ shown in the main
text. b) The hardware circuit executed in the above experiment in the
$\left|2\right\rangle$ case. The two initial rotations (X and L) on the
higher-energy qubit (top) are removed when preparing $\left|0\right\rangle$,
and the second rotation L is removed when preparing $\left|1\right\rangle$.

To measure the spurious phase shift generated by leaked higher-energy qubits
undergoing the CZ gate, we perform the experiment detailed in Figure S2a. The
higher-energy qubit in the pair is prepared in each of $\left|0\right\rangle$,
$\left|1\right\rangle$, and $\left|2\right\rangle$, while a Ramsey experiment
is performed on the lower-energy qubit with an interleaved CZ gate. On the
lower-energy qubit, we vary the tomography phase $\phi_{T}$ of the second
pulse relative to the first pulse in the Ramsey experiment to record data as
shown in Figure S2b for each input state. We fit a sinusoid to the
experimental data and extract the phase offset $\phi$. We perform this
experiment on 20 pairs of qubits, and show the empirical cumulative
distribution function of the extracted phase offsets in Figure 2e of the main
text.

##  S2 Leakage Removal Strategy Details

We studied three leakage removal strategies; No reset, MLR and DQLR. We now
describe them in greater detail.

For No reset, we add no additional operations at the end of each cycle.
Because this prepares the qubit for the next cycle in whichever state was
measured rather than deterministically in $\left|0\right\rangle$, this also
requires the redefinition of the detectors in the surface and bit-flip code
circuits: rather than comparing time-neighboring measurements, we compare
time-next-neighboring measurements on the same measure qubit to detect errors
[10]. This redefinition has an insignificant impact on code performance,
especially when compared to the studied effects of leakage, and so we neglect
it from our analysis.

For MLR, we add the multi-level reset (MLR) operation introduced in [35] on
the measure qubits at the end of each cycle. Additional pulse shaping on the
diabatic return and calibration improvements allow us to achieve gate times of
160 ns without impacting performance.

For DQLR, we first perform the MLR operation on all measure qubits, as in MLR.
Following that, we perform the DQLR procedure, which consists of a
LeakageISWAP gate between pairs of measure and data qubits, and a second reset
of the measure qubits. The LeakageISWAP gate is similar to the diabatic CZ
gate used in the surface code and bit-flip code cycles, but executes an ISWAP
gate in the $\left|11\right\rangle-\left|20\right\rangle$ subspace. We note
that this DQLR procedure relies on the high fidelity of the preceding MLR
operation; when the measure qubit is prepared in $\left|0\right\rangle$, the
LeakageISWAP gate removes $\left|2\right\rangle$ on the data qubits. Any reset
error leaving $\left|1\right\rangle$ on the measure qubits will be converted
into leakage on the data qubit by the LeakageISWAP gate. Our results show that
this error path is sufficiently low in probability so as not to increase the
leakage population on the data qubits.

Figure S3:  Cross-entropy benchmarking (XEB) of DQLR operation.  Inferred XEB
error per cycle for 9 data qubits when the DQLR operation is applied on the
data qubit, compared to XEB error per cycle when the data qubit idles for the
equivalent time. Shaded region indicates 1 SD error of inferred XEB error per
cycle. Vertical dashed lines indicate the mean XEB error per cycle over all
data qubits. (Inset) XEB circuit where random single-qubit unitary rotations
(U) are repeatedly applied to the target qubit, interleaved with a reset
operation (R), which is either DQLR or idle (Idle). The final state of the
target qubit is measured. The cross-entropy between the measured and expected
distribution of states is calculated and the resulting XEB error per cycle is
inferred.  Figure S4:  Leakage dynamics in a surface code with different
removal strategies.  Comparison of excess leakage population dynamics over 5
cycles for all qubits in a distance-3 surface code after a full
$\left|1\right\rangle\rightarrow\left|2\right\rangle$ leakage injection during
the first cycle. With No reset, leakage transport mechanisms lead to
increasing leakage population over nearly all qubits involved in the code.
With the introduction of MLR, a sink for leakage population is present on
measure qubits, mitigating the spread of leakage from leakage transport
effects. The central leakage-injected data qubit still remains significantly
leaked, even after 5 cycles. Using DQLR, leakage populations on all qubits in
the code are brought to about $1\times 10^{-3}$ or lower within 2 cycles.

Ideally, DQLR should not induce additional errors on the data qubit. In
particular, when the lower-energy measure qubit is in $\left|0\right\rangle$,
the LeakageISWAP gate should act as an identity operation on the data qubit
computational basis. However, the non-zero time to execute the DQLR procedure
introduces incoherent errors caused by relaxation, in addition to coherent
errors from miscalibration. We evaluate the impact of the DQLR procedure on
the data qubit state using cross-entropy benchmarking (XEB). The inset of
Figure S3 shows the experimental circuit used to evaluate XEB error. The upper
and lower qubits mimic the role of data and measure qubits, respectively. The
section of the circuit within the parentheses is repeated a variable number of
times, and the final state of the upper qubit is measured. The cross-entropy
of the measured and expected distributions of states is calculated as a
function of repetitions, and then the XEB error per repetition is extracted.
In a given repetition, a random unitary U is executed on the upper qubit,
followed by a reset operation R. In the case of DQLR, R is substituted with
the DQLR procedure. We compare this to the Idle case, where R is replaced with
waiting for the duration of the DQLR procedure. We carry this measurement out
over the 9 pairs of data and measure qubits corresponding to the pairings used
in the distance-3 surface code experiment. By comparing the resulting
distribution of XEB error per cycle for DQLR and Idle, we conclude that the
DQLR operation does not induce significantly more errors than idling for the
equivalent duration. Furthermore, the mean error rate of less than $2.5\times
10^{-3}$ per cycle when using the DQLR operation is low enough so that the
operation is suitable to be added to a sensitive QEC circuit such as the
surface code. We note that leakage is still a relevant consideration in this
XEB experiment, and is captured by XEB error per cycle as an incoherent error.
Thus, it is possible that the leakage removal properties of DQLR result in
underreported XEB error per cycle when compared to Idle, which allows for
leakage to accumulate over the course of the XEB circuit.

