††thanks: Equal contribution††thanks: Equal contribution

# A trapped ion quantum computer with robust entangling gates and quantum
coherent feedback

Tom Manovitz<EMAIL_ADDRESS>Yotam Shapira  Lior Gazit  Nitzan
Akerman  Roee Ozeri  Department of Physics of Complex Systems and AMOS,
Weizmann Institute of Science, Rehovot 7610001, Israel

(September 2021)

###### Abstract

Quantum computers are expected to achieve a significant speed-up over
classical computers in solving a range of computational problems. Chains of
ions held in a linear Paul trap are a promising platform for constructing such
quantum computers, due to their long coherence times and high quality of
control. Here we report on the construction of a small, five-qubit, universal
quantum computer using ${}^{88}\text{Sr}^{+}$ ions in an RF trap. All basic
operations, including initialization, quantum logic operations, and readout,
are performed with high fidelity. Selective two-qubit and single-qubit gates,
implemented using a narrow linewidth laser, comprise a universal gate set,
allowing realization of any unitary on the quantum register. We review the
main experimental tools, and describe in detail unique aspects of the
computer: the use of robust entangling gates and the development of a quantum
coherent feedback system through EMCCD camera acquisition. The latter is
necessary for carrying out quantum error correction protocols in future
experiments.

##  I Introduction

Ion traps are among the leading quantum computing platforms explored today due
to their strong isolation from the environment and naturally long coherence
times. Trapped-ion quantum systems offer extremely accurate and reliable
single qubit and many qubit operations; high fidelity state preparation and
measurement; and high-connectivity between qubits, supported by the long-range
Coulomb interaction between ions Ozeri (2011).

Several approaches are currently pursued in order to scale-up trapped-ion
quantum computing to a macroscopic number of qubits. These include shuttling
of ions between different trapping regions in a CCD-like array of traps
Kielpinski  _et al._ (2002); Wan  _et al._ (2019); Pino  _et al._ (2021a), and
the heralded entanglement of ions in different traps using Hong-Ou-Mandel
interference and detection of photons that were scattered by these ions Monroe
_et al._ (2014); Hucul  _et al._ (2015). While experimental progress along
these two lines is encouraging, large scale trapped-ion quantum computing is
yet to be demonstrated.

In the intermediate term, trapped-ion quantum computing is performed mostly
using registers of few-to-several tens of trapped-ion qubits in a single
Coulomb crystal. In these systems, different ions in the register are often
individually addressed with focused laser beams. Entanglement is generated
using the motional modes of the ion crystal as spin-dependent force mediators
between ions. The collective nature of the crystal motional modes leads to
all-to-all connectivity between ion qubits. This high connectivity was shown
to be an advantage in comparisons with other qubit technologies Linke  _et
al._ (2017). State detection of different qubits in the register is performed
in parallel by state-selective fluorescence and the imaging of these photons
on a sensitive CCD camera or an array of photon-detectors. Only in very few
such demonstrations the ability to perform mid-circuit measurement and
conditional coherent feedback was shown Egan  _et al._ (2020).

Several small trapped-ion quantum registers were demonstrated and benchmarked
in the last few years. Notable examples include Ca+ registers of optical
qubits at the University of Innsbruck and Alpine Quantum Computing Schindler
_et al._ (2013); Pogorelov  _et al._ (2021a); Yb+ registers by JQI, IonQ
Debnath  _et al._ (2016); Wright  _et al._ (2019) and the Tsinghua group Lu
_et al._ (2019); and shuttling-based trapped ion registers by NIST, ETH,
Honeywell and Mainz Home  _et al._ (2009); Negnevitsky  _et al._ (2018); Pino
_et al._ (2021b); Hilder  _et al._ (2021).

