# Parametrically-controlled microwave-photonic interface for the fluxonium

Ke Nie  Department of Physics, University of Illinois Urbana-Champaign,
Urbana, IL 61801  Aayam Bista  Department of Physics, University of Illinois
Urbana-Champaign, Urbana, IL 61801  Kaicheung Chow  Holonyak Micro $\&$
Nanotechnology Lab, University of Illinois Urbana-Champaign, Urbana, IL 61801
Wolfgang Pfaff  Department of Physics, University of Illinois Urbana-
Champaign, Urbana, IL 61801  Materials Research Laboratory, University of
Illinois at Urbana-Champaign, Urbana, IL 61801  Angela Kou  Department of
Physics, University of Illinois Urbana-Champaign, Urbana, IL 61801  Materials
Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL
61801

###### Abstract

Converting quantum information from stationary qubits to traveling photons
enables both fast qubit initialization and efficient generation of flying
qubits for redistribution of quantum information. This conversion can be
performed using cavity sideband transitions. In the fluxonium, however, direct
cavity sideband transitions are forbidden due to parity symmetry. Here we
circumvent this parity selection rule by using a three-wave mixing element to
couple the fluxonium to a resonator. We experimentally demonstrate a scheme
for interfacing the fluxonium with traveling photons through microwave-induced
parametric conversion. We perform fast reset on the fluxonium qubit,
initializing it with $>95\%$ ground state population. We then implement
controlled release and temporal shaping of a flying photon, useful for quantum
state transfer and remote entanglement. The simplicity and flexibility of our
demonstrated scheme enables fluxonium-based remote entanglement architectures.

##  I Introduction

Interfacing stationary logical qubits and flying photons can serve multiple
purposes [1], such as implementing on-demand transfer and remote entanglement
of quantum bits in a modular quantum system [2, 3], fast evacuation of qubit
excitations for initialization [4], and efficient generation of non-classical
single-photon states [5, 6, 7, 8, 9, 10, 11, 12]. In circuit quantum
electrodynamics (cQED) [13], high-fidelity qubit-photonic interfaces have been
built due to the availability of strong coupling between superconducting
circuits and microwave photons. Experiments interfacing a transmon qubit [14]
with traveling photons have demonstrated fast unconditional qubit reset [15,
16] and deterministic generation and temporal shaping of single photons for
remote entanglement [17, 18, 19].

In recent years, the fluxonium [20] has emerged as a promising superconducting
qubit due to its remarkably long coherence times [21, 22] and high gate
fidelities [23, 24]. A coherent fluxonium-photonic interface would be
beneficial for fluxonium quantum processing on multiple fronts. First,
efficient initialization of the fluxonium is important as the low transition
frequency of the fluxonium results in a highly-mixed thermal population.
Second, faithful mapping between a fluxonium excitation and a flying qubit
would allow the realization of modular quantum systems linked via traveling
microwave photons using the fluxonium. A typical method for engineering this
interface involves converting a fluxonium excitation into a fast-decaying
resonator photon through pumping of cavity sideband transitions [25, 26, 27].
Recent experiments have reported multiple ways of circumventing the parity
selection rule forbidding these direct transitions by flux-biasing the
fluxonium qubit away from half flux [25], using microwave-driven multi-photon
transitions [26], and coupling the fluxonium to a resonator via a third-order
mixing element known as the Superconducting Nonlinear Asymmetric Inductive
eLement (SNAIL) [28, 27]. The above-mentioned experiments have been solely
focused on fluxonium initialization and cooling.

In this Letter, we report an experiment interfacing the fluxonium qubit with
traveling photons using parametric conversion in a fluxonium-resonator system
that enables both fluxonium initialization and efficient generation of flying
qubits. We use a SNAIL-based resonator as an intermediary coupling element to
coherently convert between fluxonium excitations and microwave photons. We
control the coupling rate by modulating the pump amplitude, which allows for
fast switching between different quantum information tasks. In particular, we
perform fast reset of the fluxonium qubit in the strong coupling regime. We
then operate in the weak coupling regime to controllably release and
temporally shape single traveling photons. Our scheme provides a
straightforward pathway for both fluxonium initialization and for
incorporating the fluxonium qubit into remote entanglement schemes based on
traveling microwave photons.

Figure 1:  (a) Circuit schematic of the device. A fluxonium (blue) is
inductively coupled to a SNAIL resonator (purple), which couples capacitively
to a readout resonator (green). The SNAIL’s three-wave mixing term mediates
the conversion of a fluxonium excitation to a resonator photon at coupling
rate $g(t)$. The readout resonator emits photons to a coupled transmission
line (orange) with decay rate $\kappa$. Pump tones for qubit control and
parametric conversion are applied through a separate charge line (red). (b)
Energy level structure of the fluxonium-resonator system shown in (a) used for
interfacing a fluxonium excitation with a microwave photon. (c) Two-tone
spectroscopy data of the $|e0\rangle\leftrightarrow|g1\rangle$ transition near
$\Phi_{\text{ext}}^{f}/\Phi_{0}=9.5$. The change in the sign of the resonator
phase response is due to a flip in the direction of the dispersive shift.

##  II Fluxonium-SNAIL-resonator system

The main challenge faced when building a coherent fluxonium-photonic interface
is achieving strong, tunable coupling in a fluxonium-resonator system. Direct
conversion between the fluxonium and resonator photons is forbidden due to the
parity selection rule in linearly-coupled fluxonium-resonator systems. Strong
coupling is made further difficult by the large detuning between the fluxonium
computational states and the resonator states. These barriers can be overcome
by engineering third-order nonlinear coupling between the fluxonium and a
resonator via a SNAIL (first demonstrated in Ref. [27]). The SNAIL’s third-
order nonlinearity breaks the parity symmetry of the wavefunctions in the
composite Hilbert space. Furthermore, operating the SNAIL at a similar
transition frequency to the resonator while inductively coupling it to the
fluxonium through the $\phi$ operator effectively allows the SNAIL to bridge
the detuning gap between the fluxonium and the resonator. The SNAIL is then
capable of strong coupling to both the fluxonium computational states and the
resonator simultaneously, which makes it an excellent mediator for a
fluxonium-photonic interface.

Our circuit for implementing third-order nonlinear coupling of a fluxonium-
resonator system is shown in Fig. 1(a). We build a system of three coupled
modes: the fluxonium qubit mode ($\hat{f}$), an intermediary SNAIL mode
($\hat{s}$), and a readout resonator mode ($\hat{r}$) coupled to a
transmission line with loss rate $\kappa/2\pi=0.4$ MHz. The system Hamiltonian
up to third order is given by

$\hat{H}_{\text{sys}}/\hbar=\sum_{i=f,s,r}4E_{C,i}\hat{n}_{i}^{2}+\frac{1}{2}E_{L,i}\hat{\phi}_{i}^{2}-E_{J,f}\cos{(\hat{\phi}_{f}-\phi_{\text{ext}}^{f})}\\\
+c_{3}^{s}\hat{\phi}_{s}^{3}+g_{fs}\hat{\phi}_{f}\hat{\phi}_{s}+g_{sr}\hat{n}_{s}\hat{n}_{r}.$
(1)

Here $E_{C,i},E_{L,i},E_{J,i}$ are the charging, inductive, and Josephson
energies, respectively (see Appendix A for system parameters and time domain
measurements). The Cooper pair number and phase operator are denoted by
$\hat{n}_{i}$ and $\hat{\phi_{i}}$. The subscripts $f,s,r$ represent the
fluxonium qubit, the SNAIL resonator, and the readout resonator. We also
introduce $\phi_{\text{ext}}^{f}\equiv 2\pi\Phi_{\text{ext}}^{f}/\Phi_{0}$,
where $\Phi_{\text{ext}}^{f}$ is the external flux through the fluxonium
qubit. The coupling strength between each neighboring element is captured by
$g_{ij}$ in the Hamiltonian, and $c_{3}^{s}$ represents the third-order
coefficient of the SNAIL. Importantly, the SNAIL’s third-order term enables
all energy-conserving three-wave mixing processes. We can then apply a
microwave pump with frequency matching the detuning between the fluxonium mode
and the readout resonator mode to realize the effective system Hamiltonian,

$\hat{H}_{\mathrm{conv}}/\hbar=g(t)\hat{f}\hat{r}^{\dagger}+h.c..$ (2)

Here the coupling rate $g(t)$ is given by

$g(t)=6c_{3}^{s}\xi(t)\phi_{f,\mathrm{zpf}}\phi_{s,\mathrm{zpf}}\phi_{r,\mathrm{zpf}},$
(3)

with $|\xi(t)|^{2}$ being the number of pump photons in the SNAIL resonator,
and $\phi_{i,\mathrm{zpf}}$ denoting the zero-point fluctuations of the flux
associated with each mode (Appendix C). This Hamiltonian allows us to drive
transitions of the form $|i,n\rangle\leftrightarrow|j,n\pm 1\rangle,~{}i\neq
j$ and $n>0$, where $i,j$ denote any neighbouring eigenstates of the fluxonium
and $n$ denotes the number of photons in the readout resonator. By directly
driving the $|e0\rangle\leftrightarrow|g1\rangle$ transition, excitations of
the fluxonium can be parametrically converted into a photon in the readout
resonator, which is then emitted at the decay rate $\kappa$ (Fig. 1(b)).

The rate of conversion is controlled by the pump amplitude $\xi(t)$ and the
third-order coefficient $c_{3}^{s}$. A strong microwave drive on the SNAIL
induces a significant ac-Stark shift on the transition (see Appendix C for
further details on the system Hamiltonian), which complicates pulse
calibration due to the Stark shift changing non-negligibly during the pulse
rise time. We therefore choose to work with a large $c_{3}^{s}$, which can be
obtained by operating at nonzero $\Phi_{\text{ext}}^{s}$ through the SNAIL. We
flux bias the system to $\Phi_{\text{ext}}^{f}/\Phi_{\text{0}}=9.5$,
corresponding to $\Phi_{\text{ext}}^{s}/\Phi_{\text{0}}=0.27$ (Appendix A).
Spectroscopy of the desired $|e0\rangle\leftrightarrow|g1\rangle$ transition
at this flux point is shown in Fig. 1(c). We note the clear visibility of this
transition at the half-flux sweet spot of the fluxonium, indicating that the
parity selection rule has been broken [27]. The expected $c^{s}_{3}/2\pi$ is
simulated to be 350 MHz at this point.

Figure 2:  (a) The pulse sequence for characterizing coherence and tunability
of the $|e0\rangle\rightarrow|g1\rangle$ transition. A square pulse (purple)
with frequency $\omega_{d}$ is applied to the system to drive the
$|e0\rangle\rightarrow|g1\rangle$ transition. A wait time of
$t_{\text{buffer}}$ is included to ensure that the photonleaves the resonator
before readout (green). (b) Rabi oscillations of the
$|e0\rangle\leftrightarrow|g1\rangle$ transition under a strong pump tone
($g\gg\kappa$) as a function of detuning. (c) Measured qubit $|e0\rangle$
population vs. pulse duration data from (b) at zero detuning. The data is fit
using the master equation (solid). The effective qubit decay rate is given by
$\kappa/2$ in the strong coupling regime (dashed). (d) The time evolution of
the $|e0\rangle$ population under a weak pump tone ($g\ll\kappa$) as a
function of detuning between the pump and the transition frequency
$\omega_{d}-\omega_{0}$. (e) Measured qubit $|e0\rangle$ population vs. pulse
duration data from (d) at zero detuning. A simple exponential decay with
$\Gamma_{\mathrm{eff}}=4g^{2}/\kappa$ is plotted for reference (dashed).

##  III Coherent Driving

Here we demonstrate that our parametric scheme can reach different coupling
regimes useful for fast reset and generating flying photons. We characterize
the transition $|e0\rangle\leftrightarrow|g1\rangle$ by driving it with a
square pulse of varying duration and power (Fig. 2). We measure the qubit
population in time as a function of detuning $\omega_{d}-\omega_{0}$, where
$\omega_{d},\omega_{0}$ represent the driving frequency and the transition
frequency, respectively (Fig. 2(a)). As the system evolves coherently as a
superposition of $|e0\rangle$ and $|g1\rangle$, $|g1\rangle$ decays back to
$|g0\rangle$ and emits the photon field into the transmission line at an
effective decay rate depending on the relative strength of $g$ and $\kappa$.
Before readout, we wait for time $t_{\text{buffer}}=5$ $\mu$s ($\sim
10\kappa^{-1}$), to ensure that the photon has left the resonator to avoid
changes to the fluxonium readout based on the cavity state.

We first characterize the operation of driving the
$|e0\rangle\leftrightarrow|g1\rangle$ transition in the strong coupling regime
($g\gg\kappa$), which is useful for fast qubit reset. Under a strong pump
tone, the system undergoes fast Rabi oscillations between $|e0\rangle$ and
$|g1\rangle$ as shown in Fig. 2(b). We fit the data at zero detuning (Fig.
2(c)) using the system master equation (Appendix C). From this fit, we find
$g/2\pi=1.27\text{ MHz}\gg\kappa/2\pi$ with exponential decay rate
$\Gamma_{\mathrm{eff}}=\kappa/2$ given that the qubit lifetime $T_{1}\gg
1/\kappa$, which indicates that the system is in the strong coupling regime.

We then characterize the $|e0\rangle\leftrightarrow|g1\rangle$ transition in
the weak coupling regime, which is useful for efficient generation and
temporal shaping of flying photons. Under a relatively weak pump, the state
$|e0\rangle$ is slowly converted into $|g1\rangle$ with linewidth $\kappa$. In
this regime, we can tune the effective decay rate
$\Gamma_{\mathrm{eff}}=4g^{2}/\kappa$ via the strength of our pump, which
controls $g$. In Fig. 2(d), we show the qubit $|e0\rangle$ population
evolution in time as a function of detuning $\omega_{d}-\omega_{0}$ when the
$|e0\rangle\leftrightarrow|g1\rangle$ transition is driven by a relatively
weak pump tone. The data at zero detuning is plotted separately in Fig. 2(e).
A fit to the master equation (Appendix C) yields a coupling rate
$g/2\pi=0.14\text{ MHz}\ll\kappa/2\pi$. The qubit decay exhibits no
oscillations and resembles that of standard Purcell loss after the ring-down
time $1/\kappa$ with effective decay rate
$\Gamma_{\mathrm{eff}}=4g^{2}/\kappa$. Now with a well-characterized ability
to drive and tune the $|e0\rangle\leftrightarrow|g1\rangle$ transition, we can
use this transition for fast reset and controlled release of single photons.

