# Conditional not displacement: fast multi-oscillator control with a single
qubit

Asaf A. Diringer  Eliya Blumenthal1 Avishay Grinberg1 Liang Jiang2 Shay
Hacohen-Gourgy1 1Department of Physics, Technion - Israel Institute of
Technology, Haifa 32000, Israel  2Pritzker School of Molecular Engineering,
University of Chicago, Chicago, Illinois 60637, USA

###### Abstract

Bosonic encoding is an approach for quantum information processing, promising
lower hardware overhead by encoding in the many levels of a harmonic
oscillator. Scaling to multiple modes requires them to be decoupled for
independent control, yet strongly coupled for fast interaction. How to perform
fast and efficient universal control on multiple modes remains an open
problem. We develop a control method that enables fast multi-mode generation
and control of bosonic qubits which are weakly coupled to a single ancilla
qubit. The weak coupling allows for excellent independent control, despite the
weak ancilla coupling our method allows for fast control. We demonstrate our
control by using a superconducting transmon qubit coupled to a multi-mode
superconducting cavity. We create both entangled and separate cat-states in
different modes of a multi-mode cavity, showing the individual and coupled
control of the modes. We show that the operation time is not limited by the
inverse of the dispersive coupling rate, which is the typical timescale, and
we exceed it in practice by almost 2 orders of magnitude. Our scheme allows
for multi-mode bosonic codes as well as more efficient scaling of hardware.

Bosonic codes are an approach in which a quantum information is redundantly
encoded in the many levels of a quantum harmonic oscillator. They are an
alternative to the conventional approach of multiple two-level systems,
allowing for quantum error correction already at the lowest hardware level [1,
2, 3]. Concatenating these bosonic qubits into a larger error-correcting code
may provide a path to robust quantum computation with reduced hardware
overhead [4, 5, 6]. Significant progress has been made in two leading
platforms, circuit QED and Trapped Ions, where small error correction
protocols for stabilizing a bosonic qubit have been demonstrated. [7, 8, 9].
Photonic systems are also finding new and improved schemes for generating such
states [10, 11, 12].

Achieving universal control of multiple bosonic qubits in a fast and efficient
manner remains a gap to be filled. Universal control means any state can be
prepared and controlled. The question is what are the best methods to achieve
this goal, in terms of speed and efficiency. To achieve any control over a
harmonic oscillator it must be coupled to an ancilla. Universal control can be
obtained by a combination of unconditional and conditional operations
(operations conditioned on the coupled system). Typically the conditional
operations are slow as they need to be highly frequency selective, and their
durations are limited by the inverse of the interaction rate.

Figure 1: Conditional NOt Displacement (CNOD) multi-mode control (a)
Illustration of a single qubit ancilla coupled to multiple Bosonic modes. (b)
Top \- Schematic illustration of the Fourier decomposition of the anti-
symmetric pulse, these pulses are used to compose the CNOD. The anti-symmetric
pulse is a sum of two Gaussians (green), which have the same amplitude but
opposite phases, and their center frequencies are equally detuned from either
side of $\omega_{c}^{(e)}\equiv\omega_{c}-\chi$ (frequency of the EM mode when
the ancilla is excited). The Fourier transform of the sum of the two Gaussians
(qualitatively represented by the black line) is anti-symmetric with respect
to $\omega_{c}^{(e)}$ and has a node at that frequency. The result is a no-
displacement conditioned on the excited state of the ancilla. Bottom \- Two
simulated trajectories of the EM mode phase space show the dynamics of the
pulse. The initial state is cavity vacuum, where the ancilla is in the exited-
(red) and ground- (blue) state with the time evolution portrayed by the color
gradient. The dynamics show that initially, as the anti-symmetric pulse is
applied, the oscillator is largely displaced independently of the state of the
ancilla. The node present at the oscillator frequency corresponding to the
excited state of the ancilla leads to a fully destructive interference. When
the ancilla is at the ground state the interference is not fully destructive
and a finite displacement occurs. (c) Joint characteristic tomography of Bell-
cats (left) compared with ideal state (right). 2D cuts of the 4D joint
characteristic functions $[\mathcal{C}_{J}(\gamma_{A},\gamma_{B})]$ of the
Bell-cats $\left|\psi_{+}\right\rangle$. The cuts are along the planes,
Re$(\gamma_{A})$-Re$(\gamma_{B})$ and Im$(\gamma_{A})$-Im$(\gamma_{B})$, for
$\alpha_{A}\simeq\alpha_{B}\simeq 1.7$).

For example, in circuit QED the interaction is often obtained by dispersively
coupling an electromagnetic mode (EM) to an ancillary two-level system with a
coupling rate $\chi$. $\chi$ is the cross-Kerr coupling strength, and is the
frequency shift of the EM mode associated with a transition of the ancilla. In
this regime, universal control is implemented by various methods, such as the
Selective Number-dependent Arbitrary Phase (SNAP) gate, which relies on
ancilla rotations conditioned on the number of photons in the EM mode [13] or
model-based pulse shaping such as GRadient Ascent Pulse Engineering (GRAPE)
[14]. Such methods intrinsically rely on being selective on specific
frequencies to apply the conditional operations. This results in typical
operation times comparable to $1/\chi$. Hence, for speed, we desire an
interaction rate $\chi$, that is much faster than the decay rate. On the other
hand large $\chi$ leads to unwanted interactions due to higher order terms,
such as self-Kerr and Purcell decay. Furthermore for few-mode control,
frequency crowding becomes a serious issue [15, 16].

Fortunately, we can perform quantum operations much faster than $1/\chi$,
which is not a fundamental limit. By combining a fast conditional displacement
(ECD gate) with the intrinsically fast unconditional rotation of the ancilla,
universal control of a single EM mode can be much faster than $1/\chi$ [17].

Few-mode entanglement has been shown, either in the relatively large $\chi$
regime or at a slow rate [18, 19, 20, 16]. A fast and efficient method for
few-mode control that can overcome the above problems can be significant in
lowering hardware and control overhead.

Here we show how to apply universal few-mode control with a single ancilla in
the small $\chi$ regime. We develop a control approach which we refer to as
Conditional NOt Displacement (CNOD). We utilize an ancilla superconducting
qubit coupled to a multi-mode superconducting microwave cavity, and utilize it
to create both entangled cat-states and separate cat-states in two different
modes in the cavity. We show that the CNOD operation speed is not limited by
$1/\chi$, and we exceed it in practice by almost 2 orders of magnitude. We
also show how the CNOD enables us to do both local and non-local measurements,
meaning both separate and joint properties of these modes.

In typical tomography schemes the expectation values of non-local operators
are obtained by measurements of separate ancillas for each mode. The results
are compared for correlations [18, 19, 20], hence multiple ancilla and high-
fidelity single-shot ancilla measurements are required. The multi-mode CNOD
simplifies tomography. Using the CNOD we map non-local properties onto the
ancilla. Since correlations in the direct measurement of the non-local
operators are preserved during averaging, both requirements are removed in our
method. Finally we give a proof that universality can be generalized to a
multi-mode system.

The CNOD concept is general and can be understood as follows. For conditional
operations, rather than applying a narrow-band pulse at a selected frequency
to conditionally displace a mode, we apply a broad-band pulse, and do not
displace at selected frequencies. The _no displacement_ is achieved by
designing a zero amplitude at the selected frequencies in the Fourier spectrum
of the control pulse. This approach alleviates the requirement for the narrow
bandwidth of the pulse, allowing for faster operation.