Mid-circuit measurements, and the ability to coherently operate on qubits
conditioned on the measurement outcome, is an integral part of many-quantum
computing protocols, most notably quantum error-correction. Mid-circuit
measurement and coherent feedback in multi-ion crystals present several
challenges. Firstly, in state-selective fluorescence it is difficult to
prevent neighbouring, quantum-information carrying, ions from scattering
photons and thus losing coherence. In recent demonstrations this difficulty
was mitigated by hiding logic ions through encoding them on superpositions in
a meta-stable optical level Schindler  _et al._ (2013), by using different
atomic species as readout and logic ions Home  _et al._ (2009); Negnevitsky
_et al._ (2018), and by shuttling ions to separate measurement regions Home
_et al._ (2009); Pino  _et al._ (2021b); Hilder  _et al._ (2021). Secondly,
coherent feedback requires the extraction of the measurement outcome to a
classical control system in times shorter than the system’s coherence time.
Multi-ion crystals are often read-out by CCD cameras that spatially resolve
the different ions. CCD readout, however, has a non-negligible constant time-
delay determined by the transfer time of charge through the CCD. In previous
demonstrations of trapped-ions mid-circuit measurement followed by coherent
feedback, the readout was performed using a fast photo-detectors rather than a
CCD camera.

Another challenge of quantum computing in multi-ion crystals is preserving
their high fidelity when applied to ion-pairs selected from a multi-ion
crystal. The higher effective mass of the crystal and the presence of more
normal modes renders entanglement gates more susceptible to noise. Recently,
robust entanglement gates were demonstrated using coherent control and dynamic
decoupling techniques in two-ion crystals Shapira  _et al._ (2018); Webb  _et
al._ (2018); Zarantonello  _et al._ (2019); Manovitz  _et al._ (2017). The
adaptation of such robust gates to multi-ion crystals is therefore a next
important step. A recent publication on an 11-qubit quantum computer reported
the implementation of entanglement gates that are robust against trap-
frequency errors Wright  _et al._ (2019).

Here we describe a 5-qubit quantum computer built at the Weizmann Institute of
Science. Our computer is based on a single crystal of Sr+ ions. Qubits are
encoded either on optically separated orbitals or in ground-state spin states.
We demonstrate mid-circuit measurement followed by coherent feedback using an
ion resolving imaging capability on a CCD camera. Furthermore, we apply two-
qubit gates on all ion pairs, using the crystal center-of-mass mode of motion.
Our gates were engineered to be robust against gate timing errors and off-
resonance carrier coupling. Such small-scale quantum computing modules are
expected to be the building blocks of future large-scale trapped ion computers
and therefore the improvement of their performance is important.

##  II System overview

Our quantum computer is based on a chain of ${}^{88}\text{Sr}^{+}$ ions held
in a linear Paul trap. The alkaline earth ${}^{88}\text{Sr}^{+}$ possesses a
single valence electron, resulting in a simple level structure with optical
transitions at convenient visible or near-infrared wavelengths (422, 674, 1092
and 1033 nm), easily generated using diodes lasers. The relevant level
structure for ${}^{88}\text{Sr}^{+}$ includes the ground state
$5S_{\frac{1}{2}}$; two long-lived excited states $4D_{\frac{3}{2}}$ (435 ms
lifetime) and $4D_{\frac{5}{2}}$ (390 ms); and two short-lived excited states,
$5P_{\frac{1}{2}}$ (8 ns) and $5P_{\frac{3}{2}}$ (8 ns). A diagram of the
these levels is shown in Fig. 1.

Two electromagnetic coils in a Helmholtz configuration produce a constant
magnetic field of $3-5$ G in the vicinity of the ions, defining a quantization
axis and Zeeman splitting all levels. A qubit can be encoded on the ion in two
ways: A Zeeman qubit is encoded using the two Zeeman-split (2.802 MHz/G) spin
levels of the ground state $5S_{\frac{1}{2}}$ manifold, while an optical qubit
is encoded by pairing one of the states in this manifold with one of the
states of the $4D_{\frac{5}{2}}$ orbital manifold (1.68 MHz/G). We primarily
make use of the optical qubit. In order to store and protect qubit information
during measurement, we encode qubits entirely in the $4D_{\frac{5}{2}}$
orbital manifold. Moreover, in order to selectively hide qubits from
interaction with the addressing laser, we may encode these qubits on varying
pairs of states in the $5S_{\frac{1}{2}}$ and $4D_{\frac{5}{2}}$ manifolds.