##  S3 Effect of reset strategies on leakage dynamics

As we have demonstrated in the main text, leakage transport can move leakage
population from qubit to qubit in a structured circuit such as the surface
code. Once a data qubit is leaked, leakage removal strategies must be employed
or the leakage population may remain for many QEC cycles and cause additional
leakage and leakage-induced error through leakage transport. In Figure S4, we
evaluate the dynamics of leakage population in a surface code under the three
leakage removal strategies discussed in this work. We fully inject leakage at
the beginning of the first cycle by performing a
$\left|1\right\rangle\rightarrow\left|2\right\rangle$ rotation on the central
data qubit to obtain near-50% population of $\left|2\right\rangle$. We measure
excess leakage population, which is defined as the leakage population without
injection subtracted from the leakage population with injection. This allows
us to separate the contribution of intrinsic heating to leakage population
dynamics from the injected leakage population.

For No reset, leakage transport allows for leakage to move freely throughout
the qubits surrounding the central data qubit, eventually resulting in
measurable excess leakage population in nearly all of the 17 qubits involved
in the distance-3 surface code. The lack of leakage removal procedures on
either the measure qubits or data qubits leaves $T_{1}$ energy relaxation of
$\left|2\right\rangle$ as the primary mechanism for leakage population
mitigation. As we showed in Figure 1c of the main text, $T_{1}$ of
$\left|2\right\rangle$ can be longer than 10 surface code cycles, and this
number is expected to increase as qubit coherence improves and as surface code
cycle durations shorten. Hence, relying on energy relaxation is not a viable
strategy for leakage removal in QEC. This is partially addressed by MLR by
applying a multi-level reset gate on all measure qubits at the end of each
cycle. In Figure S4, the effect of this operation appears as reduced excess
leakage populations on all measure qubits at the end of every cycle. However,
leakage transport within a QEC cycle can still move leakage population beyond
nearest-neighbor qubits. This is readily observed in the excess leakage
population of the data qubit two sites away from the central data qubit, which
exhibits increasing population even though the measure qubit between the two
data qubits has its leakage population removed by the MLR operation at the end
of every cycle.

DQLR suppresses the ability of leakage to hop between data qubits by directly
removing a large fraction of all the data qubits’ leakage populations at the
end of each cycle. In particular, the central data qubit has its excess
leakage population reduced to less than 1% after the first cycle. Similarly,
other data qubits that previously leaked due to leakage transport have their
excess leakage populations reduced close to the measurement floor. The
shortened lifetime of leakage on all qubits is clearly seen for DQLR after two
QEC cycles, where excess leakage population over all qubits is less than
$1\times 10^{-3}$.

##  S4 Effect of reset strategies on QEC error detection

Figure S5:  Logical performance of bit-flip code in time.  Logical error
probability of the distance-21 bit-flip code over 60 cycles for the three
reset strategies No reset (red), MLR (green), and DQLR (blue). For MLR and
DQLR, we also execute the code while injecting 1% leakage population per
cycle. Early in the code (fewer than 5 cycles), boundary effects cause all
three reset strategies to perform similarly. As the code progresses through
more and more cycles, however, logical performance for the three strategies
diverge as leakage populations are handled differently.  Figure S6:
Distance-3 surface code detection probabilities.  Average detection
probabilities over 30 surface code cycles for weight-2 (circles) and weight-4
(squares) stabilizers for three leakage removal strategies No reset (red), MLR
(green), and DQLR (blue). Individual stabilizer detection probabilities are
shown as lighter lines.

In Figure S5, we compare the time dynamics of the bit-flip code under the
three different reset strategies presented in the main text. We execute a
distance-21 bit-flip code over 60 cycles, as described by the circuit in
Figure 4d of the main text. For MLR and DQLR, we inject both 0% and 1% leakage
population in each cycle, whereas for No reset we do not inject leakage. In
the first few cycles of code execution, differences in the logical performance
of the bit-flip code between the various leakage removal strategies and
injection populations are difficult to distinguish – we attribute this to
time-boundary effects where physical errors have not sufficiently accumulated
to manifest as a logical error, and statistical limitations where the logical
error probability is much smaller than what is resolvable by the number of
trials. However, the logical performance for the three leakage removal
strategies begins to diverge after about 10 cycles. The accumulation of
leakage on measure and data qubits causes a rapid rise of logical error
probability for No reset, where it exceeds $1\times 10^{-2}$ error by 25
cycles. This is in contrast to MLR and DQLR without leakage injection, which
continue to have less than $3\times 10^{-3}$ logical error probability through
60 cycles; DQLR has the best performance with about $1\times 10^{-3}$ logical
error probability at 60 cycles.

Turning to the cases where we inject 1% leakage population per cycle for MLR
and DQLR, we observe a notable distinction in logical error probability
scaling over cycle number. With MLR, the logical error probability rises to
about $1\times 10^{-2}$ by 30 cycles, with a similar qualitative behavior to
No reset without leakage injection. However, when we use DQLR, the code
sustains logical error probabilities of less than $5\times 10^{-3}$ over 60
cycles. This is a signature that QEC scaling in time can be more easily
achieved when using DQLR.

In Figure S6, we present the average weight-2 stabilizer detection
probabilities in addition to the average weight-4 stabilizer values already
shown in Figure 5a of the main text. Additionally, we show the detection
probabilities associated with the individual stabilizers. We can draw parallel
conclusions for the behavior of weight-2 stabilizers as we did for weight-4
stabilizers in the main text. For No reset, the average weight-2 stabilizer
detection probabilities rise and do not stabilize over the course of 30
surface code cycles, and even exhibit damped oscillations at early cycles.
When using MLR, the detection probability stability improves significantly and
the average probability only rises by about $1\times 10^{-2}$ over 10 cycles
before flattening. However, the best performance and stability is achieved
with DQLR, where the average weight-2 stabilizer detection probability remains
at 11% over the course of the entire 30-cycle experiment.