We now construct the specific conditional displacement for our circuit QED
system using the CNOD approach. We consider a two-level system of frequency
$\omega_{q}$ dispersively coupled to an EM mode of frequency $\omega_{c}$.
Working in the frame rotating with the ground state of the ancilla and the EM
mode, such a system is described by a dispersive Hamiltonian:

$H_{s}/\hbar=-\chi\left|e\right\rangle\left\langle e\right|a^{{}^{\dagger}}a,$
(1)

where $\chi$ is the coupling strength, $\left|e\right\rangle$ is the ancilla
excited state, and $a$ is the EM mode lowering operator. We start with a pulse
whose Fourier decomposition is anti-symmetric with respect to the frequency of
the EM mode when the ancilla is excited ($\omega_{c}^{(e)}$) and thus has a
zero at that frequency, as shown in Fig.1b. This anti-symmetric pulse will
generate a cavity displacement only if the ancilla is in the ground state, and
not in the excited state.

The anti-symmetric pulse time $\tau$ is small but finite, 60$\sim$290ns in our
experiment. During the operation an additional conditional rotation of the
phase space of the EM mode by $-\chi\tau$ occurs due to Hamiltonian evolution.
We cancel the state-dependent rotation of the EM mode using an unconditional
$\pi$ rotation on the ancilla. We also apply a digital rotation of the frame
of the EM mode by $-\chi\tau$ followed by a second anti-symmetric pulse,
identical to the first up to a $\pi$ phase. All together, up to a local phase,
this procedure results in a displacement of the cavity mode conditioned on the
ancilla state,

$\mathrm{CNOD}(\alpha)\equiv\left|e\right\rangle\left\langle
g\right|\mathcal{D}(\alpha/2)-\left|g\right\rangle\left\langle
e\right|\mathcal{D}(-\alpha/2)$ (2)

where $\mathcal{D}(\alpha)=e^{\alpha a^{\dagger}-\alpha^{*}a}$ is the
displacement operator, and in our case here each anti-symmetric pulse creates
a displacement of $\alpha/2$.

Figure 2: Device illustration and protocol for generating cat-states (a)
Schematic of the 3D aluminum flute cavity with a single transmon qubit ancilla
inside. The cavity is made of a single slab of high-purity aluminum. Holes
(white circles) are drilled from opposite sides, creating a seamless cavity in
the center of the aluminum block (blue shaded columns). The cavity has a
single pin-antenna (golden connector), which serves as a port for both input
and output. The pin-antenna is located at the nodes of the memory modes
TE102-Alice (red) and TE202-Bob (green), and at the anti-node of the readout
mode TE101 (not shown in illustration). The pin-antenna is used for control of
all elements as well as for readout. (b) Microwave control sequence for
generating the different cat-states and tomographically reconstructing the
characteristic functions, with parameters depending on the target state. The
first ancilla rotation $R_{\hat{n}}(\theta)$ determines phase of the cat-
state. The first CNODs generate cats entangled to the ancilla, and the
following CNODs with $\tilde{\beta_{i}}$’s set to satisfy
$\sum_{i=1,2}\tilde{\alpha_{i}}\tilde{\beta_{i}}=\pi/2$ disentangle the
ancilla from the EM modes. In the final rotation in the tomography: 1) the
choice of $\hat{y}$ or $\hat{x}$ determines if we measure the real or
imaginary part of the characteristic function respectively, 2) the $\pm$ are
interleaved to remove biasing error in the ancilla measurement. Generation
times for the Alice-, Bob-, and Bell-cats were 476ns, 620ns, and 1312ns
respectively. These include the qubit rotations $\pi$-pulses and
$\pi/2$-pulses which were 22ns and 16ns respectively. These generation times
give a speed up of 14, 75, and 36 respectively, as compared with $3\pi/\chi$.
$3\pi/\chi$ is the typical time in the standard method and also does not
include the qubit rotation times [18].

The CNOD method generalizes to a setting with a single ancilla coupled to
multiple modes. Since we are in the small $\chi$ regime the cross-Kerr between
modes is negligible, and the CNOD is achieved by applying the anti-symmetric
pulse to more modes.

Our experimental setup uses a circuit QED architecture. A transmon type
superconducting ancilla is introduced into to a 3D flute style superconducting
cavity [21], as illustrated in Fig.2a. We use the TE102-Alice and TE202-Bob as
memory modes for encoding the cat states, and the TE101 for dispersively
reading out the transmon ancilla state. In our design a single pin-antenna
sets the lifetime ($T_{1}$) of our modes (see Table 1). The pin is located at
the nodes of the Alice and Bob memory modes and at the anti-node of the TE101
readout mode. This placement strongly suppresses the coupling of the memory
modes to the transmission line while retaining a strong coupling to the
readout mode for faster ancilla readout. We achieve almost 2 orders of
magnitude difference in the coupling strength, which allows for a simplified
setup with a single port, while still providing sufficiently long memory mode
lifetime for our experiment [22].

Figure 3: Single-mode characteristic function tomography Measured real part of the single-mode characteristic functions of Alice $[\mathcal{C}_{A}(\gamma_{A})]$ (top row) and Bob $[\mathcal{C}_{B}(\gamma_{B})]$ (middle row), and calculated for an ideal state $[\mathcal{C}_{thy}(\gamma_{thy})]$ (bottom row). These are shown for the different cat-states labeled on top. Single-cat and product-cats are expected to be identical in an ideal system where Alice and Bob have completely independent control.  Mode | $\omega/2\pi$ | Mode | $T_{1}$ |  $T_{2}$(Echo) | $\chi/2\pi$
---|---|---|---|---|---
Alice | 6.56 GHz | TE102 | 60-62 $\mu s$ | - | 216-224 kHz
Bob | 8.02 GHz | TE202 | 36-38 $\mu s$ | - | 31-36 kHz
Ancilla | 5.37 GHz | - | 12-15 $\mu s$ | 12-14 $\mu s$ | -
Readout | 4.02 GHz | TE101 |  $\sim$ 600 ns | - |  $\sim$ 80 kHz
Table 1: Parameters and coherence times of the EM modes and the transmon
ancilla.

To second order our system can be described by the following Hamiltonian [23],

$\begin{split}H/\hbar=&\
\omega_{A}a^{{}^{\dagger}}a+\omega_{B}b^{{}^{\dagger}}b+\omega_{q}\left|e\right\rangle\left\langle
e\right|\\\
-&\left(\chi_{A}a^{{}^{\dagger}}a+\chi_{B}b^{{}^{\dagger}}b\right)\left|e\right\rangle\left\langle
e\right|,\end{split}$ (3)

where $\omega_{A}$, $a^{\dagger}$ and $\omega_{B}$, $b^{\dagger}$ are the
Alice and Bob mode frequencies and raising operators respectively,
$\omega_{q}$ is the transmon transition frequency, and $\chi_{A}$ and
$\chi_{B}$ are the dispersive shifts (cross-Kerr) of each mode with the
transmon ancilla. Higher order Kerr coefficients are neglected. To generate
the cat states we use CNODs and unconditional operations. We carry out the
unconditional ancilla rotations and cavity state displacements in a standard
way by applying microwave pulses with a Gaussian envelope.

For tomography we measure the complex characteristic function
Tr$[\mathcal{D}(\alpha)\rho]$ [24]. We utilize the CNOD to apply
${D}(\alpha)$. For each state, the single-mode characteristic functions of
both modes are measured independently. To quantify the entanglement between
the modes we perform a two-mode joint characteristic tomography by measuring
the function,

$\mathcal{C}_{J}=\mathrm{Tr}[\rho\mathcal{D}_{\gamma_{A}}\mathcal{D}_{\gamma_{B}}],$
(4)

where $\mathcal{C}_{J}$ is the joint charactristic function [25] in the 4D
phase space. We measure $\mathcal{C}_{J}$ by using CNODs on the two modes and
then measuring the ancilla to obtain the real and imaginary parts using the
sequence shown in Fig.2b. Both real and imaginary parts of the characteristic
function are analyzed and used to reconstruct the density matrices through
maximum likelihood estimation (MLE). This procedure allows us to estimate the
fidelities of our states. See supplementary information for exact details,
calibrations and procedures.

We start by generating independent cats. We create cat-states in each mode
while keeping the other mode in vacuum, which we refer to as single-cat. We
compare these to a two-mode product state, where a cat-state resides in each
mode, which we refer to as product-cats. In the ideal setting we expect them
to be identical.