Figure 3:  (a) Measured qubit $|g0\rangle$ population vs. reset pulse duration
for a fixed pulse amplitude. A maximum of $92\%$ fluxonium population is reset
to the ground state (dashed) before correction for qubit decay during readout.
Taking the readout error into account, the maximum population reaches 96%
(dash-dotted). A reset threshold of $99.3\%$ is predicted by numerical
simulation in this system upon having a longer reset pulse with no
$t_{\mathrm{buffer}}$ and perfect readout (dotted). Histograms of single-shot
qubit readout at thermal equilibrium (b) and at the maximum reset efficiency
(c) with $5\times 10^{5}$ data points collected are provided.

##  IV Fast Reset

Here we perform fast reset of the fluxonium in the strong coupling regime.
Specifically, we turn on a strong pump for a pulse duration $t=200$ ns for
fast evacuation of the $|e0\rangle$ population into the $|g1\rangle$ state. We
then turn off the drive and wait for $t_{\mathrm{buffer}}=5\ \mu$s to allow
the photon to leave before readout. In Fig. 3(a), we show the measured
fluxonium population in $|e0\rangle$ as a function of the reset pulse duration
for a fixed pulse amplitude. As we increase the pulse duration, the fluxonium
population in $|e0\rangle$ decreases and reaches the maximum population in
$|g0\rangle$ with pulse duration $t=200~{}$ns before the excitation gets
swapped back to $|e0\rangle$. At thermal equilibrium, the qubit
($\omega_{f}/2\pi=81$ MHz) has a probability of $55\%$ to be in the ground
state, which is equivalent to a qubit temperature of 20 mK (Fig. 3(b)). Using
our reset protocol, we measure a maximum of $92\%$ qubit population in the
ground state, which corresponds to cooling the qubit to 1.6 mK (Fig. 3(c)).
Both the long timescale and the observed inefficiency of the reset is due to
the low decay rate $\kappa/2\pi=0.4$ MHz of the resonator. The qubit heats
during $t_{\text{buffer}}$ and readout. If we correct for the readout error
induced due to the qubit population change, the reset efficiency rises to
$\sim 96\%$.

We note two possible ways to improve upon our demonstrated reset method. By
prolonging the reset pump while waiting for the photon to leave the resonator,
no $t_{\mathrm{buffer}}$ is needed and we predict the reset efficiency to
saturate at $\sim 99.3\%$ for a reset pulse of duration $\gtrsim 5\ \mu$s
based on numerical simulations assuming perfect readout (Appendix D).
Alternatively, a higher reset efficiency of $\sim 99.8\%$ can be reached in
500 ns using a resonator with $\kappa/2\pi=4$ MHz (Appendix D).

Figure 4:  (a) The pulse sequence for coherently transfering an excitation
from the fluxonium to a traveling photon. The fluxonium is initialized to be
in $|\psi\rangle=(|g0\rangle+e^{i\phi}|e0\rangle)/\sqrt{2}$ by first applying
a reset pulse followed by a Rabi drive of amplitude $\pi/2$ and phase $\phi$
on the qubit. A photon is then released by weakly pumping the fluxonium at the
$|e0\rangle\rightarrow|g1\rangle$ transition frequency such that $g\ll\kappa$.
(b) Measured in-phase quadrature of the emitted photon as a function of time,
with the fluxonium initialized to be in
$|\psi^{\pm}\rangle=(|g0\rangle\pm|e0\rangle)/\sqrt{2}$. A square pulse is
used to release a photon, and the data is fit using input-output theory
(solid). With a constant coupling, the output fields mostly follow an
exponential decay (dashed). (c) Shaping the wave packet to be a hyperbolic
secant function. The measured in-phase quadrature (dark green) and field
amplitude (light green) of the emitted photon is plotted as a function of time
with the fluxonium initialized to be in $|\psi^{+}\rangle$.

##  V Controlled Release

Equipped with the ability to initialize the fluxonium qubit state, we now
demonstrate controlled transfer of a fluxonium excitation to a flying photon
in the weak coupling regime. By modulating the amplitude and phase of the
drive, we can shape the temporal profile of the outgoing photon wavepacket,
which is useful for implementing deterministic state transfer protocols [2].

For characterization of the controlled conversion process, we analyze the
emitted photon superposition state of the
$(|0\rangle+e^{i\phi}|1\rangle)/\sqrt{2}$, which has a non-zero field
amplitude. To initialize the qubit in states
$|\psi\rangle=(|g0\rangle+e^{i\phi}|e0\rangle)/\sqrt{2}$, we apply a reset
pulse followed by a $\pi/2$ pulse of phase $\phi$ on the qubit. A traveling
photon of state $(|0\rangle+e^{i\phi}|1\rangle)/\sqrt{2}$ is then controllably
released by pumping the fluxonium at the $|e0\rangle\rightarrow|g1\rangle$
transition frequency in the regime where $g\ll\kappa$ (pulse sequence shown
Fig. 4(a)). The outgoing field is recorded using heterodyne detection, which
also captures the phase information of the emitted photon.

For initial characterization, we perform this controlled release protocol
using a simple square pulse. In Fig. 4(b), we show the measured in-phase
quadrature of the propagating fields as a function of time, with the fluxonium
initialized to be in $|\psi^{\pm}\rangle=(|g0\rangle\pm|e0\rangle)/\sqrt{2}$.
With a square pulse, the coupling rate of
$|e0\rangle\leftrightarrow|g1\rangle$ is constant, which should result in a
simple exponential damping of the output fields after some ring-up time
proportional to $\kappa^{-1}$ as shown in Fig. 4(b). In this case, the exact
response of the resonator as well as the emitted photon related by
$\hat{r}_{\text{out}}=\sqrt{\kappa}\hat{r}$ can be analytically computed using
input-output theory (Appendix C), which fits our measured data well. Here the
coupling rate $g(t)$ is fit to be $2\pi\times 0.1\ \mathrm{MHz}\ll\kappa$.
Upon shifting the phase of the qubit Rabi pulse by $\pi$, the response of the
in-phase signal also shifts correspondingly, which indicates the phase
coherence of the controlled release process.

We now demonstrate that we can dynamically tune the photon’s emission rate to
build a time-symmetric wave packet, which can be used in remote entanglement
protocols for efficient absorption by a receiver. We shape the outgoing wave
packet to a hyperbolic secant function,

$\Psi(t)=\frac{1}{4}\sqrt{\gamma_{\text{ph}}}\text{sech}(\frac{\gamma_{\text{ph}}t}{2}),$
(4)

where $\gamma_{\text{ph}}$ is the photon bandwidth (Appendix E). In Fig. 4(c),
we show the measured in-phase quadrature and field amplitude of the shaped
wave packet with the fluxonium initialized to be in $|\psi^{+}\rangle$, which
displays a clear hyperbolic secant shape. This emitted wavepacket could then
be captured by a receiver by time-reversing the emission process [2].

##  VI Outlook

We have demonstrated parametric conversion between a fluxonium qubit
excitation and a flying photon by successfully driving the
$|e0\rangle\leftrightarrow|g1\rangle$ transition. Utilizing this transition,
we have initialized the fluxonium qubit in the ground state with measured
probability $>95\%$ and predict a reset threshold in the same system of
$>99\%$ with a longer reset pulse. By using a resonator with a higher decay
rate, our simulations indicate that the fluxonium could be reset with fidelity
$\sim 99.8\%$ in 500 ns. We have also demonstrated coherent release and
temporal shaping of a flying photon detected using heterodyne measurement. By
introducing a microwave-photonic interface for the fluxonium qubit, we have
paved the way for remotely entangling fluxonium qubits via a microwave-
photonic link.

###### Acknowledgements.

We acknowledge useful discussions with Xi Cao, Michael Mollenhauer, Rafael
Goncalves, Sonia Rani, Cheeranjeev Purmessur, Supriya Mandal, Shyam Shankar,
and Srivatsan Chakram. Our traveling-wave parametric amplifier is provided by
IBM. We are grateful to Chunyang Ding and David Schuster for providing us with
PCBs. This research is carried out in part in the Materials Research Lab
Central Facilities and the Holonyak Micro and Nanotechnology Lab, University
of Illinois. This research is partially supported by the Air Force Office of
Scientific Research under award number FA9550-21-1-032. Research is also
sponsored by the Army Research Office and is accomplished under Grant Number
W911NF-23-1-0096. The views and conclusions contained in this document are
those of the authors and should not be interpreted as representing the
official policies, either expressed or implied, of the Army Research Office or
the U.S. Government.

##  Appendix A Device characterization

Figure 5:  False-colored SEM images of (a) the full device and (b) the SNAIL-
fluxonium. The SNAIL-fluxonium (purple and blue) couples to the readout
resonator (green) via shared capacitance between the SNAIL’s antenna and the
readout resonator. The readout resonator couples capacitively to the readout
line (orange). The charge line (red) couples capacitively to the other antenna
of the SNAIL resoantor.

Figure 5 shows the optical and scanning electron microscope (SEM) images of
our device. To drive the parity-forbidden transition
$|e0\rangle\leftrightarrow|g1\rangle$, we need strong coupling between the
fluxonium and the SNAIL, and between the SNAIL and the readout resonator
respectively. We realize strong coupling by using multiple SNAILs as a shared
inductance with the fluxonium superinductance. Moreover, one side of the SNAIL
capacitor is enlarged for stronger coupling capacitance with the readout
resonator and asymmetric driving of the system. A charge line weakly coupled
to the other pad of the SNAIL resonator is included for driving. The coupling
between the charge line and the SNAIL is designed as a trade off between the
need to strongly drive the parity-forbidden transitions and while still
suppressing the Purcell loss of the fluxonium excitation through the SNAIL to
the charge line. The readout resonator is coupled to the readout line in the
hanger style with relatively low decay rate $\kappa/2\pi=0.4$ MHz. We
fabricate the entire device, with the exception of Al/AlOx/Al junctions, with
tantalum on a sapphire wafer. A global coil is used to tune the flux across
the entire chip, including flux threading though the fluxonium
$\Phi_{\text{ext}}^{f}$ and the SNAIL $\Phi_{\text{ext}}^{s}$. This is made
possible by designing a large area ratio between the fluxonium and the SNAIL
$A_{f}/A_{s}\approx 35$, so that one may tune either flux independently. We
operate our device at $\Phi_{\text{ext}}^{f}/\Phi_{0}=9.5$, corresponding to
$\Phi_{\text{ext}}^{s}/\Phi_{\text{0}}=0.27$. A summary of our device’s
parameters at $\Phi_{\text{ext}}^{f}/\Phi_{0}=9.5$ are provided in Table I.
Two-tone spectroscopy data for the fluxonium qubit is shown in Fig. 6(a).

Table 1: System parameters at $\Phi_{\text{ext}}^{f}/\Phi_{0}=9.5$. Parameters
for the fluxonium and the readout resonator are extracted by fitting their
respective spectroscopy. The coupling strengths are estimated based on the
measured dispersive shift. We provide designed values for parameters we do not
measure.

Fluxonium | Readout Resonator
---|---
$E_{C,f}/2\pi$ | 0.89 GHz | $\omega_{r}/2\pi$ | 6.245 GHz
$E_{L,f}/2\pi$ | 0.47 GHz | $\kappa/2\pi$ | 0.4 MHz
$E_{J,f}/2\pi$ | 5.54 GHz | $\chi_{fr}/2\pi$ | 0.55 MHz
$\omega_{q}/2\pi$ | 81 MHz |  |
Single SNAIL | SNAIL Resonator
$\alpha$ | 0.5* | $E_{C,s}/2\pi$ | 0.39* GHz
$E_{J,s}/2\pi$ | 65* GHz | $E_{L,s}/2\pi$ | 15.4* GHz
$N_{s}$ | 3 | $c_{3}^{s}/2\pi$ | 350* MHz
$g_{fs}/2\pi$ | 500 MHz | $g_{sr}/2\pi$ | 130 MHz

  * *

These are designed parameters, as we do not perform two-tone spectroscopy on
the SNAIL mode.

Figure 6: (a) Fluxonium qubit $|g\rangle\leftrightarrow|e\rangle$ transition
frequency as a function of applied flux. Two-tone spectroscopy data is fit
using the following parameters
$\\{E_{C,f},E_{L,f},E_{J,f}\\}/2\pi=\\{0.9,0.5,5.54\\}$ GHz. (b)-(d) Time
domain measurements of $T_{1}$, $T_{2}^{R}$ and $T_{2}^{E}$ respectively.

The time domain measurements for our fluxonium qubit are shown in Fig.
6(b)-(d). A reset pulse is added for initialization before the usual $T_{1}$
and $T_{2}$ measurements. Fits of the flux-dependence of our device’s
relaxation time suggest that it is mainly limited by Purcell loss through the
internal loss of the SNAIL resonator.

##  Appendix B Experimental setup

Figure 7:  Simplified schematic of the measurement setup for the controlled
release experiment. The qubit Rabi drive is directly output by an integrated
FPGA system (_Quantum Machines, OPX+_). The pump tone for the
$|e0\rangle\leftrightarrow|g1\rangle$ transition is applied by mixing drives
from a microwave generator (_SignalCore, SC5511A_) as input LO and IQ tones
from the FPGA system. The pump signal is first amplified and then sent into
the dilution refrigerator (_Oxford, Triton 300_) along with the qubit Rabi
drive through the sample’s charge line. The output signal from the sample goes
through a circulator and is amplified by a traveling wave parametric amplifier
(TWPA) at the mixing chamber (MXC). Above the MXC, the output signal goes
through two additional circulators. At room temperature, the signal gets
amplified before demodulated down to 50 MHz. We use the same input LO
generator for output demodulation, so that the output signal is phased locked
to the output LO. The demodulated signal goes through three 41-58 MHz band-
pass filters before a final stage of amplification. The signal is then further
demodulated by analog-to-digital converters (ADC) in the FPGA system.

