Present address: ]Université Grenoble Alpes, CEA, INAC-Pheliqs, 38000
Grenoble, France Present address: ]Google, Santa Barbara CA 93117, USA.
Present address: ]Amazon AWS, San Francisco CA 94105, USA. Present address:
]Université de Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire
de Physique, F-69342 Lyon, France

# Quantum communication with itinerant surface acoustic wave phonons

É. Dumur  [  Pritzker School of Molecular Engineering, University of Chicago,
Chicago IL 60637, USA  Argonne National Laboratory, Argonne IL 60439, USA  K.
J. Satzinger  [  Pritzker School of Molecular Engineering, University of
Chicago, Chicago IL 60637, USA  Department of Physics, University of
California, Santa Barbara CA 93106, USA  G. A. Peairs  [  Pritzker School of
Molecular Engineering, University of Chicago, Chicago IL 60637, USA
Department of Physics, University of California, Santa Barbara CA 93106, USA
M.-H. Chou  Pritzker School of Molecular Engineering, University of Chicago,
Chicago IL 60637, USA  Department of Physics, University of Chicago, Chicago
IL 60637, USA  A. Bienfait  [  Pritzker School of Molecular Engineering,
University of Chicago, Chicago IL 60637, USA  H.-S. Chang  Pritzker School of
Molecular Engineering, University of Chicago, Chicago IL 60637, USA  C. R.
Conner  Pritzker School of Molecular Engineering, University of Chicago,
Chicago IL 60637, USA  J. Grebel  Pritzker School of Molecular Engineering,
University of Chicago, Chicago IL 60637, USA  R. G. Povey  Pritzker School of
Molecular Engineering, University of Chicago, Chicago IL 60637, USA
Department of Physics, University of Chicago, Chicago IL 60637, USA  Y. P.
Zhong  Pritzker School of Molecular Engineering, University of Chicago,
Chicago IL 60637, USA  A. N. Cleland  Pritzker School of Molecular
Engineering, University of Chicago, Chicago IL 60637, USA  Argonne National
Laboratory, Argonne IL 60439, USA  Corresponding author<EMAIL_ADDRESS>
###### Abstract

Surface acoustic waves are commonly used in classical electronics
applications, and their use in quantum systems is beginning to be explored, as
evidenced by recent experiments using acoustic Fabry-Pérot resonators. Here we
explore their use for quantum communication, where we demonstrate a single-
phonon surface acoustic wave transmission line, which links two physically-
separated qubit nodes. Each node comprises a microwave phonon transducer, an
externally-controlled superconducting variable coupler, and a superconducting
qubit. Using this system, precisely-shaped individual itinerant phonons are
used to coherently transfer quantum information between the two physically-
distinct quantum nodes, enabling the high-fidelity node-to-node transfer of
quantum states as well as the generation of a two-node Bell state. We further
explore the dispersive interactions between an itinerant phonon emitted from
one node and interacting with the superconducting qubit in the remote node.
The observed interactions between the phonon and the remote qubit promise
future quantum optics-style experiments with itinerant phonons.

##  I Introduction

Quantum communication is of significant interest for the generation of remote
entanglement and the secure transmission of information, as well as for
distributed quantum computing [1, 2, 3, 4, 5, 6, 7]. There are several
demonstrations of long-distance quantum communication protocols using optical
methods, in parallel with demonstrations of similar protocols using microwave-
frequency photons, including Bell state entanglement of remote qubits as well
as the transmission of multi-qubit entangled states [8, 9, 10, 11, 12, 13, 14,
15, 16]. Microwave-frequency phonons, as opposed to photons, can also be used
for quantum communication as well as for coupling hybrid quantum systems [17,
18, 19, 20], in the latter case taking advantage of the strong strain coupling
in some optical as well as atomic-scale systems. Microwave-frequency acoustic
resonators may be able to serve as very long-lived quantum memories [21].
Quantum communication protocols implemented with phonons are thus of
significant scientific as well as practical interest. Recent advances in the
quantum control of phonons include the creation and measurement of stationary
phonon quantum states [22, 23, 24], the emission and absorption of phonons in
an acoustic resonator [25], and the generation of entangled phonons in a
phonon-mediated quantum eraser experiment [26].

Here we report the experimental realization of a phonon-based quantum
communication channel, enabling the communication of quantum states via
traveling phonons linking two physically-distinct quantum nodes. The phonons
are emitted in the communication channel as short-duration acoustic pulses,
sufficiently brief that the extent of the acoustic pulses is significantly
less than the length of the channel, such that the phonons travel in a
particle-like fashion along the channel, which we term itinerant.

The experimental system is shown schematically in Fig. 1, with the physical
setup in Fig. 1a and the circuit schematic in Fig. 1b. The
$2\text{\,}\mathrm{mm}$-long phonon communication channel
($500\text{\,}\mathrm{ns}$ single-trip time) is terminated at each end by a
specially-designed unidirectional interdigitated transducer, which is in turn
connected to a superconducting qubit via a superconducting tunable coupler.
The unidirectional transducers (UDTs) differ from conventional acoustic
transducers, here emitting itinerant phonons in only one direction, as opposed
to more standard bi-directional transducers, which emit excitations equally in
two opposing directions (see Supplementary Note 1; a related but distinct
design appears in [27]). We note this device differs from the experimental
construction in e.g. Ref. [25], which uses a single bidirectional transducer
in a Fabry-Pérot cavity. In that experiment, a single phonon comprises
acoustic excitations that travel in two opposing directions to distant
acoustic mirrors, from which the excitations reflect and return to interfere
constructively at the emitting transducer, where the excitation can be
intercepted by one of two qubits. In the experiment here, two distinct
unidirectional transducers are used to link two physically separate nodes.
Each transducer is constructed to emit an acoustic excitation in only one
direction, creating a significantly more flexible and general-purpose design,
with physically separate and distinct phonon emitter and receiver.