Our constructed CNODs implement conditional displacements on each mode
independent of other one. The $\pi$-pulse in the middle of the CNOD gate
creates an echoing effect. This ensures that while we manipulate one of the
modes the other mode does not evolve up to an overall rotation, which we
negate by a digital rotation. We apply the sequences shown in Fig. 2b.
Specifically, we generate the single-cat states in either Alice
$\left|\psi_{\mathrm{s}_{\mathrm{A}}}\right\rangle=\mathcal{N}(\left|\alpha\right\rangle_{A}+\left|-\alpha\right\rangle_{A})\otimes\left|0\right\rangle_{B}$
or Bob
$\left|\psi_{\mathrm{s}_{\mathrm{B}}}\right\rangle=\mathcal{N}\left|0\right\rangle_{A}\otimes(\left|\alpha\right\rangle_{B}+\left|-\alpha\right\rangle_{B})$
separately, where $\mathcal{N}$ is a normalization factor. For this we apply
the CNODs on the relevant mode only. To generate the product-cats state
$\left|\psi_{\mathrm{p}}\right\rangle=\mathcal{N}(\left|\alpha\right\rangle_{A}+\left|-\alpha\right\rangle_{A})\otimes(\left|\alpha\right\rangle_{B}+\left|-\alpha\right\rangle_{B})$
we apply the same operations sequentially, first on Bob and then on Alice.

The real parts of the characteristic functions are displayed in Fig.3, the
imaginary part is not shown as it is very close to zero everywhere (example is
shown in the supplementary materials). The results for the single-cat states
and the corresponding parts of the product-cats state look almost identical,
as we would expect (and desire) for individual mode control. This is in
contrast to previous work where the inherited cross Kerr is substantial (large
$\chi$s) [18]. One visible discrepancy in our data is the positivity of the
single-cat vs the product-cats of Bob. The smaller positivity in the product-
cats is due to the sequential creation of the cats which leads to longer
operation time. The result is additional decoherence due to photon loss. The
second visible discrepancy is a small rotation of a couple of degrees due to
the cross-Kerr interaction during the CNOD itself. Importantly, this rotation
is independent of the state in either mode, it depends only on the CNOD being
applied and can be digitally corrected. This correction has been applied
experimentally in the other datasets shown and is detailed in the
supplementary materials.

We now proceed to the entangled cats. When applied to multiple modes
conditioned on the same ancilla, the CNOD gate allows to entangle different
modes. Combined with unconditional ancilla rotations, the set
$\\{\mathrm{CNOD}(\vec{\alpha_{i}}),R(\phi,\theta)\\}$, where $i$ is the mode
index, is universal in the control of the Hilbert space comprised of multiple
Bosonic modes and a single ancilla qubit (see supplementary materials). We
demonstrate the entangling properties of our gate set by generating the Bell-
cats state
$\left|\psi_{\mathrm{b}}\right\rangle\equiv\mathcal{N}(\left|\alpha_{A},\alpha_{B}\right\rangle+\left|-\alpha_{A},-\alpha_{B}\right\rangle)$
using the sequence illustrated in Fig.2b with
$\alpha_{A}\simeq\alpha_{B}\simeq 1.7$.

Specifically, we implement a two mode CNOD gate
$\mathrm{CNOD}(2\alpha_{A},2\alpha_{B})$, realizing the three-way entangling
operation
$\frac{1}{\sqrt{2}}(\left|g\right\rangle+\left|e\right\rangle)\left|0\right\rangle_{A}\left|0\right\rangle_{B}\rightarrow\mathcal{N}(\left|e\right\rangle\left|\alpha\right\rangle_{A}\left|\alpha\right\rangle_{B}-\left|g\right\rangle\left|-\alpha\right\rangle_{A}\left|-\alpha\right\rangle_{B})$
between the Alice and Bob modes and the ancilla. Then an unconditional ancilla
rotation followed by a second, small CNOD at the perpendicular axis
$\mathrm{CNOD}(i\beta_{A},i\beta_{B})$ with
$\sum_{i=1,2}\alpha_{i}\beta_{i}=\frac{\pi}{2}$, is used to disentangle the
ancilla. To apply the first two-mode CNOD a single-mode CNOD is applied on Bob
and then on Alice. Simultaneous application of the CNOD to both modes is
possible and faster, yet we found that it induces undesired effects for large
displacements (see supplementary materials). These are attributed to coherent
transitions at high photon numbers. It should be possible to circumvent this
restriction, yet we postpone such analyses for future work. We do use a
simultaneous application of the two-mode CNOD for the second disentangling
CNOD gate, which implements small displacements only.

We start with the characteristic functions of each individual mode separately
for the Bell-cats state
$\mathcal{C}_{i}(\gamma_{i})=Tr[\mathcal{D}(\gamma_{i})\rho]$ (i = A, B),
which are shown in the right column of Fig.3. We see in the measured single-
mode characteristic functions $\mathcal{C_{A}}$ and $\mathcal{C_{B}}$ that on
their own the state of each mode is in a statistical mixture of two different
coherent states, as expected. In the figure we can see the absence of blobs,
indicating the expected mixture.

The core features of the Bell-cat state $\left|\psi_{b}\right\rangle$ in the
4D characteristic function are found in 2D cuts along the
Re$(\gamma_{A})$-Re$(\gamma_{B})$ and Im$(\gamma_{A})$-Im$(\gamma_{B})$
planes. We plot the cuts for an ideal state and for measured data from dataset
1 (Fig.1c). We also plot for comparison, the 2D cuts for the product-cats and
adjacent to it measured data from dataset 2 (Fig.4). There are a few prominent
features of the Bell-cats in the 2D cuts, as compared with theory and the
product-cats. The Re$(\gamma_{A})$-Re$(\gamma_{B})$ cut of
$\left|\psi_{b}\right\rangle$ contains three positively valued Gaussians
spheres (a.k.a blobs). The phase is conveyed by the two blobs centered around
$(\pm 2\alpha_{A},\pm 2\alpha_{B})$, and would be absent for coherent states.
For a pure Bell-cats state, the maximum of these blobs should be 0.5, and is
$\simeq 0.36$ in dataset 2.

The fringes in the Im$(\gamma_{A})$-Im$(\gamma_{B})$ cut indicate the
correlation between the coherent states in each of the different modes. The
maximum is 0.96 compared with 1 for an ideal state. For disentangled cats we
observe the checkerboard pattern, which is a results of the product of
orthogonal fringes from the two separate cats in the product state, Fig.4d.

Using two-mode CNOD operations we can map on to the ancilla correlations
between the Alice and Bob modes and measure them directly. Specifically, we
regard each mode as a logical cat-qubit in the standard cat-codes
$\left|0_{L}/1_{L}\right\rangle\equiv\left|\alpha/-\alpha\right\rangle$, such
that
$\left|\psi_{b}\right\rangle=\mathcal{N}(\left|0_{L}0_{L}\right\rangle+\left|1_{L},1_{L}\right\rangle)$.
We directly measure expectation values of both single cat-qubits and two cat-
qubits logical operators (see supplementary materials for the exact mapping
protocol). This mapping allows us to extract the entanglement witness of our
Bell-cats by measuring 4 points only of $\mathcal{C}_{J}$ to obtain the direct
fidelity estimation [26] (2 are sufficient, yet we use 4 for symmetry). We use
dataset 2 shown in Fig.4 and obtain
$\frac{1}{4}(\langle\hat{I}\hat{I}\rangle+\langle\hat{Z}\hat{Z}\rangle+\langle\hat{X}\hat{X}\rangle-\langle\hat{Y}\hat{Y}\rangle)\approx
84\%$ against the ideal Bell-cat, surpassing the classical correlation bound
of 50%.