Figure 7 shows the simplified schematic of the measurement setup used for the
controlled-release measurement demonstrated in Section V. For this experiment,
it is important to ensure that the photon release and photon capture processes
are phase-locked. To ensure phase-locking of the release process, we apply the
Rabi pulse on the qubit directly using the I channel output from the FPGA
system, and modulate the pump tone addressing the
$|e0\rangle\leftrightarrow|g1\rangle$ transition using the same FPGA system.
To ensure phase-locking of the output signal, we use the pump signal generator
to provide the local oscillator for output demodulation. The pump tone is
amplified before sent into the dilution refrigerator. The pump tone and the
qubit Rabi drive couple to the SNAIL-fluxonium through the charge line weakly
coupled to the SNAIL’s antenna.

On the output side, the signal leaving the resonator is first routed through a
circulator before being amplified by a TWPA. The signal goes through two
additional circulators thermalized to the MXC. At room temperature, the output
signal is amplified and then demodulated down to 50 MHz. We use the same input
LO generator for output demodulation, so that the output signal is phase-
locked to the output LO. The demodulated signal goes through 41-58 MHz band-
pass filters to filter away any pump tones that leak to the readout line.
After filtering, the signal gets amplified before being demodulated by the
FPGA.

Figure 8:  (a)-(c) Simulated population in $|g0\rangle,|e0\rangle,|g1\rangle$
vs. reset pulse duration with the same parameters as Fig. 3(a), but zero
$t_{\mathrm{buffer}}$. A maximum of $\sim 99.3\%$ fluxonium population
(dashed) can be initialized to the ground state. (d)-(f) Simulated population
in $|g0\rangle,|e0\rangle,|g1\rangle$ vs. reset pulse duration with
$g/2\pi=1.27$ MHz and $\kappa/2\pi=4$ MHz. A higher resonator decay rate can
increase the ground state reset population to $\sim 99.8\%$ (dashed).

##  Appendix C System Hamiltonian

Without microwave drives, the Hamiltonian of the system consisting of the
fluxonium qubit mode ($\hat{f}$), the SNAIL resonator mode ($\hat{s}$), and
the readout resonator mode ($\hat{r}$) is given by

$\hat{H}/\hbar=\sum_{j=f,s,r}\omega_{j}\hat{j}^{\dagger}\hat{j}+\sum_{j\neq
k}\chi_{jk}\hat{j}^{\dagger}\hat{j}\hat{k}^{\dagger}\hat{k}\\\
+c_{3}^{s}\hat{\phi}_{s}^{3},$ (5)

considering only single-photon excitations of the system. Here the flux across
the SNAIL is given by the sum of the contributions from all the circuit
elements,

$\hat{\phi}_{s}=\sum_{j=f,s,r}\phi_{j,\mathrm{zpf}}(\hat{j}^{\dagger}+\hat{j}),$
(6)

where $\phi_{j,\mathrm{zpf}}$ are the quantum zero-point fluctuations of
junction flux associated with each mode obtained from Hamiltonian
diagonalization. For quantum systems involving the fluxonium qubit, it is
difficult to obtain an analytical solution for $\chi_{kl}$ and
$\phi_{j,\mathrm{zpf}}$, but one may perform numerical diagonalization to
obtain these parameters. The three-wave-mixing term in the Hamiltonian breaks
the parity selection rule, and allows one to drive the parity-forbidden
transition $|e0\rangle\leftrightarrow|g1\rangle$.

In the presence of a microwave pump acting on the SNAIL, the three-wave-mixing
and the driving terms are given by,

$\hat{H}_{\text{3WM+d}}/\hbar=c_{3}^{s}(\phi_{f,\mathrm{zpf}}(\hat{f}^{\dagger}+\hat{f})+\phi_{s,\mathrm{zpf}}(\hat{s}^{\dagger}+\hat{s})\\\
+\phi_{r,\mathrm{zpf}}(\hat{r}^{\dagger}+\hat{r}))^{3}+\epsilon(t)e^{-i\omega_{d}t}(\hat{s}^{\dagger}+\hat{s}).$
(7)

Going into the displaced frame, we introduce the displaced amplitude,

$\tilde{\xi}(t)=\xi(t)e^{-i\omega_{d}t}=\frac{\epsilon(t)e^{-i\omega_{d}t}}{i\kappa_{s}/4+\Delta_{c}}\approx\frac{\epsilon(t)e^{-i\omega_{d}t}}{\Delta_{c}},$
(8)

where the detuning $\Delta_{c}$ is the difference between the drive frequency
and the SNAIL mode frequency $\Delta_{s}=\omega_{d}-\omega_{s}$. Through the
substitution of $\hat{s}=\hat{\tilde{s}}+\tilde{\xi}(t)$ and moving into the
rotating frame, we find the conversion Hamiltonian,

$\displaystyle\hat{H}_{\text{conv}}/\hbar$
$\displaystyle=6c_{3}^{s}(\prod_{i=f,s,r}\phi_{i,\mathrm{zpf}})(\xi(t)\hat{f}\hat{r}^{\dagger}e^{-i\delta
t}+\xi^{*}(t)\hat{f}^{\dagger}\hat{r}e^{i\delta t})$
$\displaystyle=(g(t)\hat{f}\hat{r}^{\dagger}e^{-i\delta
t}+g(t)^{*}\hat{f}^{\dagger}\hat{r}e^{i\delta t}),$ (9)

where $g(t)=6c_{3}^{s}\xi(t)\prod\limits_{i=f,s,r}\phi_{i,\mathrm{zpf}}$ is
the coupling rate of the $|e0\rangle\leftrightarrow|g1\rangle$ transition, and
$\delta=\omega_{d}-(\omega_{r}-\omega_{f})$ is a small detuning between the
pump frequency and the $|e0\rangle\leftrightarrow|g1\rangle$ transition
frequency. In simulation, we find that computing the bare SNAIL phase matrix
element squared $6c_{3}^{s}\xi(t)|\langle g1|\hat{\phi}_{s}^{2}|e0\rangle|$
already gives useful guidance for choosing system parameters.

Besides $\hat{H}_{\text{conv}}$, there are additional terms corresponding to
the ac-Stark shifts generated by the microwave pump on the SNAIL,

$\hat{H}_{\mathrm{Stark}}/\hbar=|\xi(t)|^{2}(\chi_{fs}\hat{f}^{\dagger}\hat{f}+\chi_{sr}\hat{r}^{\dagger}\hat{r}).$
(10)

The Stark shifts can be compensated by modulating the pump frequency. Large-
amplitude drives result in the Stark shift changing non-negligibly during the
pulse rise time, which significantly complicates pulse calibration. We
therefore choose to operate the system at high flux quanta for a large third-
order coefficient $c_{3}^{s}$ and apply pumps of relatively low power.

Summarizing all the terms, we may write out an effective system Hamiltonian,

$\hat{H}_{\mathrm{eff}}/\hbar=\tilde{\omega}_{f}\hat{f}^{\dagger}\hat{f}+\tilde{\omega}_{r}\hat{r}^{\dagger}\hat{r}+\chi_{fr}\hat{f}^{\dagger}\hat{f}\hat{r}^{\dagger}\hat{r}\\\
+2g\cos(\omega_{d}t)(\hat{f}\hat{r}^{\dagger}+\hat{f}^{\dagger}\hat{r}),$ (11)

where $\tilde{\omega_{f}},\tilde{\omega_{r}}$ is the new fluxonium and
resonator frequency including Stark shifts. Here we omit the SNAIL mode in the
effective Hamiltonian, since the SNAIL mode should not be populated in our
experiment. We also assume $g$ to be real without loss of generality.

We perform more precise simulations of the time evolution of the given quantum
system by solving the Lindblad master equation,

$\delta\hat{\rho}=-i[\hat{H}_{\text{eff}},\hat{\rho}]+\sum_{k=g0,e0,g1}[L_{k}\rho
L_{k}-\frac{1}{2}\\{L_{k}L_{k},\hat{\rho}\\}].$ (12)

Here $\hat{\rho}$ is the system density matrix operator.
$\hat{H}_{\text{sys}}$ is defined in equation (1) and $\hat{H}_{d}$ is the
driving Hamiltonian. The jump operators
$L_{g0}=\sqrt{\Gamma_{\downarrow}}|e0\rangle\langle
g0|,L_{e0}=\sqrt{\Gamma_{\uparrow}}|g0\rangle\langle e0|,$ and
$L_{g1}=\sqrt{\kappa}|g0\rangle\langle g1|$ describe the main depolarization
paths of the quantum system.

Considering only the conversion $|e0\rangle\leftrightarrow|g1\rangle$ at
resonance, and resonator decay $\kappa$ into the readout line, the system is
well described by the Heisenberg equations of motion,

$\displaystyle\delta_{t}\hat{f}$ $\displaystyle=-ig\hat{r},$ (13)
$\displaystyle\delta_{t}\hat{r}$
$\displaystyle=-ig\hat{f}-\frac{\kappa}{2}\hat{r},$ (14)
$\displaystyle\hat{r}_{\text{out}}$ $\displaystyle=\sqrt{\kappa}\hat{r},$ (15)

where $\hat{r}_{\text{out}}$ is the outgoing photonic mode. For the case of
constant coupling rate $g$ with some initial condition $\hat{f}(0)$, one may
obtain an analytical solution. The analytical solution and discussion of the
resonant case are carefully laid out in the supplemental information of Ref.
[4] for a system of two cavity modes.

##  Appendix D Fast reset efficiency

Our reset inefficiency is mainly due to qubit $T_{1}$ depolarization. There
are three stages of depolarization (mainly heating) we need to consider for
our reset process: during pumping, during $t_{\mathrm{buffer}}$, and during
readout. Knowing the qubit depolarization rate $\Gamma$ and temperature, we
can obtain $\Gamma_{\uparrow}$ and $\Gamma_{\downarrow}$ related by the
following equation [29],

$\displaystyle\Gamma$ $\displaystyle=\Gamma_{\uparrow}+\Gamma_{\downarrow}$
(16) $\displaystyle\Gamma_{\uparrow}$
$\displaystyle=\exp{(-\hbar\omega_{f}/k_{B}T)}\Gamma_{\downarrow}.$ (17)

To get an estimate of the fluxonium heating during the 200 ns pumping time, we
solve the master equation [Eq. (12)] and find the loss to be negligible.
During $t_{\mathrm{buffer}}$, the fluxonium population heats toward thermal
equilibrium,

$P_{e}(t)=\Gamma_{\uparrow}/\Gamma+(P_{e,0}-\Gamma_{\uparrow}/\Gamma)\exp{(-\Gamma
t)},$ (18)

where $P_{e,0}$ is the $|e0\rangle$ population initially after pumping. Since
a heterodyne measurement integrates the signal over time, the final population
we obtain from readout of pulse duration ${t}_{\mathrm{readout}}$ is given by,

$\bar{P_{e}}=1/{t}_{\mathrm{readout}}\int_{t_{\mathrm{buffer}}}^{t_{\mathrm{buffer}}+t_{\mathrm{readout}}}P_{e}(t)dt,$
(19)

In our experiment, we use $t_{\mathrm{buffer}}=5\ \mu$s and
$t_{\mathrm{readout}}=10\ \mu$s, which results in $\sim$ 4% loss during
$t_{\mathrm{buffer}}$ and another time-averaged $\sim$ 4% loss during readout.

One can improve the reset efficiency by leaving the pump on while waiting for
$|g1\rangle$ decaying to $|g0\rangle$. In Fig. 8(a)-(c), we show the simulated
time evolution of the system population with a longer reset pulse, assuming
perfect readout. Population in $|g0\rangle$ reaches a steady state of $\sim
99.3\%$ shortly after 5 $\mu$s of pumping, denoting the highest possible reset
efficiency in our current system. This reset threshold is consistent with the
simple calculation,

$\displaystyle P_{g}^{\mathrm{th}}$
$\displaystyle=\frac{\Gamma_{\uparrow}+\Gamma_{\mathrm{eff}}}{\Gamma_{\uparrow}+\Gamma_{\downarrow}+\Gamma_{\mathrm{eff}}},$
(20)
$\displaystyle=\frac{\Gamma_{\uparrow}+\kappa/2}{\Gamma_{\uparrow}+\Gamma_{\downarrow}+\kappa/2}\approx
99.3\%,$ (21)

where $\Gamma_{\mathrm{eff}}$ denotes the effective decay rate of the system.
However, by leaving the pump on, the resonator photon decays at a rate of
$\kappa/2$ instead of $\kappa$, which lengthens the initialization process.
Therefore we choose to demonstrate faster reset at the expense of slightly
lower ground state initialization in the main text.

If we keep the same coupling rate $g/2\pi=1.27$ MHz but use a higher decay
rate for the resonator $\kappa/2\pi=4$ MHz, the system would be in the weak
coupling regime. In this case, the effective decay rate is given by
$4g^{2}/\kappa$, and no $t_{\mathrm{buffer}}$ is needed as the population in
$|g1\rangle$ is strongly suppressed. The $|g0\rangle$ population reaches
$\sim$99.8% after 500 ns of pumping according to numerical simulation (Fig.
8(d)-(f)). Despite the need for a slightly longer reset pulse, the total reset
time is reduced and the reset efficiency is improved drastically.

Figure 9:  (a) Coupling strength $g(t)$ as a function of pulse duration
obtained from numerical simulation (solid) and analytical solution (dashed) to
produce a hyperbolic secant wave packet with $\gamma_{\text{ph}}/2\pi=0.2$
MHz. (b) Measured fluxonium population in $|e0\rangle$ vs. pulse duration
under the pump tone shown in (a). The measured population is fit using the
master equation (solid). The qubit population saturates at around $8\%$, which
is consistent with our reset efficiency (dashed).

