# A hard open X-band RF accelerating structure made by two halves

Bruno Spataro a, Mostafa Behtouei a, Fabio Cardelli a, Martina Carillo b,
Valery Dolgashev c, Luigi Faillacea, Mauro Migliorati b, and Luigi Palumbob
aINFN, Laboratori Nazionali di Frascati, P.O. Box 13, I-00044 Frascati, Italy;
bDipartimento di Scienze di Base e Applicate per l’Ingegneria (SBAI), Sapienza
University of Rome, Rome, Italy;
cSLAC National Accelerator Laboratory ;

###### Abstract

High-gradient linacs of next generation require novel accelerating structures
which are compact, robust and cost-effective. Dedicated research and
development have been launched in the linear-collider community. This paper
focuses on the technological developments directed to show the viability of
novel welding techniques and related applications, in order to benefit from
the superior high-gradient performance of accelerating structures made of
hard-copper alloys. The structure geometry that we propose allows getting a
high longitudinal shunt impedance of the accelerating mode and increasing the
mode separation frequencies.

###### keywords:

Hard accelerating structures, Two halves X-band accelerating structure,
Particle Acceleration, Linear Accelerators, Accelerator Subsystems and
Technologies

††journal: MDPI

##  1 Activities on X band structures made of two halves

High-gradient linacs are sought for various applications, from high-energy
physics, industry and medicine, and require innovative accelerating
structures. An intense activity on research and development has been launched
in the linear-collider community, as in [1, 2, 3, 4, 5, 6]. A continuous
collaboration on the study of various geometries, materials, surface
processing techniques and technological developments of accelerating
structures has involved, for more than a decade, the SLAC National Accelerator
Laboratory in the USA, the Italian Institute of Nuclear Physics - National
Laboratories of Frascati (INFN-LNF), and the High Energy Accelerator Research
Organization (KEK) in Japan [2].

This paper is part of this study and it focuses on the technological
developments directed to show the viability of novel welding techniques [3, 5,
6], and related applications, in order to benefit from the superior high-
gradient performance of accelerating structures made of hard-copper alloys.
The technological activity of testing high-gradient RF sections, in particular
at 11.424 GHz, began at INFN-LNF in the framework of the above-mentioned
collaboration, and it is related to the investigation of breakdown mechanisms,
which limit the high gradient performance of these structures. It consists in
the design, construction and high-power experimental tests of standing wave
(SW) 11.424 GHz (X-band) accelerating cavities with different materials and
methods. Figure 1 shows the two halves of the X-band assembled structure under
RF low level test.

The goal of the collaboration is to assess the maximum sustainable gradients
with extremely low probability of RF breakdown in normal-conducting high-
gradient RF cavities. The most common bonding techniques, used worldwide, are
high-temperature brazing and diffusion bonding. The brazing and the diffusion
bonding are performed inside a high-temperature furnace. On the other hand,
experimental results with hard copper cavities, conducted at SLAC, CERN and
KEK [1, 2] have shown that hard materials sustain higher accelerating
gradients for the same breakdown rate. Therefore it would be better to avoid
high temperature processing of the cavities to benefit from superior high
gradient performance of hard copper alloys.

In this framework, we are conducting experiments that involve the Electron
Beam Welding (EBW) and Tungsten Inert Gas (TIG) processes, which allow us to
build practical, multi-cell structures made with hard copper alloys, in order
to increase their RF performance against the soft ones. For this purpose, open
structures made of two halves have been investigated and fabricated. Details
on the fabrication procedure of “split” X Band structures made of two halves,
or “open”structures, by using the TIG method are given in ref [7].

In this paper, we present the RF characterization and low-power RF tests of a
TIG welded two-halves split hard-copper structure [8, 9] that will be
consequently employed for high-gradient tests and for the study of the RF
breakdown physics. To this aim, the structure geometry that we propose (and
shown in Fig. 2 -¿ cavity design) allows getting a high longitudinal shunt
impedance of the accelerating mode, increasing the mode separation
frequencies, and improving the operation vacuum level.

Figure 1: Picture of the two halves structure under test.

In addition, intense beam currents and multibunch operation are essential
features, for example for increasing the luminosity of a linear collider, but
beam current wakefields and coupled-bunch mode instabilities, which mostly
arise from the parasitic modes of the accelerating structures, can put a limit
to the accelerator performance. Hence our main interest is also to detune the
cavity in order to reduce the beam instability by using a novel simpler
technique and dedicated absorbers of the higher order modes (HOMs). The two
cavity halves are aligned and clamped together by means of male-female
matching surface, and the clamping is obtained with stainless screws and TIG
welded at COMEB [10] on the outer surface. Preliminary low-power RF
measurements are in agreement with the simulated ones. The estimation of mode
separation is described later in this paper. The activity to fabricate a
section with 4 quadrants is in progress, too.

##  2 Low level RF measurements

It consists of three cells: the middle one is the high gradient cell, while
the first and third ones are end-cells. The peak on-axis electric field in the
middle cell has to be two times higher than the end-cells.

Figure 3 reports one-quarter of two half structure sketch for the simulation
studies performed with ANSYS HFSS software [11].

Figure 2: one-quarter of two half structure sketch for the simulation studies
by using ANSYS HFSS software.

Figure 3: Longitudinal field profile of the structure modes: a) $0$ mode , b)
$\pi/2$ mode, c) $\pi$ mode

Figures 3a, 3b and 3c show the longitudinal electric field profile of the $0$,
$\pi/2$ and $\pi$ modes. The simulated frequencies spectrum is reported in
Figure 5.

Figure 5 shows the measured frequencies spectrum and longitudinal field
profile of the $\pi$-mode obtained with the bead pull technique by using two
antennas.

Figure 4: Simulated structure frequencies spectrum

Figure 5: Measured frequency spectrum and longitudinal field profile of the
$\pi$-mode

The numerical estimations of mode frequencies and the longitudinal field
profile of the $\pi$-mode are in good agreement with the experimental ones.

In Table 1 we report frequencies and quality factors of the resonant modes,
obtained from simulations, for the split open structure in comparison with the
same cavity designed with the conventional “closed” approach.

Mode | F [MHz] | F [MHz] | Q* | Q*
---|---|---|---|---
| (two halves) | (closed structure) | (two halves) | (closed structure)
$0$ | 10,749 | 10,760 | 10,520 | 10,615
$\pi/2$ | 10,979 | 10,984 | 10,213 | 10,306
$\pi$ | 11,418 | 11,420 | 10,512 | 10,610

*Quality factor

Table 1:  Comparison between mode frequencies and quality factors of the
closed structure and two halves one.

The frequency separation between the $\pi$ \- $\pi/2$ and $\pi/2$ – $0$ modes
is about 440 MHz and 230 MHz, respectively, in the case of the two halves
structure, while it is equal to about 435 MHz and 225 MHz for the closed one.
The two-halves structure cell-to-cell coupling coefficient is estimated to be
6.1% , while the one for the other structure is 6 %. It is important to notice
that the amount of frequency separation of each mode is proportional to the
cavity form factor Rsh/Q [12] which depends on the cell-to-cell coupling
coefficient. In addition, from a comparison between the two halves structure
and the closed one, we also observe that the frequency separation is wider in
the case of the two-halves structure since the capacitive effect is stronger
with respect to that of the closed cavity. In particular, the 0-mode of the
open cavity is the most affected with a deviation which is 11 MHz greater with
respect to the homologous separation in the closed one. By increasing the
number of splittings, we are able to increase the mode frequency separation
accordingly, too.

As additional information, the frequency separation for this structure results
to be 25% larger than the one achieved in other standard brazed structures
[10] with a rough hard edge geometry but without splitting [13]. The measured
quality factor of the $\pi$ mode of the two-halves structure, obtained by
using two antennas, is about Q $\sim$ 6900, which is 30 % lower with respect
to the numerical predictions due to the bad relatively poor electrical contact
during low-power measurements. Our next step is to repeat the RF
characterizations after having welded the two halves and also by using the
input power mode launcher.

We have also scheduled to construct another structure made of 4 quadrants, in
order to confirm the expected wider amount of mode frequency separation.
Moreover, for reasons related to both the feasibility of the hard structure
and a high shunt impedance, we believe that we need to find a compromise
between the of detuning effects and the number of the structure subdivisions.
This approach can be extended to the cavity resonators used in circular
accelerators, in order to eliminate or strongly reduce the longitudinal
coupled-bunch instability associated with the accelerating mode.

##  3 Possible applications of split open structures

The next generation of accelerators is highly demanding in terms of maximizing
accelerating gradients, minimizing overall machine length and cost, improving
the beam quality, reducing beam loading effects, and so on. In case of
circular accelerators, one important mechanism which produces beam instability
and power losses consists in the possibility of the charged particle to excite
high order resonant fields in RF accelerating cavities when passing through
them. This is possible only for those accelerating structures having resonant
modes in frequency ranges which overlap beam Fourier spectrum. If the beam
loading on a large electron /positrons storage ring is strong, the
accelerating mode of a normal conducting cavity gives rise to a coupled-bunch
instability.

Additionally, in recent years, we have seen a revolution in high brightness
electron beams because of the maturation of RF photo-injector performance. In
order to improve the electron beam brightness, larger mode separation is also
needed. Also in pre-buncher and chopper cavities of linear accelerator,
operating at high currents, an RF detuning is required in order to reduce the
interaction with the higher order modes which affect the beam quality. These
problems can be cured by using the split open structure approach which gives
also the possibility of improving the vacuum level of at least a factor ten
with respect the usual solution by inserting an additional vacuum chamber
connected to the beam pipe. For a fixed geometry of the structure (which means
a fixed form factor ratio Rsh/Q ) and for a given beam current and cavity peak
voltage, the frequency detuning effect can be determined, for example, as
shown in [12]. Intense investigations on the absorbers for damping higher
order modes, are in progress, too.

## Acknowledgment

The authors would like to thank James Rosenzweig from UCLA for his useful
discussion and helpful advice.

## References

  * [1] /http://clic-study.org/S.
  * [2] V.A. Dolgashev et al.,“High-Gradient RF Tests of Welded X-band Accelerating Cavities”, PRAB July 2021
  * [3] V. Dolgashev et al“Innovative compact braze-free accelerating cavity”, Journal of Instrumentation 13 (09), P09017.
  * [4] V. Dolgashev et al, “Materials and technological processes for high-gradient accelerating structures: new results from mechanical tests of an innovative braze-free cavity”, Journal of Instrumentation 15 (01), P01029.
  * [5] F. Wang, C. Adolphsen, and C. Nantista, Performance limiting effects in x-band accelerators, Phys. Rev. ST Accel. Beams 14, 010401 (2011).
  * [6] V. A. Dolgashev, High gradient, x-band and above, metallic rf structures, 2nd European Advanced Accelerator Concepts Workshop (EAAC 2015) (2015)
  * [7] B. Spataro et al.,“Innovative compact braze-free accelerating structure”, Jinst September 19- 2018
  * [8] Zha, Hao, and Alexej Grudiev. “Design of the Compact Linear Collider main linac accelerating structure made from two halves.”Physical Review Accelerators and Beams 20.4 (2017): 042001.
  * [9] Zha, Hao, A. Grudiev, and V. Dolgashev. “RF design of the CLIC structure prototype optimized for manufacturing from two halves.”Proc. 6th Int. Particle Accelerator Conf.(IPAC’15). 2015.
  * [10] http://www.comeb.it/
  * [11] https://www.ansys.com/
  * [12] C.A.J. Van der Ger,“Bunching of intense electron beam extracted from a trioded gun modulated at 1 GHz”, NIM A 307 (1991) 553-561
  * [13] B. Spataro et al.“Technological issues and high gradient test results on X band molybdenum accelerating structures”, NIM A 657 (2011) 114-121


11institutetext: Department of Space, Earth and Environment, Chalmers
University of Technology, Onsala Space Observatory, 439 92 Onsala, Sweden
11email<EMAIL_ADDRESS>22institutetext: Royal Observatory of
Belgium, Ringlaan 3, 1180 Brussels, Belgium 33institutetext: Instituto de
Radioastronomía y Astrofísica, UNAM, Ant. carretera a Pátzcuaro 8701, Ex-Hda.
San José de la Huerta, Morelia 58089, Mich., Mexico 44institutetext:
Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr.
25, 6020 Innsbruck, Austria 55institutetext: Universidad Católica del Norte,
Instituto de Astronomía, Av. Angamos 0610, Antofagasta, Chile 66institutetext:
European Southern Observatory, Karl Schwarzschild str. 2, 85748 Garching,
Germany 77institutetext: Space Radio-Diagnostics Research Centre, University
of Warmia and Mazury, Prawocheńskiego 9, 10-720 Olsztyn, Poland
88institutetext: Facultad de Ingeniería y Arquitectura, Universidad Central de
Chile, Av. Francisco de Aguirre 0405, La Serena, Coquimbo, Chile
99institutetext: Jodrell Bank Centre for Astrophysics, Alan Turing Building,
University of Manchester, Manchester, M13 9PL, UK