##  S5 Fitting techniques for logical error probability

We employ phenomenological models to fit experimental and simulated data. The
models are implemented on the logical error per cycle $\varepsilon$ of the QEC
code. In experimental and simulated data, we measure logical error probability
$p_{L}$ after $n$ QEC cycles. The conversion between $\varepsilon$ and $p_{L}$
after $n$ cycles is given by

$\displaystyle\varepsilon$
$\displaystyle=\frac{1-\left(1-2p_{L}\right)^{\frac{1}{n}}}{2},$ (S1)
$\displaystyle p_{L}$
$\displaystyle=\frac{1-\left(1-2\varepsilon\right)^{n}}{2}.$ (S2)

With respect to injected error population $P$, $\varepsilon$ can be modeled as
a power law with an offset,

$\displaystyle\varepsilon\left(P\right)$
$\displaystyle=a\left(P+P_{0}\right)^{b},$ (S3)

where $a$, $b$, $P_{0}$ are phenomenological free parameters. We do not
ascribe a physical meaning to $P_{0}$ even though it may appear to be
“intrinsic” error at $P=0$. We use this power law model to fit the
experimental data in Figures 4a and 5b of the main text, as well as the
numerical simulations in Figure 4a of the main text.

In order to model $1/\Lambda_{5/7}$ as a function of injected leakage
population $P_{L}$ in Figure 5c of the main text, we take the inverse of the
ratio between distance-5 and distance-7 logical error per cycle,

$\displaystyle\Lambda_{5/7}\left(P_{L}\right)$
$\displaystyle=\frac{\varepsilon_{5}\left(P_{L}\right)}{\varepsilon_{7}\left(P_{L}\right)}.$
(S4)

For logical error probability $p_{L}$ with respect to cycle $n$ of a bit-flip
code operated well below threshold and in the presence of leakage dynamics
(Figure S5), we use a Gompertz model to describe logical error rate
$\varepsilon$,

$\displaystyle\varepsilon\left(n\right)$
$\displaystyle=a\exp\left(-b\exp\left(-cn\right)\right),$ (S5)

where $a$, $b$, and $c$ are phenomenological free parameters. A Gompertz model
can partly capture the transient dynamics of $\varepsilon$ at small $n$, where
time-boundary effects and still-increasing leakage populations make
$\varepsilon$ highly dependent on $n$. At large $n$, $\varepsilon$ tends to a
constant as leakage populations stabilize.

##  S6 Surface code simulations well below threshold

Figure S7:  Surface code $p_{ij}$ correlations.  a) Average autocorrelated
$p_{ij}$ matrices $\bar{p}_{t,t^{\prime}}$ of $X$ and $Z$ stabilizers under
different leakage removal strategies. Correlations along the upper and lower
diagonals ($p_{t,t^{\prime}}$ where $|t-t^{\prime}|=1$) represent independent
timelike errors, whereas non-local correlations ($p_{t,t^{\prime}}$ where
$|t-t^{\prime}|>1$) can be primarily attributed to leakage-induced correlated
errors. For No reset, non-local correlations intensify as the code is executed
in time, suggesting increasing leakage-induced correlated errors from growing
leakage population. With MLR, non-local correlations are reduced but remain,
which are manifestations of data qubit leakage-induced correlated errors. For
DQLR, complete leakage removal over all qubits results in suppression of non-
local correlations to about $1\times 10^{-3}$ or lower. b) (Left) Averaged
autocorrelation $\bar{p}_{t,t^{\prime}};s-s^{\prime}=0$ over all stabilizers
for cycles $t$ and $t^{\prime}$ in 19–29, inclusive, under different leakage
removal strategies. Average non-local correlation magnitudes do not exceed
$2\times 10^{-3}$ for DQLR, whereas No reset and MLR have exponential decays
in correlation magnitude with respect to correlation distance. The inset shows
an example of a distance-4 long-time error edge on a detection graph. (Right)
Averaged nearest-neighbor correlation $\bar{p}_{t,t^{\prime}};s-s^{\prime}=1$
over all stabilizers, extracted under the same conditions as the averaged
autocorrelation values. The inset shows an example of a distance-4 long-
diagonal error edge on a detection graph.

To gain insight into the future importance of leakage removal for scaling
quantum error correction, we performed numerical simulations of distance-5 and
distance-7 surface codes and evaluated their logical performance subject to
different levels of leakage injection carried out using the circuit shown in
the inset of Figure 5b of the main text. We performed these simulations using
a Kraus operator simulation detailed in [17]. We include operators that
accurately reflect leakage transport, leakage phase errors, and the leakage
removal parameters for both the MLR and DQLR strategies, but do not include
any sources of leakage in the baseline error model at zero leakage injection.
The error model for the baseline simulations is detailed in Table S1. We
repeat these simulations with varying amounts of injected leakage population
under two leakage removal strategies, MLR and DQLR, which is presented in the
Figure 5c of the main text.

We fit the data from the numerical simulations in Figure 5c of the main text
with a power law ratio model (Equation S4). Additionally, we fit the data for
DQLR with a line. The high degree of linearity ($R^{2}=0.983$,
$1/\Lambda_{5/7}\approx 111\times P_{L}+0.2$) suggests that the surface code
error budget $1/\Lambda_{5/7}$ for leakage under DQLR can be linearized and
modeled as an uncorrelated error source.