The ions are spatially confined to a one dimensional chain by a linear Paul
trap. A detailed description of our trap and vacuum system may be found in
Akerman  _et al._ (2011); Akerman (2012). The trap RF is at approximately 21
MHz; secular (harmonic) trapping frequencies are approximately 1-1.5 MHz along
the trap axis, corresponding to typical inter-ion spacing of about 3-4
micrometers; and 3-4 MHz along the radial axes. In order to load ions into the
trap, neutral strontium vapor, produced by heating nearby ovens, is photo-
ionized in the trapping region through a two-photon process: the $5s^{2}$
${}^{1}S_{0}$ ground state is excited to the $5s5p$ ${}^{1}P_{1}$ state and
then to the self-ionizing $5p^{2}$ ${}^{1}D_{2}$ state, using corresponding
461 nm and 405 nm lasers.

Optical control of the ions is realized using lasers at wavelengths of 422
$(5S_{\frac{1}{2}}\leftrightarrow 5P_{\frac{1}{2}})$, 674
$(5S_{\frac{1}{2}}\leftrightarrow 4D_{\frac{5}{2}})$, 1033
$(4D_{\frac{5}{2}}\leftrightarrow 5P_{\frac{3}{2}})$ and 1092 nm
$(4D_{\frac{3}{2}}\leftrightarrow 5P_{\frac{1}{2}})$. Lasers enter the vacuum
chamber through fused-silica windows on the sides or the top of the chamber.
All lasers are generated by diodes in an ECDL configuration and are locked to
stable external references. Of particular importance is the 674 nm laser,
which is used to coherently control the optical qubit. This laser is first
locked to a ultra-low expansion (ULE) Fabry-Perot cavity with high servo
bandwidth; in order to filter out the lock servo bumps, the cavity
transmission is amplified by a slave diode, and the output is then locked to
an additional temperature-stabilized ULE Fabry-Perot cavity, generating
narrow-linewidth light Peleg  _et al._ (2019).

Figure 1: ${}^{88}\text{Sr}^{+}$ level structure. (a) A 422 nm laser excites
the dipole transition from the ground state to the short lived (8 ns)
$5P_{\frac{1}{2}}$ state, used for Doppler cooling, EIT cooling, optical
pumping, and state-selective fluorescence state detection. A 674 narrow-
linewidth laser drives the narrow, dipole-forbidden transition between the
ground state and the long-lived (390 ms) excited state $4D_{\frac{5}{2}}$,
used for sideband cooling and coherent control of the optical qubit. 1092 and
1033 nm lasers are used to repump the $4D_{\frac{3}{2}}$ and
$4D_{\frac{5}{2}}$ levels, correspondingly. (b) A magnetic field Zeeman splits
all orbitals, including the ground and $4D_{\frac{5}{2}}$ orbitals which are
used to encode an optical qubit. Typically, the qubit is encoded on
$|1\rangle=|5S_{\frac{1}{2}},\frac{1}{2}\rangle$
$|0\rangle=|4D_{\frac{5}{2}},\frac{3}{2}\rangle$, as shown.

The ion chain is typically Doppler cooled to a temperature of a few mK using
the 422 nm laser, together with a 1092 nm laser to repump the long-lived
$4D_{\frac{3}{2}}$ orbital. The chain may then be further cooled to the ground
state of several, or all, motional modes using sideband and EIT cooling.
Sideband cooling is used to cool axial motional modes, and is performed by
tuning the 674 nm laser near the red-sideband resonance of the relevant mode,
together with the 1033 nm repump and 422 nm optical pumping lasers. Several
modes may be cooled in parallel by driving the 674 nm laser using multiple
frequencies. EIT cooling is used to cool all radial modes in parallel. A 422
nm $\sigma^{+}$-polarized pump and $\pi$-polarized probe isolate an effective
$\Lambda$-type 3-level system with a $k$-vector that is perpendicular to the
trap axis and equally overlaps both radial trapping axes. Blue detuned from
the transition, the pump creates a narrow asymmetric (Fano) resonance when
addressed by the probe beam, enabling efficient cooling with low background
scattering Lechner  _et al._ (2016).