##  Appendix E Shaping wave packets

To generate our desired wave packet [Eq. (4)], the coupling rate between the
fluxonium qubit and the readout resonator can be solved analytically [30] and
should be modulated as

$g(t)=\frac{\gamma_{\text{ph}}}{4\cosh{(\frac{\gamma_{\text{ph}}t}{2}})}\frac{1-e^{\gamma_{\text{ph}}t}+(1+e^{\gamma_{\text{ph}}t})\kappa/\gamma_{\text{ph}}}{\sqrt{(1+e^{\gamma_{\text{ph}}t})\kappa/\gamma_{\text{ph}}-e^{\gamma_{\text{ph}}t}}},$
(22)

where the photon bandwidth bandwidth $\gamma_{\text{ph}}$ should be smaller
than resonator decay rate $\kappa$.

While the analytical solution is specific for our chosen wave packet shape,
numerical computation can be employed for different target shapes. With the
equations of motion for the system, we may integrate equation (13), which
gives

$\displaystyle\hat{f}(t)$
$\displaystyle=\hat{f}(0)-i\int_{0}^{t}g(t)\hat{r}(t)dt.$ (23)

Substituting it into equation (C9), we obtain

$\displaystyle\delta_{t}\hat{r}=-ig(t)[\hat{f}(0)-i\int_{0}^{t}g(t)\hat{r}(t)dt]-\frac{\kappa}{2}\hat{r}(t).$
(24)

We can rewrite the integral in terms of Riemann sum for numerical simulation
purpose

$\displaystyle i\int_{0}^{t}g(t)\hat{r}(t)dt$ $\displaystyle\approx
i\sum_{i=0}^{\frac{t}{\Delta t}-1}g(t_{i})\hat{r}(t_{i})\Delta t$
$\displaystyle\delta_{t}\hat{r}+\frac{\kappa}{2}\hat{r}(t)$
$\displaystyle\approx-ig(t)[\hat{f}(0)-i\sum_{i=0}^{\frac{t}{\Delta
t}-1}g(t_{i})\hat{r}(t_{i})\Delta t].$ (25)

Reorganizing the terms, we come to an expression for the coupling rate $g(t)$
as follows,

$\displaystyle
g(t)=-\frac{\delta_{t}\hat{r}+\frac{\kappa}{2}\hat{r}(t)}{i\hat{f}(0)+\sum_{i=0}^{\frac{t}{\Delta
t}-1}g(t_{i})\hat{r}(t_{i})\Delta t}.$ (26)

Given a target wave packet $\hat{r}_{\text{out}}(t)$, we solve for the
coupling strength at any time using equation (26). In the end, we can check if
our simulated coupling strength allows for full evacuation of fluxonium
excitation by making sure

$\hat{f}(0)=i\int_{0}^{t}g(t)\hat{r}(t)dt.$ (27)

In Fig. 9(a), we show both numerically and analytically computed coupling
strength for producing a hyperbolic secant wave packet with
$\gamma_{\text{ph}}/2\pi=0.2$ MHz. Both methods yield coupling strength $g(t)$
consistent with each other. In Fig. 9(b), we show the time evolution of the
fluxonium population in $|e0\rangle$ under our simulated pump tone without
state initialization. The fluxonium population decays from thermal equilibrium
and saturates at $8\%$, consistent with our reset fidelity as well as
numerical fit using the master equation. This indicates that our applied pump
pulse amplitude evacuates a fluxonium excitation, the fidelity of which is
limited by our device’s $T_{1}$ decay during state initialization and
controlled release processes due to our low decay rate $\kappa$. By simply
designing a higher $\kappa$, one can achieve more efficient controlled release
process without additional tuning of design parameters.

## References

  * DiVincenzo [2000] D. P. DiVincenzo, The physical implementation of quantum computation, Fortschritte der Physik 48, 771–783 (2000).
  * Cirac  _et al._ [1997] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Quantum state transfer and entanglement distribution among distant nodes in a quantum network, Phys. Rev. Lett. 78, 3221 (1997).
  * Korotkov [2011] A. N. Korotkov, Flying microwave qubits with nearly perfect transfer efficiency, Physical Review B 84, 10.1103/physrevb.84.014510 (2011).
  * Pfaff  _et al._ [2017] W. Pfaff, C. J. Axline, L. D. Burkhart, U. Vool, P. Reinhold, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, Controlled release of multiphoton quantum states from a microwave cavity memory, Nature Physics 13, 882 (2017).
  * Keller  _et al._ [2004] M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther, Continuous generation of single photons with controlled waveform in an ion-trap cavity system, Nature 431, 1075 (2004).
  * Barros  _et al._ [2009] H. G. Barros, A. Stute, T. E. Northup, C. Russo, P. O. Schmidt, and R. Blatt, Deterministic single-photon source from a single ion, New Journal of Physics 11, 103004 (2009).
  * Stute  _et al._ [2013] A. Stute, B. Casabone, B. Brandstätter, K. Friebe, T. E. Northup, and R. Blatt, Quantum-state transfer from an ion to a photon, Nature Photonics 7, 219–222 (2013).
  * Schupp  _et al._ [2021] J. Schupp, V. Krcmarsky, V. Krutyanskiy, M. Meraner, T. Northup, and B. Lanyon, Interface between trapped-ion qubits and traveling photons with close-to-optimal efficiency, PRX Quantum 2, 020331 (2021).
  * Houck  _et al._ [2007] A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Generating single microwave photons in a circuit, Nature 449, 328–331 (2007).
  * Kuhn  _et al._ [2002] A. Kuhn, M. Hennrich, and G. Rempe, Deterministic single-photon source for distributed quantum networking, Physical Review Letters 89, 10.1103/physrevlett.89.067901 (2002).
  * McKeever  _et al._ [2004] J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, Deterministic generation of single photons from one atom trapped in a cavity, Science 303, 1992 (2004).
  * Mücke  _et al._ [2013] M. Mücke, J. Bochmann, C. Hahn, A. Neuzner, C. Nölleke, A. Reiserer, G. Rempe, and S. Ritter, Generation of single photons from an atom-cavity system, Physical Review A 87, 10.1103/physreva.87.063805 (2013).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Reviews of Modern Physics 93, 10.1103/revmodphys.93.025005 (2021).
  * Koch  _et al._ [2007] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the cooper pair box, Physical Review A 76, 10.1103/physreva.76.042319 (2007).
  * Magnard  _et al._ [2018] P. Magnard, P. Kurpiers, B. Royer, T. Walter, J.-C. Besse, S. Gasparinetti, M. Pechal, J. Heinsoo, S. Storz, A. Blais, and A. Wallraff, Fast and unconditional all-microwave reset of a superconducting qubit, Physical Review Letters 121, 10.1103/physrevlett.121.060502 (2018).
  * Zhou  _et al._ [2021] Y. Zhou, Z. Zhang, Z. Yin, S. Huai, X. Gu, X. Xu, J. Allcock, F. Liu, G. Xi, Q. Yu, H. Zhang, M. Zhang, H. Li, X. Song, Z. Wang, D. Zheng, S. An, Y. Zheng, and S. Zhang, Rapid and unconditional parametric reset protocol for tunable superconducting qubits, Nature Communications 12, 10.1038/s41467-021-26205-y (2021).
  * Pechal  _et al._ [2014] M. Pechal, L. Huthmacher, C. Eichler, S. Zeytinoğlu, A. A. Abdumalikov, S. Berger, A. Wallraff, and S. Filipp, Microwave-controlled generation of shaped single photons in circuit quantum electrodynamics, Phys. Rev. X 4, 041010 (2014).
  * Campagne-Ibarcq  _et al._ [2018] P. Campagne-Ibarcq, E. Zalys-Geller, A. Narla, S. Shankar, P. Reinhold, L. Burkhart, C. Axline, W. Pfaff, L. Frunzio, R. Schoelkopf, and M. Devoret, Deterministic remote entanglement of superconducting circuits through microwave two-photon transitions, Physical Review Letters 120, 10.1103/physrevlett.120.200501 (2018).
  * Kurpiers  _et al._ [2018] P. Kurpiers, P. Magnard, T. Walter, B. Royer, M. Pechal, J. Heinsoo, Y. Salathé, A. Akin, S. Storz, J.-C. Besse, S. Gasparinetti, A. Blais, and A. Wallraff, Deterministic quantum state transfer and remote entanglement using microwave photons, Nature 558, 264–267 (2018).
  * Manucharyan  _et al._ [2009] V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Fluxonium: Single cooper-pair circuit free of charge offsets, Science 326, 113–116 (2009).
  * Nguyen  _et al._ [2019] L. B. Nguyen, Y.-H. Lin, A. Somoroff, R. Mencia, N. Grabon, and V. E. Manucharyan, High-coherence fluxonium qubit, Phys. Rev. X 9, 041041 (2019).
  * Somoroff  _et al._ [2023] A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. Kuzmin, and V. E. Manucharyan, Millisecond coherence in a superconducting qubit, Phys. Rev. Lett. 130, 267001 (2023).
  * Ding  _et al._ [2023] L. Ding, M. Hays, Y. Sung, B. Kannan, J. An, A. Di Paolo, A. H. Karamlou, T. M. Hazard, K. Azar, D. K. Kim, B. M. Niedzielski, A. Melville, M. E. Schwartz, J. L. Yoder, T. P. Orlando, S. Gustavsson, J. A. Grover, K. Serniak, and W. D. Oliver, High-fidelity, frequency-flexible two-qubit fluxonium gates with a transmon coupler, Phys. Rev. X 13, 031035 (2023).
  * Moskalenko  _et al._ [2022] I. N. Moskalenko, I. A. Simakov, N. N. Abramov, A. A. Grigorev, D. O. Moskalev, A. A. Pishchimova, N. S. Smirnov, E. V. Zikiy, I. A. Rodionov, and I. S. Besedin, High fidelity two-qubit gates on fluxoniums using a tunable coupler, npj Quantum Information 8, 10.1038/s41534-022-00644-x (2022).
  * Wang  _et al._ [2024] T. Wang, F. Wu, F. Wang, X. Ma, G. Zhang, J. Chen, H. Deng, R. Gao, R. Hu, L. Ma, Z. Song, T. Xia, M. Ying, H. Zhan, H.-H. Zhao, and C. Deng, Efficient initialization of fluxonium qubits based on auxiliary energy levels (2024), arXiv:2402.06267 [quant-ph] .
  * Zhang  _et al._ [2021] H. Zhang, S. Chakram, T. Roy, N. Earnest, Y. Lu, Z. Huang, D. Weiss, J. Koch, and D. I. Schuster, Universal fast-flux control of a coherent, low-frequency qubit, Physical Review X 11, 10.1103/physrevx.11.011010 (2021).
  * Vool  _et al._ [2018] U. Vool, A. Kou, W. Smith, N. Frattini, K. Serniak, P. Reinhold, I. Pop, S. Shankar, L. Frunzio, S. Girvin, and M. Devoret, Driving forbidden transitions in the fluxonium artificial atom, Physical Review Applied 9, 10.1103/physrevapplied.9.054046 (2018).
  * Frattini  _et al._ [2017] N. E. Frattini, U. Vool, S. Shankar, A. Narla, K. M. Sliwa, and M. H. Devoret, 3-wave mixing josephson dipole element, Applied Physics Letters 110, 10.1063/1.4984142 (2017).
  * Krantz  _et al._ [2019] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Applied Physics Reviews 6, 10.1063/1.5089550 (2019).
  * Gheeraert  _et al._ [2020] N. Gheeraert, S. Kono, and Y. Nakamura, Programmable directional emitter and receiver of itinerant microwave photons in a waveguide, Phys. Rev. A 102, 053720 (2020).


loops of $M_{\mathcal{S}}$, so $L$ consists of the singletons in
$\mathcal{S}$. We will proceed inductively by considering subgraphs of $G$
with increasing numbers of vertices. For the base case, we begin with the set
$L$ of loops of $M$. The restriction $M_{\mathcal{S}}|L$ can be represented by
the $3\times|L|$ zero matrix.

For the inductive step, suppose that we have shown that
$M_{\mathcal{S}}|V(G_{0})$ is realizable over $\mathbb{R}$ for some subgraph
$G_{0}$ of $G$ containing the vertices in $L$. Let $v\in V(G)\backslash
V(G_{0})$. We will show that the restriction
$M_{\mathcal{S}}|(V(G_{0})\cup\\{v\\})$ is obtained from
$M_{\mathcal{S}}|(V(G_{0}))$ either by adding $v$ as a coloop or by freely
adding $v$ to a flat. Thus, Lemma 5.5 or Lemma 5.6 implies that
$M_{\mathcal{S}}|(V(G_{0})\cup\\{v\\})$ is realizable over the reals. We
consider the following cases.

  * (1)

There is a vertex $w\in V(G_{0})$ such that $v$ and $w$ are in the same cloud.

  * (2)

Case (1) does not hold, but there are vertices $w,x\in V(G_{0})\backslash L$,
not in the same cloud, such that $\\{v,w,x\\}$ is an unblocked set.

  * (3)

Neither Case (1) nor Case (2) holds.

For Case (1), note that $C\cup L$ is a flat of rank $1$ for every cloud $C$.
Therefore, if $C$ is the cloud containing $v$ and $w$, then we obtain
$M_{\mathcal{S}}|(V(G_{0})\cup\\{v\\})$ by freely adding $v$ to the flat
$C\cup L$.

In Case (2), the fact that $w$ and $x$ are not in the same cloud and the fact
that neither $w$ nor $x$ is in $L$ imply that $r(\\{w,x\\})=2$. The fact that
$\\{v,w,x\\}$ is an unblocked set implies that $r(\\{v,w,x\\})=2$ also. Thus,
$v$ is in the closure of $\\{w,x\\}$ in the matroid $M_{\mathcal{S}}$. Since
Case (1) does not hold, $M_{\mathcal{S}}|(V(G_{0})\cup\\{v\\})$ is obtained by
adding $v$ freely to the closure of $\\{w,x\\}$.