We use this device to demonstrate two-node quantum state transfers as well as
the phonon-mediated deterministic generation of an entangled Bell state,
representing a significant advance over prior work, in which a single
transducer was coupled to a Fabry-Pérot acoustic cavity formed by two acoustic
mirrors [22, 23, 24, 25, 26]. We also realize a single-phonon interferometer,
using one qubit to emit and detect a traveling phonon, where the phonon is
used to probe the state of the second qubit, effectively demonstrating the
dispersive interaction of a photon (localized in the remote qubit) and a
traveling phonon. Finally, we demonstrate a Ramsey interferometer, using the
second qubit to detect the presence of a traveling phonon emitted by the first
qubit, thus interchanging the roles of the qubits in the previous experiment
and demonstrating the versatility of this architecture.

## RESULTS

Figure 1:  Quantum delay line. a Schematic representation of the two quantum
communication nodes and the phononic quantum channel. Each node comprises a
superconducting qubit, a tunable coupler that allows shaping phonon release
and capture, and a unidirectional phonon transducer. Details in the Methods
section. b Circuit diagram of the assembled device. Each qubit $Q_{j}$ is
excited through its dedicated $Q_{xy~{}j}$ microwave line and frequency-
controlled through a separate $Q_{z\,j}$ flux-bias line, and each tunable
coupler $G_{k}$ is controlled via its associated $G_{z\,k}$ flux-bias line.

### Phonon-mediated quantum state transfer

We first probe the interaction between the qubits and the phonon channel, as
shown in Fig. 2a. We excite $Q_{1}$ with a $\pi$ pulse, then set its coupler
$G_{1}$ to an intermediate coupling, sufficient that $Q_{1}$’s relaxation is
dominated by phonon emission. We set $Q_{2}$’s coupler $G_{2}$ off during this
measurement, so that $Q_{2}$ does not interact with the traveling phonon. For
frequencies inside the transducer’s active band, from 3.87 to 4.01 GHz, where
the emission is almost entirely unidirectional itinerant phonons, we observe a
time-delayed revival of qubit $Q_{1}$’s excited state population
$P_{e}^{Q_{1}}$ at times that are multiples of the phonon round-trip time
$\tau_{\mathrm{RT}}\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, each
revival corresponding to the traveling phonon reflecting off the other
transducer before re-exciting $Q_{1}$. Outside the unidirectional band, we see
a complex structure in $P_{e}$ as a function of frequency and interaction
time, with broad swings of width $\sim$150\text{\,}\mathrm{MHz}$$ superposed
with narrow oscillations of width $\sim$7\text{\,}\mathrm{MHz}$$. The broad
swings and finer details are in accordance with expectations (see
Supplementary Note 1) [28].

The itinerant phonon experiments are performed at the working frequency
$\omega_{Q_{1,2}}^{\mathrm{uni}}/2\pi=$3.976\text{\,}\mathrm{GHz}$$, inside
the unidirectional band. By working outside this band, we can explore the
regime where the transducers are effectively bidirectional, using the second
working frequency
$\omega_{Q_{1,2}}^{\mathrm{bi}}/2\pi=$4.102\text{\,}\mathrm{GHz}$$. These
frequencies are marked by the dashed white and red lines, respectively, in
Fig. 2a.

To maximize the efficiency of phonon-mediated quantum state transfers, we need
to carefully shape the emission and absorption of the phonon wave packet,
which is done by time-dependent control of the coupling between the qubit and
its transducer [29, 15, 11, 12, 13, 14, 25]. We experimentally optimize the
transfer efficiency, with results shown in Fig. 2b for both the unidirectional
(left) and bidirectional (right) regimes. The transfer starts with the shaped
emission of a phonon, shown by the decrease of $Q_{1}$’s excited state
population with the expected time dependence. Both qubits then remain in their
ground states until the phonon reaches $Q_{2}$, which absorbs the itinerant
phonon, following the expected time dependence, and ultimately reaching a
plateau once the transfer is complete. The total transfer takes
$\sim$700\text{\,}\mathrm{ns}$$, including the $\sim$500\text{\,}\mathrm{ns}$$
phonon travel time. The final $Q_{2}$ population reaches a maximum of
$68\text{\,}\mathrm{\char 37\relax}$ for the unidirectional transfer, limited
mostly by phonon loss in the channel. For the bidirectional transfer, the
final $Q_{2}$ population reaches $15\text{\,}\mathrm{\char 37\relax}$, $4.5$
times less than the unidirectional population, which is
$12\text{\,}\mathrm{\char 37\relax}$ higher than the ideal value,
demonstrating good agreement with theory and excellent unidirectionality for
the transducer design. We simulate the transfer process using a cascaded
quantum input-output model [25] (solid green line). From this model we
estimate that phonon loss reduces the final unidirectional transfer efficiency
by $27\text{\,}\mathrm{\char 37\relax}$, and the finite $Q_{1}$ and $Q_{2}$
coherence times reduce the fidelity by $1\text{\,}\mathrm{\char 37\relax}$ and
$2\text{\,}\mathrm{\char 37\relax}$, respectively. We note that an equivalent
photon travel time would require a $\sim$100\text{\,}\mathrm{m}$$ long coaxial
cable, illustrating the very long delays achievable with phonon-based quantum
channels.

In Fig. 2c, we show quantum process tomography for both regimes. For the
unidirectional process, we find a process fidelity of
$\mathcal{F}^{\mathrm{uni}}=$82\pm 0.3\text{\,}\mathrm{\char 37\relax}$$,
while for the bidirectional regime, the process fidelity is limited to
$\mathcal{F}^{\mathrm{bi}}=$39\pm 0.3\text{\,}\mathrm{\char 37\relax}$$. We
compare these experimental process fidelities with predictions, and find trace
distances
$d=\sqrt{\mathrm{Tr}~{}(\chi_{\mathrm{exp}}-\chi_{\mathrm{sim}})^{2}}=0.07$
and $0.3$ for the unidirectional and bidirectional regimes. The contrast in
fidelities and trace distances underlines the importance of the unidirectional
transducers.