# The heart of Sakurai’s Object revealed by ALMA

Daniel Tafoya 11 Peter A. M. van Hoof 22 Jesús A. Toalá 33 Griet Van de Steene
22 Suzanna Randall 66 Ramlal Unnikrishnan 11 Stefan Kimeswenger 4455 Marcin
Hajduk 77 Daniela Barría 88 Albert Zijlstra 99

We present high angular-resolution observations of Sakurai’s object using the
Atacama Large Millimeter Array, shedding new light on its morpho-kinematical
structure. The millimetre continuum emission, observed at an angular
resolution of 20 milliarcsec (corresponding to 70 AU), reveals a bright
compact central component whose spectral index indicates that it composed of
amorphous carbon dust. Based on these findings, we conclude that this emission
traces the previously suggested dust disc observed in mid-infrared
observations. Therefore, our observations provide the first direct imaging of
such a disc. The H12CN($J$=4$\rightarrow$3) line emission, observed at an
angular resolution of 300 milliarcsec (corresponding to 1000 AU), displays
bipolar structure with a north-south velocity gradient. From the position-
velocity diagram of this emission we identify the presence of an expanding
disc and a bipolar molecular outflow. The inclination of the disc is
determined to be $i$=72∘. The derived values for the de-projected expansion
velocity and the radius of the disc are $v_{\rm exp}$=53 km s-1 and $R$=277
AU, respectively. On the other hand, the de-projected expansion velocity of
the bipolar outflow detected in the H12CN($J$=4$\rightarrow$3) emission of
approximately 1000 km s-1. We propose that the molecular outflow has an
hourglass morphology with an opening angle of around 60∘. Our observations
unambiguously show that an equatorial disc and bipolar outflows formed in
Sakurai’s object in less than 30 years after the born-again event occurred,
providing important constraints for future modelling efforts of this
phenomenon.

###### Key Words.:

Stars: low-mass — Stars: winds, outflows — (Stars:) binaries: general — (ISM:)
planetary nebulae: general — (ISM:) planetary nebulae: individual: Sakurai’s
Object

##  1 Introduction

Stars with masses up to 8 $M_{\odot}$ shed most of their hydrogen envelope
during the Asymptotic Giant Branch (AGB) phase. As hydrogen and helium burning
ceases in the layers around the core, they transition to the early white dwarf
sequence where they spend the remainder of their lives as white dwarfs (WDs).
Theory predicts that up to one-quarter of these stars experience a ‘final
helium shell flash’ during this stage, where the residual helium spontaneously
ignites under degenerate conditions (Iben et al. 1983; Blöcker 2001; Herwig
2005), which triggers a so-called ‘born-again event’. During the born-again
event, the remaining hydrogen envelope mixes into the stellar core and is
burned, leading to a new H-poor and 13C-rich shell ejection. After this, the
star undergoes a double loop in the Hertzsprung-Russell diagram (Lawlor &
MacDonald 2003; Hajduk et al. 2005) before helium burning stops, and the star
re-enters the WD sequence as an H-poor star. The born-again event is thought
to be a likely path for the formation of hydrogen-deficient stars such as non-
DA white dwarfs, PG 1159, and [WC] stars, as well as some R CrB stars (Werner
& Herwig 2006). Because born-again objects evolve on very short timescales, it
is extremely rare to observe the process in action. In fact, this has been
achieved for only two cases: V605 Aql, which experienced a final helium shell
flash at the beginning of the twentieth century, and V4334 Sgr (also known as
Sakurai’s object), which was caught undergoing the born-again behaviour in the
mid 1990’s (Duerbeck et al. 1996). Consequently, the born-again event is still
very poorly understood, despite its importance to stellar evolution and
evolved stellar populations.

Sakurai’s object, whose assumed distance from the Sun is 3.5 kpc (Hinkle et
al. 2020), is the only star to have been monitored with modern facilities
throughout most of the born-again process. At the time of discovery, Sakurai’s
object had evolved from the early WD sequence to a born-again red giant just a
few years earlier. Soon after that, it became completely enshrouded by a thick
layer of dust (Kimeswenger et al. 1997). Early optical observations already
provided indications that the morphology of the ejecta was bipolar with a
thick obscured disc or torus (Tyne et al. 2000; Kerber et al. 2002; Evans et
al. 2006; van Hoof et al. 2007). From MiD-Infrared/VLT Interferometer
(MIDI/VLTI) observations it was possible to derive a model of the morphology
of the ejecta and determine that it contains a dusty disc seen almost edge-on,
efficiently screening the central source (Chesneau et al. 2009). In addition,
Gemini Near InfraRed Imager/ALTtitude conjugate Adaptive optics for the
InfraRed unit (NIRI/ALTAIR) and Near-Infrared Integral Field Spectrometer
(NIFS) images of Sakurai’s object show a pair of lobes expanding with a
velocity of $\sim$300 km s-1, with the north-east and south-west lobes moving
away from and toward us, respectively (Hinkle & Joyce 2014; Hinkle et al.
2020). While all these observations point towards some form of bipolar
structure, they only give a partial and limited view of the situation. Since
it is expected that a considerable fraction of the ejected material has cooled
down and condensed into molecular gas and dust, observations at longer
wavelengths are essential to getting the full picture. In this letter we
present high angular-resolution Atacama Large Millimeter Array (ALMA)
observations to uncover the morphology of the cool ejecta in the youngest
born-again star.

##  2 Observations

We conducted observations of the continuum and molecular line emissions of
Sakurai’s object using ALMA. The project IDs associated with the observations
used in this work are 2017.1.00017.S, 2018.1.00088.S (PI: P. van Hoof), and
2018.1.00341.S (PI: D. Tafoya). All observations were performed using the ALMA
12 m array, which included 43-47 antennas. The relevant parameters of the
observations are summarised in Table LABEL:Tab:1. The data were calibrated and
imaged using the ALMA pipeline, which is included in the Common Astronomy
Software Application (CASA; McMullin et al. 2007).

For the continuum emission, we included data from all the observations.
However, for the molecular line emission, we only used the data from the
observations of project 2018.1.00341.S. The continuum emission was obtained
separately for each spectral window using line-free channels. The high
angular-resolution continuum image, shown in Fig. 1, was made only from the
observations carried out on 22 June, 2019. The continuum emission from four
spectral windows centred at 224.0 GHz, 226.0 GHz, 240.0 GHz, and 242.0 GHz was
combined, resulting in a total bandwidth of 6.73 GHz. The multi-scale
algorithm was used in the cleaning process with the CASA task ‘tclean’. The
robust parameter of the Briggs weighting scheme and the pixel size of the
image were set to 0.5 and 0.004 arcsecs, respectively. The resulting root mean
square (rms) of the continuum image is approximately 44 $\mu$Jy beam-1, with a
beam size of 0.021 $\times$ 0.019 arcsec (P.A.111The position angle (P.A.),
measured north through east, relative to the north celestial pole. $-$84.1∘).

The spectral data includes emission of the H12CN($J$=4$\rightarrow$3),
H13CN($J$=4$\rightarrow$3), and CO($J$=3$\rightarrow$2) lines. The spectral
setup of these observations consists of four 1.875 GHz wide spectral windows
centred at frequencies 343.009 GHz, 344.967 GHz, 355.009 GHz, and 356.897 GHz.
Each spectral window has 1917 channels with a width of 976.562 kHz. The
channel maps were created manually using a robust parameter of 0.0 and a pixel
size of 0.06 arcsecs. The typical rms noise in the individual 0.8 km s-1 wide
channels is 3 mJy beam-1, with a beam size of approximately 0.33 $\times$ 0.30
arcsec (P.A. $\sim$73∘).

Figure 1: ALMA continuum emission of Sakurai’s object at 233 GHz. The emitting
regions consists of a bright compact central component and faint extended
structures elongated in the northeast-southwest direction. A Gaussian fit to
the central component is shown as a grey ellipse. The black cross indicates
the continuum peak position at (J2000) R.A.=17h52m32$\hbox to0.0pt{.\hss}^{\rm
s}$6990 $\pm$ 0$\hbox to0.0pt{.\hss}^{\rm s}$0002, Dec.=$-$17∘41′7$\hbox
to0.0pt{.\hss}^{\prime\prime}$915 $\pm$ 0$\hbox
to0.0pt{.\hss}^{\prime\prime}$003\. The rms noise level of the image is 44
$\mu$Jy beam-1. The horizontal bar indicating the linear scale of the image
assumes a distance of 3.5 kpc to the source. The synthesised beam of the ALMA
observations is shown in the bottom-left corner and its parameters are:
$\theta_{\rm beam}$$=$21$\times$19 milliarcsec, P.A.$=$$-$84.1∘.

##  3 Results and discussion

###  3.1 Continuum emission: The dust disc of Sakurai’s object

The circumstellar dust of Sakurai’s object has been extensively studied at IR
wavelengths (e.g. Geballe et al. 2002; Tyne et al. 2002; Käufl et al. 2003;
Chesneau et al. 2009; Hinkle & Joyce 2014; Hinkle et al. 2020; Evans et al.
2020, 2022, and references therein). In contrast, observations of the dust at
sub-millimetre or longer wavelengths remain limited, with only a few instances
reported (e.g. Evans et al. 2004). From an analysis of the evolution of the
circumstellar dust in Sakurai’s object over an $\sim$20 year period, Evans et
al. (2020) find that, overall, the mid-IR dust emission can be adequately
modelled as originating from a blackbody that has cooled from $T_{\rm
dust}$=1200 K in 1998 to $T_{\rm dust}$=180 K in 2016. Additionally, the dust
mass has increased from $M_{\rm dust}\sim 10^{-9}\leavevmode\nobreak\
M_{\odot}$ to $M_{\rm dust}\sim 10^{-5}\leavevmode\nobreak\ M_{\odot}$ in the
same period. While the mid-IR emission indicates a decline of the dust
temperature, the increase of the continuum emission in the near-IR emission
suggests the formation of new hot dust (Hinkle et al. 2020; Evans et al. 2020,
2022). Thus, a multi-temperature model to fit both cooling, expanding dust and
a hotter near-IR excess is required, and observations at millimetre (mm) and
sub-millimetre (sub-mm) wavelengths are invaluable to constrain physical
parameters.

Figure 2: Spectral Energy Distribution of the ALMA continuum emission of
Sakurai’s object. The dashed line indicates a fit using a power law function,
as described in the main text. The error bars indicate the nominal uncertainty
in the absolute calibration of the ALMA observations of 5%.

In Fig. 1 we present an image of the millimetre (1.3 mm) continuum emission of
Sakurai’s object with an unprecedented angular-resolution of 20 milliarcsec
(70 AU). The image reveals an emitting region consisting of a bright compact
central component and faint extended structures elongated towards the
northeast-southwest direction. The total flux at 233 GHz is 25 mJy and the
peak of the emission is located at (J2000) R.A.=17h52m32$\hbox
to0.0pt{.\hss}^{\rm s}$6990 $\pm$ 0$\hbox to0.0pt{.\hss}^{\rm s}$0002,
Dec.=$-$17∘41′7$\hbox to0.0pt{.\hss}^{\prime\prime}$915 $\pm$ 0$\hbox
to0.0pt{.\hss}^{\prime\prime}$003\. A Gaussian fit to the bright compact
central component gives a deconvolved size of 44$\pm$3$\times$27$\pm$2
milliarcsec (154$\pm$10$\times$95$\pm$7 AU) with a P.A. of 90∘$\pm$6∘, shown
as a grey ellipse in Fig 1. The peak brightness temperature of this component
is $T_{\mathrm{B}}$=217 K. The faint extended emission lies along
P.A.$\approx$60∘ with a total length of approximately 230 milliarcsec (800
AU). Both the major axis of the bright compact central component and the axis
of elongation of the faint extended structures are rotated compared to the
bipolar outflow seen in near-infrared images, P.A.=21∘$\pm$5∘ (Hinkle et al.
2020), suggesting that the ALMA continuum is not tracing this bipolar outflow.
In fact, the bright compact central component is oriented nearly
perpendicularly to the bipolar outflow. Furthermore, its brightness
temperature is in excellent agreement with the temperature ($T\approx 200$ K)
of the cold dust disc hinted at by mid-IR observations (Chesneau et al. 2009;
Hinkle et al. 2020; Evans et al. 2020). This strongly suggests that the
millimetre continuum emission of the bright compact central component is
produced by such a cold dust disc222The structure referred to as a ‘disc’ may
also exhibit a torus-like morphology, but for consistency with the
nomenclature in the literature, it will be referred to as a disc unless
otherwise indicated., implying that our ALMA image is the first direct image
of it.