Now we consider Case (3). Since Case (1) does not hold, Proposition 6.8
implies that $M_{\mathcal{S}}$ has no circuit of size $2$ consisting of $v$
and some vertex of $G_{0}$. Since Case (2) does not hold, Proposition 6.8
implies that $M_{\mathcal{S}}$ has no circuit of size $3$ consisting of $v$
and two vertices of $G_{0}$. Therefore, the only circuits of $M_{\mathcal{S}}$
containing $v$ and some vertex of $G_{0}$ are sets of size $4$. Thus, the
matroid $M_{\mathcal{S}}|(V(G_{0})\cup\\{v\\})$ is obtained either by freely
adding $v$ to the ground set $V(G_{0})$ (if $r(V(G_{0}))=3$) or by adding $v$
as a coloop (if $r(V(G_{0}))<3$). ∎

##  7 Hypergraphs $\Delta^{s,t}$

Here, we study hypergraph ideals which arise naturally in the study of
conditional independence statements. We begin with notation and the general
setup of the problem. We denote the $k\times\ell$ matrix of integers

$\displaystyle\mathcal{Y}=(\mathcal{Y}_{i,j})_{i,j}=\begin{bmatrix}1&k+1&\dots&(\ell-1)k+1\\\
2&k+2&\dots&(\ell-1)k+2\\\ \vdots&\vdots&\ddots&\vdots\\\ k&2k&\dots&\ell
k\end{bmatrix}\ .$ (3)

For each $i\in[k]$ and $j\in[\ell]$, the rows and columns of $\mathcal{Y}$ are
denoted

$R_{i}=\\{\mathcal{Y}_{i,1},\mathcal{Y}_{i,2},\dots,\mathcal{Y}_{i,\ell}\\}=\\{i,k+i,\dots,(\ell-1)k+i\\},$
$C_{j}=\\{\mathcal{Y}_{1,j},\mathcal{Y}_{2,j},\dots,\mathcal{Y}_{k,j}\\}=\\{(j-1)k+1,(j-1)k+2,\dots,(j-1)k+k\\}.$

For each $s$ and $t$ with $s\leq k$ and $t\leq\ell$, we define $\Delta^{s,t}$
be the following collection of subsets of $[k\ell]$,

$\Delta^{s,t}=\bigcup_{1\leq i\leq k}\binom{R_{i}}{t}\cup\bigcup_{1\leq
j\leq\ell}\binom{C_{j}}{s}.$

If the values of $s$ and $t$ have been fixed, then we simply write $\Delta$
for $\Delta^{s,t}$. We are interested in studying the ideals
$I_{\Delta^{s,t}}$ for $d\geq\max\\{s,t\\}$ as these are examples of
conditional independence ideals with hidden variables.

######  Example 7.1.

Let $k=4,\ell=7,s=2$ and $t=3$. We have

$\mathcal{Y}=\begin{bmatrix}1&5&9&13&17&21&25\\\ 2&6&10&14&18&22&26\\\
3&7&11&15&19&23&27\\\
4&8&12&16&20&24&28\end{bmatrix}\quad\textrm{and}\quad\Delta^{2,3}=\left\\{\binom{\\{1,5,9,13,17,21,25\\}}{3}\cup\dots\cup\binom{\\{25,26,27,28\\}}{2}\right\\}.$

Calculating the dependent matroids for $\Delta^{2,3}$, we find that all such
matroids are point and line configurations. In Table 1, there are $10$
combinatorial types of configurations which appear as these matroids.
Explicitly, these are the point and line configurations which have at most $4$
lines and at most $7$ points.

$k$ $\ell$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $\cdots$ | $\cdots$ | $\infty$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$2$ | $2$ | $2$ | $3$ | 4 | $4$ | $4$ | $4$ | $4$ | $4$ | $4$ | $4$ | $4$ | $4$
$3$ | $2$ | $2$ | $3$ | $5$ | $7$ | $8$ | 9 | $9$ | $9$ | $9$ | $9$ | $9$ | $9$
$4$ | $2$ | $2$ | $3$ | $6$ | $10$ | $13$ | $20$ | $23$ | $24$ | 25 | $25$ | $25$ | $25$
Table 1: The number of combinatorial types of configurations appearing among
dependent matroids for $\Delta^{2,3}$. Note that increasing $\ell$ leads to
having arbitrarily many irreducible components for the variety
$V_{\Delta^{2,3}}$.

######  Remark 7.2.

The minimal prime decomposition of $I_{\Delta^{s,t}}$ has been extensively
studied in [HHH+10, Rau13, CMR20, PS19]. In each case, we find a matroidal
description of the prime components. In Table 2 we give a unified perspective
on these results where the minimally dependent matroids can be uniquely
identified by their loops. In the case $k=s=2$, it is possible to generalize
the prime components of $I_{\Delta^{s,t}}$, described in [CMM21] for the
$t=3$, to all $t\geq 3$. A complete description of this can be found in our
forthcoming work. In this case the minimally dependent matroids are given by
configurations of points lying in $(t-1)$-dimensional affine subspaces. When
$s=2$, $t=3$ and $k,\ell$ are arbitrary, the dependent matroids for
$\Delta^{s,t}$ are given by configurations with at most $k$ points and $\ell$
lines. The specific case with $d=k=s=t=3$ and $\ell=4$ is given in Example
3.12. The ideal $I_{\Delta^{s,t}}$ has two components both of which are
matroid varieties corresponding to the configurations in Figure 2.

$(k,\ell,s,t)$ |  Minimally dependent matroids $M_{\mathcal{S}}$ for $\Delta^{s,t}$ |  Simplification
---|---|---
$(k,\ell,2,2)$ |  The parallel classes of $M_{\mathcal{S}}$ are the connected components of $\Delta^{s,t}\backslash\mathcal{S}$. $M_{\mathcal{S}}$ is minimally dependent if for any $\mathcal{T}\subseteq\mathcal{S}$, there exists a parallel class of $M_{\mathcal{T}}$ which contains at least two distinct parallel classes of $M_{\mathcal{S}}$. |  $M_{\mathcal{S}}$: Uniform matroid on the ground set of parallel classes of $M_{\mathcal{S}}$.
$(k,\ell,2,\ell)$ |  $\mathcal{S}=\emptyset$: $M_{\emptyset}$ is the uniform matroid on $[k\ell]$ of rank $\ell-1$.
$\mathcal{S}\neq\emptyset$: $M_{\mathcal{S}}$ is minimally dependent if $|\mathcal{S}\cap R_{i}|=1$ for each row $R_{i}$ and $\mathcal{S}\cap C_{j}\neq\emptyset$ for at least two distinct columns $C_{j}$ of $\mathcal{Y}$. The parallel classes of $M_{\mathcal{S}}$ are $C_{j}\backslash\mathcal{S}$ for each column $C_{j}$. |  $M_{\emptyset}$: Already simplified.
$M_{\mathcal{S}}$: Uniform matroid on the ground set of parallel classes of
$M_{\mathcal{S}}$.
$(2,\ell,2,3)$ |  $\mathcal{S}:$ a minimal set from [CMM21, Definition 3.15]. $M_{\mathcal{S}}$ is the matroid of a point and line configuration. The parallel classes of $M_{\mathcal{S}}$ are: $\mathcal{C}=\bigcup C_{j}$ where the union is taken over all columns $C_{j}$ of $\mathcal{Y}$ such that $C_{j}\cap\mathcal{S}=\emptyset$, and the sets $C_{j}\backslash\mathcal{S}$ where $C_{j}\cap\mathcal{S}\neq\emptyset$. The circuits of size $3$ in $M_{\mathcal{S}}$ are given by $3$-subsets of the rows $R_{1}\backslash\mathcal{S}$ and $R_{2}\backslash\mathcal{S}$ respectively. |  $M_{\mathcal{S}}$: Matroid of a point and line configuration with at most two lines intersecting at $\mathcal{C}$ (possibly empty set).
Table 2: The minimally dependent matroids for $\Delta^{s,t}$, which are
uniquely determined by their loops $\mathcal{S}\subseteq[k\ell]$.

######  Example 7.3.

Let $s=2$ and $t=3$. For each $2\leq k\leq 4$ and $3\leq\ell$ we calculate the
number of possible combinatorial types of points and line configurations which
appear among dependent matroids for $\Delta^{s,t}$. These calculations are
displayed in Table 1. Note that with $12$ points, we observe all combinatorial
types of configurations with at most $4$ lines. So, in order to determine the
components of the hypergraph variety $V_{\Delta^{s,t}}$, we need only show
that a finite number of point and line configurations have irreducible
varieties.

### 7.1 Grid matroids of small rank.

Although the minimal matroids for $\Delta^{s,t}$ are known for some specific
values of $s$ and $t$, it seems difficult to determine the minimal matroids
for arbitrary values of $s$ and $t$. However, there is no difficulty if $d$ is
sufficiently small. We first recall the notion of affine matroids.

######  Definition 7.4.

Let $\mathbb{F}$ be a field, and let $S=\\{v_{1},\ldots,v_{k}\\}$ be a
collection of (not necessarily distinct) vectors in $\mathbb{F}^{d-1}$. Let
$v_{i}^{\prime}\in\mathbb{F}^{d}$ be the vector whose first coordinate is $1$
and whose other coordinates are those of $v_{i}$.

  * •

The collection $S$ is _affinely dependent_ if $k>0$ and there are elements
$a_{1},\ldots,a_{k}\in\mathbb{F}$ that are not all $0$ with
$\sum_{i=1}^{k}a_{i}v_{i}={\bf 0}$ and $\sum_{i=1}^{k}a_{i}=0$. This is
equivalent to the condition that $\\{v_{1}^{\prime},\ldots,v_{k}^{\prime}\\}$
is linearly dependent.

  * •

A matroid $M$ on the set $[n]$ is _affine_ over the field $\mathbb{F}$ if
there is a function $\phi:[n]\rightarrow\mathbb{F}^{d-1}$ such that
$X\subseteq[n]$ is an independent set in $M$ if and only if $\phi(X)$ is
affinely independent. In this case, $M$ has rank at most $d$ and can be
realized by a $d\times n$ matrix whose columns are
$v_{1}^{\prime},\ldots,v_{n}^{\prime}$. An affine matroid must be loopless,
and it is simple if and only if $\phi$ is injective. If $F$ is a flat of $M$
of rank $t$, then there is a $(t-1)$-dimensional affine subspace of
$\mathbb{F}^{d-1}$ whose intersection with $[n]$ is $\phi(F)$.

######  Theorem 7.5.

Let $s,t,k,\ell,d$ be positive integers such that $3\leq s\leq t\leq\ell$,
$s\leq k$, and $t\leq d\leq s+t-3$. Then
$\mathcal{C}=\min(\Delta^{s,t}\cup\binom{[k\ell]}{d+1})$ is the collection of
circuits of an $\mathbb{R}$-realizable matroid on $[k\ell]$ of rank $d$. This
is the unique minimal matroid for $\Delta^{s,t}$ in this case.

###### Proof.

One can check fairly easily that $\mathcal{C}$ satisfies circuit elimination
and therefore is the collection of circuits of a matroid. However, this will
be unnecessary because we will prove this theorem by constructing a
realization for an affine matroid over $\mathbb{R}$ whose circuits are
$\mathcal{C}$. Because $\mathcal{C}$ is itself the collection of circuits of a
matroid $M$, there can be no matroid $N\neq M$ with rank at most $d$ and
ground set $[k\ell]$ such that
$\Delta^{s,t}\subseteq\mathcal{D}(N)\subseteq\mathcal{D}(M)$. Thus, $M$ is the
unique minimal matroid for $\Delta^{s,t}$ with $d\leq s+t-3$.

To construct a realization for an affine matroid over $\mathbb{R}$ whose
circuits are $\mathcal{C}$, we must define a function
$\phi:[k\ell]\rightarrow\mathbb{R}^{d-1}$. In such a matroid, each $R_{i}$
must be a flat of rank $t-1$ and each $C_{j}$ must be a flat of rank $s-1$. A
flat of rank $d-m$ in an affine matroid over $\mathbb{R}$ corresponds to an
affine subspace of $\mathbb{R}^{d-1}$ of dimension $d-m-1$. For each such
subspace, there are $m$ distinct affine hyperplanes in $\mathbb{R}^{d-1}$
whose intersection is the subspace. Therefore, each $\phi(R_{i})$ must be
defined by the intersection of a collection $\mathcal{H}_{R_{i}}$ of $d-t+1$
affine hyperplanes. These hyperplanes are defined by the following equations,
with the matrix $A_{i}=[a_{p,q}]$ having full row rank.

$\begin{array}[]{lccclcl}a_{1,1}x_{1}&+&\cdots&+&a_{1,d-1}x_{d-1}&=&c_{1}\\\
&&&&&\vdots&\\\
a_{d-t+1,1}x_{1}&+&\cdots&+&a_{d-t+1,d-t+1}x_{d-1}&=&c_{d-t+1}\\\ \end{array}$

We choose $k$ such collections of hyperplanes
$\mathcal{H}_{R_{1}},\mathcal{H}_{R_{2}},\ldots,\mathcal{H}_{R_{k}}$ such that
the rank-$(t-1)$ subspaces they define are in “general position”. That is, for
every $n<t$, the intersection of every collection of $n$ such subspaces is a
subspace of rank $t-n$. Similarly, each $\phi(C_{j})$ must be defined by the
intersection of a collection $\mathcal{H}_{C_{j}}$ of $d-s+1$ hyperplanes
defined by the following equations, with the matrix $B_{j}=[b_{p,q}]$ having
full row rank.