Figure 4: Joint characteristic tomography of Bell-cats compared with product-
cats 2D cuts of the 4D joint characteristic functions
$[\mathcal{C}_{J}(\gamma_{A},\gamma_{B})]$ for the Bell-cats (a)-(b), and for
product-cats (c)-(d). Cuts are along the planes,
Re$(\gamma_{A})$-Re$(\gamma_{B})$ ((a) and (c)) and
Im$(\gamma_{A})$-Im$(\gamma_{B})$ ((b) and (d)). 1D cuts from the 2D planes
corresponding to the dashed lines ((e)-(f)).

We employed our CNOD operations in a multi-mode cavity to create single-,
product-, and Bell-cats states. Our CNOD method allowed fast operation in the
weak coupling regime in a multi-mode cavity. The times for generating the
different cats (Fig. 2) give speedups of almost two orders of magnitude as
compared with $\frac{3\pi}{\chi}$ from previous work [18]. In practice we were
limited by the available amplifiers, further optimization may yield faster
operations. Using the anti-symmetric shape for our pulses should make the CNOD
rather robust. We can compare with a similar method shown on a single mode
[17]. There they used for the cavity pulses approximate delta functions with a
waiting time between them. Our anti-symmetric pulses require less peak power
(see supplementary materials). We leave the detailed analysis, comparisons,
and further investigation of the speed limit to future work.

The combination of weak coupling and fast operation enabled by the CNOD method
was the key to the high quality state generation in a multi-mode setting,
despite the rather pedestrian cavity photon life-times. The photon lifetimes
are the main contributor to the infidelity and can be improved by a lot, these
were set by the pin-antenna coupling. Using a single pin-antenna for both
memory (input) and readout (output) modes serves as a double sided sword. It
simplifies the setup, however in practice there is a limit on the coupling
ratio of readout to memory modes. Increasing the ratio far beyond the current
2 orders of magnitude may prove rather challenging in this setup. Sacrificing
the setup simplicity and separating the readout and memory can give more than
an order of magnitude lifetime improvement [16].

Our results here, together with our proof that a single ancilla weakly coupled
to a multi-mode system allows for universal control, present an efficient
method for control of modules with few modes. This creates a path to efficient
multi-mode continuous variable encoding [27]. Extending beyond a few modes
could be achieved by coupling such modules through a control bus and
additional coupling elements [28]. Overall our approach can help reduce
hardware overhead for Bosonic encoding.

## Acknowledgments

This research was supported by the Israeli Science Foundation (ISF), Pazi
foundation, and Technion’s Helen Diller Quantum Center. The work made use of
the Micro Nano Fabrication Unit at the Technion. L.J. acknowledges support
from the ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399,
FA9550-21-1-0209), NSF (OMA-1936118, ERC-1941583, OMA-2137642), NTT Research,
and the Packard Foundation (2020-71479). We thank A. Turner, B. Katzir and D.
Oren Caspi for useful discussions.

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Supplemental Materials: Fast and simple method for quantum control of multiple
bosonic modes

##  I Experimental setup

The experimental system wiring schematically represented in Fig. S1. The
readout and the Alice and Bob storage oscillators are modes of the microwave
flute cavity [29] machined out of a single slab of high-purity N5 Aluminum.
Following Ref. [30], the seamless cavity was then etched in Transene A
Aluminum etchant for 4 hours.

As a two level ancilla, we use the two lowest levels of a transmon which was
fabricated by Aluminum deposition on resist patterns formed by electron beam
lithography, with a layer of ZEON ZEP 520A resist on a layer of MicroChem 8.5
MMA EL11 resist on top of a silicon substrate. Development of the resist was
done at room temperature for MMA and at 0C for ZEP. The Al/AlOx/Al Josephson
junction was fabricated using a bridge-free process [31].

To couple the ancilla to the oscillators, the edge of the silicone chip
opposite to the transmon is clamped as the superconducting circuit is inserted
through a 12mm tunnel of with diameter of 4mm into the cavity.

Figure S1: Schematic illustration of the experimental setup. The flute cavity
and the coupled transmon ancilla are placed at the bottom stage of a dilution
refrigerator and shielded by a Cryoperm-Copper-Tin shield. We use an
Operator-X (OPX) by Quantum Machines, to generate low frequency pulses with
arbitrary shapes. We mix the low frequency signals with high frequency signals
from our generators. We send all the control signals, for the transmon and the
memory modes, and the readout signals for the readout mode, down a single
attenuated line. The readout signal is reflected through the flute cavity,
amplified, and digitized at room temperature.

##  II Full Hamiltonian and parameters

During the application the CNOD gate, our system reaches large photon numbers.
Therefore, the Hamiltonian providing an effective description of our undriven
system should include higher order terms and is given by

$\begin{split}H/\hbar=&\
\omega_{A}a^{{}^{\dagger}}a+\omega_{B}b^{{}^{\dagger}}b+\omega_{q}\left|e\right\rangle\left\langle
e\right|+\omega_{R}r^{{}^{\dagger}}r\\\
-&\left(\chi_{A}a^{{}^{\dagger}}a+\chi_{B}b^{{}^{\dagger}}b-\chi_{R}r^{{}^{\dagger}}r\right)\left|e\right\rangle\left\langle
e\right|\\\
+&\frac{K_{A}}{2}a^{{}^{\dagger}}a^{{}^{\dagger}}aa-\frac{K_{B}}{2}b^{{}^{\dagger}}b^{{}^{\dagger}}bb-
K_{AB}a^{{}^{\dagger}}ab^{{}^{\dagger}}b\\\
-&\left(\frac{K_{R}}{2}r^{{}^{\dagger}}r+K_{AR}a^{{}^{\dagger}}a+K_{BR}b^{{}^{\dagger}}b\right)r^{{}^{\dagger}}r,\end{split}$
(S1)

where $a,b$ and $r$ are the bosonic annihilation operators of the oscillator
like Alice, Bob and readout mode respectively and
$\left|e\right\rangle\left\langle e\right|$ is the excited state of the
transmon ancilla which is treated as a two-level system. The Kerr
nonlinearties are estimated by $K_{ij}=\frac{\chi_{i}\chi_{j}}{2K_{t}}$ where
$K_{t}\approx 194$ MHz is the transmon anharmonicity.

##  III Ancilla to cat mapping

We show the mapping of an ancilla state on to the Bloch sphere of single mode
cat code, which is generalized to multiple modes. For an initial state,

$\left|\psi_{i}\right\rangle=\left(\cos\left(\frac{\theta}{2}\right)\left|g\right\rangle-i\sin\left(\frac{\theta}{2}\right)e^{i\phi}\left|e\right\rangle\right)\otimes\left|\vec{0}\right\rangle$
(S2)

where $\vec{0}$ is the vacuum of all modes. Applying a CNOD gate followed by a
$\mathcal{R}_{\hat{y}}(\pi/2)$ rotation we get,

$\frac{1}{\sqrt{2}}\left(\cos\left(\frac{\theta}{2}\right)\left(\left|e,\frac{\vec{\alpha}}{2}\right\rangle-\left|g,\frac{\vec{\alpha}}{2}\right\rangle\right)+i\sin\left(\frac{\theta}{2}\right)e^{i\phi}\left(\left|g,-\frac{\vec{\alpha}}{2}\right\rangle+\left|e,-\frac{\vec{\alpha}}{2}\right\rangle\right)\right)$
(S3)

To disentangle the modes from the ancilla we perform a second CNOD gate with
$i\vec{\beta}$,

$\\-\frac{e^{i\vec{\alpha}\cdot\vec{\beta}/4}}{\sqrt{2}}\left(\cos\left(\frac{\theta}{2}\right)\left(\left|g,\frac{\vec{\alpha}}{2}-i\frac{\vec{\beta}}{2}\right\rangle
e^{-i\vec{\alpha}\cdot\vec{\beta}/2}+\left|e,\frac{\vec{\alpha}}{2}+i\frac{\vec{\beta}}{2}\right\rangle\right)+i\sin\left(\frac{\theta}{2}\right)e^{i\phi}\left(\left|g,-\frac{\vec{\alpha}}{2}-i\frac{\vec{\beta}}{2}\right\rangle-\left|e,-\frac{\vec{\alpha}}{2}+i\frac{\vec{\beta}}{2}\right\rangle
e^{-i\vec{\alpha}\cdot\vec{\beta}/2}\right)\right)$ (S4)