Early analysis of IR observations indicated that the circumstellar dust of
Sakurai’s object primarily consisted of graphitic carbon (Eyres et al. 1998;
Tyne et al. 2002). Subsequent studies using Spitzer Space Telescope and James
Clerk Maxwell Telescope (JCMT) data suggested that the graphitic carbon
fraction decreased, while the amorphous carbon fraction increased (Evans et
al. 2004, 2006). Chesneau et al. (2009) noted the absence of spectral features
in the MIDI IR spectrum, which they interpreted as indicative of the dominance
of amorphous carbon grains in the dust composition. Furthermore, Evans et al.
(2020) reported evidence for weak 6–7$\mu$m absorption, which they attribute
to hydrogenated amorphous carbon formed in material ejected by Sakurai’s
object at early stages after the born again event.

While all of this evidence appears to suggest that the dust is predominantly
composed of amorphous carbon, there remains a need for additional constraints
on the properties of the dust grains. Some of these constraints can be
obtained from the analysis of data taken within the far-infrared to millimetre
spectral range. In particular, at mm and sub-mm wavelengths, valuable insights
into the characteristics of the emitting dust can be gained by assuming
optically thin emission and fitting the observed flux densities to a power law
of the form $S_{\nu}\propto\nu^{2+\beta}$, where $\beta$ represents the dust
opacity index. For amorphous carbon, $\beta\approx 0.7-1$, whereas for
graphitic carbon, $\beta\approx 2$ (e.g. Mennella et al. 1995, 1998). From our
ALMA observations we obtained the continuum flux densities of Sakurai’s object
for the frequency range 212-357 GHz. The fluxes were measured by integrating a
region containing emission exceeding 3 times the rms noise level of the image,
and they are listed in Table 2 and shown in Fig. 2. A power law fit to the
data gives an opacity index $\beta$=0.53$\pm$0.03. It is noteworthy that, up
to this point, this value represents the most precisely determined dust
opacity index for this source within the mm and sub-mm wavelengths range. This
value is close to the one expected for the sample BE of amorphous carbon
analogues studied by Mennella et al. (1998), albeit being slightly smaller.
The reason for this could be, as Evans et al. (2004) points out, that the
emission is produced by large grains and/or or there has been an increase in
the population of hotter grains (which have flatter emissivities).

Assuming isothermal dust with a temperature $T_{\rm dust}=217$ K (i.e.
assuming that the peak continuum emission is optically thick) and that most of
the dust emission is optically thin, the mass of the dust can be obtained
using the following expression:

$M_{\rm dust}=\frac{S_{\nu}\,D^{2}}{\kappa_{\nu}\,B_{\nu}(T_{\rm dust})},$

where $S_{\nu}$ is the flux density of the continuum emission, $D$ is the
distance to the source, $\kappa_{\nu}=\kappa_{0}(\nu/\nu_{0})^{\beta}$ is the
dust absorption coefficient and $B_{\nu}(T)$ is the Planck function. Dust
absorption coefficient of amorphous carbon in the literature range from
$\kappa_{300\,{\rm GHz}}\sim 0.9\,\text{cm}^{2}\,\text{g}^{-1}$ up to
$\kappa_{300\,{\rm GHz}}\sim 90\,\text{cm}^{2}\,\text{g}^{-1}$ (Draine & Lee
1984; Ossenkopf & Henning 1994; Mennella et al. 1998; Suh 2000). Thus, using
the dust opacity index obtained from our ALMA observations ($\beta=0.53\pm
0.03$, see Fig. 2), the calculated mass of amorphous carbon dust in Sakurai’s
object is $6\times 10^{-3}\,M_{\odot}$ and $6\times 10^{-5}\,M_{\odot}$ for
$\kappa_{300\,{\rm GHz}}\sim 0.9\,\text{cm}^{2}\,\text{g}^{-1}$ and
$\kappa_{300\,{\rm GHz}}\sim 90\,\text{cm}^{2}\,\text{g}^{-1}$, respectively.
Estimations of the dust mass based on mid-IR observations range from $1.8$ to
$6\times 10^{-5}\,M_{\odot}$ (Chesneau et al. 2009; Hinkle et al. 2020; Evans
et al. 2020). Thus, in order to obtain a dust mass consistent with these
previous estimations, it is necessary to consider amorphous carbon dust with a
relatively large absorption coefficient in the sub-mm wavelength regime.

Figure 3: Sakurai’s object H12CN($J$=4$\rightarrow$3),
H13CN($J$=4$\rightarrow$3) and CO($J$=3$\rightarrow$2) line emission. The
green line is the spectrum of the emission line detected previously with APEX.
The blue line is the spectrum observed with ALMA. The spectral resolution of
the APEX and ALMA observations is 20 km s-1 and 0.8 km s-1, respectively. The
bottom axis indicates the local standard of rest (LSR) velocity and the top
axis indicated the velocity offset from the systemic velocity assumed to be
125 km s-1. Figure 4: Spatial-kinematical distribution of the
H12CN($J$=4$\rightarrow$3) line emission in Sakurai’s object. Left: Spatial
distribution of the centroid positions of the H12CN($J$=4$\rightarrow$3) line
emission at each individual velocity channel. Right: Position-velocity diagram
of the H12CN($J$=4$\rightarrow$3) line emission centroids along the
Declination axis. The vertical and horizontal dashed lines indicate the
velocity offset and Declination offset, respectively, over which an inversion
of the velocity gradient is observed. The error bars correspond to the
uncertainties of the centroid positions. The axis indicating the linear scale
assume a distance of 3.5 kpc to the source. Figure 5: Bipolar outflow and
expanding disc around Sakurai’s object. Left: H12CN($J$=4$\rightarrow$3) line
emission and 233 GHz continuum emission. The dashed lines delineate an
hourglass morphology hinted at by the continuum emission. Right: Hourglass
model for the spatial-kinematical distribution of the bipolar outflow in
Sakurai’s object. The right and top axis indicating the linear scale assume a
distance of 3.5 kpc to the source.

###  3.2 Line emission: an expanding disc and a fast bipolar outflow in
Sakurai’s object

Carbon-rich molecules (CN, C2, CO) were detected in the ejecta of Sakurai’s
object as early as one year after it was discovered to be undergoing the born-
again event (Eyres et al. 1998). Subsequently, H-containing species (e.g. HCN
and C2H2) were also found in the ejected material (Evans et al. 2006). All of
these molecular species were detected against the dust continuum emission,
probing primarily the molecular gas along the line-of-sight direction. More
recently, Tafoya et al. (2017) detected for the first time emission of
H12CN($J$=4$\rightarrow$3), H13CN($J$=4$\rightarrow$3), and
H12CN($J$=2$\rightarrow$1) lines, and van Hoof et al. (2018) presented semi-
resolved ALMA images of the line of emission of CN, CO, and HC3N. While these
observations suggest that the CO molecules are situated in proximity to the
dusty region (Eyres et al. 2004; Worters et al. 2009; van Hoof et al. 2018),
the HCN and CN are believed to reside in bipolar lobes (Evans et al. 2006; van
Hoof et al. 2018). However, due to the limited angular resolution of previous
observations, it has not been possible to discern the specific regions from
which the molecular emission originates.

To further investigate the molecular component of the ejecta around Sakurai’s
object, we conducted ALMA observations targeting the
H12CN($J$=4$\rightarrow$3), H13CN($J$=4$\rightarrow$3), and
CO($J$=3$\rightarrow$2) line emissions. These observations provided an angular
resolution of approximately 300 milliarcsec ($\sim$1000 AU) and a spectral
resolution of 0.8 km s-1. Figure 3 shows the spectra obtained from the ALMA
observations, alongside the spectra acquired using the Atacama Pathfinder
Experiment (APEX) in 2013 and 2016 (Tafoya et al. 2017). Due to the lower S/N
of the APEX spectra, accurately measuring any changes in the brightness of the
H12CN($J$=4$\rightarrow$3) and H13CN($J$=4$\rightarrow$3) lines between the
APEX and ALMA observations is challenging. However, the spectrum in the right
panel of Fig. 3 clearly shows that the CO($J$=3$\rightarrow$2) line, observed
in the ALMA data, was either not detected or was significantly weaker in the
APEX observations. This could be the result of either the recent formation of
significant amounts of CO molecules or changes in the physical conditions of
the emitting gas. We note that although CO molecules were found in the newly
formed ejecta of Sakurai’s object (Eyres et al. 1998; Pavlenko et al. 2004),
they were detected through absorption features in the near-infrared spectra,
which is sensitive to even small amounts of CO. Consequently, the cause of the
observed enhancement in the CO($J$=3$\rightarrow$2) line intensity remains
uncertain.

The widths of the molecular lines are notably broad. The velocity offset range
covered by the H12CN($J$=4$\rightarrow$3) line extends from $-$350 km s-1 to
300 km s-1, with the red-shifted portion even exceeding the bandwidth of our
spectral window. In contrast, the H13CN($J$=4$\rightarrow$3) and
CO($J$=3$\rightarrow$2) lines are blended, preventing us from spatially
separating their emissions. Furthermore, the blue-shifted part of the
CO($J$=3$\rightarrow$2) line also extends beyond the limits of our spectral
window. Despite these limitations, the significantly higher spectral
resolution of the ALMA observations enables the identification of distinct
velocity components for each molecule. Additionally, the achieved high S/N
allows for precise determination of the centroid of the molecular emission in
each velocity channel, surpassing the size of the synthesised beam (Condon
1997). Specifically, the uncertainties of the centroid positions are obtained
as $\sigma_{\rm centroid}$=$({\sigma_{\theta}^{2}+\sigma_{\rm
BP}^{2}})^{1/2}$, where $\sigma_{\theta}$$\approx$$\theta_{\rm beam}$/(2 SNR)
and $\sigma_{\rm BP}$=$\theta_{\rm beam}$($\sigma_{\phi}$/360∘). In these
expressions $\theta_{\rm beam}$ is the beam size and $\sigma_{\phi}$ is phase
noise of the bandpass calibrator (e.g. Zhang et al. 2017). For our
observations, $\theta_{\rm beam}$=300 milliarcsecs and $\sigma_{\phi}$=0.08∘.

In Fig. 4 we present the spatial distribution and a position-velocity (PV)
diagram of the H12CN($J$=4$\rightarrow$3) line emission. The emission exhibits
a clear north-south velocity gradient, with the blue-shifted and red-shifted
components positioned to the south and north, respectively, of the peak of the
continuum emission. The spatial distributions and PV diagrams of the
H13CN($J$=4$\rightarrow$3) and CO($J$=3$\rightarrow$2) lines show similar
characteristics to those of the H12CN($J$=4$\rightarrow$3) line, but they
emitting region seems to be more compact. This aligns perfectly with the
direction and velocity gradient of the near-infrared outflow (Hinkle et al.
2020), confirming that the H12CN($J$=4$\rightarrow$3) line emission indeed
traces the bipolar outflow of Sakurai’s object.

Upon closer examination of the PV diagram, it is evident that there is a
distinctive change in the velocity gradient. In the central region, where
velocity offsets range from $-$50 to 50 km s-1 (see right panel of Fig. 4),
the slope is inverted. The horizontal dashed lines in the PV diagram indicate
that this emission extends over approximately 50 milliarcsec ($\sim$175 AU),
which is comparable to the size of the dust disc observed in the ALMA
continuum image. This particular pattern of an inverted velocity gradient is
commonly observed in systems that exhibit both a bipolar outflow and an
expanding equatorial disc (e.g. Alcolea et al. 2007). Thus, it is likely that
the inverted velocity gradient is produced by the expansion of the equatorial
disc seen in the millimetre continuum image (see Fig.1).