$\begin{array}[]{lccclcl}b_{1,1}x_{1}&+&\cdots&+&b_{1,d-1}x_{d-1}&=&c_{d-t+2}\\\
&&&&&\vdots&\\\
b_{d-s+1,1}x_{1}&+&\cdots&+&b_{d-s+1,r-1}x_{d-1}&=&c_{2d-s-t+2}\\\
\end{array}$

Choosing the subspaces defined by
$\mathcal{H}_{R_{1}},\mathcal{H}_{R_{2}},\ldots,\mathcal{H}_{R_{k}},\mathcal{H}_{C_{1}},\mathcal{H}_{C_{2}},\ldots,\mathcal{H}_{C_{\ell}}$
will ensure that every subset of $\phi([k\ell])$ that should be affinely
dependent is indeed so. However, we must also ensure that every subset of
$\phi([k\ell])$ that should be affinely independent is so. That is, for every
subset $X$ with $|X|\leq d$ such that no $t$-subset of any $R_{i}$ or
$s$-subset of any $C_{j}$ is contained in $X$, $\phi(X)$ must be affinely
independent. To do this, we will choose the collections
$\mathcal{H}_{C_{1}},\mathcal{H}_{C_{2}},\ldots,\mathcal{H}_{C_{\ell}}$
successively. Suppose we have chosen the collections
$\mathcal{H}_{C_{1}},\ldots,\mathcal{H}_{C_{j-1}}$ and now must choose
$\mathcal{H}_{C_{j}}$. If $X$ is a subset of $C_{1}\cup\dots\cup C_{j}$ such
that $\phi(X)$ is to be affinely independent, then $\phi(X-C_{j})$ is a basis
for a subspace $S$ of $\mathbb{R}^{d-1}$. We must choose $\mathcal{H}_{C_{j}}$
so that, for each $i\leq k$, the intersection of all hyperplanes in
$\mathcal{H}_{R_{i}}\cup\mathcal{H}_{C_{j}}$ is a subspace of rank
$d-(d-s+1)-(d-t+1)=d-(2d-s-t+2)=s+t-d-2$ that avoids all such subspaces $S$.

The requirement that this subspace has rank $s+t-d-2$ is equivalent to the
matrix $\left[\begin{array}[]{c}A_{i}\\\ \hline\cr B_{j}\\\
\end{array}\right]$ having full row rank. This subspace must be nonempty since
$R_{i}\cap C_{j}\neq\emptyset$. Thus, we must have $s+t-d-2\geq 1$, which is
true because $d\leq s+t-3$. The requirement of avoiding all subspaces defined
by independent sets can be achieved as there are only finitely many such
subspaces to avoid but infinitely many collections to choose to be
$\mathcal{H}_{C_{j}}$. ∎

### 7.2 General grid matroids.

Given a collection $\mathcal{D}$ of subsets of a ground set $E$, we wish to
find the matroids that are minimally dependent for $\mathcal{D}$. However, for
our purposes here, it is necessary to find the matroids minimally dependent
for $\mathcal{D}$ among the class of realizable matroids.

Some work toward these ideas has been done by Martí-Farré [MF14]. Below, we
recall the algorithmic procedure used in [MF14] to obtain minimally dependent
matroids. This algorithm is not quite completely satisfactory from our
perspective for two reasons. First, the algorithm does not take realizability
into account at all. Martí-Farré and de Mier worked with realizability in
[MFdM15], but as they said there, “The problems under consideration are far
from being solved.” Second, although the algorithm is guaranteed to give all
of the minimally dependent matroids, it may also give additional matroids that
are not minimally dependent. One must compare the matroids given by the
algorithm to determine which ones are minimally dependent.

In fact, we will use the algorithm of Martí-Farré to prove a result that
illustrates the difficulties of determining the minimal matroids for
$\Delta^{s,t}$ in general. First, we recall some terminology and notation,
much of it coming from [MF14]. A _clutter_ on a set $\Omega$ is a collection
of subsets of $\Omega$ such that no set is contained in another. (The term
_clutter_ is another word for what we have been calling a _simple
hypergraph_.) If $\Lambda$ is a clutter on $\Omega$, let
$\Lambda^{+}=\\{A\subseteq\Omega:A_{0}\subseteq A\textnormal{ for some
}A_{0}\in\Lambda\\}$. If $\Upsilon$ is any collection of subsets of $\Omega$,
then recall from Section 2 that $\min(\Upsilon)$ denotes the clutter of
inclusion-wise minimal sets in $\Upsilon$. For $B\subseteq\Omega$, define

$I_{\Lambda}(B)=\bigcap_{A\in\Lambda,A\subseteq B}A.$

It follows from the circuit elimination axiom that $\Lambda$ is the collection
of circuits of a matroid if and only if $I_{\Lambda}(A_{1}\cup
A_{2})=\emptyset$ for all $A_{1},A_{2}\in\Lambda$ with $A_{1}\neq A_{2}$. To
describe the algorithm, we use the notation of Martí-Farré and de Mier in
[MFdM17]. Let $\Lambda$ be a clutter with $A_{1},A_{2}\in\Lambda$. An
$\alpha_{1}$-transformation of $\Lambda$ is

$\alpha_{1}(\Lambda;A_{1},A_{2})=\begin{cases}\min(\Lambda\cup\\{A_{1}\cap
A_{2}\\})&\textnormal{if }I_{\Lambda}(A_{1}\cup A_{2})\neq\emptyset,\\\
\Lambda&\textnormal{otherwise.}\\\ \end{cases}$

The $\alpha_{2}$\- and $\alpha_{3}$-transformations of $\Lambda$ are

$\alpha_{2}(\Lambda)=\min(\Lambda\cup\\{(A_{1}\cup
A_{2})\backslash\\{x\\}:A_{1},A_{2}\in\Lambda,A_{1}\neq A_{2},x\in A_{1}\cap
A_{2}\\})\textnormal{ and}$ $\alpha_{3}(\Lambda)=\min(\Lambda\cup\\{(A_{1}\cup
A_{2})\backslash I_{\Lambda}(A_{1}\cup A_{2}):A_{1},A_{2}\in\Lambda,A_{1}\neq
A_{2}\\}).$

The following result is proved in [MF14, Theorem 13].

######  Theorem 7.6.

Let $\Lambda\neq\\{\emptyset\\}$ be a clutter on a finite set $\Omega$ and let
$M$ be a minimally dependent matroid for $\Lambda$ whose collection of
circuits is $\mathcal{C}(M)$. There is a sequence of clutters
$\Lambda=\Lambda_{0},\Lambda_{1},\dots,\Lambda_{r}=\mathcal{C}(M)$ such that,
for each $i\geq 1$, we have $\Lambda_{i-1}^{+}\subsetneq\Lambda_{i}^{+}$ and
such that $\Lambda_{i}$ is either an $\alpha_{1}$-, $\alpha_{2}$-, or
$\alpha_{3}$-transformation of $\Lambda_{i-1}$.

So, in order to obtain all minimally dependent matroids for a clutter, it
suffices to perform every possible combination of $\alpha_{1}$-,
$\alpha_{2}$-, and $\alpha_{3}$-transformations on the clutter. Then, one must
compare the resulting matroids and discard any matroid whose collection of
dependent sets strictly contains the collection of dependent sets of another
matroid obtained from the algorithm. Although the elements of the ground set
of $\Delta^{s,t}$ are the integers $1,2,\dots,\ell k$, it will be convenient
to think of each element as an ordered pair $(i,j)$, where $(i,j)$ is the
unique element of $R_{i}\cap C_{j}$. In the remainder of this section, we show
that every matroid can be obtained as a restriction of a matroid obtained from
some $\Delta^{s,t}$ by $\alpha_{1}$\- and $\alpha_{2}$-transformations. We
prove the following result.

######  Theorem 7.7.

Let $M$ be a matroid on ground set $[n]$. There are positive integers
$s,t,k,\ell$ and a sequence of clutters
$\Delta^{s,t}=\Lambda_{0},\Lambda_{1},\dots,\Lambda_{r}$ on the set $[k\ell]$
such that, for each positive integer $i\leq r$, $\Lambda_{i}$ is either an
$\alpha_{1}$\- or an $\alpha_{2}$-transformation of $\Lambda_{i-1}$ and
$\Lambda_{r}$ is the collection of circuits of a matroid with a restriction
isomorphic to $M$.

We will prove Theorem 7.7 after giving some definitions and proving some
lemmas below. We remark that these lemmas and Theorem 7.7, as well as their
proofs, are true if one replaces $\alpha_{2}$-transformations with
$\alpha_{3}$-transformations. We will use $\alpha_{1}$\- and
$\alpha_{2}$-transformations to obtain a matroid whose restriction to
$\\{(1,j):1\leq j\leq n\\}$ is isomorphic to $M$. That isomorphism will be
$j\rightarrow(1,j)$ for each $j\in[n]$. Thus, to simplify notation, we
identify the element $(1,j)\in R_{1}$ with the element $j\in[\ell]$.
(Similarly, we identify subsets of $R_{1}$ with subsets of $[\ell]$.)

######  Definition 7.8.

Let $t\geq 3$, and let $M$ be a matroid of rank $t-1$ on $[n]$. Let $c$ be the
number of loops of $M$. Let $c^{\prime}=\max\\{c,t-1\\}$ and
$n^{\prime}=n+c^{\prime}-c$. Let $M^{\prime}$ be the matroid on $[n^{\prime}]$
obtained from $M$ by adding $c^{\prime}-c$ loops. Let $\ell=n^{\prime}+2(t-2)$
and $M^{+}$ be the matroid on $[\ell]$ obtained from $M^{\prime}$ by freely
adding $\ell-n^{\prime}$ elements to the flat $[n^{\prime}]$.

Note that $c^{\prime}$ is the number of loops of both $M^{\prime}$ and
$M^{+}$. Also note that, since the rank of $M^{\prime}$ is $t-1$, there are at
least $t-1$ non-loop elements of $M^{\prime}$. Therefore, $n^{\prime}\geq
t-1+c^{\prime}\geq 2(t-1)$.

######  Definition 7.9.

Let $M$ be a matroid of rank $t-1$ on ground set $[n]$, where $t\geq 3$. Let
$\ell=n^{\prime}+2(t-2)$, while $s=3$ and $k=5$. For a non-negative integer
$p\leq t-1$, let $\Lambda_{p}(M)$ be the clutter on $[k\ell]$ consisting of
the inclusion-wise minimal sets among

  * (1)

$3$-subsets of $C_{j}$, where $1\leq j\leq\ell$,

  * (2)

$t$-subsets of $R_{i}$, where $1\leq i\leq k$,

  * (3)

subsets of $[n^{\prime}]\subseteq R_{1}$ of size at least $(t-p)$ that are
circuits of $M^{\prime}$,

  * (4)

sets of the form $A\cup\\{x\\}\subseteq R_{1}$, where
$x\in\\{\ell-2p+1,\ell-2p+2\\}$ and $A$ is a $(t-p-1)$-subset of $[\ell-2p]$
containing exactly one circuit of $M^{\prime}$, and

  * (5)

sets of the form $H_{i}\cup K_{j}$ where $H_{i}\subseteq R_{i}\backslash
C_{j}$ and $K_{j}\subseteq C_{j}\backslash R_{i}$, with $|H_{i}|=t-1$ and
$|K_{j}|=2$.

We define a _cross set_ to be a set of the form described in (5).

Note that $\Lambda_{0}(M)=\alpha_{2}(\Delta^{s,t})=\alpha_{3}(\Delta^{s,t})$
and the members of $\Lambda_{p}(M)$ must be inclusion-wise minimal. Hence, not
all sets listed under (1)-(5) above are necessarily members of
$\Lambda_{p}(M)$. Therefore, we need the following result.

######  Lemma 7.10.

Every subset of $R_{1}$ of size at least $t-p$ that is a circuit of
$M^{\prime}$ is a member of $\Lambda_{p}(M)$.

###### Proof.

It suffices to show that no member of $\Lambda_{p}(M)$ is properly contained
in a circuit of $M^{\prime}$. Other than circuits of $M^{\prime}$, the only
members of $\Lambda_{p}(M)$ that are contained in $R_{1}$ are $t$-subsets of
$R_{1}$ and the sets described in (4) of Definition 7.9. No circuit of
$M^{\prime}$ has size greater than $t$. Therefore, no $t$-subset of $R_{1}$ is
properly contained in a circuit of $M^{\prime}$. If $p=t-1$, then $t-p-1=0$.
Since the empty set is never a circuit of a matroid, none of the sets
described in (4) of Definition 7.9 exist. If $p\leq t-2$, then
$\ell-2p+1\geq\ell-2(t-2)+1=n^{\prime}+2(t-2)-2(t-2)+1>n^{\prime}$. Since
$[n^{\prime}]$ is the ground set of $M^{\prime}$, no set described in (4) is
contained in a circuit of $M^{\prime}$. ∎

######  Lemma 7.11.

Let $M$ be a matroid of rank $t-1$, where $t\geq 3$. Let $1\leq p\leq t-1$.
There is a sequence of clutters
$\Lambda_{p-1}(M)=\Lambda_{p,0},\Lambda_{p,1},\dots,\Lambda_{p,q}=\Lambda_{p}(M)$,
where $\Lambda_{p,i}$ is an $\alpha_{1}$-transformation of $\Lambda_{p,i-1}$.

###### Proof.

Note that $\Lambda_{p}(M)=\min(\Lambda_{p-1}(M)\cup\Gamma)$, where $\Gamma$ is
the collection of

  * •

$(t-p)$-subsets of $[n]\subseteq R_{1}$ that are circuits of $M^{\prime}$ and

  * •

sets of the form $A\cup\\{x\\}$, where $x\in\\{\ell-2p+1,\ell-2p+2\\}$ and $A$
is a $(t-p-1)$-subset of $[\ell-2p]$ containing exactly one circuit of
$M^{\prime}$.

By induction, assume
$\Lambda_{p,i-1}=\min(\Lambda_{p-1}(M)\cup\Gamma^{\prime})$, where
$\Gamma^{\prime}\subseteq\Gamma$. Let $X\in\Gamma\backslash\Gamma^{\prime}$.
We proceed by taking cases on $p$.