Since $|\vec{\beta}|<<|\vec{\alpha}|$ we can approximate the state as,

$\approx-\frac{e^{i\vec{\alpha}\cdot\vec{\beta}/4}}{\sqrt{2}}\left(\cos\left(\frac{\theta}{2}\right)\left|g,\frac{\vec{\alpha}}{2}\right\rangle
e^{-3i\vec{\alpha}\cdot\vec{\beta}/4}+\cos\left(\frac{\theta}{2}\right)\left|e,\frac{\vec{\alpha}}{2}\right\rangle
e^{i\vec{\alpha}\cdot\vec{\beta}/4}+i\sin\left(\frac{\theta}{2}\right)e^{i\phi}\left|g,-\frac{\vec{\alpha}}{2}\right\rangle
e^{i\vec{\alpha}\cdot\vec{\beta}/4}-\left|e,-\frac{\vec{\alpha}}{2}\right\rangle
e^{-i3\vec{\alpha}\cdot\vec{\beta}/4}\right)$ (S5)

where we used
$\left\langle\delta|\eta\right\rangle=e^{iIm\left(\delta\eta^{*}\right)}e^{-\frac{1}{2}\left|\delta-\eta\right|^{2}}\approx
e^{iIm\left(\delta\eta^{*}\right)}$. Using
$\vec{\beta}\cdot\vec{\alpha}=\frac{\pi}{2}$ up to a global phase we get,

$\begin{split}\left|+i\right\rangle\otimes\left(\cos\left(\frac{\theta}{2}\right)\left|\frac{\vec{\alpha}}{2}\right\rangle-\sin\left(\frac{\theta}{2}\right)e^{i\phi}\left|-\frac{\vec{\alpha}}{2}\right\rangle\right)\end{split}$
(S6)

where
$\left|+i\right\rangle=\frac{\left|g\right\rangle+i\left|e\right\rangle}{\sqrt{2}}$.

##  IV Calibrations of pulses and cat phase

In this section, we describe the different methods that we used to calibrate
the CNOD amplitude and acquired geometric phase, and to measure the parameters
of the system.

We calibrated the CNOD amplitude by measuring a single axis (1D sweep) of the
characteristic function of a vacuum state (example shown in Fig. S2a). We fit
the data to the expected Gaussian shape, and thus we directly get the
amplitude scale.

A finite thermal population inside each mode would result in an inaccurate
calibration, as the Gaussian at the origin of the characteristic function of a
thermal state is narrower than that of a vacuum state. When we performed the
procedure MLE and reconstructed the density matrix of the different prepared
states, we found the inaccuracy in pulse amplitude to be small. If we were to
take a finite thermal state the calculated fidelities would be a bit higher
due to the narrowing effect of finite temperature. Our results are obtained by
assuming a vacuum state, thus the fidelities that we show here are a lower
bound of the true fidelities.

We calibrated the geometric phase that is acquired due to a CNOD by performing
the following sequence: $\mathcal{R}_{\hat{y}}(\pi/2)$, CNOD$(\gamma)$,
CNOD$(-\gamma)$, and either $\mathcal{R}_{\hat{y}}(\pi/2)$ or
$\mathcal{R}_{\hat{x}}(\pi/2)$ followed by a projective measurement of
$\sigma_{z}$ of the transmon ancilla. The two CNODs of opposite directions
cancel each other out, and leave the memory mode in the vacuum state, but
still impart a geometric phase on the transmon ancilla. By measuring both
$\sigma_{x}$ and $\sigma_{y}$, we are able to extract the acquired phase, as
shown in Fig. S2b.

Figure S2: Calibration of CNOD amplitude and acquired geometric phase. (a) The
characteristic function of a vacuum state along an arbitrary axis of length
$2\gamma_{\mathrm{max}}$. By fitting the data to the expected Gaussian of
variance $2$, we calibrate the maximal displacement amplitude
$\gamma_{\mathrm{max}}$. (b) The acquired geometric phase as function of the
displacement amplitude $\gamma$. We scale the displacement amplitude only by
scaling the anti-symmetric pulse amplitude while keeping the pulse time fixed.
We use a spline fit to extrapolate the phase for any magnitude of $\gamma$.

To calibrate the disentangling pulse amplitude, we sweep over $\beta$, the
amplitude of the disentangling CNOD$(\mathrm{i}\beta)$, and measure the purity
of the transmon ancilla. For this procedure we measure $\sigma_{z}$,
$\sigma_{x}$ and $\sigma_{y}$, but in practice the information about the
purity lies in the $x$-axis alone. At first, we pick the amplitude that
corresponds to the highest purity. Then we use it to create a cat-state and
measure its characteristic function along the displaced axis. We fine tune
$\beta$ to minimize asymmetry around $0$, thus maximizing disentanglement.

Figure S3: Calibration of CNOD amplitude used for disentanglement of the cat.
The expectation value of $\sigma_{x}$ of the transmon ancilla, that is
proportional to the readout amplitude, after creation of a cat. The maximal
value of $|\langle\sigma_{x}\rangle|$ indicates a maximum of purity and
therefore corresponds to the ideal disentangling displacement amplitude.

We accurately measure both the frequency and the lifetime of each mode using a
sequence that is similar to Ramsey interferometry. The sequence is composed of
$\mathcal{D}(\gamma)$, wait a time of $t$, $\mathcal{R}_{\hat{y}}(\pi/2)$,
CNOD$(-\mathrm{Exp}(\mathrm{i}\delta\omega t)\gamma)$ and
$\mathcal{R}_{\hat{y}}(-\pi/2)$ followed by a projective measurement of
$\sigma_{z}$ of the transmon ancilla (similar to what that was done by [8]).
We introduce $\delta\omega$ to add artificial detuning and allow for more
precise measurement of frequency. The results of one such measurement are
shown in Fig. S4. As the initial coherent state rotates and decays (assuming
negligible dephasing) the measurement data is expected to behave according to

$\mathcal{C}\left(e^{\mathrm{i}\delta\omega
t}\gamma\right)\propto\cos\left(A(\gamma)e^{-t/2T_{1}}\sin\left(\left(\delta\omega+\Delta\omega\right)t\right))\right)$
(S7)

where $A$ is some scaling coefficient, $T_{1}$ is the mode’s single photon
decay rate, $\Delta\omega$ is the detuning between the rotating frame and the
mode’s resonant frequency. In addition, this procedure enables precise
measurements of the cross-Kerr between the memory modes, and between each mode
and the transmon ancilla. For example, performing this procedure with the
transmon at the excited state would decrease $\Delta\Omega$ by $\chi$.

Figure S4: characteristic function of a decaying coherent state.
$\mathcal{C}\left(e^{\mathrm{i}\delta\omega t}\gamma\right)$ versus the
evolution time, starting with coherent state $\Ket{\gamma}$. The oscillations
are induced by both artificial detuning and detuning between the rotating
frame and the resonance frequency of the mode. The decay is dominated by the
single photon lifetime of the mode. We fit the data to eq. S7.

##  V characteristic function tomography

To reconstruct the prepared state in Alice and Bob, we performed
characteristic function tomography. We measured both the single-mode
characteristic function for each mode and 2D cuts of the 4D joint-
characteristic function.