For an expanding circular thin disc of radius $R$, whose polar axis has an
inclination $i$ with respect to the line of sight, that is $i$=0∘ and $i$=90∘
correspond to the disc seen face-on and edge-on, respectively, the projected
minor axis in angular units, $\theta_{\rm disc}$, and its line-of-sight
velocity, $v_{\rm los}$, are given by: $\theta_{\rm disc}=2R{\rm cos}\,i/D$
and $v_{\rm los}=v_{\rm exp}{\rm sin}\,i$, respectively, where $D$ is the
distance to the source and $v_{\rm exp}$ is the expansion velocity of the
disc. Assuming a constant expansion velocity, the radius of the disc can be
calculated as $R=v_{\rm exp}\,\tau$, where $\tau$ is the time since the
ejection of the molecular gas. Thus, the inclination of the disc can be
obtained from the following expression:

${\rm tan}\,i=3.0\left[\frac{v_{\rm los}}{\rm
50\,km\,s^{-1}}\right]\left[\frac{\tau}{\rm
25\,years}\right]\left[\frac{\theta_{\rm disc}}{\rm
50\,milliarcsec}\right]^{-1}\left[\frac{D}{\rm 3.5\,kpc}\right]^{-1}.$

From our observations we find that $v_{\rm los}$=50 km s-1 and $\theta_{d}$=50
milliarcsec, and considering that the ejection of the material occurred around
25 years ago, the resulting inclination of the disc is $i$=72∘. This result
confirms that the equatorial disc is seen almost edge-on, completely obscuring
the central star (Chesneau et al. 2009; Hinkle et al. 2020). The de-projected
expansion velocity and the radius of the disc are $v_{\rm exp}$=53 km s-1 and
$R$=277 AU, respectively.

We note that the derived inclination of the disc from our ALMA observations is
opposite to the one deduced from the VLTI observations (Chesneau et al. 2009).
While the VLTI observations suggest that the northern side of the disc is
tipped toward us, which would mean that the northern edge of the expanding
disc is red-shifted, the ALMA observations yield the opposite for the inner 50
milliarcsec: blue-shifted emission toward the north and red-shifted emission
toward the south (cf. right panel of Fig. 4). Furthermore, the inclination
obtained from the VLTI observations implies that the northern part of the
outflow should be red-shifted, which is opposite to the velocity gradient seen
from the near-infrared and ALMA observations. Thus, we conclude that the
southern side of the disc is the one tipped toward us.

The derived inclination provides an estimate of the de-projected expansion
velocity of the bipolar outflow detected through the
H12CN($J$=4$\rightarrow$3) emission, which is approximately 1000 km s-1,
assuming a fairly collimated structure. Such high expansion velocities for the
bipolar outflow of Sakurai’s object are not unexpected. For instance, when
correcting for the inclination, the expansion velocity of the He I 10830 $\AA$
emission line observed with the Gemini NIFS shows a similar value. Also,
Kerber et al. (2002) concluded that the bipolar outflow, traced by the [N II]
$\lambda$6583 line emission, exhibited an expansion velocity of about 800 km
s-1. It is worth noting that such a high expansion velocity implies that the
observed lobes should be more than four times larger than observed, assuming
that the molecular bipolar outflow formed at the same time as the rest of the
material was ejected. This could indicate that the H12CN($J$=4$\rightarrow$3)
emission does not fully trace the entirety of the molecular outflow but only
the regions where the excitation conditions required to observe HCN molecules
are optimal. On the other hand, we cannot dismiss the possibility that the
outflow traced by the H12CN($J$=4$\rightarrow$3) emission has a relatively
wide opening angle. In such case, the expansion velocity of the molecular
material would be lower, which could explain the smaller observed size of the
molecular lobes.

The left panel of Fig. 5 shows the continuum emission superimposed on the
spatial distribution of the H12CN ($J$=4$\rightarrow$3) line emission. Towards
the north, the continuum emission shows two protrusions that appear to be
tracing the base of an hourglass-shaped outflow (indicated by the dashed
lines). On the southern side, the continuum emission exhibits another such
protrusion. A sketch of the inferred geometry and orientation of the material
around Sakurai’s object is shown in the right panel of Fig. 5. As suggested by
the protrusions in the continuum emission, the bipolar outflow lobes have an
opening angle of around 60∘ and their walls seem to be denser than the regions
close to the polar axis. In addition, as mentioned above, the spatio-
kinematical configuration of the molecular emission is the same as the one
derived from observations of the He I 10830Å and the [C I] 9850Å lines (Hinkle
et al. 2020). Thus, it is possible that the expanding atomic gas forms a more
collimated outflow that entrains the surrounding molecular material around it,
forging the molecular outflow.

##  4 Concluding remarks

It is remarkable that most known objects that are thought to have experienced
a born-again event (A30, A78, V605 Aql, Sakurai’s object), have a strikingly
similar morphology, namely an expanding equatorial disc and a bipolar outflow
(Toalá et al. 2015; Hinkle et al. 2020; Tafoya et al. 2022; Rodríguez-González
et al. 2022). The formation of this particular morphology has most often been
attributed to a stellar/sub-stellar binary companion. For example A30, a
common-envelope phase following the born-again event has been proposed to
explain the observed physical structure and abundances of the ejecta around
the central star (Rodríguez-González et al. 2022). Tafoya et al. (2022)
proposed a similar common-envelope scenario to account for the expanding
equatorial structure and bipolar outflow seen in their ALMA observations of
V605 Aql. In contrast, standard models of the born-again event are based on
single-star evolution that would result in a simple, spherical morphology of
the ejecta. This has so far been observed in only one case, HuBi 1 (Toalá et
al. 2021; Rechy-García et al. 2020). It is difficult to explain the more
complex morphology observed for the other observed objects without invoking
common envelope evolution, which would necessitate a paradigm shift since
close-binary interactions would certainly influence the physics of the born-
again process.

Our ALMA observations have revealed, in unprecedented detail, the morpho-
kinematical structure of the material surrounding the youngest born-again
star, allowing us to study the initial stages of the ejection process. These
results set stringent observational constraints for future models that will
need to be consistent not only with the observed chemistry, but also the
physical structure and intriguing morphology observed in the different stages
of the born-again process.

###### Acknowledgements.

This paper makes use of the following ALMA data: ADS / JAO.ALMA # 2017.1.
00017.S, 2018.1.00088.S and 2018.1.00341.S. ALMA is a partnership of ESO
(representing its member states), NSF (USA) and NINS (Japan), together with
NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ. DT acknowledges support from Onsala Space
Observatory for the provisioning of its facilities support. The Onsala Space
Observatory national research infrastructure is funded through Swedish
Research Council grant No 2017-00648.

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##  Appendix A ALMA Observations

Table 1: Parameters of the ALMA observations Observation date | Frequencya | Flux & Bandpass calibrator | Phase calibrator | Bandwidthb | rmsc | Beam
---|---|---|---|---|---|---
(YYYY-MM-DD) | (GHz) | (name, Jy) | (name, mJy) | (GHz) | ($\mu$Jy beam-1) | (mas$\times$mas,degree)c
Project code: 2017.1.00017.S
2018-09-28 | 257.049 | J1924-2914, 2.977 | J1743-1658, 142.5$\pm$0.8 | 1.86 | 130 | 268$\times$218, 69.7
| 258.911 | 2.963 | 142.2$\pm$0.8 | 0.91 | 130 | 264$\times$214, 69.6
| 272.041 | 2.868 | 138.6$\pm$0.8 | 0.82 | 160 | 252$\times$204, 68.8
| 273.904 | 2.855 | 139.1$\pm$0.7 | 1.64 | 160 | 251$\times$204, 68.6
Project code: 2018.1.00088.S
2019-05-01 | 212.043 | J1924-2914, 2.746 | J1743-1658, 143.4$\pm$0.9 | 1.52 | 31 | 515$\times$454, $-$79.9
| 214.933 | 2.723 | 142.8$\pm$1.0 | 1.86 | 30 | 514$\times$452, $-$81.1
| 227.357 | 2.634 | 139.6$\pm$1.0 | 0.89 | 37 | 487$\times$431, $-$81.4
| 230.038 | 2.616 | 138.6$\pm$1.0 | 0.86 | 40 | 482$\times$425, $-$81.4
2019-03-31[2019-04-11]d | 220.945 | J1924-2914[J1517-2422], 2.900[2.453] | J1743-1658, 157.0$\pm$0.8[145.0$\pm$0.5] | 1.16 | 48 | 792$\times$571, $-$88.1
| 223.944 | J1924-2914[J1517-2422], 2.875[2.443] | 155.9$\pm$0.8[144.3$\pm$0.5] | 1.85 | 41 | 780$\times$560, $-$88.5
| 236.454 | J1924-2914[J1517-2422], 2.776[2.402] | 151.9$\pm$0.9[140.0$\pm$0.6] | 1.21 | 54 | 745$\times$533, $-$88.9
| 238.941 | J1924-2914[J1517-2422], 2.757[2.394] | 151.2$\pm$0.7[139.5$\pm$0.6] | 1.85 | 52 | 735$\times$531, $-$88.9
2019-04-30[2019-05-01]d | 250.982 | J1924-2914, 2.483 | J1743-1658, 130.8$\pm$0.9[126.4$\pm$0.9] | 1.25 | 57 | 492$\times$403, 89.2
| 252.850 | 2.472 | 130.2$\pm$0.9[126.5$\pm$1.0] | 1.41 | 57 | 486$\times$400, 88.9
| 265.808 | 2.400 | 126.9$\pm$0.7[123.7$\pm$1.0] | 1.08 | 70 | 472$\times$412, $-$66.9
| 267.677 | 2.390 | 126.4$\pm$0.7[122.9$\pm$0.7] | 1.19 | 71 | 458$\times$381, 89.8
2019-06-22 | 224.000 | J1924-2914, 2.913 | J1743-1658, 157.0$\pm$1.0 | 1.75 | 73 | 25$\times$24, 48.2
| 226.000 | 2.898 | 156.1$\pm$0.6 | 1.48 | 73 | 24$\times$23, $-$82.8
| 240.000 | 2.795 | 152.6$\pm$0.8 | 1.75 | 77 | 23$\times$22, $-$89.4
| 242.000 | 2.782 | 151.8$\pm$0.7 | 1.75 | 84 | 23$\times$21, $-$87.2
Project code: 2018.1.00341.S
2019-04-21 | 343.009 | J1924-2914, 2.051 | J1742-1517, 149.2$\pm$0.4 | 1.37 | 240 | 335$\times$303, 77.3
| 344.967 | 2.044 | 148.7$\pm$0.4 | 0.79 | 260 | 336$\times$309, 72.0
| 355.009 | 2.008 | 147.8$\pm$0.5 | 0.76 | 290 | 326$\times$298, 73.8
| 356.897 | 2.002 | 147.1$\pm$0.5 | 1.36 | 280 | 323$\times$293, 76.8
aCentral frequency of the spectral window.
bEffective bandwidth of the spectral window used to produce the continuum
images.
crms noise over the effective bandwidth used to produce the continuum images.
dmas=milliarcsec.
eData taken at two different dates were used to produce the continuum images.
Table 2: Frequency and Flux Density Measurements

Frequency (GHz)aa$a$Central frequency of the bandwidth used to create the continuum images. | Flux Density (mJy)bb$b$Flux density obtained by integrating a region containing emission exceeding 3 times the rms noise level of the image.The nominal uncertainty in the absolute calibration of the ALMA observations of 10% is adopted.
---|---
212.071 | 19.91
214.908 | 20.45
220.918 | 23.81
223.913 | 24.15
224.006 | 24.57
225.733 | 23.57
227.537 | 24.01
230.150 | 24.60
236.392 | 28.63
238.911 | 29.10
239.974 | 29.40
242.482 | 30.13
250.953 | 31.08
252.820 | 32.45
257.927 | 35.36
259.805 | 35.88
265.775 | 37.04
267.645 | 37.80
272.886 | 40.02
274.756 | 42.68
342.960 | 69.70
344.387 | 70.27
355.186 | 76.75
356.856 | 78.11
333


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§ GFLOW AND DETERMINISM

Here, we introduce the notation and concepts needed to state and prove the fact that gflow is a sufficient condition for uniform, strong and stepwise determinism of a measurement pattern. Then we give the detailed statement along with the proof.

Throughout this appendix, $\simeq$ shall denote equality up to a global phase; that is, for linear maps $L$ and $K$, we have $L\simeq K$ iff there is an angle $\al\in[0,2\pi)$ such that $L=e^{i\al}K$.

Recall from Definition <ref> that the linear map implemented by a particular set of measurement outcomes in a pattern is called a branch of the pattern. We begin by making this more precise.

Let $\pat$ be a measurement pattern with $n$ measurement commands. A branch of the pattern is a sequence $p$ of length $n$ whose entries are binary digits $0$ and $1$ together with a linear map $\intf{\pat}_p$, where each measurement is replaced with an outcome corresponding to either $0$ or $1$ as indicated by the corresponding entry in $p$.

If $s$ is a binary sequence with $k$ entries where $k\leq n$, we may also define an `intermediate pattern' $\intf{\pat}_s$ by replacing the first $k$ measurements in $\pat$ by an outcome corresponding to the entries in $s$.