Table S1:  Hypothetical device error model for simulations carried out in Figure 5c of the main text. Parameter | Value
---|---
Single qubit gate Pauli error | $2\times 10^{-4}$
CZ gate Pauli error | $1\times 10^{-3}$
Readout and reset error | $1\times 10^{-2}$
Idling Pauli error from relaxation | $3\times 10^{-3}$
from dynamical decoupling | $1\times 10^{-3}$
Qubit $T_{1}$ | 75 $\mu$s
Qubit $T_{2}$ | 75 $\mu$s
Single qubit gate time | 15 ns
CZ gate time | 25 ns
Combined readout and reset time | 300 ns

##  S7 Error correlations in the surface code experiment

One technique to evaluate error correlations in the surface code is to employ
$p_{ij}$ correlation matrices [39, 35, 14, 17]. By analyzing the data
presented in Figure S6 with $p_{ij}$ correlation matrices, we can elucidate
the presence of error correlations in space and time over the course of the
QEC experiment.

For $p_{ij}$ matrices, $i$ and $j$ correspond to nodes in the detection graph,
each of which have stabilizer (or space, $s$) and time ($t$) coordinates;
$i=\left(s,t\right)$, $j=\left(s^{\prime},t^{\prime}\right)$. To first focus
on correlations in time, we average the matrix elements $p_{ij}$ with
$s=s^{\prime}$ over all $t$ and $t^{\prime}$, producing autocorrelation
matrices $\bar{p}_{t,t^{\prime}}$, defined as

$\displaystyle\bar{p}_{t,t^{\prime}}$
$\displaystyle=\frac{\sum\limits_{s=s^{\prime}}{p_{ij}}}{\sum\limits_{s=s^{\prime}}1}.$
(S6)

The autocorrelation matrix $\bar{p}_{t,t^{\prime}}$ reports the average of the
probabilities of error graph edges between nodes $i$ and $j$ associated to the
same stabilizer $s$ and for arbitrary time separation $|t-t^{\prime}|$.

In Figure S7a, we show averaged autocorrelation matrices
$\bar{p}_{t,t^{\prime}}$ for both $X$\- and $Z$-basis stabilizers over the 30
cycle experiment under the three leakage removal strategies investigated in
this work. $Z$-basis stabilizers do not report detection probabilities at the
time-boundary cycles 0 and 30 and thus do not have $\bar{p}_{t,t^{\prime}}$
elements associated with those cycles. Independent Pauli errors present
themselves as detection events on consecutive cycles $|t-t^{\prime}|=1$. In
such an ideal setting with only fully independent errors, we expect non-local
correlations to vanish, i.e. $\bar{p}_{t,t^{\prime}}=0$ where
$|t-t^{\prime}|>1$. Leakage and other time-non-local error sources will break
this assumption and force those $\bar{p}_{t,t^{\prime}}$ elements to be non-
zero.

For No reset, non-local correlations immediately appear at cycle 1 and
increase in intensity over the course of the 30-cycle experiment. The high
values for non-local $\bar{p}_{t,t^{\prime}}$ matrix elements greater than 1%
at large correlation distances $|t-t^{\prime}|$ indicate that non-local
correlations are significant contributors to logical performance degradation,
and that QEC cannot be practically scaled in time under these conditions [27].

For MLR, non-local correlations are visibly reduced when compared to No reset.
Still, an impactful degree of non-local correlation remains, primarily
stemming from data qubit leakage and resultant leakage-induced correlated
errors, such as CZ gate phase shifts highlighted in Figure 2e of the main
text.

For DQLR, nearly all non-local correlations are heavily suppressed in the
experiment. This is seen by very small correlation magnitudes less than 0.2%
at correlation distances greater than 1. Qualitatively, the effective
suppression of time-non-local correlations suggests we are much closer to
fulfilling the QEC requirement of uncorrelated errors. Furthermore, this is
evidence that our DQLR procedure does not introduce additional unwanted
correlations within the experiment.

We isolate and average $\bar{p}_{t,t^{\prime}}$ matrix elements for cycles $t$
and $t^{\prime}$ in 19–29, inclusive, in Figure S7b. This offers a
quantitative profile to the degree of non-local correlations present in the
surface code as a function of time correlation distance $t-t^{\prime}$, which
correspond to the probability of encountering _long-time error edges_ in the
detection graph. Again, under ideal circumstances with fully independent
errors, the correlation strength $\bar{p}_{t,t^{\prime}}$ should be 0 at all
$t-t^{\prime}$ greater than 1. DQLR most closely approximates this condition,
where $|\bar{p}_{t,t^{\prime}}|$ exceeds $2\times 10^{-3}$ only for distance-2
correlation and otherwise is below 0.2% for all distances up through 10. The 1
SD error bars for DQLR suggest that we cannot resolve variations in
$|\bar{p}_{t,t^{\prime}}|<1\times 10^{-3}$ for these non-local correlations.
However, for MLR, $|\bar{p}_{t,t^{\prime}}|$ is about an order of magnitude
higher at 1% for distance-2 correlation, and slowly decays with distance,
remaining above 0.1% even at distance-10. Finally, No reset never has
$\bar{p}_{t,t^{\prime}}<1\%$ for any of the correlation distances considered
here.

In order to evaluate correlations across space, we can perform a similar
analysis but now with nearest-neighbor stabilizers,

$\displaystyle p_{t,t^{\prime}}$
$\displaystyle=\frac{\sum\limits_{s-s^{\prime}=1}{p_{ij}}}{\sum\limits_{s-s^{\prime}=1}1}.$
(S7)

The resulting average time correlations, which indicate the probability of
encountering a _long-diagonal error edge_ , are shown in the right panel of
Figure S7b. By only considering time correlation distances of
$t-t^{\prime}>1$, we primarily exclude contributions from CZ gate errors,
which manifest as _diagonal edges_ at $s-s^{\prime}=1$ and $t-t^{\prime}=1$
[17]. The remaining correlations at $t-t^{\prime}>1$ then predominantly arise
from leakage and crosstalk and should be zero under ideal circumstances
without these error sources.