Case 1. Assume that $p=1$. Let $m_{1},m_{2}\in[\ell]\backslash X$. (This is
possible because $\ell\geq t+1$ if and only if $n^{\prime}\geq-t+5$. We have
$t\geq 3$, implying that $3t\geq 9$. This implies $n^{\prime}\geq
2t-2\geq-t+7$.) Let $K_{m_{1}}=\\{(2,m_{1}),(3,m_{1})\\}$ and
$K_{m_{2}}=\\{(4,m_{2}),(5,m_{2})\\}$. Note that $X\cup K_{m_{1}}$ and $X\cup
K_{m_{2}}$ are cross sets and therefore members of $\Lambda_{p,0}$. Also note
that $X\cup K_{m_{1}}$ and $X\cup K_{m_{2}}$ are the only members of
$\Lambda_{p,i-1}$ contained in $X\cup K_{m_{1}}\cup K_{m_{2}}$. Therefore,
$\alpha_{1}(\Lambda_{p,i-1};X\cup K_{m_{1}},X\cup
K_{m_{2}})=\min(\Lambda_{p,i-1}\cup\\{X\\})$. By iterating this process, we
obtain $\Lambda_{1}(M)$.

Case 2. Assume that $p>1$. By Definition 7.9, we see that
$X\cup\\{\ell-2(p-1)+1\\}\in\Lambda_{p-1}(M)$ and
$X\cup\\{\ell-2(p-1)+2\\}\in\Lambda_{p-1}(M)$. Also note that
$X\cup\\{\ell-2(p-1)+1\\}$ and $X\cup\\{\ell-2(p-1)+2\\}$ are the only members
of $\Lambda_{p,i-1}$ contained in $X\cup\\{\ell-2(p-1)+1,\ell-2(p-1)+2\\}$.
This follows from the fact that all members of $\Lambda_{p,i-1}$ are
inclusion-wise minimal (so $X\cup\\{\ell-2(p-1)+1,\ell-2(p-1)+2\\}$ is not a
member of $\Lambda_{p,i-1}$) and the fact that $t\geq 3$ (so
$\\{\ell-2(p-1)+1,\ell-2(p-1)+2\\}$ is not a member of $\Lambda_{p,i-1}$).
Therefore,
$\alpha_{1}(\Lambda_{p,i-1};X\cup\\{\ell-2(p-1)+1\\},X\cup\\{\ell-2(p-1)+2\\})=\min(\Lambda_{p,i-1}\cup\\{X\\})$.
By iterating this process, we obtain $\Lambda_{p}(M)$. ∎

###### Proof of Theorem 7.7.

First we consider the case where the rank of $M$ is at most $1$. Let $c$ be
the number of loops of $M$. Let $s=t=k=2$ and $\ell=\max\\{c+1,n\\}$. For
$i\leq c$, let
$\Lambda_{i}=\alpha_{1}(\Lambda_{i-1};\\{(1,i),(1,\ell)\\},\\{(1,i),(2,i)\\})=\Lambda_{i-1}\cup\\{(1,i)\\}$.
The result is that $\\{(1,j)\\}\in\Lambda_{c}$ for every $j\leq c$. Now, let
$\Lambda_{c+1}=\alpha_{2}(\Lambda_{c})$. This is the union of $\Lambda_{c}$
with the collection of $2$-subsets of $\\{(2,j):1\leq
j\leq\ell\\}\cup\\{(1,j):c+1\leq j\leq\ell\\}$. This is the collection of
circuits of a matroid whose restriction to $\\{(1,j):1\leq j\leq n\\}$ is
isomorphic to $M$.

Now let $M$ be a matroid on $[n]$ of rank $t-1$, where $t\geq 3$. Let $c$,
$c^{\prime}$, $n^{\prime}$, $M^{\prime}$, $\ell$, and $M^{+}$ be as in
Definition 7.8, while $s=3$ and $k=5$. As noted above,
$\alpha_{2}(\Delta^{s,t})=\Lambda_{0}(M)$. By repeated use of Lemma 7.11, we
can obtain $\Lambda_{t-1}(M)$ from $\Lambda_{0}(M)$ by a sequence of
$\alpha_{1}$-transformations.

Now, let $\Gamma$ be the clutter consisting of all $2$-subsets of columns
$C_{j}$ for $1\leq j\leq\ell$ other than the subsets containing a loop of
$M^{\prime}$. We will construct a sequence of clutters
$\Lambda_{t-1}(M)=\Lambda^{\prime}_{0},\Lambda^{\prime}_{1},\dots,\Lambda^{\prime}_{q}=\min(\Lambda_{t-1}(M)\cup\Gamma)$
such that $\Lambda^{\prime}_{i}$ is an $\alpha_{1}$-transformation of
$\Lambda^{\prime}_{i-1}$. By induction, assume
$\Lambda^{\prime}_{i-1}=\min(\Lambda_{t-1}(M)\cup\Gamma^{\prime})$, where
$\Gamma^{\prime}\subseteq\Gamma$. Let $X\in\Gamma\backslash\Gamma^{\prime}$.
Then there are positive integers $j,m_{1},m_{2}$ such that
$X=\\{(m_{1},j),(m_{2},j)\\}$. Let
$\\{m_{3},m_{4}\\}\subseteq\\{2,3,4,5\\}\backslash\\{m_{1},m_{2}\\}$. Let
$K_{m_{3}}\subseteq R_{m_{3}}\backslash C_{j}$ and $K_{m_{4}}\subseteq
R_{m_{4}}\backslash C_{j}$ with $|K_{m_{3}}|=|K_{m_{4}}|=t-1$. Moreover, we
can choose $K_{m_{3}}$ and $K_{m_{4}}$ so that no column contains an element
of both. (This is possible because $\ell\geq 2(t-1)+1$ if and only if
$n^{\prime}\geq 3$. This is true because $n^{\prime}\geq 2(t-1)\geq 4$.) Note
that the only members of $\Lambda^{\prime}_{i-1}$ contained in $X\cup
K_{m_{3}}\cup K_{m_{4}}$ are the cross sets $X\cup K_{m_{3}}$ and $X\cup
K_{m_{4}}$. Therefore, $\alpha_{1}(\Lambda^{\prime}_{i-1};X\cup
K_{m_{3}},X\cup K_{m_{4}})=\min(\Lambda^{\prime}_{i-1}\cup\\{X\\})$. By
iterating this process, we obtain $\min(\Lambda_{t-1}(M)\cup\Gamma)$. Let
$\Lambda^{\prime\prime}=\min(\Lambda_{t-1}(M)\cup\Gamma)$. By Lemma 7.10, the
circuits of $M^{\prime}$ are members of $\Lambda_{t-1}(M)$. The circuits of
$M^{+}$ that are not circuits of $M^{\prime}$ are the $t$-subsets of $R_{1}$
not containing a circuit of $M^{\prime}$. Thus, the members of
$\Lambda^{\prime\prime}$ are:

  * •

$2$-subsets of $C_{j}$, where $1\leq j\leq\ell$, other than the subsets
containing a loop of $M^{\prime}$,

  * •

subsets of $R_{1}$ that are circuits of $M^{+}$, and

  * •

$t$-subsets of $R_{i}$, where $2\leq i\leq k$.

Recall that $c$ is the number of loops of $M$ and $c^{\prime}$ is the number
of loops of $M^{\prime}$. Without loss of generality, assume that the loops of
$M^{\prime}$ are $\\{n-c+1,n-c+2,\dots,n^{\prime}\\}$.

Let $\Psi$ be the clutter consisting of singleton sets $\\{(i,j)\\}$, where
$2\leq i\leq 5$ and either $1\leq j\leq n-c$ or $n^{\prime}+1\leq j\leq\ell$.
(So $j$ is not a loop of $M^{+}$.) We will construct a sequence of clutters
$\Lambda^{\prime\prime}=\Lambda^{\prime\prime}_{0},\Lambda^{\prime\prime}_{1},\dots,\Lambda^{\prime\prime}_{z}=\min(\Lambda^{\prime\prime}\cup\Psi)$
such that $\Lambda^{\prime\prime}_{i}$ is an $\alpha_{1}$-transformation of
$\Lambda^{\prime\prime}_{i-1}$. By induction, assume
$\Lambda^{\prime\prime}_{i-1}=\min(\Lambda^{\prime\prime}\cup\Psi^{\prime})$,
where $\Psi^{\prime}\subseteq\Psi$. Let
$\\{(i,j)\\}\in\Psi\backslash\Psi^{\prime}$. Let
$A_{1}=\\{(i,j)\\}\cup\\{(i,m):n-c+1\leq m\leq n-c+t-1\\}$ and let
$A_{2}=\\{(i,j),(1,j)\\}$. Since $n^{\prime}\geq n-c+t-1$, no element of
$A_{1}\cup A_{2}$ is a singleton set in $\Psi$. Thus, $A_{1}$ and $A_{2}$ are
the only members of $\Lambda^{\prime\prime}_{i-1}$ contained in $A_{1}\cup
A_{2}$ and
$\alpha_{1}(\Lambda^{\prime\prime}_{i-1};A_{1},A_{2})=\min(\Lambda^{\prime\prime}\cup\\{(i,j)\\})$.
By iterating this process, we obtain $\min(\Lambda^{\prime\prime}\cup\Psi)$.

Let $\Lambda^{\prime\prime\prime}=\min(\Lambda^{\prime\prime}\cup\Psi)$. The
members of $\Lambda^{\prime\prime\prime}$ are

  * •

singleton sets $\\{(i,j)\\}$ where $2\leq i\leq 5$ and either $1\leq j\leq
n-c$ or $n^{\prime}+1\leq j\leq\ell$,

  * •

$2$-subsets of $C_{j}\backslash\\{(1,j)\\}$, where $n-c+1\leq j\leq
n^{\prime}$,

  * •

$t$-subsets of $\\{(i,j):n-c+1\leq j\leq n^{\prime}\\}$ for some fixed $i$
with $2\leq i\leq 5$, and

  * •

subsets of $R_{1}$ that are circuits of $M^{+}$.

Note that each member of $\Lambda^{\prime\prime\prime}$ is either contained in
$R_{1}$ or disjoint from $R_{1}$. For every clutter, a process of repeated
$\alpha_{2}$-transformations on the clutter must eventually terminate in a
clutter that is the collection of circuits of a matroid. Therefore, repeated
$\alpha_{2}$-transformations on $\Lambda^{\prime\prime\prime}$ will result in
a clutter that is the collection of circuits of a matroid $N$ whose circuits
are either contained in $R_{1}$ or disjoint from $R_{1}$. Since the members of
$\Lambda^{\prime\prime\prime}$ contained in $R_{1}$ are the circuits of
$M^{+}$, circuit elimination implies that the circuits of $N$ contained in
$R_{1}$ are precisely the circuits of $M^{+}$. In particular, the restriction
of $N$ to $\\{(1,j):1\leq j\leq n\\}$ is isomorphic to $M$. ∎

Theorem 7.7 applies to all matroids, regardless of rank. However, in the case
where the matroid is simple and has rank at most $3$, the result can be
illustrated more clearly using a different procedure from the one used in the
proof of the theorem. We use point and line configurations. Consider the Fano
matroid whose point and line configuration is the Fano plane given in Figure
3. It is well-known that the Fano matroid is not realizable over any field of
characteristic other than $2$; therefore, it is not $\mathbb{C}$-realizable.
In the case where $n=7$ and

$\mathcal{D}=\left\\{L_{1}=\\{1,2,4\\},L_{2}=\\{1,3,6\\},L_{3}=\\{1,5,7\\},L_{4}=\\{2,3,5\\},L_{5}=\\{2,6,7\\},L_{6}=\\{3,4,7\\},L_{7}=\\{4,5,6\\}\right\\}\cup\binom{[n]}{4},$

the only minimally dependent matroid with respect to $\min(\mathcal{D})$ is
the Fano matroid, which is not relevant for our purposes because it is not
$\mathbb{C}$-realizable. However, this can arise as the simplification of a
matroid coming from a point and line configuration with $s=2$ and $t=3$.

Indeed, consider the case $k=\ell=7$. Let $(i,j)$ be the unique element
contained in $R_{i}\cap C_{j}$. Iteratively performing
$\alpha_{1}$-transformations (defined earlier in this subsection), we can
obtain from the clutter $\Delta^{2,3}$ a clutter that contains all of the
singletons in $R_{i}$ except for the elements $(i,j)$ such that $j\in L_{i}$.
In particular, the members of this clutter are the following:

  * •

all singletons except for the elements $(i,j)$ such that $j\in L_{i}$,

  * •

$3$-subsets of the form $\\{(i,j)\in R_{i}:j\in L_{i}\\}$, and

  * •

$2$-subsets of $C_{j}$ of the form $\\{(i_{i},j),(i_{2},j)\\}$, where $j\in
L_{i_{1}}\cap L_{i_{2}}$.

Then, by performing two transformations (both $\alpha_{2}$-transformations,
both $\alpha_{3}$-transformations, or one of each), we obtain a clutter that
is the collection of circuits of a matroid whose simplification is the Fano
matroid and whose rank-$1$ flats each have three non-loop elements. The second
transformation is needed to include the Fano matroid’s circuits of size $4$.
(This works because, for each $4$-element circuit $C$ of the Fano matroid,
there is a fifth element $x$ such that $C\cup\\{x\\}$ is the union of two
$3$-element circuits whose intersection is $x$.)

##  8 Conditional independence models

Conditional independence (CI) models play an important role in algebraic
statistics [Stu05]. Given a collection of random variables and knowledge of
the conditional dependencies, or independencies, among them, we can ask what
are the distributions that satisfy them. In a more general setting, some of
the random variables appearing in a CI model can be prescribed as unobserved
(or hidden). Our goal is to determine when certain constraints on the observed
variables arise from conditions on the hidden variables [SA15]. This problem
can be restated algebraically by noting that probability distributions
satisfying CI statements are the solutions of certain polynomial equations
[DSS09, Sul18] which generate the so-called CI ideal. The distributions
satisfying a given collection of CI statements can be recovered by
intersecting the CI ideal with the probability simplex. When there are no
hidden variables, these polynomials are binomials and their associated ideals
are well-studied; see e.g. [Fin11, HHH+10, Rau13, ST13]. However, in the
presence of hidden variables, the polynomials become far more complicated of
arbitrarily high degrees and very difficult to calculate; see e.g. [PS19,
CMR20].