We may now state various forms of determinism (Definition <ref>) using the notation of the above definition. A pattern $\pat$ with $n$ measurement commands is deterministic if for any branches $p$ and $q$ there is a complex number $c$ such that $\intf{\pat}_p=c\intf{\pat}_q$. It is strongly deterministic if for any branches $p$ and $q$ we have $\intf{\pat}_p\simeq\intf{\pat}_q$. It is stepwise deterministic if the pattern $\intf{\pat}_s$ is deterministic for any binary sequence $s$ with at most $n$ entries. Recall that a pattern is uniformly deterministic if it is deterministic for any choice of measurement angles in the measurement commands appearing in $\pat$.

It will be useful to encode the measurement planes using following functions for any  $(G,I,O,\ld)$.

\begin{align*}
\beta_X,\beta_Z : \comp O &\rightarrow\{0,1\} \\
\beta_X(i) &= \begin{cases} 1 & \text{if } \ld(i)\in\{\XYm,\XZm\} \\
0 & \text{if } \ld(i)=\YZm \end{cases} \\
\beta_Z(i) &= \begin{cases} 1 & \text{if } \ld(i)\in\{\XZm,\YZm\} \\
0 & \text{if } \ld(i)=\XYm \end{cases}
\end{align*}

The following observation will be the starting point for our proof of the main theorem.

Let $s_i$ be the binary digit indicating the measurement outcome of qubit $i$. Then[Note that the pattern below is not runnable, as the corrections depend on a future measurement outcome.]

\begin{equation*}
\intf{M^{\ld(i), \al(i)}_i [X_i]^{\beta_Z(i)s_i} [Z_i]^{\beta_X(i)s_i}}_p \simeq \bra{+_{\ld(i),\al(i)}}_i
\end{equation*}

for both branches $p$.

If $p=(0)$, so that $s_i=0$, this is clear, as the corrections vanish while there is nothing to correct. Hence suppose $p=(1)$. We have three cases:

* $\ld(i) = \XYm$, so that $\beta_Z(i)=0$ and $\beta_X(i)=1$,

* $\ld(i) = \XZm$, so that $\beta_Z(i)=\beta_X(i)=1$,

* $\ld(i) = \YZm$, so that $\beta_Z(i)=1$ and $\beta_X(i)=0$.

Recalling that the Pauli operators act on the basis qubits as

\begin{align*}
X\ket{0}=\ket{1}\quad &X\ket{1}=\ket{0} \\
Z\ket{0}=-\ket{0}\quad &Z\ket{1}=\ket{1}
\end{align*}

we compute in each case, starting from the (dual of the) left-hand side:

\begin{align*}
1. && Z_i(\ket 0 - e^{i\al(i)}\ket 1) &= -(\ket 0 + e^{i\al(i)}\ket 1) \\
&&& \simeq \ket{+_{\XYm,\al(i)}}_i, \\
2. && Z_iX_i\left(\sin\left(\frac {\al(i)} 2\right)\ket 0 - \cos\left(\frac {\al(i)} 2\right)\ket 1\right) &= \sin\left(\frac {\al(i)} 2\right)\ket 1 + \cos\left(\frac {\al(i)} 2\right)\ket 0 \\
&&& = \ket{+_{\XZm,\al(i)}}_i, \\
3. && X_i\left(\sin\left(\frac {\al(i)} 2\right)\ket 0 - i\cos\left(\frac {\al(i)} 2\right)\ket 1\right) &= \sin\left(\frac {\al(i)} 2\right)\ket 1 - i\cos\left(\frac {\al(i)} 2\right)\ket 0 \\
&&&\simeq \cos\left(\frac {\al(i)} 2\right)\ket 0 + i\sin\left(\frac {\al(i)} 2\right)\ket 1 \\
&&&= \ket{+_{\YZm,\al(i)}}_i. \qedhere
\end{align*}

Let $(G,I,O)$ be an open graph. View $(V,I,O)$ as the register of a measurement pattern as in Definition <ref>. Define the following command for each $i\in\comp I$:

$$K_i\coloneqq X_i\prod_{j\in N_G(i)}Z_j.$$

We call $K_i$ the graph stabiliser at qubit $i$ for reasons explained below. Observe that the commands $Z_j$ in the product act on different qubits and hence commute with each other. Thus the product does not depend on the order, so that $K_i$ is well-defined.

The reason for calling the $K_i$ graph stabilisers is the following lemma.

For any open graph $(G,I,O)$ and for any $i\in\comp I$ with graph stabiliser $K_i$ we have

$$K_iE_GN_{\comp I} = E_GN_{\comp I}.$$

Notation: Given a command $C$ in a measurement pattern with register $V$, for any subset of qubits $S\sse V$ we write $C_{S}\coloneqq\prod_{i\in S}C_i$ if the right-hand side is well-defined (in particular, if it does not depend on the order in which the different operations are applied). If all the $C_i$'s are moreover Pauli $X$ or $Z$ operators, and $s_i$ is a binary digit corresponding to the measurement outcome of qubit $i$, we may treat $C_{S}$ as a joint correction by writing $C_{S}^{s_i}\coloneqq\prod_{i\in S}[C_i]^{s_i}$. Note that the graph stabiliser may now be written as $K_i = X_iZ_{N_G(i)}$.

We are now ready to state and prove the central theorem about gflow.

Let $\Gamma = (G,I,O, \ld)$ be a . If $\Gamma$ has a gflow $\gflow = (g,\prec)$, then for any set of measurement angles $\alpha:\comp O\rightarrow[0,2\pi)$ the following pattern is runnable as well as strongly and stepwise deterministic:

\begin{equation}\label{pattern}
\pat_{\Gamma,\gflow,\al}\coloneqq\prod^\prec_{i\in \comp{O}} \left( X_{g(i)\setminus\{i\}}^{s_i}Z_{\odd{}{g(i)}\setminus\{i\}}^{s_i} M^{\lambda(i), \al(i)}_i\right) E_GN_{\comp{I}},
\end{equation}

where $\prod^\prec$ denotes concatenation of commands following the order $\prec$. Moreover, $\pat_{\Gamma,\gflow,\al}$ realizes the linear map associated with $\Gamma$ at angles $\al$ (Definition <ref>) in the sense that for any branch $p$ we have

\begin{equation}\label{eqn:unit_embedding}
\intf{\pat_{\Gamma,\gflow,\al}}_p\simeq\prod_{i\in \comp{O}}\bra{{+_{\lambda(i), \al(i)}}}_i E_G\phZ {\comp{I}}{}.
\end{equation}

Note that the pattern in equation (<ref>) is uniformly deterministic, as it is (strongly) deterministic independently of the choice of $\al$. It can be shown <cit.> that any strongly deterministic pattern implements a unitary embedding. Thus the map in equation (<ref>) is guaranteed to be one. In particular, if $|I|=|O|$, then the map is unitary.

First observe that $\pat_{\Gamma,\gflow,\al}$ is indeed runnable; the conditions <ref> and <ref> ensure that no command acts on a qubit that has been measured[We remark that the condition (<ref>) would fail to guarantee this.], and all the other conditions in Definition <ref> are easily verified from the shape of the pattern.

For strong determinism, it is sufficient to show the claimed equality (<ref>), as this shows that any two branches are equal (up to a global phase). The proof of the equality is based on Lemma <ref> and the following observation. Using the fact that the Pauli matrices are involutive, we show that for all $i\in\comp O$,

\begin{align}\label{eqn:identity_Z}
\prod_{u\in g(i)}Z_{\eve{}{g(i)}\cap N_G(u)} &= I, \\ \nonumber
\prod_{u\in g(i)}Z_{\odd{}{g(i)}\cap N_G(u)} &= Z_{\odd{}{g(i)}},
\end{align}

where $I$ is the identity matrix. The first equality follows by taking $u\in g(i)$ and $j\in N_G(u)$ such that $j$ is in the even neighbourhood of $g(i)$. Then the number of neighbours of $j$ in $g(i)$ is even, so that $Z_j$ appears in the product an even number of times. For the second equality, take $u\in g(i)$ and $j\in N_G(u)$ such that $j$ is in the odd neighbourhood of $g(i)$. Then the number of neighbours of $j$ in $g(i)$ is odd, so that $Z_j$ appears in the product an odd number of times.

Thus we compute from equation (<ref>):

\begin{align*}
& \intf{\pat_{\Gamma,\gflow,\al}}_p \\
&= \prod^\prec_{i\in \comp{O}} \intf{ M^{\lambda(i), \al(i)}_i X_{g(i)\setminus\{i\}}^{s_i} Z_{\odd{}{g(i)}\setminus\{i\}}^{s_i} }_p E_G N_{\comp{I}} && \textrm{(s1)} \\
&= \prod^\prec_{i\in \comp{O}} \intf{ M^{\lambda(i), \al(i)}_i \cx{i}{2\beta_Z(i)s_i} \cz{i}{2\beta_X(i)s_i} X_{g(i)\setminus\{i\}}^{s_i} Z_{\odd{}{g(i)}\setminus\{i\}}^{s_i} }_p E_G N_{\comp{I}} && \textrm{(s2)} \\
&\simeq \prod^\prec_{i\in \comp{O}} \left( \bra{+_{\lambda(i), \al(i)}}_i \cx{i}{\beta_Z(i)s_i} \cz{i}{\beta_X(i)s_i} X_{g(i)\setminus\{i\}}^{s_i} Z_{\odd{}{g(i)}\setminus\{i\}}^{s_i} \right) E_G N_{\comp{I}} && \textrm{(s3)} \\
&\simeq \prod^\prec_{i\in \comp{O}} \left( \bra{+_{\lambda(i), \al(i)}}_i X_{g(i)}^{s_i} Z_{\odd{}{g(i)}}^{s_i} \right) E_G N_{\comp{I}} && \textrm{(s4)} \\
&= \prod^\prec_{i\in \comp{O}} \left( \bra{+_{\lambda(i), \al(i)}}_i \prod_{u\in g(i)}\cx{u}{s_i} \prod_{v\in g(i)}Z_{\odd{}{g(i)}\cap N_G(v)}^{s_i} \prod_{w\in g(i)}Z_{\eve{}{g(i)}\cap N_G(w)}^{s_i} \right) E_G N_{\comp{I}} && \textrm{(s5)} \\
&\simeq \prod^\prec_{i\in \comp{O}} \left( \bra{+_{\lambda(i), \al(i)}}_i \prod_{u\in g(i)} \left( \cx{u}{s_i} Z_{\odd{}{g(i)}\cap N_G(u)}^{s_i} Z_{\eve{}{g(i)}\cap N_G(u)}^{s_i} \right) \right) E_G N_{\comp{I}} && \textrm{(s6)} \\
&= \prod^\prec_{i\in \comp{O}} \left( \bra{+_{\lambda(i), \al(i)}}_i \prod_{u\in g(i)} \left( \cx{u}{s_i} Z_{N_G(u)}^{s_i} \right) \right) E_G N_{\comp{I}} && \\
&= \prod^\prec_{i\in \comp{O}} \left( \bra{+_{\lambda(i), \al(i)}}_i \prod_{u\in g(i)}K_u^{s_i} \right) E_G N_{\comp{I}} && \\
&= \prod^\prec_{i\in \comp{O}} \bra{+_{\lambda(i), \al(i)}}_i E_G N_{\comp{I}}. && \textrm{(s7)}
\end{align*}

Here we used:

* The fact that the correction terms do not act on the vertex $i$.

* Involutivity of the Pauli matrices.

* Lemma <ref> and that $X_i$ and $Z_i$ anti-commute.

* Conditions <ref>-<ref>. For example, if $\ld(i)=YZ$, then $\beta_Z(i)=1$ and by <ref> $i\in g(i)$, so that $\cx{i}{s_i}X_{g(i)\setminus\{i\}}^{s_i}=X_{g(i)}^{s_i}$, while $\beta_X(i)=0$ and again by <ref> $i\notin\odd{}{g(i)}$, so that $Z_{\odd{}{g(i)}\setminus\{i\}}=Z_{\odd{}{g(i)}}$. The cases of other measurement planes are similar.

* Equations (<ref>) and the definition of $X_{g(i)}^{s_i}$.

* The facts that $X_i$ and $Z_i$ anti-commute, and that commands acting on different qubits commute.

* Lemma <ref> and <ref> in order to commute the graph stabilisers past the liner maps.

Thus the pattern is strongly deterministic.