Similar to the average correlation $\bar{p}_{t,t^{\prime}}$ for
$s-s^{\prime}=0$, we observe the same hierarchy of correlation strengths for
$s-s^{\prime}=1$ under No reset, MLR, and DQLR. No reset performs the worst
with correlation strengths at about 1% for all correlation distances studied
here. MLR performs considerably better, but still has measurable correlations
at time distances as large as 6 cycles. Given that we desire zero correlation
at all time distances, DQLR most closely approximates this condition, with
$\left|\bar{p}_{t,t^{\prime}}\right|$ at 0.1% for distance-2 correlations, and
otherwise well below 0.1% for longer time distances.

These observed correlation strengths suggest that if scalable QEC requires
near-independent errors, complete leakage removal on all qubits must be
carried out in some form. The DQLR strategy presented in this work provides
one pathway to reaching that requirement. As a corollary, we show that No
reset and MLR cannot support this requirement under existing leakage rates and
transport mechanisms.


12021202013421asna.202100000

11institutetext:  Institute of Astronomy and National Astronomical
Observatory, Bulgarian Academy of Sciences,
Tsarigradsko Shose 72, BG-1784 Sofia, Bulgaria 22institutetext: Departamento
de Física, Escuela Politécnica Superior de Jaén, Universidad de Jaén, Campus
Las Lagunillas, A3,
23071, Jaén, Spain

…

# Radius, rotational period and inclination of the Be stars in the Be/gamma-
ray binaries MWC 148 and MWC 656††thanks: Data from TESS and Rozhen

R. K. Zamanov Corresponding author<EMAIL_ADDRESS>K. A. Stoyanov 11 J.
Martí 22 V. D. Marchev 11 Y. M. Nikolov 11

(2020 November 3; 2020 February 3)

###### Abstract

Using TESS photometry and Rozhen spectra of the Be/$\gamma$-ray binaries MWC
148 and MWC 656, we estimate the projected rotational velocity (${v}\sin i$),
the rotational period (Prot), radius (R1), and inclination ($i$) of the mass
donor. For MWC 148 we derive P${}_{\rm rot}=1.10\pm 0.03$ d, R${}_{\rm
1}=9.2\pm 0.5$ R⊙, $i=40^{\circ}\pm 2^{\circ}$, and ${v}\sin i=272\pm 5$ km
s-1. For MWC 656 we obtain P${}_{\rm rot}=1.12\pm 0.03$ d, R${}_{\rm 1}=8.8\pm
0.5$ R⊙, $i=52^{\circ}\pm 3^{\circ}$, and ${v}\sin i=313\pm 3$ km s-1. For MWC
656 we also find that the rotation of the mass donor is coplanar with the
orbital plane.

###### keywords:

Stars: emission-line, Be – binaries: spectroscopic – Gamma rays: stars –
Stars: individual: MWC 148, MWC 656

##  1 Introduction

The $\gamma$-ray binaries are a recently established and rare subclass of the
high-mass X-ray binaries with most of their luminosity output being radiated
above 1 MeV (Dubus 2013; Chernyakova & Malyshev 2020). They are composed of an
OBe donor star and a neutron star or a black hole (Mirabel 2012). The
mechanism responsible for the high-energy emission in these systems is still a
subject of debate. The $\gamma$-rays could be produced either by accretion-
driven jets, or by the rotation-powered strong pulsar winds interacting with
the nearby medium (Dubus 2006; Romero et al. 2007; Massi & Jaron 2013), and/or
by a neutron star in the propeller regime (Wang & Robertson 1985).

So far, seven systems have been confirmed as $\gamma$-ray binaries - PSR
B1259-63 (Aharonian et al. 2005a), LS 5039 (Aharonian et al. 2005b), LS I +61
303 (Albert et al. 2006), HESS J0632+057 (Aharonian et al. 2007), 1FGL
J1018.6-5856 (Corbet et al. 2011), PSR J2032+4127 (Lyne et al. 2015) and
LMC-P3 (Corbet et al. 2016). The nature of the compact object is already known
in PSR B1259-63 and PSR J2032+4127, where radio observations confirm that they
contain neutron stars (Johnston et al. 1992; Abdo et al. 2009). Based on the
donor star and the presence of a circumstellar disc, two subgroups of
$\gamma$-ray binaries are proposed. The first subgroup harbours an O-type
donor star and shows a single-peak profile in their $\gamma$-ray light-curve.
The second subgroup contains an OBe star and shows several peaks in their
light-curves, occasionally correlated with the times when the compact object
crosses and in some cases truncates the circumstellar disc of the donor star
(Paredes & Bordas 2019).

MWC 148 (HD 259440) was identified as the optical counterpart of the variable
TeV source HESS J0632+057 (Aharonian et al. 2007) and also detected in the GeV
domain (Li et al. 2017). The system consists of a B0Vpe star (Casares et al.
2012) and a compact object, with an orbital period Porb = 315${}^{+6}_{-4}$ d
(Aliu et al. 2014).

MWC 656 (HD 215227) is the suspected optical counterpart of the $\gamma$-ray
source AGL J2241+4454 detected by the $AGILE$ satellite above 100 MeV
(Lucarelli et al. 2010; Williams et al. 2010). It is the first discovered
binary containing a black hole as a companion of a Be star (Casares et al.
2014). MWC 656 was only occasionally detected at GeV energies (Aleksić et al.
2015). The black hole nature of the compact object renders it similar to the
typical $\gamma$-ray binaries. The orbital period of the system, obtained by
optical photometry and later confirmed by radial velocity measurements, is
P${}_{\rm orb}=60.37\pm 0.04$ d (Williams et al. 2010; Casares et al. 2014).

Here, using photometric and spectral observations, we estimate the rotational
period, the radius and the inclination of the mass donors in the
Be/$\gamma$-ray binaries MWC 148 and MWC 656.