The measurement procedure that we used is composed of a sequence of
$\mathcal{R}_{\hat{y}}(\pi/2)$, CNOD${}^{(A)}(\gamma_{A})$,
CNOD${}^{(B)}(\gamma_{B})$, and either $\mathcal{R}_{\hat{y}}(\pi/2)$ or
$\mathcal{R}_{\hat{x}}(\pi/2)$ followed by a projective measurement of
$\sigma_{z}$ of the transmon ancilla. This sequence allows us to measure the
real part or the imaginary part, respectively, of
$\mathcal{C}(\gamma_{A},\gamma_{B})$. We used CNODs for the measurement
procedure longer than for the state preparation in order to reduce the effect
of higher order terms such as self- and cross-Kerr on the measurement outcome.
We included rotations in two directions that map
$\mathcal{C}(\gamma_{A},\gamma_{B})$ to $-\sigma_{z}$, and summed the results
with the appropriate sign. The expectation value for each point was averaged
out of 4000 experiment runs.

We apply a couple of required actions in our post-processing procedure. One,
the acquired geometric phase due to the CNOD, which was calibrated prior to
running the experiment (see section IV), and two the characteristic function
was digitally rotated accordingly. In addition, we shifted both $\gamma_{A}$
and $\gamma_{B}$ by a real constant, this is to account for the imperfect
disentanglement (see section V.1). An example of this post processing
procedure is shown in Fig. S5.

Figure S5: Post processing of a joint characteristic function tomography. The
shift in $\gamma_{A}$ and $\gamma_{B}$ due to the imperfect disentanglement is
corrected as indicated by the dashed lines in the real part of
$\mathcal{C}(\gamma)$. The two symmetric blobs in the imaginary part are
corrected by accounting for the geometric phase that is acquired by the CNOD.
As a result, the imaginary part of the processed data almost vanishes.

###  V.1 Imperfect disentanglement

The disentanglement procedure is imperfect and improves exponentially in cat
size. In this work we used relatively smaller cats, and so we calculated the
error due to the imperfect disentanglement. The prominent effect is a shift in
the characteristic function expected from the experiment. This shift is
accounted for in the processing. The exact state of the system after the
disentangling pulse is (section III,

$\frac{1}{\sqrt{2}}\left(\cos\left(\frac{\theta}{2}\right)\left(\left|g,\frac{\alpha}{2}-i\frac{\beta}{2}\right\rangle
e^{-i\alpha\beta/2}+\left|e,\frac{\alpha}{2}+i\frac{\beta}{2}\right\rangle\right)+i\sin\left(\frac{\theta}{2}\right)e^{i\phi}\left(\left|g,-\frac{\alpha}{2}-i\frac{\beta}{2}\right\rangle-\left|e,-\frac{\alpha}{2}+i\frac{\beta}{2}\right\rangle
e^{-i\alpha\beta/2}\right)\right)$ (S8)

If the disentanglement was perfect then after a $\pi/2$ rotation around the
X-axis we would end up in the product state of a cat with the qubit in the
ground state. Then to measure the characteristic function we would apply the
regular procedure, starting with a $\pi/2$ rotation around the Y-axis. Since
the disentanglement is not perfect, after this procedure we remain with small
residual entanglement. This state is,

$\left(\left|-i\right\rangle\otimes\left|\psi_{-i}\right\rangle-\left|+i\right\rangle\otimes\left|\psi_{+i}\right\rangle\right)$
(S9)

where

$\left|\psi_{-i}\right\rangle=\frac{1}{\sqrt{2}}\left(\cos\left(\frac{\theta}{2}\right)e^{-i\alpha\beta/2}\left|\frac{\alpha}{2}-i\frac{\beta}{2}\right\rangle+i\sin\left(\frac{\theta}{2}\right)e^{i\phi}\left|-\frac{\alpha}{2}-i\frac{\beta}{2}\right\rangle\right)$
(S10)

$\left|\psi_{+i}\right\rangle=\frac{1}{\sqrt{2}}\left(\cos\left(\frac{\theta}{2}\right)\left|\frac{\alpha}{2}+i\frac{\beta}{2}\right\rangle-i\sin\left(\frac{\theta}{2}\right)e^{i\phi}e^{-i\alpha\beta/2}\left|-\frac{\alpha}{2}+i\frac{\beta}{2}\right\rangle\right)$
(S11)

and $\alpha\beta=\pi/2$. We then apply a CNOD($\gamma$). Doing a partial trace
over the EM mode we get,

$\begin{split}Tr_{\mathrm{EM}}\left(D\left(\frac{\gamma}{2}\sigma_{z}\right)\left(\left|-i\right\rangle\otimes\left|\psi_{-i}\right\rangle-\left|+i\right\rangle\otimes\left|\psi_{+i}\right\rangle\right)\left(\left\langle-i\right|\otimes\left\langle\psi_{-i}\right|-\left\langle+i\right|\otimes\left\langle\psi_{+i}\right|\right)D^{\dagger}\left(\frac{\gamma}{2}\sigma_{z}\right)\right)=&\\\
\frac{1}{2}(I+\langle z(\gamma)\rangle\sigma_{z}+\langle
y(\gamma)\rangle\sigma_{y}+\langle x(\gamma)\rangle\sigma_{x})\end{split}$
(S12)

For $\phi=0$ and $\theta=-\pi/2$ we get $\langle z(\gamma)\rangle=\langle
y(\gamma)\rangle=0$, as expected for the characteristic function of the
corresponding cat. $\langle x(\gamma)\rangle$ corresponds to the real part of
the characteristic function, and given by,

$\begin{split}&\ \
e^{-\frac{1}{2}\left|\gamma\right|^{2}}\left(-\cos\left(Im\left(\alpha\gamma\right)\right)\sin\left(Im\left(i\beta\gamma\right)\right)+e^{-\frac{1}{2}\left|\alpha\right|^{2}}\sinh\left(Re\left(\alpha\gamma\right)\right)\cos\left(Im\left(i\beta\gamma\right)\right)\right)\\\
&+\ \
e^{-\frac{1}{2}\left(\left|\beta\right|^{2}+\left|\gamma\right|^{2}\right)}\cosh\left(Re\left(i\beta\gamma\right)\right)\left(\cos\left(Im\left(\alpha\gamma\right)\right)+e^{-\frac{1}{2}\left|\alpha\right|^{2}}\cosh\left(Re\left(\alpha\gamma+i\beta\gamma\right)\right)\right)\end{split}$
(S13)

This results in a small shift in the real axis compared with the exact
characteristic function,

$\frac{e^{-\frac{1}{2}\left|\gamma\right|^{2}}}{1+e^{-\frac{1}{2}\left|\alpha\right|^{2}}}\left(\cos\left(Im\left(\gamma\alpha\right)\right)+e^{-\frac{1}{2}\left|\alpha\right|^{2}}\cosh\left(Re\left(\gamma\alpha\right)\right)\right)$
(S14)

###  V.2 Density matrix reconstruction

We reconstruct the density matrix using a maximum likelihood estimation (MLE)
method [32]. For measured expectation values
$\langle\mathcal{D}(\lambda)\rangle_{\textrm{meas}}$ of a set of $\lambda$’s,
the likelihood function to be minimized is given by

$\mathcal{L}(\rho_{\mathrm{MLE}})=\sum_{\lambda}\frac{\left|\mathrm{Tr}[\mathcal{D}(\lambda)\rho_{\mathrm{MLE}}]-\langle\mathcal{D}(\lambda)\rangle_{\textrm{meas}}\right|}{\delta\langle\mathcal{D}(\lambda)\rangle_{\textrm{meas}}}$
(S15)

where $\rho_{\textrm{MLE}}$ is the reconstructed density matrix, and
$\delta\langle\mathcal{D}(\lambda)\rangle_{\textrm{meas}}$ is the standard
error of the measured expectation value.