It remains to show that $\pat_{\Gamma,\gflow,\al}$ is stepwise deterministic. Thus consider a subgraph of $\Gamma$ (with $I$, $O$ and $\ld$ restricted appropriately) obtained by removing some minimal vertices in the order $\prec$. The interpretation is that those vertices have already been measured. By conditions <ref> and <ref>, for any vertex $i$ in the subgraph, the correction set $g(i)$ and its odd neighbourhood do not contain any of the removed vertices, and are thus themselves contained in the subgraph. Therefore we may restrict $g$ to the subgraph, and the odd neighbourhood of each $g(i)$ in the subgraph will be equal to its odd neighbourhood in the full graph. Thus the conditions <ref>, <ref> and <ref> are satisfied by the subgraph. If we also restrict the partial order to the subgraph, we find that the conditions <ref> and <ref> are satisfied. Hence the gflow of the full graph restricts to a gflow on the subgraph. Therefore, since the measurements in $\pat_{\Gamma,\gflow,\al}$ follow the order $\prec$, we may think of each measurement as removing a minimal vertex in $\Gamma$. As we have just seen, the graph with some minimal vertices removed has a gflow given by restriction of the original gflow, so that repeating the preceding parts of the proof will yield another (strongly) deterministic pattern. This pattern is equal to the proper sub-pattern of $\pat_{\Gamma,\gflow,\al}$ which does not involve the already-performed measurement and its corresponding corrections. Therefore, the original pattern is stepwise deterministic.

§ PROOFS FOR SECTION <REF>

The following lemma records how the odd neighbourhood of an arbitrary set of vertices changes under local complementation.

Given a graph $G=(V,E)$, a subset $A\subseteq V$ and a vertex $u \in V$, we have

$$\odd {G\star u} A=\begin{cases}\odd G A \symd (N_G(u)\cap A) & \text{if $u\notin \odd G A$}\\ \odd G {A} \symd (N_G(u)\setminus A)&\text{if $u\in \odd G A$}.\end{cases}$$

Now we can prove the main lemma.

We divide the proof into three parts, first proving that $g'(u)$ satisfies the required properties, then proving that $g'(v)$ for $v\neq u$ satisfies <ref> and <ref>, and finally showing that $g''(v)$ satisfies the conditions <ref>–<ref>. Each part is further subdivided based on the values of $\lambda(u)$ or $\lambda(v)$.

Part 1: Note that condition <ref> is trivially satisfied by $g'(u)$. We now show that $g'(u)$ satisfies the remaining conditions, subdividing into two cases based on $\lambda(u)$.

Case 1a: If $\ld(u)\in\{\XYm,\XZm\}$, then $u\in\odd{G}{g(u)}$ and $\ld'(u)\in\{\XYm,\XZm\}$ with $\ld'(u)\neq\ld(u)$.
We have $\odd{G\star u}{\{u\}} = N_{G\star u}(u) = N_G(u)$.

\begin{align}
\odd{G\star u}{g'(u)} &= \odd{G\star u}{g(u)\symd\{u\}} \nonumber \\
&= \odd{G\star u}{g(u)} \symd \odd{G\star u}{\{u\}} \nonumber \\
&= \odd{G}{g(u)} \symd (N_G(u)\setminus g(u)) \symd N_G(u) \nonumber \\
&= \odd{G}{g(u)} \symd (N_G(u)\cap g(u)), \label{eq:ld_XYXZ}
\end{align}

where the third equality uses Lemma <ref>.
Hence we have

* $u\in\odd{G\star u}{g'(u)}$ since $u\in\odd{G}{g(u)}$ and $u\notin N_G(u)$,

* $u\in g'(u)$ if and only if $u\notin g(u)$ (this is desired because $\ld'(u)\neq \ld(u)$); together with the previous item this yields Condition <ref> or <ref>, as appropriate,

* if $v\in\odd{G\star u}{g'(u)}$, then either $v\in\odd{G}{g(u)}$ or $v\in g(u)$, so either $u=v$ or $u\prec v$; this is Condition <ref>.

So $g'(u)$ indeed satisfies all the required conditions.

Case 1b: If $\ld(u)=\YZm$, then $u\notin\odd{G}{g(u)}$, hence

\begin{equation}\label{eq:ld_YZ}
\odd{G\star u}{g'(u)} = \odd{G\star u}{g(u)} = \odd{G}{g(u)} \symd (N_G(u)\cap g(u)),
\end{equation}

again using Lemma <ref>.
This implies

* $u\notin\odd{G\star u}{g'(u)}$ because $u\notin\odd{G}{g(u)}$ and $u\notin N_G(u)$,

* $u\in g'(u)$ because $u\in g(u)$; together with the previous item this yields Condition <ref>,

* if $v\in\odd{G\star u}{g'(u)}$, then either $v\in\odd{G}{g(u)}$ or $v\in g(u)$, so either $u=v$ or $u\prec v$; this is Condition <ref>.

So $g'(u)$ indeed satisfies all the required conditions.

Part 2:
We show that the correction set $g(v)$ for any $v\neq u$ satisfies Conditions <ref> and <ref> of the gflow definition.
The proof is split into subcases depending on whether $u\in\odd{G}{g(v)}$.

Case 2a:
If $u\notin\odd{G}{g(v)}$, first note that

\[
\odd{G\star u}{g'(v)} = \odd{G\star u}{g(v)} = \odd{G}{g(v)} \symd (N_G(u)\cap g(v))
\]

Hence if $w\in\odd{G\star u}{g(v)}$, then either $w\in\odd{G}{g(v)}$ or $w\in g(v)$, so $v=w$ or $v\prec w$; this is Condition <ref>.
Condition <ref> is trivially still satisfied.

Case 2b:
If $u\in\odd{G}{g(v)}$, first note that $v\prec u$.
If $w\in g'(v)$, then $w\in g(v)$ or $w\in g'(u)$ or $w=u$, and in each case either $w=v$ or $v\prec w$, this is Condition <ref>.
Furthermore, we have

\begin{align*}
\odd{G\star u}{g'(v)} &= \odd{G\star u}{g(v) \symd g'(u) \symd \{u\}} \\
&= \odd{G\star u}{g(v)} \symd \odd{G\star u}{g'(u)} \symd N_G(u) \\
&= \odd{G}{g(v)} \symd (N_G(u)\setminus g(v)) \symd \odd{G}{g(u)} \symd (N_G(u)\cap g(u)) \symd N_G(u) \\
&= \odd{G}{g(v)} \symd (N_G(u)\cap g(v)) \symd \odd{G}{g(u)} \symd (N_G(u)\cap g(u))
\end{align*}

where the third step uses the property that $\odd{G\star u}{g'(u)} = \odd{G}{g(u)} \symd (N_G(u)\cap g(u))$ for any $\ld(u)$, which follows from combining (<ref>) and (<ref>).
Hence if $w\in\odd{G\star u}{g'(v)}$, then at least one of the following holds:

* $w\in\odd{G}{g(v)}$, so $v=w$ or $v\prec w$

* $w\in g(v)$, so $v=w$ or $v\prec w$

* $w\in\odd{G}{g(u)}$, so $u=w$ or $u\prec w$; in both cases $v\prec w$ since $v\prec u$

* $w\in g(u)$, so $u=w$ or $u\prec w$; in both cases $v\prec w$ since $v\prec u$.

In each case, Condition <ref> is satisfied.

Part 3:
Finally we show that the correction set $g(v)$ for any $v\neq u$ satisfies Conditions <ref>, <ref> or <ref> of gflow.
The proof is split into subcases depending on whether $v\in N_G(u)$.

Case 3a: Suppose $v\in N_G(u)$ and distinguish cases according to $\ld(v)$.

* Suppose $\ld(v)=\XYm$, then $v\notin g(v)$ and $v\in\odd{G}{g(v)}$. Furthermore, $\ld'(v)=\XYm$.
We have

* $v\in\odd{G\star u}{g(v)}$ since $v\in\odd{G}{g(v)}$ and $v\notin g(v)$,

* $v\notin g'(v)$,

which together give Condition <ref>.

* Suppose $\ld(v)=\XZm$, then $v\in g(v)$ and $v\in\odd{G}{g(v)}$.
Furthermore, $\ld'(v) = \YZm$.
We have

* $v\notin\odd{G\star u}{g(v)}$ since $v\in\odd{G}{g(v)}$ and $v\in N_G(u)\cap g(v)$,

* $v\in g'(v)$,

which together give Condition <ref>.

* Suppose $\ld(v)=\YZm$, then $v\in g(v)$ and $v\notin\odd{G}{g(v)}$.
Furthermore, $\ld'(v) = \XZm$.
We have

* $v\in\odd{G\star u}{g(v)}$ since $v\notin\odd{G}{g(v)}$ and $v\in N_G(u)\cap g(v)$,

* $v\in g'(v)$,

which together give Condition <ref>.

Case 3b: Suppose $v\notin N_G(u)$ and distinguish cases according to $\ld(v)$.

* Suppose $\ld(v)=\XYm$, this case is analogous to the corresponding one in Subcase 3a, so Condition <ref> is satisfied.

* Suppose $\ld(v)=\XZm$, then $v\in g(v)$ and $v\in\odd{G}{g(v)}$.
Furthermore, $\ld'(v) = \XZm$.
We have

* $v\in\odd{G\star u}{g(v)}$ since $v\in\odd{G}{g(v)}$ and $v\notin N_G(u)\cap g(v)$,

* $v\in g'(v)$,

which together give Condition <ref>.

* Suppose $\ld(v)=\YZm$, then $v\in g(v)$ and $v\notin\odd{G}{g(v)}$.
Furthermore, $\ld'(v) = \YZm$.
We have

* $v\notin\odd{G\star u}{g(v)}$ since $v\notin\odd{G}{g(v)}$ and $v\notin N_G(u)\cap g(v)$,

* $v\in g'(v)$,

which together give Condition <ref>.

Recall that $G\land uv=G\star u\star v\star u$. Hence by applying Lemma <ref> three times we get the correct underlying graph, which has a gflow. It remains to show that the labels change in the manner stated.
Let us denote the labelling function associated with $\Gp$ by $\ld'$ (as in Lemma <ref>). Similarly, we denote the labelling and correction set functions for $\Gpp$ by $\ld''$ and $g''$. We then compute:

\[
\ld''(v) = \begin{cases} \XZm &\text{if } \ld'(v)=\XYm \text{ iff } \ld(v)=\XYm \\
\XYm &\text{if } \ld'(v)=\XZm \text{ iff } \ld(v)=\YZm \\
\YZm &\text{if } \ld'(v)=\YZm \text{ iff } \ld(v)=\XZm, \end{cases}
\]

so that

\[
\hat\ld(v) = \begin{cases} \XYm &\text{if } \ld''(v)=\XYm \text{ iff } \ld(v)=\YZm \\
\YZm &\text{if } \ld''(v)=\XZm \text{ iff } \ld(v)=\XYm \\
\XZm &\text{if } \ld''(v)=\YZm \text{ iff } \ld(v)=\XZm, \end{cases}
\]

and by symmetry the same holds for $u$.
For $w\in\comp O\setminus\{u,v\}$ we have

\[
\ld''(w) = \begin{cases} \YZm &\text{if } w\in N_{\Gp}(v) \text{ and } \ld'(w)=\XZm \\
\XZm &\text{if } w\in N_{\Gp}(v) \text{ and } \ld'(w)=\YZm \\
\ld'(w) &\text{otherwise}. \end{cases}
\]

Note that $w\in N_{\Gp}(v)$ iff $w\in N_G(v)\symd N_G(u)$, whence, denoting complementation in the vertex set by an overline, we obtain

\begin{align*} \ld''(w) &= \begin{cases} \YZm &\text{if}\quad \begin{aligned} &w\in \comp{N_G(v)}\cap N_G(u), \ld(w)=\YZm, \text{ or} \\
&w\in N_G(v)\cap\comp{N_G(u)}, \ld(w)=\XZm, \text{ or} \\
&w\in N_G(v)\cap N_G(u), \ld(w)=\XZm \end{aligned} \\ \\
\XZm &\text{if}\quad \begin{aligned} &w\in \comp{N_G(v)}\cap N_G(u), \ld(w)=\XZm, \text{ or} \\
&w\in N_G(v)\cap\comp{N_G(u)}, \ld(w)=\YZm, \text{ or} \\
&w\in N_G(v)\cap N_G(u), \ld(w)=\YZm \end{aligned} \\ \\
\ld(w) &\text{otherwise}, \end{cases} \\
&= \begin{cases} \YZm &\text{if } w\in N_G(v) \text{ and } \ld(w)=\XZm \\
\XZm &\text{if } w\in N_G(v) \text{ and } \ld(w)=\YZm \\
\ld(w) &\text{otherwise}. \end{cases} \end{align*}

Similarly, we have

\[
\hat\ld(w) = \begin{cases} \YZm &\text{if } w\in N_{\Gpp}(u) \text{ and } \ld''(w)=\XZm \\
\XZm &\text{if } w\in N_{\Gpp}(u) \text{ and } \ld''(w)=\YZm \\
\ld''(w) &\text{otherwise}. \end{cases}
\]

Noting that $w\in N_{\Gpp}(u)$ iff $w\in N_{\Gp}(u)\symd N_{\Gp}(v)$ iff $w\in N_G(v)$ we get

\[
\hat\ld(w) = \begin{cases} \YZm &\text{if } w\in N_{G}(v) \text{ and } \ld(w)=\YZm \\
\XZm &\text{if } w\in N_{G}(v) \text{ and } \ld(w)=\XZm \\
\ld(w) &\text{otherwise}, \end{cases}
\]

which reduces to $\hat\ld(w)=\ld(w)$.