Figure 2: Phonon-mediated quantum state transfer and process tomography. a
Measured $Q_{1}$ excited state population $P_{e}^{Q_{1}}$ as a function of
time and $Q_{1}$ bare frequency, with coupler $G_{1}$ at an intermediate
coupling $\kappa_{1}/2\pi=$2.4\text{\,}\mathrm{MHz}$$ (measured at
$3.976\text{\,}\mathrm{GHz}$) and $G_{2}$ set to zero coupling. In this
configuration, $Q_{1}$’s energy relaxation is dominated by phonon emission via
$\mathrm{UDT}_{1}$, followed by traveling phonon dynamics. The white and red
dashed lines indicate the unidirectional and bidirectional working
frequencies, respectively (see text); inset shows the qubit excitation and
measurement pulse sequence. b Quantum state transfer via a traveling phonon at
the unidirectional (left) and bidirectional (right) operating frequencies.
$Q_{2}$’s final population is $4.5$ times smaller for the bidirectional
transfer compared to the unidirectional transfer, in line with simulations.
Green solid lines are from a master equation simulation. Inset: Pulse
sequence. For either process, $Q_{1}$’s emission rate is set to
$\kappa_{c}^{\mathrm{uni|bi}}/2\pi=10|$6\text{\,}\mathrm{MHz}$$, corresponding
to a $81|$138\text{\,}\mathrm{ns}$$ full-width-at-half-maximum (FWHM) phonon
wavepacket. c Quantum process tomography for the unidirectional and
bidirectional regimes, with process fidelities of
$\mathcal{F}_{\mathrm{uni}}=\mathrm{Tr}~{}(\chi_{\mathrm{exp}}\cdot\chi_{\mathrm{ideal}})=82\pm$0.3\text{\,}\mathrm{\char
37\relax}$$ and $\mathcal{F}_{\mathrm{bi}}=39\pm$0.3\text{\,}\mathrm{\char
37\relax}$$, respectively. Red solid lines show values expected for an ideal
transfer; black dashed lines show master equation simulations, taking into
account finite qubit coherence and phonon channel losses. Uncertainties are
standard deviations from the mean.

### Traveling phonon-mediated remote entanglement

We further explore the capabilities of itinerant phonon communication by
performing a phonon-mediated remote entanglement of the two qubits, shown in
Fig. 3. The protocol is similar to that for the quantum state transfer, except
here we calibrate the emission pulse to only emit $Q_{1}$’s excitation as a
phonon with a probability of 1/2, meaning that immediately following the
‘half-emission,’ with qubit $Q_{2}$ in the ground state, the system is ideally
in the state $(|e0g\rangle+|g1g\rangle)/\sqrt{2}$ (writing the state
$|Q_{1}\,\gamma\,Q_{2}\rangle$ where $\gamma$ represents the itinerant
phonon). During the time the emitted ‘half-phonon’ travels along the phonon
channel, $Q_{1}$’s remaining excitation decays following $Q_{1}$’s intrinsic
$T_{1}$ time, with $Q_{1}$’s coupling to the channel set to zero. The
traveling half-phonon is then captured by $Q_{2}$, generating a Bell state
$|\psi\rangle=(|eg\rangle+e^{\mathrm{i}\varphi}|ge\rangle)/\sqrt{2}$ between
the two qubits, with $\varphi$ a relative phase.

Figure 3a shows the time-dependent qubit state populations $P_{e}$ for each
qubit, which agree well with a master equation simulation. Following capture
of the half-phonon, we perform quantum state tomography at time
$t_{\mathrm{m}}=$750\text{\,}\mathrm{ns}$$; these measurements are used to
reconstruct the two-qubit density matrix $\rho$ shown in Fig. 3b. We find a
Bell state fidelity
$\mathcal{F}_{\mathrm{Bell}}=\mathrm{Tr}~{}(\rho_{\mathrm{ideal}}\cdot\rho)=$72\text{\,}\mathrm{\char
37\relax}$$ and a concurrence $\mathcal{C}=$0.53\text{\,}$$, close to the
master equation simulation results, with a trace distance
$d^{\mathrm{Bell}}=\sqrt{\mathrm{Tr}~{}(\rho_{\mathrm{exp}}-\rho_{\mathrm{sim}})^{2}}=$0.13\text{\,}$$.

Figure 3: Phonon-mediated Bell state generation. a Inset: Pulse sequence for
Bell state generation. Main panel: Excited state probabilities for $Q_{1}$
(red) and $Q_{2}$ (blue) as a function of time. Green lines are results from a
master equation simulation. The final state is analyzed at
$t_{m}=$725\text{\,}\mathrm{n}\mathrm{s}$$ (gray dashed line). b Bell state
density matrix, absolute values, without readout correction, measured at
$t_{\mathrm{m}}$. Red solid lines show values expected for an ideal Bell
state; black dashed lines show simulation results including qubit coherence
and phonon channel losses.

### Phonon-qubit dispersive interaction

Sensing traveling phonons without absorbing them would provide a highly useful
capability, as would being able to use a traveling phonon as a probe of a
remote quantum system, which we explore in a pair of related experiments.
First, we use a traveling phonon as a probe of a remote quantum two-level
system, shown in Fig. 4a. We use qubit $Q_{1}$ as the emitter and receiver of
a “half-phonon” that is detected interferometrically [25, 26] when returning
to $Q_{1}$. This allows us to measure how the phase of the traveling phonon is
affected by interacting dispersively with qubit $Q_{2}$, which serves as a
stand-in for a generic quantum system.

The pulse sequence for this state detection is shown to the right in Fig. 4a:
We first prepare $Q_{1}$ in its excited state, and emit a half-phonon, which
reflects from the distant transducer, whose coupling to $Q_{2}$ is turned on
during the reflection process, and the half-phonon interacts with $Q_{1}$ on
its return. During the half-phonon transit, we briefly shift $Q_{1}$’s
frequency so that $Q_{1}$’s excited state acquires a relative phase $\varphi$,
yielding an interferometric interaction with the returning half-phonon, either
interfering constructively to return $Q_{1}$ towards its excited state, or
destructively and having $Q_{1}$ emit its remaining energy and relax to its
ground state. In Fig. 4a, we show the final $Q_{1}$ population as a function
of the phase $\varphi$ (blue points), showing a characteristic interference
pattern with a visibility of $32\text{\,}\mathrm{\char 37\relax}$.