Table 1: Spectroscopic observations of MWC 148 and MWC 656. In the table are given the date of observations, UT start, exposure time in seconds, signal-to-noise ratio at about 6600 Å, EW(H$\alpha$), FWHM($H\alpha$), EW(FeII5316), and FWZI(FeII5316).  Object | Date-obs | exp-time | S/N | EW(H$\alpha$) | FWHM(H$\alpha$) | EW(FeII5316) | FWZI(FeII) |
---|---|---|---|---|---|---|---|---
| yyyy-mm-dd:hh:mm | [sec] |  | [Å] | [km s-1] | [Å] | [km s-1] |
MWC 148 | 2019-02-20T18:45 | 2400 | 75 | -46.9 | 411 | -0.59 | 672 |
MWC 148 | 2019-12-06T01:45 | 2400 | 95 | -46.4 | 405 | -0.60 | 711 |
MWC 148 | 2020-01-16T21:16 | 3600 | 70 | -44.8 | 418 | -0.59 | 683 |
MWC 148 | 2020-09-06T02:01 | 1800 | 60 | -43.7 | 412 | -0.57 | 709 |
MWC 656 | 2019-08-22T20:10 | 3600 | 90 | -20.9 | 493 | -0.48 | 705 |
MWC 656 | 2019-12-05T17:48 | 2400 | 100 | -21.6 | 491 | -0.54 | 722 |
MWC 656 | 2020-09-05T21:50 | 3600 | 64 | -21.7 | 490 | -0.60 | 700 |

Figure 1:  The emission lines $H\alpha$ and FeII 5316 in the spectra of MWC
148 (left panel) and MWC 656 (right panel).

##  2 Observations

We use both space photometry and ground-based spectra.

###  2.1 TESS photometric data

The Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015) is a
space-based optical telescope launched in 2018 with the primary mission to
perform an all-sky survey to search for transiting exoplanets. In order to
fulfill the mission, the sky is divided into a number of sectors, each of
which corresponds to the total field of view of all four cameras of the
telescope – $24^{\circ}\times 96^{\circ}$. Each sector is observed for
approximately 27 days at a cadence of 2 minutes. Simple aperture photometry is
applied to each of the data files to obtain a barycentered light curve file of
the selected object. The bandpass of TESS is centered on the classical Ic
filter, but it is wider and spans from 6000 Å to 10 000 Å, in other words the
telescope observes the red and the near-infrared emissions of the stars. The
light curves of MWC 656 and MWC 148 were downloaded from the Mikulski Archive
for Space Telescopes archives111https://archive.stsci.edu/tess/.

###  2.2 Rozhen spectral data

The spectral observations were obtained with the ESPERO spectrograph of the
2.0m RCC telescope at Rozhen National Astronomical Observatory located in
Rhodope Mountain, Bulgaria. ESPERO is a fiber-fed Echelle spectrograph giving
a dispersion of 0.06 Å px-1 and resolving power $\sim 30000$ at 6560 Å (Bonev
et al. 2017). The spectral processing and measurements of the spectral lines
are performed using standard routines provided by IRAF (Tody 1993). On each
spectrum we measure the equivalent width EW(H$\alpha$) and the full width at
half maximum (FWHM) of the H$\alpha$ emission line, equivalent width EW(5316)
and the full width at zero intensity (FWZI) of the FeII 5316 Å line. FWZI is
the full width of the emission line at the continuum level. Examples of the
emission lines are presented in Fig. 1. They are normalized to the local
continuum. The typical errors of our measurements are $\pm 5$% in
EW(H$\alpha$), $\pm 7$ km s-1 in FWHM(H$\alpha$), $\pm 10$% in EW(FeII5316),
$\pm 40$ km s-1 in FWZI(FeII). The log of observations is given in Table 1.

Traces of residual wings due to photospheric absorption were not detected and
the interpolated continuum was taken as the baseline during the measurements
of EW and FWHM. The equivalent width is measured using $splot$ routine in IRAF
by marking two continuum points around the line to be measured. The linear
continuum is subtracted and the flux is determined by simply summing the
pixels with partial pixels at the ends. The method calculates the area under
the profile irrespective of its shape (e.g. Mathew & Subramaniam 2011). The
FWHM is measured by identifying the points of the emission line profile where
the intensity is equal to one half of the peak intensity, as shown in Fig. 1
of Glebocki et al. (1986). The horizontal distance between this two points was
measured. This measurement also does not depend on the profile shape.

##  3 Analysis methodology

In this section we give the equations connecting the relevant parameters of
the primary components on which our estimates are based. For the Be stars,
Hanuschik (1989) gives a relation between projected rotational velocity
(${v}\sin i$), FWHM(H$\alpha$), and EW(H$\alpha$). We use his relation in the
form

${v}\sin i=0.813\>(\rm FWHM\>10^{0.08\log EW(H\alpha)\ -0.14}-70),$ (1)

where FWHM and $v\sin i$ are measured in km s-1; EW(H$\alpha$) is in Å.

The Fe II lines are optically thin and their profiles reflect the Keplerian
rotation in the innermost part of the Be disc (Hanuschik 1996). The
inclination of the Be star is connected with the full width at zero intensity
of the FeII lines and its radius:

$\frac{\rm FWZI}{2\sin i}=\left(\frac{\rm GM_{1}}{(1+\epsilon)\rm
R_{1}}\right)^{1/2},\\\ $ (2)

where G is the gravitational constant, M1 is the mass of the Be star, R1 is
its radius, $i$ is the inclination of the Be star to the line of sight,
$\epsilon$ is a dimensionless parameter, $\epsilon\geq 0$. Eq. 2 represents
the Keplerian motion in the disc and is a modification of that used in Sect.
6.1 of Casares et al. (2012). The region where the Fe II lines are produced
can be extended down to the very surface of the Be star or close to it. The
parameter $\epsilon$, for which we adopt $0\leq\epsilon<0.1$, represents how
close to the surface of star the emission at FWZI of the FeII lines is formed.

The rotational period of the Be star is also connected with the above
parameters:

${\rm P_{\rm rot}}=\frac{2\pi R_{1}}{{v}\sin i}\sin i.$ (3)

The rotational periodicity is probably due to the interaction between the
magnetic field of the Be star and its circumstellar disc or the presence of
some physical feature, such as a spot or cloud, co-rotating with the star
(Smith, Henry & Vishniac 2006). The rotational periods in the Be/$\gamma$-ray
binaries are expected to be of order 1 day (Zamanov et al. 2016).