Qubit initialization and measurement are realized using the 422 nm laser,
together with the 1092 repump laser to prevent pumping into the long-lived
$4D_{\frac{3}{2}}$ orbital. The qubits are prepared in the qubit ground state
$|1\rangle=|5S_{\frac{1}{2},+\frac{1}{2}}\rangle$ by optical pumping using a
$\sigma^{+}$ polarized 422 nm beam. This process prepares $|1\rangle$ with an
error smaller than $10^{-3}$ within $10$ $\mu$s. Strong measurement of the
optical qubit is achieved by state-selective fluorescence: ions are
illuminated with the light, which scatters off ions occupying $|1\rangle$ but
not off those occupying the qubit excited state
$|0\rangle=|4D_{\frac{5}{2},+\frac{3}{2}}\rangle$ (or any other state in the
$|4D_{\frac{5}{2}}\rangle$ manifold). The scattered fluorescence is imaged
onto an electron-multiplying CCD (EMCCD) camera, from which the state is
determined for each ion separately. Fluorescence is collected for 1 ms, giving
rise to an error of $2.6\times 10^{-3}$ due to the finite lifetime of
$|0\rangle$.

Coherent operations are performed with either the 674 nm laser (optical qubit)
or an RF magnetic field (Zeeman qubit). The latter is induced by running RF
current through an electrode which is 2 mm away from the ions, and is used to
drive transitions between Zeeman states within the $5S_{\frac{1}{2}}$ and
$4D_{\frac{5}{2}}$ manifolds. A narrow-linewidth Peleg  _et al._ (2019) 674 nm
laser, illuminating the ions through a global and an individual addressing
path, is used to drive coherent optical qubit operations. The global path
overlaps all ions in the chain with approximately uniform intensity. By
resonantly addressing the ions, this path can be used to apply global unitary
rotations of the form,

$U_{G}(\phi,\theta)=\exp{\left(i\frac{\theta}{2}\sum_{i}\sigma^{\phi}_{i}\right)}.$
(1)

Here $\sigma^{\phi}_{i}=\cos{(\phi)}X_{i}+\sin{(\phi)}Y_{i}$, while $X_{i}$
and $Y_{i}$ are the standard Pauli operators acting on the optical qubit
subspace of ion $i$. Using the Mølmer-Sørensen interaction, the global path
can also be used to generate global entangling operations of the form,

$U_{MS}(\phi,\theta)=\exp{\left(i\frac{\theta}{2}\sum_{i<j}\sigma^{\phi}_{i}\sigma^{\phi}_{j}\right)}.$
(2)

The individual addressing path is tightly focused onto a single ion at a time,
and can be quickly steered between ions using a pair of acousto-optic
deflectors (AODs) in an XY (perpendicular) configuration. For this path, the
laser is tuned off resonance to generate a light shift, and can therefore
realize the operations,

$U_{i}(\theta)=\exp{\left(i\frac{\theta}{2}Z_{i}\right)}.$ (3)

Here $Z_{i}$ is the according Pauli operator acting on the optical qubit
subspace of ion $i$. Together, these $U_{G}(\phi,\theta)$,
$U_{MS}(\phi,\theta)$ and $U_{i}(\theta)$ operations constitute a universal
gate set on the optical qubit register. Below we describe our implementation
of entangling and individual addressing operations in more detail.

##  III Individual qubit addressing and detection

Universal quantum computation requires the ability to both operate on, and
measure, the state of any specific qubit in the register. In ion traps, the
inter-ion spacing of a few micrometers James (2000) is larger than optical
wavelengths, implying that one can use focused laser beams for resolving the
position of each ion in the chain Schindler  _et al._ (2013); Debnath  _et
al._ (2016); Pogorelov  _et al._ (2021b). However, while larger than optical
wavelengths, inter-ion spacing is not _much_ larger, meaning high numerical
apertures are necessary and no more than minor aberrations can be tolerated
for individual addressing to work with sufficiently low cross-talk error.
Other approaches to individual addressing include spectrally resolving ions
Mintert and Wunderlich (2001); Johanning  _et al._ (2009); Timoney  _et al._
(2011), coupling ions to spatially varying oscillating fields Navon  _et al._
(2013), or shuttling them to separate trapping regions Kielpinski  _et al._
(2002); Wan  _et al._ (2019); Pino  _et al._ (2021a)).