We repeat the experiment with $Q_{2}$ excited by a $\pi$ pulse at the
beginning of the experiment, with experiment otherwise unchanged; the results
are shown in Fig. 4a (salmon points). There are three effects on the
oscillation pattern: A slight increase in the oscillation minima, attributed
to a decrease of the phonon coherence [25] in its interaction with $Q_{2}$; a
more marked reduction of visibility attributed to an inadequate absorption of
the phonon wave packet; and, most significantly, a phase shift of
$\Delta\varphi_{\mathrm{exp}}=0.40\pi$ attributed to the dispersive
interaction between $Q_{2}$ and the traveling half-phonon, close to our fit-
free simulated value of $\Delta\varphi_{\mathrm{sim}}=0.41\pi$ (see
Supplementary Note 9). This last effect points to the interesting possibility
of using phonons as dispersive probes of other quantum systems.

In a separate experiment, shown in Fig. 4b, we swap the roles of the qubits,
so $Q_{2}$ is now used as a dispersive probe for the phonon released by
$Q_{1}$, using a Ramsey fringe measurement of $Q_{2}$. The pulse sequence is
shown to the right in Fig. 4b, where $Q_{2}$ is placed in the state
$(|g\rangle+e^{\mathrm{i}\theta}|e\rangle)/\sqrt{2}$ by the initial $\pi/2$
rotation, performed about an axis rotated in the $x-y$ plane of the Bloch
sphere by $\theta$, and the $\theta$-dependent evolution of $Q_{2}$ is
compared for where $Q_{1}$ is not excited (no probe phonon) to where $Q_{1}$
is excited and $Q_{2}$ interacts with the subsequently-released traveling
phonon. In the latter case the Ramsey fringe visibility is reduced, which we
attribute to leakage from $Q_{2}$ into the phononic channel, but we again
observe a significant phase shift, here as high as
$\Delta\theta_{\mathrm{exp}}=0.95\,\pi$ close to our simulation
$\Delta\theta_{\mathrm{sim}}=0.99\,\pi$.

Figure 4: Phonon interferomeric probe. Dispersive state-dependent
interferometric probe. a Schematic roles played by each element. b Blue
(salmon) points show the $\varphi$ dependence of $Q_{1}$’s final $P_{e}$ when
$Q_{2}$ is in $|g\rangle$ ($|e\rangle$), showing a dependence of the phonon
phase on $Q_{2}$’s state, with a shift of $\sim 0.4\pi$ rad. c Qubit pulse
sequences (see text for details). d Schematic roles played by each element for
the dispersive phase-dependent interferometric probe. e Ramsey interference in
$Q_{2}$ (blue circles) reveals $Q_{2}$’s dependence on the relative phase with
the phonon. When repeating the same protocol with no initial $\pi$ pulse on
$Q_{1}$, we measure the Ramsey interference of $Q_{2}$ (salmon squares),
shifted by $\pi$ compared to the first measurement. Blue (salmon) points show
the $\varphi$ dependence of $Q_{1}$’s final $P_{e}$ when $Q_{2}$ is in
$|g\rangle$ ($|e\rangle$), showing a dependence of the phonon phase on
$Q_{2}$’s state, with a shift of $\sim 0.4\pi$ rad. f Qubit pulse sequences,
similar to c except $Q_{2}$ is always placed in
$(|g\rangle+e^{\mathrm{i}\theta}|e\rangle)/\sqrt{2}$ and the
$\theta$-dependent $P_{e}^{Q2}$ is measured.

## DISCUSSION

In conclusion, we demonstrate controlled phonon-mediated quantum state
transfer and remote entanglement between two quantum nodes, each node
comprising a superconducting qubit with a time-variable coupler, using
individual itinerant SAW phonons traveling in an acoustic transmission line
after a controlled, on-demand release, followed by capture. Using this
architecture, we also demonstrate the dispersive interaction between an
itinerant phonon and a superconducting qubit. These results have been made
possible by the integration of broadband, highly unidirectional transducer in
a $2\text{\,}\mathrm{mm}$ long phonon communication channel, as well as the
use of a quantum state protocol requiring tunable coupling to each qubit node
[2]. Achieving a quite impressive quantum state transfer fidelity of $82\pm
0.2\text{\,}\mathrm{\char 37\relax}$, limited by the loss in the phonon
channel, this platform paves the way for quantum-optics-like experiments
realized with individual phonons instead of photons.

## METHODS

###  I.1 Device fabrication and characterization

The device used in these experiments comprises two dies, a sapphire die with
the two superconducting qubits ($Q_{1}$ and $Q_{2}$), and their associated
tunable couplers ($G_{1}$ and $G_{2}$, respectively), as well as control and
readout wiring, and a lithium niobate die with the phononic channel and the
two unidirectional transducers. The two dies are fabricated separately then
flip-chip assembled [30]. The full circuit schematic is shown in Fig. 1b,

The acoustic die is fabricated using a single layer of
$\sim$25\text{\,}\mathrm{nm}$$ thick aluminium patterned by PMMA liftoff on a
LiNbO3 wafer, $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick. The central
part of the acoustic device is the $\ell=$2\text{\,}\mathrm{mm}$$-long
phononic channel, with width $W=$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
terminated at each end by a unidirectional transducer (UDT1,2).

The unidirectional transducers (UDTs) are described more completely in
Supplementary Note 1. Briefly, the two (nominally identical) unidirectional
transducers (UDTs) each comprise a standard bi-directional interdigitated
transducer (IDT) combined with an acoustic mirror (a reflective grating). The
IDT emits equal-amplitude acoustic excitations in opposite directions, one
towards and the other away from the second UDT. The acoustic mirror, placed
immediately adjacent to the IDT on the side opposite the second UDT, reflects
its incident excitation back towards the second UDT, such that it interferes
constructively with the other excitation. Each UDT is coupled inductively to
one of the two qubits.

We have separately characterized similar IDT-mirror designs, where in the
frequency band from about $3.85$ to $4\text{\,}\mathrm{GHz}$, excellent
directionality is achieved, with emission from the UDT almost entirely
directed away from the mirror. Typical directivities are greater than
$20\text{\,}\mathrm{dB}$. Outside this unidirectional band, the mirrors are
less effective and the devices emit more strongly in both directions [27].