The applied methodology involves the following steps:

  1. 1.

A periodogram analysis of the TESS data is performed to estimate Prot;

  2. 2.

The parameter ${v}\sin i$ is estimated using Eq. 1 and the data in Table 1;

  3. 3.

A mass value for the Be star is adopted according to its spectral type;

  4. 4.

Using Eq. 2 and Eq. 3, the values of R1 and $i$ are calculated for the
primaries of MWC 148 and MWC 656.

The period-search methods applied to the TESS photometry were the phase
dispersion minimization, PDM (Stellingwerf 1978) and the CLEAN algorithm
(Roberts, Lehar & Dreher 1987).

##  4  MWC 148

TESS photometry for MWC 148, in the interval JD 2451468.2 – JD 2451490.0, can
be accessed under Input Catalogue ID 234929785 and is plotted in Fig. 5. The
periodogram analysis is presented in the left panel of Fig. 5, where the PDM
statistic (theta) and the CLEAN component amplitude are plotted. Over the
entire data set, the analysis yields P${}_{\rm rot}=1.0908\pm 0.0002$ d. The
periodogram analysis of the data in the $({\rm JD}-2450000)$ time interval
1468 – 1490 gives a clear period of 1.09 $\pm 0.03$ days, in the interval 1468
– 1477 a clear period of 1.131 $\pm 0.025$ days, and in the interval 1478 –
1490 a clear period of 1.075 $\pm 0.025$ days. A visual inspection of the data
gives us the possibility to select the parts of the light curve where this
periodicity is most clearly visible. Using days 1468 – 1474, we find 1.156 d,
and for days 1479 – 1487 we find 1.101 d. The light curve for days 1468 – 1474
is plotted in the right panel of Fig. 5. We consider that the rotational
period of the Be star in MWC 148 is in the range $1.09\leq\rm P_{rot}<1.16$ d.

Figure 2: TESS light curve of MWC 148.

Figure 3: Left panel: periodogram analysis for MWC 148 – theta and power
versus period. The right panel represents the light curve during 1468 - 1474
folded with a 1.156 d period.

Figure 4: TESS light curve of MWC 656.

Figure 5: Left panel: periodogram analysis for MWC 148. The most significant
period is 0.559 days. The right panel represents the light curve during 1751 -
1763 folded with a 1.117 d period, which is probably the rotational period of
the B star.

Using Eq. 1 and the values given in Table 1, we find ${v}\sin i=272\pm 5$ km
s-1. Casares et al. (2012) give a spectral type B0Vpe. A B0V star is expected
to have on average mass M${}_{1}=15.0\pm 0.5$ M⊙ (Hohle et al. 2010). For MWC
148, Aragona et al. (2010) derived M${}_{1}=13.2-19.0$ M⊙ from spectral model
fits, which is in agreement with the adopted M${}_{1}=15.0$ M⊙. Adopting
FWZI($\lambda 5316$) $\approx 694$ km s-1, and ${\rm v}\sin i=272$ km s-1, we
find $i=40^{\circ}\pm 2^{\circ}$ and R${}_{1}=9.2\pm 0.5$ R⊙.

It is worth noting for comparison that (i) a B0V star is expected to have
R${}_{1}=7.2$ $R_{\odot}$ (Straizys & Kuriliene 1981); (ii) Casares et al.
(2012) give FWZI($\lambda 5018$) $\sim 1300$ km s-1 and ${v}\sin i=373$ km
s-1; (iii) ${v}\sin i=430$ km s-1 (Gutiérrez-Soto et al. 2007) and ${v}\sin
i=500$ km s-1 (Aragona et al. 2010) are reported for this object.

##  5  MWC 656

TESS photometry for MWC 656 (HD 215227), in the interval JD 2451738 – 2451763,
can be accessed under Input Catalogue ID 153880067 (Fig. 5). The data gap is
due to the telescope being repointed to transfer data to Earth at this time.
The periodogram analysis is shown in the left panel of Fig. 5 with the same
kind of plots as for the previous source. Over the entire data set, the
analysis yields P${}_{\rm rot}=0.5586\pm 0.0002$ d. In contrast, we do not
find a clear period in the $({\rm JD}-2450000)$ interval 1738 – 1751 days.
However in the interval 1751.6 – 1762.7 days, a clear period of 0.557 $\pm
0.025$ days is detected. Looking at the interval 1760-1763 days, we see that
there is a repetition of strong and weak maxima, which probably means that the
rotational period is doubled P${}_{\rm rot}=1.117$ d.

For the primary, Williams et al. (2010) estimated T${}_{\rm eff}=19000\pm
3000$ K, $\log g=3.7\pm 0.2$, M${}_{1}=7.7\pm 2.0$ M⊙ , and R${}_{1}=6.6\pm
1.9$ R⊙. Casares et al. (2014) found that the mass donor is a giant (B1.5-2
III) and gave a mass range 10–16 M⊙. From the results of Hohle et al. (2010),
such a star is expected to have $8.0<\rm M_{1}<10.0$ M⊙.

Using the relation between ${v}\sin i$, FWHM($H\alpha$), and EW(H$\alpha$)
(Eq. 1), and our values (see Table 1), we find ${v}\sin i=313\pm 3$ km s-1,
which is similar to the values $330\pm 30$ km s-1 (Casares et al. 2014),
$262\pm 26$ km s-1 (Yudin 2001), and $330\pm 50$ km s-1 (Williams et al.
2010).

We measure FWZI(FeII 5316) = 12.5-12.9 Å$=709\pm 12$ km s-1, which is below
the value FWZI(FeII 5018)$\sim 1000$ km s-1 (Casares et al. 2012).