Let $X,Y_{1},Y_{2}$ be observed and $H_{1},H_{2}$ be hidden random variables
taking values in the finite sets
$\mathcal{X},\mathcal{Y}_{1},\mathcal{Y}_{2},\mathcal{H}_{1},\mathcal{H}_{2}$
of cardinalities $|\mathcal{X}|=d$, $|\mathcal{Y}_{1}|=k$,
$|\mathcal{Y}_{2}|=\ell$, $|\mathcal{H}_{1}|=s-1$, $|\mathcal{H}_{2}|=t-1$.
Consider the CI model given by:

$\displaystyle\mathcal{C}:\ \
{X}\perp\\!\\!\\!\perp{Y_{1}}\mid\\{Y_{2},H_{1}\\}\quad\textrm{and}\quad{X}\perp\\!\\!\\!\perp{Y_{2}}\mid\\{Y_{1},H_{2}\\}.$
(4)

Then the CI ideal associated to $\mathcal{C}$ is precisely the hypergraph
ideal $I_{\Delta^{s,t}}$ in §7; see [CMM21, CMR20].

######  Example 8.1.

Let $d=k=s=t=3$ and $\ell=4$. The joint distribution of $Y_{1}$ and $Y_{2}$
has state space $\mathcal{Y}=\mathcal{Y}_{1}\times\mathcal{Y}_{2}$ which is
identified with the $3\times 4$ matrix $\mathcal{Y}$ with values in the set
$[12]$ as in (3). In this case, the CI ideal
$I_{\Delta}\subseteq\mathbb{C}[P]:=\mathbb{C}[p_{x,y}:x\in\mathcal{X},y\in\mathcal{Y}]$
is the hypergraph ideal from Example 3.12. The ideal $I_{\Delta}$ has two
prime components. One component, associated to the matroid $M_{0}$ from
Example 3.12, is generated by all $3$-minors of the matrix of variables $P$.
This is the ideal associated to the CI statements
${X}\perp\\!\\!\\!\perp{\\{Y_{1},Y_{2}\\}}\mid H_{1}$ and
${X}\perp\\!\\!\\!\perp{\\{Y_{1},Y_{2}\\}}\mid H_{2}$. Note that $H_{1}$ and
$H_{2}$ are hidden random variables taking the same number of values, and so
the ideals do not distinguish them. The other prime component $I_{M}$ is the
ideal of the configuration described in Example 3.12 consisting of $12$ points
and $7$ lines. Besides the rank constraints given by the original CI model,
the ideal $I_{M}$ contains geometric constraints which are satisfied by all
distributions which do not lie in $V_{M_{0}}$.

We note that for large values of $d$, understanding the algebraic properties
of the ideal $I_{\Delta^{s,t}}$ and its (primary) decomposition is hard. In
particular, Theorem 7.7 shows that, up to simplification, any matroid can
appear among the dependent matroids for $\Delta^{s,t}$. And, in the example
above, some high-degree polynomials may appear in the generating sets of the
primary components of $I_{\Delta^{s,t}}$. However, for $d\leq s+t-3$ we have
shown in Theorem 7.5 that $\Delta^{s,t}$ has a unique minimally dependent
matroid $M$. So, for a generic point $p$ in the variety $V_{\Delta^{s,t}}$,
the additional polynomial constraints on $p$ arise from geometric constraints
on realizations of $M$.

######  Remark 8.2.

Suppose that $H_{1}$ and $H_{2}$ are both constant. The intersection axiom
states that any distribution satisfying $\mathcal{C}$ in (4) generically
satisfies ${X}\perp\\!\\!\\!\perp{\\{Y_{1},Y_{2}\\}}$. Here, genericity means
that the distributions have non-zero probabilities. The intersection axiom has
been studied for various CI models; see e.g. [HHH+10, Rau13, CMR20, CMM21,
PS19]. We note that our family of hypergraph varieties include all these cases
as examples. In particular, the corresponding ideal $I_{\Delta^{s,t}}$ has a
distinguished prime component with a particular statistical significance,
since the distributions which lie inside do not contain any structural zeros
[Stu02]. Importantly, if we assume, without loss of generality, that $s\leq t$
then this prime ideal can be realized as the CI ideal of
${X}\perp\\!\\!\\!\perp{\\{Y_{1},Y_{2}\\}}\mid H_{2}$. We may therefore deduce
a hidden variable version of the intersection axiom as follows:

$\mathcal{C}=\\{{X}\perp\\!\\!\\!\perp{Y_{1}}\mid\\{H_{1},Y_{2}\\},\
{X}\perp\\!\\!\\!\perp{Y_{2}}\mid\\{Y_{1},H_{2}\\}\\}\implies{X}\perp\\!\\!\\!\perp{\\{Y_{1},Y_{2}\\}\mid
H_{2}}.$

Finally, we give connections of our work to an interesting conjecture by
Matúš; see [Mat99].

######  Conjecture 8.3 ([Mat99]).

For any discrete conditional independence model $\mathcal{C}$, there exists a
distribution $p\in V(J_{\mathcal{C}})$ such that all joint probabilities of
$p$ are rational.

In Theorem 7.7, we have seen that for large enough $s,t,k,\ell$, any matroid
may appear among the dependent matroids for $\Delta^{s,t}$. A natural approach
is to carefully choose additional conditional independence and dependence
statements for the model (4), in order to guarantee that any distribution $p$
satisfying $\mathcal{C}$ is a realization of a given, realizable, matroid.
Note that there exist matroids that are not realizable over the rationals but
are realizable over a real field extension. Hence, this might lead to a
characterization of CI models with hidden variables for which Conjecture 8.3
does not hold.

Acknowledgement. We would like to thank Leonid Monin for helpful
conversations. We would like to thank the anonymous referees for their
insightful comments. Clarke, Mohammadi and Motwani were supported by the
grants G0F5921N (Odysseus programme) and G023721N from the Research Foundation
- Flanders (FWO), and the UGent BOF grant STA/201909/038. Clarke was partially
supported as an overseas researcher under Postdoctoral Fellowship of Japan
Society for the Promotion of Science (JSPS).

## References

  * [BC03] Winfried Bruns and Aldo Conca.  “Gröbner bases and determinantal ideals.”  Commutative Algebra, Singularities and Computer Algebra 179 (2003): 9–66.
  * [BS89] Jürgen Bokowski and Bernd Sturmfels.  “Computational synthetic geometry.”  Lecture Notes in Math. 1355 (1989).
  * [CLO15] David A. Cox, John Little, and Donal O’Shea.  “Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (4th edition).”  Undergrad. Texts Math. (2015).
  * [CMM21] Oliver Clarke, Fatemeh Mohammadi, and Harshit J Motwani.  “Conditional probabilities via line arrangements and point configurations.”  Linear Multilinear Algebra (2021): 1–33.
  * [CMR20] Oliver Clarke, Fatemeh Mohammadi, and Johannes Rauh.  “Conditional independence ideals with hidden variables.”  Adv. in Appl. Math. 117 (2020): 102029.
  * [Cra65] Henry H. Crapo.  “Single-element extensions of matroids.”  Journal of Research of the National Bureau of Standards, Section B, 69B (1965): 55–65.
  * [DSS09] Mathias Drton, Bernd Sturmfels, and Seth Sullivant.  “Lectures on Algebraic Statistics.”  Oberwolfach Semin. 39 (2009).
  * [EHHM13] Viviana Ene, Jürgen Herzog, Takayuki Hibi, and Fatemeh Mohammadi.  “Determinantal facet ideals.”  Michigan Math. J. 62 (2013): 39–57.
  * [Fin11] Alex Fink.  “The binomial ideal of the intersection axiom for conditional probabilities.”  J. Algebraic Combin. 33 (2011): 455–463.
  * [GGMS87] Israel M. Gelfand, R. Mark Goresky, Robert D. MacPherson, and Vera V. Serganova.  “Combinatorial geometries, convex polyhedra, and schubert cells.”  Adv. Math. 63 no. 3 (1987): 301–316.
  * [GS] Daniel R. Grayson and Michael E. Stillman.  “Macaulay2.” A software system for research in algebraic geometry.  Available at https://faculty.math.illinois.edu/Macaulay2/.
  * [Har13] Robin Hartshorne.  “Algebraic geometry.”  Grad. Texts in Math. 52 (2013).
  * [HHH+10] Jürgen Herzog, Takayuki Hibi, Freyja Hreinsdóttir, Thomas Kahle, and Johannes Rauh.  “Binomial edge ideals and conditional independence statements.”  Adv. in Appl. Math. 45 no. 3 (2010): 317–333.
  * [HS04] Serkan Hoşten and Seth Sullivant.  “Ideals of adjacent minors.”  J. Algebra, 277 no. 2 (2004): 615–642.
  * [KLS13] Allen Knutson, Thomas Lam and David E. Speyer.  “Positroid varieties: juggling and geometry.”  Compos. Math 149 no. 10 (2013): 1710–1752.
  * [LV13] Seok Hyeong Lee and Ravi Vakil.  “Mnëv-Sturmfels universality for schemes.”  Clay Mathematics Proceedings 18, A celebration of algebraic geometry, (2013): 457–468.
  * [Mat99] František Matúš.  “Conditional independences among four random variables III: Final conclusion.”  Combin. Probab. Comput. 8 no. 3 (1999): 269–276.
  * [MF14] Jaume Martí-Farré.  “From clutters to matroids.”  Electron. J. Combin., 21 no. 1 (2014): P1.11.
  * [MFdM15] Jaume Martí-Farré and Anna de Mier.  “Completion and decomposition of a clutter into representable matroids.”  Linear Algebra Appl. 472 (2015): 31–47.
  * [MFdM17] Jaume Martí-Farré and Anna de Mier.  “Transformation and decomposition of clutters into matroids.”  Adv. Math. 312 (2017): 286–314.
  * [Mnë85] Nikolai E. Mnëv.  “On manifolds of combinatorial types of projective configurations and convex polyhedra.”  Soviet Math. Doklady 32 (1985): 335–337.
  * [Mnë88] Nikolai E. Mnëv.  “The universality theorems on the classification problem of configuration varieties and convex polytopes varieties.”  Lecture Notes in Math. 1346 Topology and geometry—Rohlin seminar (1988): 527–543.
  * [MR18] Fatemeh Mohammadi and Johannes Rauh.  “Prime splittings of determinantal ideals.”  Comm. Algebra 46 no. 5 (2018): 2278–2296.
  * [Oh09] Suho Oh.  “Combinatorics of Positroids.”  Discrete Math. Theor. Comput. Sci. DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (2009).
  * [Oxl11] James Oxley.  “Matroid Theory.”  Second edition. Oxf. Grad. Texts Math (2011).
  * [PS19] Gerhard Pfister and Andreas Steenpass.  “On the primary decomposition of some determinantal hyperedge ideal.”  J. Symbolic Comput. 103 (2019): 14–21.
  * [PW70] Mike J. Piff and Dominic J.A. Welsh.  “On the vector representation of matroids.”  J. Lond. Math. Soc. (2) 2 (1970): 284–288.
  * [Rau13] Johannes Rauh.  “Generalized binomial edge ideals.”  Adv. in Appl. Math. 50 no. 3 (2013): 409–414.
  * [RG99] Jürgen Richter-Gebert.  “The universality theorems for oriented matroids and polytopes.  Contemp. Math. 223 (1999): 269–292.
  * [RG11] Jürgen Richter-Gebert.  “Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry.”  Springer Berlin Heidelberg, 2011.
  * [Ryb11] Grigori L. Rybnikov.  “On the fundamental group of the complement of a complex hyperplane arrangement.”  Funct. Anal. Appl. 45 no. 2 (2011): 137–148.
  * [SA15] Bastian Steudel and Nihat Ay.  “Information-theoretic inference of common ancestors.”  Entropy 17 no. 4 (2015): 2304–2327.
  * [SJS17] Meera Sitharam, Audrey St. John, and Jessica Sidman.  “Handbook of Geometric Constraint Systems Principles (1st edition).”  Chapman & Hall/CRC Math. (2017).
  * [ST13] Irena Swanson and Amelia Taylor.  “Minimal primes of ideals arising from conditional independence statements.”  J. Algebra 392 (2013): 299–314.
  * [Stu89] Bernd Sturmfels.  “On the matroid stratification of Grassmann varieties, specialization of coordinates, and a problem of N. White.”  Adv. Math. 75 no. 2 (1989): 202–211.
  * [Stu90] Bernd Sturmfels.  “Gröbner bases and Stanley decompositions of determinantal rings.”  Math. Z. 205 no. 1 (1990): 137–144.
  * [Stu02] Bernd Sturmfels.  “Solving systems of polynomial equations.”  CBMS Reg. Conf. Ser. Math. 97 (2002).
  * [Stu05] Milan Studený.  “Probabilistic conditional independence structures.”  Information Science and Statistics Springer, London, (2005).
  * [Stu08] Bernd Sturmfels.  “Algorithms in invariant theory.”  Texts Monogr. Symbol. Comput. (2008).
  * [STW21] Jessica Sidman, Will Traves, and Ashley Wheeler.  “Geometric equations for matroid varieties.”  J. Combin. Theory Ser. A 178 (2021): 105360.
  * [Sul18] Seth Sullivant.  “Algebraic Statistics.”  Grad. Stud. Math. (2018).

Authors’ addresses:

Department of Mathematics, University of Bristol, Bristol, UK
E-mail address<EMAIL_ADDRESS>
School of Mathematics, University of Bristol, Bristol, UK
Heilbronn Institute for Mathematical Research, Bristol, UK
Current address: Department of Mathematics, Vanderbilt University, Nashville,
Tennessee, USA
E-mail address<EMAIL_ADDRESS>
Department of Computer Science, KU Leuven, Celestijnenlaan 200A, B-3001
Leuven, Belgium
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven,
Belgium
Department of Mathematics and Statistics, UiT – The Arctic University of
Norway, 9037 Tromsø, Norway
E-mail address<EMAIL_ADDRESS>
Department of Mathematics, Ghent University, Gent, Belgium
New address: Departments of Computer Science and Mathematics, KU Leuven,
Celestijnenlaan 200A, B-3001 Leuven, Belgium
E-mail address<EMAIL_ADDRESS>