The superconducting qubit die is fabricated on
$430\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick sapphire using standard
lithographic processing [15]. The qubits $Q_{1,2}$ are tunable xmon-style
qubits [31, 32], where each qubit’s frequency is controlled by a flux line
$Q_{\mathrm{z1,z2}}$, and excited using a capacitively-coupled microwave line
$Q_{\mathrm{xy1,xy2}}$. Each qubit is coupled to the SAW chip through a
superconducting tunable coupler $G_{1,2}$, whose coupling is controlled [33]
using external flux lines $G_{\mathrm{z1,z2}}$. Qubit states are inferred from
standard dispersive measurements using a separate readout resonator for each
qubit. The readout resonators are connected to a common readout line; more
details are given in the Supplementary Note 1.

The qubits are characterized with their couplers turned off (see Supplementary
Notes 6 and 7). At the qubit idle frequency
$\omega_{\mathrm{idle}}/(2\pi)\sim$4.3\text{\,}\mathrm{GHz}$$, we find the
qubits have an energy relaxation time
$T_{1}=$57\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ ($Q_{1}$) and
$38\text{\,}\mathrm{\SIUnitSymbolMicro s}$ ($Q_{2}$), with a coherence time
$T_{2}^{\mathrm{Ramsey}}=$1.11\text{\,}\mathrm{\SIUnitSymbolMicro s}$$
($Q_{1}$) and $0.88\text{\,}\mathrm{\SIUnitSymbolMicro s}$ ($Q_{2}$) (most
likely limited by flux noise as the qubits are tuned far away from their flux-
insensitive point). These times demonstrate the potential for excellent qubit
coherence when using a flip-chip assembly [30].

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request. Correspondence and requests for
materials should be addressed to A. N. Cleland (anc@uchicago.edu).

## ACKNOWLEDGEMENTS

Devices and experiments were supported by the Air Force Office of Scientific
Research and the Army Research Laboratory, and material for this work was
supported by the Department of Energy (DOE). É.D. was supported by LDRD funds
from Argonne National Laboratory, K.J.S. was supported by NSF GRFP (NSF
DGE-1144085) and A.N.C. was supported by the DOE, Office of Basic Energy
Sciences. This work was partially supported by the UChicago MRSEC (NSF
DMR-2011854), AFOSR under award FA9550-20-1-0270, the NSF QLCI for HQAN (NSF
Award 2016136) and made use of the Pritzker Nanofabrication Facility, which
receives support from SHyNE, a node of the National Science Foundation’s
National Nanotechnology Coordinated Infrastructure (NSF NNCI ECCS-2025633).

## COMPETING INTERESTS

The authors declare no competing financial or non-financial interests.

## AUTHOR CONTRIBUTIONS

É.D. designed, and fabricated the devices. É.D. performed the experiment and
analyzed the data. K.J.S., G.A.P. and M.-H.C. participated to the design
process of the unidirectional transducer. É.D., K.J.S., A.B., H.-S.C., J.G.,
Y.P.Z. developed the fabrication process of the superconducting circuit. É.D.,
K.J.S. and A.B. wrote code to model surface acoustic wave. A.N.C. advised on
all efforts. All authors contributed to discussion and production of the
manuscript.

Supplementary Discussion for “Quantum communication with itinerant surface
acoustic wave phonons”

##  II Device details and new unidirectional transducer design

Figure 5: Device construction. (a) Micrograph of the assembled flip-chip
device bonded in its sample holder. (b, c) Optical micrograph of the sapphire
qubit die with the two qubits ($Q_{1,2}$, blue), connected to their tunable
couplers ($G_{1,2}$, purple). The sapphire die is electrically coupled to the
lithium niobate flip-chip die through overlapping inductive leads (orange).
(d, e) Optical micrograph of the lithium niobate die with two unidirectional
transducers ($\mathrm{UDT}_{1,2}$, green).

The acoustic die is fabricated using a single layer of
$\sim$25\text{\,}\mathrm{nm}$$ thick aluminium patterned by PMMA liftoff on a
LiNbO3 wafer, $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick. The central
part of the acoustic device is the $\ell=$2\text{\,}\mathrm{mm}$$-long
phononic channel, with width $W=$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
terminated at each end by a novel unidirectional transducer (UDT1,2), see Fig.
5.

The two (nominally identical) unidirectional transducers (UDTs) each comprise
two main components: A standard bi-directional interdigitated transducer (IDT)
combined with an acoustic mirror. The IDT emits equal-amplitude acoustic
excitations in opposite directions, one towards ($T$) and the other away ($A$)
from the second UDT. The acoustic mirror, placed immediately adjacent to the
IDT on the side opposite the second UDT, reflects its incident excitation $A$
back towards the second UDT, such that it interferes constructively with the
excitation $T$.

The IDT itself comprises a double-electrode tapered IDT [28] with
$N^{\mathrm{IDT}}=24$ cells, designed for a center wavelength
$\lambda_{0}^{\mathrm{IDT}}=$975\text{\,}\mathrm{nm}$$. The acoustic mirror
comprises a tapered open-circuit reflective grating [28] with
$N^{\mathrm{mirror}}=488$ electrodes, placed $500\text{\,}\mathrm{nm}$ from
the last finger of the IDT, with a design center wavelength
$\lambda_{0}^{\mathrm{mirror}}=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The
grating center wavelength is slightly larger than the IDT center wavelength to
compensate for frequency-loading of the IDT by the superconducting circuit.
Each IDT is inductively coupled to one of the qubits on the sapphire chip.

We have separately characterized similar IDT-mirror designs, where in the
frequency band from about $3.85$ to $4\text{\,}\mathrm{GHz}$, excellent
directionality is achieved, with emission from the UDT almost entirely
directed away from the mirror. Typical directivities are greater than
$20\text{\,}\mathrm{dB}$. Outside this unidirectional band, the mirrors are
less effective and the devices emit more strongly in both directions [27].