Adopting M${}_{1}=10.0\pm 0.5$ M⊙, ${v}\sin i=313$ km s-1, and
$FWZI(FeII)=709$ km s-1, the system of Eqs. 2 and 3 can be solved. As a
result, we find R${}_{1}=8.8\pm 0.5$ R⊙ and $i=52^{\circ}\pm 2^{\circ}$. The
obtained value of R1 agrees with the estimates of the average radius for a
B1.5-2 III star ($8.3-8.8$ R⊙, Straizys & Kuriliene 1981).

For MWC 656, from the radial velocity measurements, Casares et al. (2014)
obtained $M_{1}\sin^{3}i_{\rm orb}=5.83\pm 0.70$. For M${}_{1}=10$ M⊙, this
gives $53^{\circ}<i_{\rm orb}<60^{\circ}$. It appears that the orbital plane
and the equatorial plane of the Be star are practically coplanar within $\pm
4^{\circ}$.

##  6 Discussion

In Be/X-ray binaries the primary is a rapidly rotating Be star with mass $\sim
10$ M⊙ and the secondary is a neutron star or a black hole. The secondary mass
is expected to be $\sim 2$ M⊙ for a neutron star, and $\sim 5$ M⊙ in the case
of black hole. The orbital periods are in the range 10 – 400 d (Reig 2011).
The Be/$\gamma$-ray binaries are a subgroup of the Be/X-ray binaries and
should have similar binary parameters. They are a product of the evolution of
a binary containing two moderately massive stars, which undergoes mass
transfer from the originally more massive star towards its companion (Pols et
al. 1991; Negueruela 2007). The eccentricities in these systems are caused by
a kick to the compact object during the supernova explosion that formed it
(e.g. Martin et al. 2009).

HESS J0632+057 (MWC 148) produces non-thermal radio, X-ray, GeV and very high-
energy gamma-ray emission. The non-thermal emission is modulated with the 315
days orbital period and has a a peculiar light curve containing two peaks,
separated by a dip – a sharp peak with a short decay before the apastron
passage and a broad (several tens of days) secondary peak after the apastron
passage in X-rays and TeV (Aliu et al. 2014, Archer et al. 2020). The non-
thermal activity before and around apastron can be linked to (i) the
accumulation of non-thermal particles in the vicinity of the binary, and the
sudden drop of the emission before apastron is produced by the disruption of
the two-wind interaction structure, allowing these particles to escape
efficiently (Bosch-Ramon et al. 2017) or (ii) an accumulation of hot shocked
plasma in the inner spiral arm, later released when the spiral arm is
disrupted in the periastron-apastron direction (Barkov & Bosch-Ramon 2018).

For MWC 148, two solutions for the orbit indicate that it is highly eccentric:
$e=0.83\pm 0.08$ (Casares et al. 2012) and $e=0.64\pm 0.29$ (Moritani et al.
2018). In our previous paper (Zamanov et al. 2017), we assumed the value of
$i$, on the basis of the strong resemblance of the optical emission lines
between MWC 148 and the bright Be star $\gamma$ Cas (Zamanov, Stoyanov & Martí
2016), for which the inclination is $i=43^{\circ}\pm 3^{\circ}$ (Poeckert &
Marlborough 1978; Clarke 1990). Our result here for MWC 148 ($i=40^{\circ}\pm
2^{\circ}$) confirms the assumption that one of the reasons for this
similarity is the inclination.

MWC 656 is faint in X-rays and it reaches the faintest X-ray luminosities ever
detected in stellar-mass black holes (Ribó et al. 2017). It may not
continuously emit in $\gamma$-rays (Alexander & McSwain 2016). For this
binary, Casares et al. (2014) found that the mass of the black hole is in the
range 3.8 – 6.9 $M_{\odot}$. The orbital eccentricity is $e$ = 0.10 $\pm$
0.04. It is suggested that the warp and precession observed in Be star discs
may be caused by a misalignment between the spin axis of the Be star and the
orbit of the binary companion (Martin et al. 2011). Our results indicate that
there is no misalignment in MWC 656, and consequently warping and precession
should not be observed.

The velocity kick from the supernova has two effects – it renders the orbit
eccentric and it misaligns the orbit with respect to the spin axis of the Be
star. In systems that experience low velocity kicks, the misalignments tend to
be small (Martin et al. 2011). The low orbital eccentricity and the alignment
between the spin axis of the primary and the axis of the binary orbit,
indicates that the compact object in MWC 656 was born with low kick velocity.

##  7 Conclusions:

We evaluated some parameters for the mass donor stars in the Be/$\gamma$-ray
binaries MWC 148 and MWC 656. For MWC 148, we estimate ${v}\sin i=272\pm 5$ km
s-1, P${}_{\rm rot}=1.10\pm 0.03$ d, R${}_{1}=9.2\pm 0.5$ R⊙, and
$i=40^{\circ}\pm 2^{\circ}$. For MWC 656, we obtain ${v}\sin i=313\pm 3$ km
s-1, P${}_{\rm rot}=1.12\pm 0.03$ d, R${}_{1}=8.8\pm 0.5$ R⊙, and
$i=52^{\circ}\pm 3^{\circ}$. These parameters should be useful for future
accurate modeling of these systems.

###### Acknowledgements.

This work is supported by Bulgarian National Science Fund – project
K$\Pi$-06-H28/2 ”Binary stars with compact object”. JM acknowledges support by
grant PID2019-105510GB-C32 / AEI / 10.13039/501100011033 from the Agencia
Estatal de Investigación of the Spanish Ministerio de Ciencia, Innovación y
Universidades, and by Consejería de Economía, Innovación, Ciencia y Empleo of
Junta de Andalucía as research group FQM- 322, as well as FEDER funds. YMN
acknowledges grant DCM 577/17.08.2018 from National Research Programme ”Young
scientists and postdoctoral students” of the Bulgarian Ministry of Education
and Science. We are grateful to Mike Bode (Liverpool John Moores University)
for critical reading of the manuscript and the anonymous referee for useful
comments.

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