§ FINDING MAXIMALLY DELAYED GFLOW IN MULTIPLE MEASUREMENT PLANES

The original algorithm for finding a maximally delayed gflow from Ref. [Mhalla and Perdrix, 2008]
is restricted to patterns where all measurements are in the -plane.
Here we extend this procedure to measurements in the and -planes,
in the manner of Ref. [Browne et al., 2007].
We will always assume that any connected component of a labelled open graph contains an input or an output.
This is because any connected component containing neither inputs nor outputs contributes only an irrelevant scalar factor to a measurement-based computation.
In the presence of gflow, this assumption implies the following stronger property.

Let $(G,I,O,\ld)$ be a labelled open graph with the property that any connected component of $G$ intersects $I\cup O$.
Suppose furthermore that $(G,I,O,\ld)$ has gflow.
Then any isolated vertex is an output.

Let $(g,\prec)$ be a gflow on $(G,I,O,\ld)$, and let $v\in V$ be arbitrary.
We will show that either

* $v\in O$, or

* $v\in\comp{O}$ and there exists $w\in V$ with $w \sim v$.

In the first case, there is nothing to prove.
In the second case, consider the different measurement planes.

* Suppose $\ld(v)=\XYm$, then $v\notin g(v)$ and $v\in\odd{}{g(v)}$ by <ref>.
This implies there exists $w\in g(v)\cap N_G(v)$.

* Suppose $\ld(v)=\XZm$, then $v\in g(v)$ and $v\in\odd{}{g(v)}$ by <ref>.
This again implies that there exists $w\in g(v)\cap N_G(v)$.

* Finally, suppose $\ld(v)=\YZm$, then $v\in g(v)$ and $v\notin\odd{}{g(v)}$ by <ref>.
Note that $v\notin I$ since inputs cannot appear in correction sets.
Also, $v$ is not an output, but its connected component must intersect $I\cup O$, so $v$ must have a neighbour $w$.

If $(g,\prec)$ is a maximally delayed gflow of $(G,I,O,\ld)$, then $V_0^\prec = O$.

To show that $V_0^\prec \sse O$:
Assume towards a contradiction that there exists $u\in V_0^\prec \setminus O$.
Then $u \in \max_\prec V$ by Definition <ref>
and $g(u)$ is either $\set{u}$ or $\emptyset$
by <ref> of Definition <ref>.

* If $g(u) = \emptyset$ then $u \notin \odd{}{g(u)}$ and
$u \notin g(u)$, contradicting all of <ref>–<ref>, so the  could not have gflow.

* Thus $g(u)=\{u\}$.
Now, since $u$ is not an output, Lemma <ref> implies $u$ has a neighbour; call this neighbour $v$.
But $u\in g(u)$ and $v\sim u$ imply $v \in \odd{}{g(u)}$ and therefore $u \prec v$ by <ref>.

This contradicts the property that $u\in \max_\prec V$.
Therefore such a $u$ cannot exist and instead $V_0^\prec \sse O$.

To show that $O \sse V_0^\prec$ we assume that $O \setminus V_0^\prec\neq\emptyset$.
Let $\prec'=\prec\setminus (O \times V)$,
which considers all outputs to be maximal.
Then $(g,\prec')$ is a gflow of $(G,I,O,\ld)$:

* Conditions <ref> and <ref> of Definition <ref> are satisfied by $\prec'$ because
elements of $O$ are not in the domain of $g$

* Conditions <ref>–<ref> are unaffected because the image of $g$ has not changed.

Thus, $(g,\prec')$ is a gflow of $(G,I,O,\ld)$.
Moreover, for any $k$, $\bigcup_{i=1}^k V_i^\prec \sse \bigcup_{i=1}^k V_i^{\prec'}$,
and $\abs{V_0^\prec} < \abs{V_0^{\prec'}}$, thus $(g,\prec')$ is more delayed than $(g,\prec)$,
which is a contradiction.

If $(g,\prec)$ is a maximally delayed gflow of $(G,I,O,\ld)$,
then $(\tilde{g},\tilde{\prec})$ is a maximally delayed gflow of $(G,I,O \cup V_1^\prec,\ld)$,
where $\tilde{g}$ is the restriction of $g$ to the domain $V\setminus (V_0^\prec\cup V_1^\prec)$ and $\tilde{\prec}=\prec\setminus (V_1^\prec\times V_0^\prec)$.

First show that $(\tilde{g},\tilde{\prec})$ is a gflow:

* <ref> for $\tilde{\prec}$: If $v \in V \setminus (V_0^\prec\cup V_1^\prec)$ and $w\in \tilde{g}(v)$, then $w\in g(v)$ and thus $v=w$ or $v \prec w$
(<ref> for $\prec$.)
Therefore $v=w$ or $v \,\tilde{\prec}\, w$.

* <ref> for $\tilde{\prec}$: If $v \in V \setminus (V_0^\prec\cup V_1^\prec)$ and  $w\in\odd{}{\tilde{g}(v)}$, then $w\in\odd{}{g(v)}$ and thus $v=w$ or $v\prec w$ (<ref> for $\prec$.)
Again $v=w$ or $v\,\tilde{\prec}\, w$.

* <ref> <ref> <ref> for $\tilde{\prec}$: Since $\tilde{g}=g$ on $V\setminus (V_0^\prec\cup V_1^\prec)$, these conditions still hold for all $v\in V\setminus (V_0^\prec\cup V_1^\prec)$.

Now to show that $(\tilde{g},\tilde{\prec})$ is maximally delayed:
Assume for a contradiction that there exists a gflow $(g',\prec')$ which is more delayed than $(\tilde{g},\tilde{\prec})$,
i.e. that $\abs{\bigcup_{i=0}^k V_i^{\prec'}} \geq \abs{\bigcup_{i=0}^k V_i^{\tilde\prec}}$,
where the inequality is strict at some $k=j$.
Extend $(g', \prec')$ on $(G,I,O \cup V_1^\prec,\ld)$ to $(g'',\prec'')$ on $(G,I,O,\ld)$ where

\begin{align}
g''(v) &= \begin{cases} g'(v) &\text{if } v\in V\setminus (V_0^\prec\cup V_1^\prec) \\ g(v) &\text{if } v\in V_1^\prec \end{cases} \\
\prec'' &= \prec' \cup \{(u,v)\mid u\in V_1^\prec \wedge u\prec v\}
\end{align}

We can relate $(g',\prec')$ to $(g'', \prec'')$ and $(\tilde g, \tilde \prec)$ to $(g, \prec)$ via

\begin{align}
\bigcup_{i=0}^k V_i^{\prec'} &= \bigcup_{i=0}^{k+1} V_i^{\prec''} &
\bigcup_{i=0}^k V_i^{\tilde\prec} &= \bigcup_{i=0}^{k+1} V_i^{\prec} &
V_0^{\prec} & = V_0^{\prec''}
\end{align}

Therefore $\abs{\bigcup_{i=0}^{k} V_i^{\prec''}} \geq \abs{\bigcup_{i=0}^{k} V_i^{\prec}}$
and the inequality is strict at $k=j+1$.
This is a contradiction of maximality of $(g, \prec)$, so $(\tilde{g},\tilde{\prec})$ must be maximally delayed.

If $(g,\prec)$ is a maximally delayed gflow of a labelled open graph $(G,I,O,\ld)$, then $V_1^\prec = V_1^{\prec,\XYm} \cup V_1^{\prec,\XZm} \cup V_1^{\prec,\YZm}$, where

\begin{align*}
V_1^{\prec,\XYm} &= \{u\in \comp{O} \mid \ld(u)=\XYm \wedge \exists K\sse O \text{ s.t.\ } \odd{}{K}\cap \comp{O} = \{u\}\} \\
V_1^{\prec,\XZm} &= \{u\in \comp{O} \mid \ld(u)=\XZm \wedge \exists K\sse O \text{ s.t.\ } \odd{}{K\cup\{u\}}\cap \comp{O} = \{u\}\} \\
V_1^{\prec,\YZm} &= \{u\in \comp{O} \mid \ld(u)=\YZm \wedge \exists K\sse O \text{ s.t.\ } \odd{}{K\cup\{u\}}\cap \comp{O} = \emptyset\}.
\end{align*}

We first show that if $(g,\prec)$ is a maximally delayed gflow, then for any $u\in V_1^\prec$, we have $g(u)\sse O\cup\{u\}$.
This is because if $v\in g(u)$ and $v\neq u$, then $u\prec v$ by property <ref>.
Furthermore, by the definition of $V_1^\prec$, if $u\prec v$ then $v\in V_0^\prec$, and therefore $v \in O$ by Lemma <ref>.
Combined with properties <ref> and <ref>–<ref>, this shows that
$\odd{}{g(u)}\cap \comp{O} = \{u\}$ if $\ld(u)\in\{\XYm,\XZm\}$,
and $\odd{}{g(u)}\cap \comp{O} = \emptyset$ if $\ld(u)=\YZm$.
Therefore $V_1^\prec \sse V_1^{\prec,\XYm} \cup V_1^{\prec,\XZm} \cup V_1^{\prec,\YZm}$.

To prove that any vertex satisfying one of the three conditions

* $\ld(u)=\XYm$ and $\exists K\sse O \text{ s.t.\ } \odd{}{K}\cap \comp{O} = \{u\}$

* $\ld(u)=\XZm$ and $\exists K\sse O \text{ s.t.\ } \odd{}{K\cup\{u\}}\cap \comp{O} = \{u\}$

* $\ld(u)=\YZm$ and $\exists K\sse O \text{ s.t.\ } \odd{}{K\cup\{u\}}\cap \comp{O} = \emptyset$

must be in $V_1^\prec$, proceed by contradiction.
As in the proof of Ref. <cit.>, we show that a gflow cannot be maximally delayed
if there exists some vertex $w$ which satisfies one of conditions 1–3 but is not in $V_1^\prec$.

Suppose $(g,\prec)$ is a maximally delayed gflow and there exists $w\in V\setminus V_0^\prec$
such that $w$ satisfies condition $j$, where $j\in\{1,2,3\}$, and let $K$ be the set identified in that condition.

\begin{align}
g'(u) := \begin{cases}
K &\text{if } u=w \text{ and } \ld(w)=\XYm \\
K\cup\{w\} &\text{if } u=w \text{ and } \ld(w)\in\{\XZm,\YZm\} \\
g(u) &\text{otherwise.}
\end{cases}
\end{align}

Let $\prec'$ be the partial order defined by[The proof of Ref. <cit.> is missing the third case of the definition of $\prec'$,
but it is actually required in that proof as well.]

\begin{align}
u\prec' v \text{ if } \begin{cases}
u\neq w \text{ and } u\prec v \\
u=w \text{ and } v\in K \\
u=w \text{ and } v\in \odd{}{g'(w)} \setminus \set{w}
\end{cases}
\end{align}

Then property <ref> of the gflow definition holds by the properties of $(g,\prec)$ and the second case of the definition of $\prec'$.
Property <ref> holds by the properties of $(g,\prec)$ and the third case of the definition of $\prec'$.
Properties <ref>–<ref> are satisfied for all $v\in V\setminus\{w\}$ since $g$ has not changed for those vertices.
For $w$ itself, only one of properties <ref>–<ref> is relevant, and it is satisfied by condition $j$.