As mentioned in the main text, from measurements of the response of a single
qubit coupled to its transducer, shown in Fig. 1 in the main text, we estimate
the unidirectional band of the transducers to be about
$140\text{\,}\mathrm{MHz}$ wide, centered at $3.95\text{\,}\mathrm{GHz}$, in
accordance with room-temperature measurements of similar devices. Outside the
unidirectional active band, we see a complex structure in $P_{e}$ as a
function of frequency and interaction time, with broad swings of width
$\sim$150\text{\,}\mathrm{MHz}$$ superposed with narrow oscillations of width
$\sim$7\text{\,}\mathrm{MHz}$$. The broad swings and finer details are in
accordance with expectations, including simulations in which we construct the
detailed response of the transducers and the channel using $P$-matrix theory
[28].

We note that inside the active unidirectional transducer band, a small feature
appears at $3.92\text{\,}\mathrm{GHz}$, which we interpret as a destructive
interference between the transducer and its adjacent mirror, which impedes
phonon emission over a bandwidth of $\sim$15\text{\,}\mathrm{MHz}$$. This
allows us to extract the effective distance between the transducer and the
mirror $d_{\mathrm{IDT-mirror}}=$140\text{\,}\mathrm{nm}$$, which is larger
than the design value of $50\text{\,}\mathrm{nm}$, perhaps due to an under-
compensation of the effect of the transducer and mirror tapers. This
interference however has no effect on measurements performed away from this
frequency.

The superconducting qubit die is fabricated on
$430\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick sapphire using standard
lithographic processing [15]. The qubits $Q_{1,2}$ are tunable xmon-style
qubits [31, 32], where each qubit’s frequency is controlled by a flux line
$Q_{\mathrm{z1,z2}}$, and excited using a capacitively-coupled microwave line
$Q_{\mathrm{xy1,xy2}}$. Each qubit is coupled to the SAW chip through a
superconducting tunable coupler $G_{1,2}$, whose coupling is controlled [33]
using external flux lines $G_{\mathrm{z1,z2}}$. Qubit states are inferred from
standard dispersive measurements using a separate readout resonator for each
qubit. The readout resonators are connected to a common readout line.

##  III Flux crosstalk calibration

All experiments rely on a precise control of the tunable coupler settings and
the qubit frequencies. Tuning is achieved by applying local flux bias using
microwave cables connected to room temperature digital electronics. We note
that an appreciable level of cross-talk was measured between the each flux
control line of our chip and the flux-tunable elements. This crosstalk was
calibrated by measuring the frequency shift of each qubit due to each flux
lines. We then assumed that the crosstalk was symmetric between each coupler-
qubit pair, for example $Q_{1}$ to $G_{1}$, allowing us to obtain a crosstalk
matrix which was used through linear inversion to cancel the crosstalk.

##  IV Calibration of qubit frequency shifts due to tunable couplers

The coupler design[33] is an inductive $\pi$ network whose center inductance
is a Josephson junction, whose phase and thus effective inductance can be
tuned from a finite inductance to a very large positive inductance and then
immediately to a large negative inductance and back to a finite (negative)
inductance by applying an external flux. This results in varying the coupling
from a finite value through zero to a (negative) finite value. Tuning the
coupler however affects its associated qubit frequency, as the inductance of
the coupler determines in part the qubit frequency. We calibrate the flux
required on the qubit flux bias line to cancel this induced frequency shift.
By doing this for both qubit-coupler pairs, we ensure control of the qubit
frequency during any experiment.

##  V Phononic channel loss characterization

To estimate the phononic channel loss, we must ensure there is no parallel
loss from the qubits. This is achieved by turning off the coupling between
each qubit and its associated transducer. With the coupler off, we measure the
intrinsic qubit lifetime; see Table. 1. We next have to calibrate the qubits
and couplers to achieve an optimal shaped phonon envelope. This calibration is
checked by monitoring the qubit excited state population time dependence
$P_{e}(t)$.

Given the emitted phonon amplitude envelope $c_{\mathrm{out}}(t)$, the
corresponding qubit excited state population $P_{\mathrm{e}}(t)$ can be
determined, in the absence of other qubit loss processes. Experimentally, we
compare the measured $P_{\mathrm{e}}^{\mathrm{m}}(t)$ to the expected
$P^{\mathrm{e}}_{\mathrm{e}}(t)$ for each $c_{\mathrm{out}}(t)$. This process
is relatively straightforward in the unidirectional regime but somewhat more
challenging in the bidirectional regime, likely due to the slower emission
rate in the less symmetrical admittance environment.

We then measure the qubit excited state population while sweeping the delay
between the emission and absorption; this is shown in Fig. 6. We observe a
mathematically geometrical damping of the qubit excited state revival as
function of the number of round-trips. From this measurement we extract a
mechanical loss of
$\alpha=$173\text{\,}\mathrm{Np}\text{\,}{\mathrm{m}}^{-1}$$.

Figure 6: Optimized multiple phonon emission and absorption from $Q_{1}$,
left, and $Q_{2}$, right. In this measurement the losses in the phononic
channel are the dominant loss. From a simple exponential fit we extract a mean
$T_{\mathrm{saw}}=$1.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, which
corresponds to a loss rate
$\alpha=$173\text{\,}\mathrm{Np}\text{\,}{\mathrm{m}}^{-1}$$ and a free space
velocity $v_{\mathrm{f}}=$3863\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$.

##  VI Phonon thermalization

Figure 7: Qubit cooling and thermometry. (a) Pulse sequences showing the
phonon-based cooling procedure followed by a qubit excited state measurement.
(b) Measurement of the qubit state populations $P_{g}$ and $P_{e}$, for
$Q_{1}$ on the left and $Q_{2}$ on the right. The upper (lower) panels show
the population of the qubit ground (excited) state, respectively. Blue circles
and orange squares show the measurement results with and without phononic
cooling. Solid lines show simple cosine fits used to extract the relative
amplitude of each curve, from which the qubit temperatures are deduced.