Once the density matrix $\rho$ is reconstructed, we are able to calculate its
fidelity with respect to a target state $\sigma$, according to

$\mathcal{F}(\rho,\sigma)=\left(\mathrm{Tr}\left[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right]\right)^{2}$
(S16)

##  VI Multi-mode displacements

###  VI.1 Displaced frame

It is instructive to analyze the relations of the CNOD gate in the frame of
the semi-classical phase space trajectories as defined in Ref. [17].
Specifically, for a single mode, a transformation by a time dependent unitary
$U=D\left(\alpha\left(t\right)\right)=exp\left[\alpha\left(t\right)a^{{}^{\dagger}}-\alpha^{*}\left(t\right)a\right]$
is applied. As with the ECD gate of Ref. [17], the CNOD gate includes both the
phase space echo $\alpha\rightarrow-\alpha$ and qubit echo
$\left|g\right\rangle\leftrightarrow\left|e\right\rangle$ at time T.
Therefore, many of their conclusions for a single mode, such as the
significant reduction of the effect of terms proportional to sign$(\alpha)$,
can be extended to the CNOD.

We expand the discussion to a multi-mode setup and consider the effects of
higher order terms which include operators of multiple modes, specifically the
cross-Kerr term. In the two-mode displaced frame, corresponding to a
simultaneous application of the CNOD gate on both modes, the cross Kerr term
is given by

$\displaystyle a^{\dagger}ab^{\dagger}b\rightarrow$
$\displaystyle(a^{{}^{\dagger}}+\alpha^{*})(a+\mathcal{\alpha})(b^{{}^{\dagger}}+\beta^{*})(b+\beta)$
$\displaystyle=$ $\displaystyle
a^{{}^{\dagger}}ab^{{}^{\dagger}}b+\left|\alpha\right|^{2}b^{{}^{\dagger}}b+\left|\beta\right|^{2}a^{{}^{\dagger}}a$
$\displaystyle+$ $\displaystyle\left|\alpha\right|^{2}\beta
b^{{}^{\dagger}}+\alpha\left|\beta\right|^{2}a^{{}^{\dagger}}+h.c.$

As with the single-mode higher order terms, we notice that some of the
elements in the transformed cross-Kerr term should be greatly reduced by the
phase space echoing. The remaining non-negligible terms,
$|\alpha|^{2}b^{\dagger}b$ and $|\beta|^{2}a^{\dagger}a$ correspond in the
former (later) to a shift in the frequency of the Bob (Alice) mode which
depends only on the displaced frame of Alice (Bob). For the application of
CNOD gate to only one of the modes, we notice that this term effects the other
mode just by an unconditional rotation of it’s phase space.

###  VI.2 Simultaneous multi-mode CNOD

Even with the addition of higher order terms, our effective description of the
system with an almost dispersive Hamiltonian is expected to eventually
breakdown as the implementations of the CNODs populate oscillator states with
increasing photon numbers [33, 34, 35]. The expected behaviour is that at
large photon numbers higher order transitions, where the transmon ancilla is
excited beyond the $\left|g\right\rangle,\left|e\right\rangle$ manifold, are
induced. Such higher order nonlinear transitions are expected to limit the
Hilbert space accessible to the CNOD and thus reducing the speedup by which it
implements conditional displacements.

To estimate the speed and displacement amplitudes of the various
implementations of the CNOD gate at which such additional decoherence is
induced, we measure the depolarization of the ancilla during the CNOD gates.
We start by preparing the transmon in
$(\left|g\right\rangle+\left|e\right\rangle)$ state and apply the conditional
displacements $\mathrm{CNOD}(\gamma_{A},\gamma_{B})$,
$\mathrm{CNOD}(-\gamma_{A},-\gamma_{B})$, with a fast ancilla $\pi$ rotation
in between. Finally, a tomographic measurement of the transmon qubit is
performed, similar to the measurement of the characteristic function. The
normalized purity of the qubit subspace,
$\left\langle{P_{s}}\right\rangle=\left\langle{\sigma_{z}}\right\rangle^{2}+\left\langle{\sigma_{x}}\right\rangle^{2}+\left\langle{\sigma_{y}}\right\rangle^{2}$
is displayed in Fig. S6 as a function of the two-mode displacement amplitudes
$\gamma_{A}$ and $\gamma_{B}$.

Figure S6: Measured normalized purity $\left\langle{P_{s}}\right\rangle$ as a
function of $\gamma_{A}$ and $\gamma_{B}$, the final displacement amplitude
and the Alice and Bob mode respectively. The anti-symmetric pulses of which
comprise the CNOD are applied simultaneously to the two modes by two pulses
with a carrier frequency corresponding to the of each of the modes when the
ancilla is excited. The two pulses have the same envelope which is composed of
a Gaussian of standard deviation of $\sigma=144$ ns and total length 0f
$2\sigma=288$ which is multiplied by $\sin\left(\Delta(t-t_{0})\right)$ with
$\Delta\approx 10$ Hz and $t_{0}=144$ ns is half the pulse duration, making
anti-symmetric in both time and frequency domain. The results are averaged
2000 times for each of the tomographic measurements. The results are
normalized by the purity at zero displacements to account for the decoherence
of the ancilla which is not related to the memory modes.

We find that by applying the CNOD to both modes simultaneously the ancilla
exhibits a significant depolarization at certain displacement amplitudes. The
abruptness of these disappearance lines and the reemergence of the ancilla at
larger displacement amplitudes, may indicate that this phenomena is related to
resonant transitions to higher levels or residual entanglement with the cavity
modes. Indeed, one could use the available sweet spots in the large,
simulations displacements for multi-mode state manipulation. However, in this
work we chose to implement the large, multi-mode CNODs, such as the first CNOD
in the generation of the Bell-cat described in fig. 2b, by a pair of
consecutive, single-mode CNODs separated by an ancilla $\pi$ pulse. We
postpone to future work an analysis of the physical mechanisms leading to such
transitions and if they could be avoided by temporal shaping of the anti-
symmetric pulses from which the CNOD is composed.

On the other hand, we notice that simultaneous application of small
conditional displacements does not lead to additional ancila decoherence.
Therefore, small, two-mode CNOD displacements, such the second disentangling-
CNOD in fig. 2b required for the generation of the Bell-cat, were applied
simultaneously to both modes.

Furthermore, in the cuts of Fig. S6 dedicated to a single mode application of
a CNOD, we find that additional decoherence might occur for the larger
displacement. However the effect is very small and the continuous increase in
the decoherence indicates that they are most likely related to qubit dephasing
induced by photon loss which increase with the size of the overall conditional
displacement. A similar mechanism, combined with the absence for the
correction of the unconditional phase space rotation discussed in the previous
subsection, could explain continuous increase in dephasing observed for very
large simultaneous CNODs.

Finally, it is interesting to note that for our current setup and at the range
of the single-mode CNOD implementations we used, we did not observe these
effects. Something that should be looked into in the future.

##  VII Multi-mode Universality

One way to implement universal control of a single Harmonic oscillator is by
producing any Unitary evolution generated by Hamiltonians polynomial in
$X=(1/\sqrt{2})(a^{\dagger}+a)$ and $P=(i/\sqrt{2})(a^{\dagger}-a)$ [36, 37].
By including Unitaries generated by Hamiltonians of the form
$X^{j}P^{k}\sigma_{i}$, [17] extends the definition of universal control to
include an oscillator and a qubit. In the same work it was shown that the set
of operations equivalent to
$\\{\mathrm{CNOD}(\vec{\alpha_{i}}),R(\phi,\theta)\\}$ are sufficient to
obtain such universal control.

Extending a Hilbert space comprised of multiple oscillators, universal conrol
requires unitaries generated by Hamiltonians compose of arbitrary polynomials
of the oscillators ${X_{i},P_{i}}$. However, when universal control can be
implemented separately on each oscillator, it is sufficient to include the
ability to apply a beam splitter interaction of the form
$B_{ij}=P_{m}X_{n}-X_{m}P_{n}$.