Now, assume towards a contradiction that $w\notin V_1^\prec$, i.e. assume that there exists $k>1$ such that $w\in V_k^\prec$.
Our construction of $\prec'$ in conjunction with condition $j$ implies that $w \in V_1^{\prec'}$.
For any $u\in V$ define the depth of $u$, written $d(u)$, as the index $k$ such that $u \in V_k^\prec$,
and define $d'(u)$ as the index $k$ such that $u \in V_k^{\prec'}$.
Our assumption implies that $d'(w) < d(w)$,
and our construction of $\prec'$ implies that $d'(u) \leq d(u)$ for all vertices $u$.
This shows that:

\begin{align}
V_0^{\prec} & = V_0^{\prec'} &
\abs{V_1^{\prec}} & < \abs{V_1^{\prec'}} &
\abs{\bigcup_{i=0}^k V_i^{\prec}} &\leq \abs{\bigcup_{i=0}^{k} V_i^{\prec'}}
\end{align}

Therefore $(g',\prec')$ is more delayed than $(g,\prec)$, contradicting the assumption that $(g,\prec)$ is maximally delayed.
Hence any vertex $w$ satisfying one of conditions 1–3 must be in $V_1^\prec$,
i.e. $V_1^{\prec,\XYm} \cup V_1^{\prec,\XZm} \cup V_1^{\prec,\YZm} \sse V_1^\prec$.

We combine Lemma <ref> with Lemma <ref>
to construct a partial converse to Lemma <ref>;
if we know that $(\tilde g, \tilde \prec)$ is maximally delayed
on $(G, I, O \cup V_1^\prec, \ld)$,
(with $V_1^\prec$ defined as in Lemma <ref>)
we want to show that $(g, \prec)$ is maximally delayed on $(G, I, O, \ld)$.

Suppose $(g, \prec)$ is a gflow of $(G, I, O, \lambda)$
such that $V_0^\prec = O$ and $V_1^\prec = V_1^{\prec,\XYm} \cup V_1^{\prec,\XZm} \cup V_1^{\prec,\YZm}$.
Now if $(\tilde g, \tilde \prec)$ defined as in Lemma <ref> is a maximally delayed gflow for $(G,I,O \cup V_1^\prec,\ld)$,
then $(g, \prec)$ is also maximally delayed.

Recall from Lemma <ref>
that $\tilde{g}$ is the restriction of $g$ to the domain
$V\setminus (V_0^\prec\cup V_1^\prec)$ and $\tilde{\prec}=\prec\setminus (V_1^\prec\times V_0^\prec)$.
Working towards a contradiction, suppose that $(g', \prec')$ is a gflow of $(G, I, O, \lambda)$ which is more delayed than
$(g, \prec)$.
By Lemmas <ref> and <ref>
we know that $V_0^\prec = V_0^{\prec'}$
and that $V_1^\prec = V_1^{\prec'}$.
Let $j$ be such that:

\begin{align}
\abs{V^\precprimed_j} &< \abs{V^{\prec}_j} &
\forall k \quad \abs{\bigcup_{i=0}^k V_i^{\precprimed}} &\leq \abs{\bigcup_{i=0}^{k} V_i^{\prec}}
\end{align}

Such a $j$ must exist, and be greater than 1, because $(g', \prec')$ is more delayed that $(g, \prec)$.
Define $(\tilde g', \tilde \prec')$ analogously to $(\tilde g, \tilde \prec)$:

\begin{align}
\tilde g' &= \text{ the restriction of $g'$ to } V\setminus (V_0^\precprimed\cup V_1^\precprimed) \\
\tilde{\prec}'&=\precprimed\setminus (V_1^\precprimed\times V_0^\precprimed)
\end{align}

By Lemma <ref> $(\tilde g', \tildeprecprimed)$ is maximally delayed
for $(G, I, O \cup V_1^\prec, \lambda)$,
and by our assumption so is $(\tilde g, \tilde \prec)$.
Our contradiction will be shown by noting that $(\tilde g', \tildeprecprimed)$
is more delayed than $(\tilde g, \tilde \prec)$:

\begin{align}
V_j^\prec &= V_{j-1}^{\tilde \prec} & V_j^\precprimed &= V_{j-1}^{\tildeprecprimed}
& \abs{V^\precprimed_j} &< \abs{V^{\prec}_j} &\implies \abs{V^\tildeprecprimed_{j-1}} &< \abs{V^{\tilde \prec}_{j-1}}
\end{align}

\begin{align}
\forall k \quad  \abs{\bigcup_{i=0}^k V_i^{\precprimed}} &\leq \abs{\bigcup_{i=0}^{k} V_i^{\prec}}
& \implies  \quad \forall k \quad  \abs{\bigcup_{i=0}^{k-1} V_i^{\tildeprecprimed}} &\leq \abs{\bigcup_{i=0}^{k-1} V_i^{\tilde \prec}}
\end{align}

This contradicts $(\tilde g, \tilde \prec)$ being maximally delayed,
and therefore $(g, \prec)$ must be maximally delayed.

We now adapt the main gflow theorem and proof from Ref. <cit.>
to our multi-planar setting.
The idea is to iteratively construct the layers $V_k^\prec$,
finding candidate correction sets via matrix manipulation in $\mathbb F_2$.

There exists an algorithm that decides whether a given
has an (extended) gflow, and that outputs such a gflow if it exists.
Moreover, this output gflow is maximally delayed,
and takes a number of steps that is polynomial in the number of vertices of the .

Let $(G,I,O,\ld)$ be a .
The algorithm $(M, I, O, \ld)$ (Algorithm <ref>),
where $M$ is the adjacency matrix of $G$,
finds a maximally delayed gflow and returns true if one exists and returns false otherwise.

* Correctness of gflow:
At the $k^{th}$ recursive call,
the set $C$ constructed by the algorithm in the  loop at line <ref> corresponds to the layer $V^\prec_k$
of the partition induced by the returned strict partial order (Definition <ref>).
The construction of $K'$ corresponds to
finding a correcting set in $\bigcup_{j \le k} V^\prec_j$
that has the desired properties for its odd neighbourhood
(as specified in conditions <ref>–<ref> of the gflow definition,
see Lemma <ref>.)
Furthermore the restriction $K' \sse O'$ ensures condition <ref>, thus
if the algorithm returns a solution, then the solution satisfies the conditions of a gflow.

* Maximality of gflow:
By induction on the number of non-output vertices, we prove that if
the given labelled open graph has a gflow, then the algorithm outputs a
maximally delayed gflow.
Assume that the given labelled open graph has a gflow.

* The base case is when there is no non-output vertex,
and here there is no correction needed:
The empty flow $(g,\emptyset)$ (where $g$ is a function with an empty
domain) is a maximally delayed gflow.

* For the inductive step suppose that there exist some non-output vertices;
according to Lemma <ref> the elements of $V^\prec_1$ are those that satisfy one of
the tests inside the loop at line <ref>.
Thus, after the loop (line <ref>),
Since the existence of a gflow is assumed,
$V^\prec_1$ cannot be empty (all non-output vertices have to be corrected),
thus the algorithm is called recursively
on the labelled open graph $(G, I, O \cup V^\prec_1, \lambda)$ which has fewer non-output vertices.
Lemma <ref> ensures the existence of a gflow in
$(G,I,O \cup V^\prec_1, \ld)$,
and by induction we know that the recursive call returns
a maximally delayed gflow for $(G,I,O \cup V^\prec_1, \ld)$.
By Lemma <ref>
we know that this construction for $V_0^\prec$ and $V_1^\prec$
yields a maximally delayed gflow for $(G,I,O, \ld)$.

* Complexity:
First form the matrix $A$ from the adjacency matrix $M$
by removing those rows that correspond to output vertices,
and keeping all columns that correspond to outputs that are not also inputs
(i.e. keep candidate vertices for $K$).
Notice that finding a solution for $K$ inside the  loop at line <ref>
consists of solving a system $Ax=b_j$ in $\mathbb F_2$,
where $b_j$ is either a column vector with a 1 corresponding to the vertex $u$
(in the and cases), or $b_j$ is the column vector of all 0s (in the case.)
The resulting column vector $x$ has a 1 in position $x_v$ if $v \in K$ and a 0 otherwise.

$\ell' = \abs{O \setminus I}$, and
$j$ to range over the $n - \ell$ vertices in $\comp O$.
In order to solve these $n-\ell$ equations
the block matrix

\begin{align}
\quad A &\vert& b_1 &\vert& \ldots &\vert& b_{n-\ell} \quad
\end{bmatrix}
\end{align}

of dimensions $(n-\ell)\times (\ell' + n - \ell)$
is transformed into an upper
triangular form,
for instance using Gaussian elimination,
within $O(n^3)$ operations.
Then for each $b_j$ a back substitution within $O(n^2)$
operations is used to find $x_i$, if it exists, such that
$Ax_j=b_j$ (see Ref. [Bard, 2006]). The back substitutions costs $O(n^3)$ operations at each
call of the function. Since there are at most $n$ recursive calls, the
overall complexity is $O(n^4)$.

§ PSEUDOCODE FOR ALGORITHMS

Extended Generalized Flow
[1]
GFLOW$M$, I, O, $\ld$

$v\in$ O
$d(v) \gets 0$ Outputs have depth 0
GFLOWAUX$M$,  I, O, $\ld$, 1 Start the recursive process of finding $V_j^\prec$

GFLOWAUX$M$, I, O, $\ld$, k, d
$O' \gets O \setminus I$
$C \gets \emptyset$
$u \in \comp{O}$
$\ld (u) =$
$K' \gets$ Solution for $X \sse O'$ where $\odd{}{K} \cap \comp{O} = \set{u}$
$\ld (u) =$
$K' \gets \; \set{u} \; \cup$ (Solution for $K \sse O'$ where $\odd{}{K \cup{u}} \cap \comp{O} = \set{u}$)
$\ld (u) =$
$K' \gets \; \set{u} \; \cup$ (Solution for $K \sse O'$ where $\odd{}{K \cup{u}} \cap \comp{O} = \emptyset$)
$K'$ exists
$C \gets C \cup \set{u}$
$g(u) \gets K'$ Assign a correction set for $u$
$d(u) \gets k$ Assign a depth value for $u$
$C = \emptyset$
$O = V$
return (true, g, d) Halt, returning maximally delayed $g$ and depth values $d$
return (false, $\emptyset$, $\emptyset$) Halt if no gflow exists
return GFLOWAUX$M$, $\ld$, I, O $\cup$ C, k+1, d

Circuit Extraction
[1]
Extract$D$input is MBQC diagram $D$
Init empty circuit $C$
$G,I,O\gets $ Graph$(D)$get the underlying graph of $D$
$F \gets O$ initialise the frontier

$D,C \gets$ ProcessFrontier($D,F,C$)

$\exists v\in D\setminus F$there are still vertices to be processed
$D,F,C \gets $ UpdateFrontier$(D,F,C)$
$v\in F$ the only vertices still in $D$ are in $F$
$v$ connected to input via Hadamard
$C\gets$ Hadamard(Qubit($v$))
Perm $\gets$ Permutation of Qubits of $F$ to qubits of inputs step 5 of Section <ref>
swap$(q_1,q_2)$ in PermutationAsSwaps(Perm)
$C\gets$ swap$(q_1,q_2)$
return $C$

ProcessFrontier$D,F,C$ Corresponds to step 1 of Section <ref>
$v\in F$
$v$ has local Cliffords
$C\gets $ Cliffords(Qubit($v$))
Remove Cliffords from $v$ to output
edge between $v$ and $w$ in $F$
$C\gets $ CZ(Qubit($v$), Qubit($w$))
Remove edge between $v$ and $w$
return $D,C$

UpdateFrontier$D,F,C$ Corresponds to steps 3,4 of Section <ref>
$N\gets $ Neighbours$(F)$
$M \gets$ Biadjacency$(F,N)$
$M^\prime \gets $ GaussReduce$(M)$
Init $vs$ initialise empty set $vs$
row $r$ in $M^\prime$
sum$(r) == 1$there is a single 1 on row $r$
Set $v$ to vertex corresponding to non-zero column of $r$
Add $v$ to $vs$ $v$ will be part of the new frontier
$vs$ is empty
There is vertex connected to $F$
$v \gets$ vertex connected to $F$
$w \gets$ a neighbour of $w$ in $F$
$D\gets $ Pivot($D$,$v$,$w$)
$D,C \gets$ ProcessFrontier$(D,F,C)$
return $D,F,C$
$M \gets $  Biadjacency($F,ws$) smaller biadjacency matrix
$(r_1,r_2) \in $ GaussRowOperations$(M)$
$C\gets $ CNOT(Qubit$(r_1)$, Qubit$(r_2)$)
Update $D$ based on row operation
$v\in vs$ all $v$ now have a unique neighbour in $F$
$w \gets $ Unique neighbour of $v$ in $F$
$C\gets $ Hadamard(Qubit$(w)$)
$C\gets $ Phase-gate$(Phase(v),Qubit($w$))$
Remove $w$ from $F$
Add $v$ to $F$
$D,C \gets$ ProcessFrontier$(D,F,C)$
return $D,F,C$