We typically find that our superconducting qubits have higher quiescent
excited state populations than indicated by the environmental temperature, and
higher than that of the phononic devices. For every measurement reported in
the main text and in the supplementary, we used a cooling procedure that takes
advantage of the lower temperature phonon environment. We set the qubit
couplers to maximum coupling, and bring each qubit into resonance with their
respective UDT at $\sim$4.15\text{\,}\mathrm{GHz}$$, outside of the
unidirectional bandwidth. The qubits are left in this state for
$\sim$300\text{\,}\mathrm{ns}$$, long enough for the qubits to relax to the
environmental phononic temperature.

To measure the effect of this cooling protocol, we implement the experiment
described in Ref. [34] to measure the qubit excited state population, with and
without the cooling procedure. The pulse sequence and results are shown in
Fig. 7. We measure that cooling procedure consistently lowers the qubit
excited state population from $1\text{\,}\mathrm{\char 37\relax}$
($7\text{\,}\mathrm{\char 37\relax}$) to $0.5\text{\,}\mathrm{\char 37\relax}$
($0.7\text{\,}\mathrm{\char 37\relax}$) for $Q_{1}$ ($Q_{2}$).

##  VII Multiplexed qubit readout: Visibility and readout correction

Figure 8: Multi-qubit readout measurement. Each measurement consists of
placing the qubit in its ground or excited state, followed by a measurement of
the amplitude and phase of the readout signal. This figure shows the typical
qubit ground and excited state measurement, after optimization of the control
and readout sequences.

The qubit ground, excited and third states ($|g\rangle,|e\rangle$ and
$|f\rangle$) are discriminated using a dispersive readout scheme, where the
frequency shift induced by each qubit $Q_{1,2}$ on its dedicated readout
resonator $R_{1,2}$ is used to infer the qubit state. These are single-shot
measurements that ideally leave the qubit in its measured state post-
measurement. Each readout resonator is inductively-coupled to a Purcell filter
[35, 36] to allow for a stronger qubit-resonator coupling without degrading
the qubit lifetime, allowing for a faster qubit state readout. The Purcell
filter consists of a short-circuited $\lambda/2$ coplanar waveguide resonator,
with an under-coupled input port and an over-coupled output port, such that
the ratio of the coupling quality factors
$Q_{\mathrm{in}}/Q_{\mathrm{out}}>100$ by design. This maximizes the readout
signal coupled into the common output transmission line that goes to the
first-stage amplifier.

The output port is connected to a double junction isolator (Low Noise Factory)
placed on the dilution refrigerator mixing chamber stage, followed by a high
electron mobility transistor (HEMT) amplifier (Low Noise Factory) at the
$4\text{\,}\mathrm{K}$ stage. Further amplification is provided by room
temperature amplifiers (MITEQ Corp.). Detection is performed using a
heterodyne scheme implemented using custom electronics with a
$1\text{\,}\mathrm{GHz}$ sampling rate and a $250\text{\,}\mathrm{MHz}$
bandwidth, which extracts the amplitude and phase, or equivalently the $I$ and
$Q$ quadrature coordinates, of the readout signal. Qubit state discrimination
uses a threshold optimization technique shown in Fig. 8, where the $I-Q$ plane
is divided into two parts and each measured $I-Q$ coordinate pair is assigned
to the qubit ground $|g\rangle$ or excited $|e\rangle$ state.

Typical single qubit readout fidelities in this experiment were
$\mathcal{F}_{\mathrm{readout}}^{Q_{1}}\sim$95\text{\,}\mathrm{\char
37\relax}$$ and
$\mathcal{F}_{\mathrm{readout}}^{Q_{2}}\sim$94\text{\,}\mathrm{\char
37\relax}$$. We chose to measure both qubits simultaneously using multiplexed
readout signals, with a single $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$
readout pulse applied and measured through the readout line. We characterize
the readout visibility by measuring the matrix $V$ defined as
$P_{\mathrm{meas}}=VP_{\mathrm{exp}}$, the transformation between the measured
probability vector and the probability vector expected given the control
sequence, in the two-qubit basis
$\\{|gg\rangle,|ge\rangle,|eg\rangle,|ee\rangle\\}$. Typical readout
characterizations are averaged $20,000$ times. A typical visibility matrix is
shown here:

$V=\left[\begin{array}[]{cccc}0.959&0.015&0.025&<0.001\\\
0.031&0.946&0.001&0.023\\\ 0.033&0.001&0.949&0.018\\\
0.001&0.036&0.031&0.932\\\ \end{array}\right]$ (1)

An ideal $V$ matrix is the identity matrix, with a trace equal to $4$. In this
example, the trace is $3.79$, which is equivalent to a total visibility of
$94.65\text{\,}\mathrm{\char 37\relax}$ for the multiplexed readout. Through
linear inversion of this matrix we apply a readout correction to our
measurement, giving $P_{\mathrm{exp}}$ in terms of $P_{\mathrm{meas}}$. We
note that this correction should not affect the quantum process fidelities
calculated in the main text, which we have checked separately.

By contrast, for the Bell state fidelity, this readout correction does affect
the result, and the results reported in the main text are not corrected in
this way, as stated there.

For completeness, we note that using the readout correction increases the
state fidelity from $72\text{\,}\mathrm{\char 37\relax}$ to
$79\text{\,}\mathrm{\char 37\relax}$. Finally, the qubit excited state
populations reported in the main text are calculated using
$P_{e}^{Q_{1}}=P_{|eg\rangle}+P_{|ee\rangle}$ and
$P_{e}^{Q_{2}}=P_{|ge\rangle}+P_{|ee\rangle}$.

##  VIII Device parameters

We summarize the device parameters in Table. 1.