Working with the generators of the universal control for each single mode,
i.e. $X_{m}^{j}P_{m}^{k}\sigma_{i}$, we expand our set by using the
commutators $[X_{m}^{2}\sigma_{x},P_{n}\sigma_{y}]\propto
X_{m}^{2}P_{n}\sigma_{z}$ for $m\neq n$. To generate the required beam
splitter term, which does not include the Pauli operator, we next commute
$[X_{m}^{2}P_{n}\sigma_{z},X_{m}\sigma_{z}]\propto X_{m}P_{n}$. As discussed
in the main text, application of the CNOD on a single mode allows for
universal control of each single oscillator, independent of and without
effecting the other mode. Thus, with the addition of the single ancilla
unconditional rotation $R_{\phi}(\theta)$, the combination of the single modes
$\mathrm{CNOD}(\gamma_{i})$ enables universal control of Hilbert space
comprised of multiple oscillators coupled to a single qubit.

##  VIII Measurement of non local operators

To quantify the entanglement and fidelity of our state to the Bell-cat, it is
sufficient to measure the expectation value of the logical cat-code operator
$\mathcal{F}=\frac{1}{4}\left(\left\langle II\right\rangle+\left\langle
ZZ\right\rangle+\left\langle XX\right\rangle-\left\langle
YY\right\rangle\right)$ which for a general form of the two modes cat-code
state
$\left|\psi_{G}\right\rangle=a\left|\alpha_{A},\alpha_{B}\right\rangle+b\left|\alpha_{A},-\alpha_{B}\right\rangle+c\left|-\alpha_{A},\alpha_{B}\right\rangle+d\left|-\alpha_{A},-\alpha_{B}\right\rangle$
is given by $\frac{1}{4}\left(|a|^{2}+|d|^{2}+ad^{*}+a^{*}d\right)$. We
proceed to show that by multi-mode CNODs we are able to map the expectation
values of the 2-mode operators of the cat qubits onto the single ancilla.
Starting with the ancilla at the ground state we apply two $\pi/2$ pulses
around the Y-axis which are separated by
CNOD$\left(i\frac{\pi}{4\alpha_{A}},-i\frac{\pi}{4\alpha_{B}}\right)$ and
obtain

$\begin{split}U_{II+ZZ}\left|g\right\rangle\otimes\left|\psi_{G}\right\rangle\approx\left|g\right\rangle\otimes\left(a\left|\alpha_{A},\alpha_{B}\right\rangle+d\left|-\alpha_{A},-\alpha_{B}\right\rangle\right)+\left|-i\right\rangle\otimes\left(b\left|\alpha_{A},-\alpha_{B}\right\rangle\right)+\left|+i\right\rangle\otimes\left(c\left|-\alpha_{A},\alpha_{B}\right\rangle\right)\end{split}$
(S17)

where
$U_{II+ZZ}=R_{\frac{\pi}{2},\hat{y}}\cdot\mathrm{CNOD}\left(i\frac{\pi}{4\alpha_{A}},-i\frac{\pi}{4\alpha_{B}}\right)\cdot
R_{\frac{\pi}{2},\hat{y}}$ and we assume
$\frac{\pi}{4\alpha_{A}},\frac{\pi}{4\alpha_{A}}<<1$. We conclude by
measurement in the Z basis of the ancilla, resulting in the expectation value
$\left\langle
U_{II+ZZ}^{{}^{\dagger}}\sigma_{z}U_{II+ZZ}\right\rangle=\left|a\right|^{2}+\left|d\right|^{2}$
which is equal the expectation value $\left\langle
II\right\rangle+\left\langle ZZ\right\rangle$ of the cat qubits. Next, a
similar procedure where CNOD gate is replaced by
CNOD$\left(2\alpha_{A}.2\alpha_{B}\right)$ is use to obtain

$\begin{split}U_{XX-YY}\left|\psi_{G}\right\rangle=&\ \
-\frac{1}{\sqrt{2}}\left|-\right\rangle\otimes\left(a\left|2\alpha_{A},2\alpha_{B}\right\rangle+b\left|2\alpha_{A},0\right\rangle+c\left|0,2\alpha_{B}\right\rangle\right)\\\
&\ \
-\frac{1}{\sqrt{2}}\left|+\right\rangle\otimes\left(b\left|0,-2\alpha_{B}\right\rangle+c\left|-2\alpha_{A},0\right\rangle+d\left|-2\alpha_{A},-2\alpha_{B}\right\rangle\right)\\\
&\ \
-\frac{1}{2}\left(\left(a+d\right)\left|g\right\rangle+\left(a-d\right)\left|e\right\rangle\right)\otimes\left|0,0\right\rangle\end{split}$
(S18)

where $U_{XX-
YY}=R_{\frac{\pi}{2},\hat{y}}\cdot\mathrm{CNOD}\left(2\alpha_{A}.2\alpha_{B}\right)\cdot
R_{\frac{\pi}{2},\hat{y}}$ and we assumed $\alpha_{A},\alpha_{B}$ are large so
we may neglect the cross terms. The corresponding measurement of the ancilla
results we obtain $\left\langle U_{XX-YY}^{{}^{\dagger}}\sigma_{z}U_{XX-
YY}\right\rangle=\frac{a^{*}d+ad^{*}}{2}=\frac{1}{2}\left(\left\langle XX-
YY\right\rangle\right)$. Thus, combining and rescaling reuslts meausred
following these two sets of operations, we are able to estimate the
entanglement of the two cat qubits by

$\mathcal{F}=\left\langle
U_{II+ZZ}^{{}^{\dagger}}\sigma_{z}U_{II+ZZ}\right\rangle+2\left\langle U_{XX-
YY}^{{}^{\dagger}}\sigma_{z}U_{XX-YY}\right\rangle.$ (S19)

##  IX State Fidelty

The Bell-cats in the table are each compared to the density matrix of the
fully mixed logical cat state
$\frac{1}{2}\mathcal{D}(\delta)(\left|e^{i\theta_{ph}}\alpha\right\rangle\left\langle
e^{i\theta_{ph}}\alpha\right|+\left|-e^{i\theta_{ph}}\alpha\right\rangle\left\langle
e^{i\theta_{ph}}\alpha\right|)\mathcal{D}^{\dagger}(\delta)$ while the product
and single-mode states are compared to
$\mathcal{N}(\mathcal{D}(\delta)\left|e^{i\theta_{ph}}\alpha\right\rangle+e^{i\phi_{rot}}\left|-e^{i\theta_{ph}}\alpha\right\rangle)$.
We did a basic step of optimization, where we allow the cat-states being
compared to have small deviations in the parameters $\alpha$, $\delta$, and
$\phi$. These could be easily corrected. The main infidelity comes from photon
loss.

Mode | State type | Fidelity | $\alpha$ | $\delta$ | $\theta$ | $\phi_{rot}$
---|---|---|---|---|---|---
| Bell-cats | 0.89 | 1.69 | -0.02-0.01i | - | 0.00
Alice | product-cats | 0.83 | 1.67 | 0.01+0.06i | 0.21 | 0.00
| single-cat | 0.87 | 1.68 | -0.04+0.05i | 0.336 | 0.00
| Bell-cats | 0.92 | 1.75 | -0.16 +0.05i | - | 0.00
Bob | product-cats | 0.77 | 1.71 | -0.1+0.05i | -0.06 | 0.25
| single-cat | 0.86 | 1.75 | -0.1+0.02i | 0.02 | 0.12
Table 1: Fidelity of prepared states

##  X CNOD Vs ECD

It is instructive to qualitatively compare the application of our CNOD method
to the recently demonstrated ECD method [17]. For this we plot the normalized
drive amplitude for each method. For the example given in Fig. S7 we use the
pulses applies on the Bob mode to create the cat state; and plot the ECD gate
of same duration to achieve the same target state with parameters similar to
those used in Ref. [17] (displacement with a gaussian having $\sigma$=11ns and
pulse duration of 44ns). The most prominent difference is the required peak
power in the CNOD method vs the ECD method, which amounts to 9.44 dB for the
given example. In addition the ECD is not perfectly anti-symmetric which leads
to more subtle differences. An in depth comparison or unification of the two
methods is left for future work.

Figure S7: A comparison between drive amplitudes of CNOD and ECD.