Purcell filter |
---|---
Idle frequency [$\text{\,}\mathrm{GHz}$] | $5.118$
Coupling quality factor | 10
Readout resonators | $R_{1}$ | $R_{2}$
Idle frequency [$\text{\,}\mathrm{GHz}$] | $5.364$ | $5.318$
Internal quality factor ($\times 10^{3}$) | $84$ | $86$
Coupling quality factor ($\times 10^{4}$) | $4$ | $9$
Qubits | $Q_{1}$ | $Q_{2}$
Maximum frequency $f\left(\Phi=0\Phi_{0}\right)$ [$\text{\,}\mathrm{GHz}$] | 5.03 | 5.18
Idle frequency [$\text{\,}\mathrm{GHz}$] | $4.221$ | $4.359$
Anharmonicity [$\text{\,}\mathrm{MHz}$] | $198$ | $198$
Thermal population without phononic thermalization | $0.0137$ | $0.0698$
Thermal population with phononic thermalisation | $0.0047$ | $0.0021$
Visibility | $94\text{\,}\mathrm{\char 37\relax}$ | $94\text{\,}\mathrm{\char 37\relax}$
Qubit capacitance (design value) [$\text{\,}\mathrm{fF}$] | $90$ | $90$
Qubit SQUID inductance [$\text{\,}\mathrm{nH}$] | $10.8$ | $10.1$
Qubit coherence at idle frequency |  |
Lifetime, $T_{1}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $57\pm 15$ | $38\pm 5$
Ramsey dephasing time, $T_{2}^{\mathrm{R}}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $1.11\pm 0.06$ | $0.88\pm 0.04$
Spin-Echo dephasing time, $T_{2}^{\mathrm{e}}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $3.8\pm 0.1$ | $3.3\pm 0.1$
Qubit coherence at bidirectional frequency, $4.102\text{\,}\mathrm{GHz}$ |  |
Lifetime, $T_{1}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $38\pm 15$ | $31\pm 9$
Ramsey dephasing time, $T_{2}^{\mathrm{R}}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $0.95\pm 0.08$ | $0.62\pm 0.06$
Spin-Echo dephasing time, $T_{2}^{\mathrm{e}}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $2.48\pm 0.06$ | $1.68\pm 0.16$
Qubit coherence at unidirectional transfer frequency, $3.976\text{\,}\mathrm{GHz}$ |  |
Lifetime, $T_{1}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $51\pm 9$ | $33\pm 5$
Ramsey dephasing time, $T_{2}^{\mathrm{R}}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $0.79\pm 0.08$ | $0.55\pm 0.26$
Spin-Echo dephasing time, $T_{2}^{\mathrm{e}}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | $2.48\pm 0.1$ | $2.26\pm 0.28$
Couplers | $G_{1}$ | $G_{2}$
Coupler inductance [$\text{\,}\mathrm{pH}$] | $910$ | $873$
Flip chip mutual (design) [$\text{\,}\mathrm{pH}$] | $160$ | $160$
Grounding inductance (design) [$\text{\,}\mathrm{pH}$] | $480$ | $480$
UDT | IDT | Mirror
Number of cells (design) | $20$ | $488$
Aperture [$\text{\,}\mathrm{\SIUnitSymbolMicro m}$] (design) | $150$ | $150$
Metallization ratio (SEM) | $0.52$ | $0.79$
Reflectivity | $0.009$ | $-0.60i$
Wavelength [$\text{\,}\mathrm{\SIUnitSymbolMicro m}$] (design) | $0.975$ | $1$
$\Delta v/v$ | $3.44\text{\,}\mathrm{\char 37\relax}$ | $2.70\text{\,}\mathrm{\char 37\relax}$
Effective IDT-mirror distance | $140\text{\,}\mathrm{nm}$ |
Free space losses | $173\text{\,}\mathrm{Np}\text{\,}{\mathrm{m}}^{-1}$ |
Table 1: Summary of device parameters

##  IX Experimental setup

Figure 9: Electronics and wiring diagram of the experimental setup.

The assembled flip-chip, shown in the main text in Fig. 1, is wire bonded with
25 $\mu$m diameter aluminum wirebonds to an aluminum sample holder with copper
traces. The sample holder is placed inside a cryogenic magnetic shield and
connected to the rest of the wiring using non-magnetic coaxial cables. The
full wiring and electronics are shown in Fig. 9. All experimental measurements
are performed using a standard commercial dry dilution refrigerator with a
base temperature below $7\text{\,}\mathrm{mK}$.

##  X Estimation of the dispersive shift

The dispersive shift happening between the itinerant phonon and the qubit is
estimated through a circuit model taking into account the full circuit i.e.
the qubit, the coupler, the flip-chip connection and the UDT. At this point,
all circuit parameters have been set from design or obtained through fitting
of other experiments. From the interaction duration $\Delta
t=$200\text{\,}\mathrm{ns}$$ for Fig. 4a and $\Delta
t=$190\text{\,}\mathrm{ns}$$ for Fig. 4b and the phonon frequency
$\omega_{\mathrm{phonon}}/(2\pi)=$3.976\text{\,}\mathrm{GHz}$$ we get a
dispersive shift of $\chi^{\mathrm{exp}}=$1.00\text{\,}\mathrm{MHz}$$ for Fig.
4a and $\chi^{\mathrm{exp}}=$2.63\text{\,}\mathrm{MHz}$$ for Fig. 4b. From our
model we estimate that during the interaction $Q_{2}$ frequency
$\omega_{Q_{2}}/(2\pi)=$4.190\text{\,}\mathrm{GHz}$$ and its decay rate into
the phononic channel $\kappa_{Q_{2}}/(2\pi)=$6\text{\,}\mathrm{MHz}$$ for Fig.
4a and $\kappa_{Q_{2}}/(2\pi)=$7.65\text{\,}\mathrm{MHz}$$ for Fig. 4b.
Approximating the UDT response to a harmonic oscillator, we obtain from the
same circuit model a phononic emission rate
$\kappa_{\mathrm{UDT}}/(2\pi)=$147\text{\,}\mathrm{MHz}$$, yielding through a
resonant Purcell effect [37] to an effective qubit-itinerant phonon coupling
$g/(2\pi)=$14.85\text{\,}\mathrm{MHz}$$ Fig. 4a and
$g/(2\pi)=$23.72\text{\,}\mathrm{MHz}$$ for Fig. 4b. From this we obtain a
dispersive shift of $\chi^{\mathrm{sim}}=$1.03\text{\,}\mathrm{MHz}$$ for Fig.
4a and $\chi^{\mathrm{sim}}=$2.63\text{\,}\mathrm{MHz}$$ for Fig. 4b.

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