We performed both qubit-resolving state measurements as well as individual
addressing for single qubit rotations using the same high-resolution optical
system. Qubits were measured by collecting state-selective fluorescence on the
$5S_{\frac{1}{2}}\leftrightarrow 5P_{\frac{1}{2}}$ transition at 422 nm, which
was imaged onto an EMCCD camera (Andor iXon Ultra 897). We individually
addressed qubits by shining 674 nm light, near-resonant with the optical
$5S_{\frac{1}{2}}\leftrightarrow 4D_{\frac{5}{2}}$ transition, in a tightly
focused beam that spatially overlaps only a single ion at a time. The laser
beam is quickly steered between ions using two orthogonal acousto-optic
deflectors (AODs).

The imaging system objective (LENS-optics) has an effective focal length of 30
mm and a NA of 0.34, and is designed to correct for chromatic aberrations of
its lenses as well as the vacuum chamber fused-silica windows at both relevant
wavelengths (422 nm for detection and 674 nm for addressing). The objective is
mounted on a Newport xyz translation stage for positioning, which is itself
connected to a custom-made holder that is tightly anchored to the vacuum
chamber flange. This allows the objective to be positioned a few mm away from
the vacuum chamber window, which is sunken into the chamber and enables an
effective working distance of 30 mm from the ions. Anchoring the objective to
the chamber minimizes vibrations relative to the ion position, suppressing
beam pointing noise, which can be significant for tightly focused beams. The
objective is designed to work with the ions in its focal-plane using a
$\infty-f$ configuration. We slightly shift the objective away from this
working point so that it directly images the ions on the EMCCD at a distance
of approximately $120$ cm. This enables imaging the ions without adding any
additional optical elements, thus reducing aberrations.

Apart from the camera, all other optical elements are positioned on an optical
breadboard mounted directly above the trap by five aluminum legs. Diagonal
brace beams are added where possible to mitigate table vibrations. A 1.5-inch
diameter hole is bored through the table, directly above the objective and the
trap. A pair of mirrors change the optical beam path from the horizontal path
on the breadboard onto the vertical path descending into the trap, and vice
versa. Near these mirrors, a longpass dichroic mirror separates the detection
and addressing beam paths Manovitz (2016).

###  III.1 Individual addressing

The narrow linewidth 674 nm light is passed through a double-pass AOD and a
single pass AOM that provide amplitude and phase control. The light is then
delivered with an optical fiber to the optical breadboard on which it passes
through a pair of Isomet OAD948-633 AODs, held in an XY configuration. The
zero-order diffraction of the second AOD is picked up by a detector; the
signal from this detector is sent to a PID that controls an attenuator on the
RF for the single pass AOM in the 674 nm individual addressing line, and is
used to stabilize the laser power. The diffracted light passes through a pair
of lenses, followed by the dichroic mirror, and is focused onto the ions by
the objective lens.

The AODs were chosen to support the contradicting demands of high resolution
and short switching time. The switching time $\tau_{s}$ is determined by the
propagation time of the acoustic wave through the laser waist of size $d$ on
the AOD: $\tau_{s}=d/v$, where $v$ is the acoustic velocity. The resolution
$N$, defined as the number of resolved optical spots that the deflector can
scan, is given by $N=\tau_{s}\Delta f$, where $\Delta f$ is the frequency
bandwidth of the AOD. Hence, resolution can be increased at the cost of longer
switching times. The Isomet AODs provide a bandwidth $\Delta f=45$ MHz and an
acoustic velocity of $v=650$ m/s. Hence, a spot size of 1.1 mm defines
switching time $\tau\approx 1.7\mu s$ and resolution $N\approx 77$.

The AODs are set in an XY configuration so that the beam can be scanned across
a plane, rather than a line, enabling precise positioning of the beam focus on
each ion. The orientation relative to the trap is such that the ion chain
corresponds to a $45^{\circ}$ angle with respect to axes defined the XY
configuration, and opposite frequencies are added to the two AODs in order to
scan across the ion chain. Such a configuration simplifies control, as the 674
nm frequency does not need to be corrected when the focus position is changed
from one ion to another. Taking the XY configuration into account, the
resolution is effectively increased to $N\approx 108$. These parameters are
sufficient, as they support low time overhead compared to single qubit gates
$(\sim 10\mu s)$, and can easily accommodate a large ion chain $(<30\mu m)$.

Figure 2: a) Beam intensity profile of the individual addressing beam along
the ion chain axis, determined by measuring the position-dependent
$\Omega_{ls}$, which is proportional to the intensity. We measure a $1/e^{2}$
waist diameter of $2w_{0}=3.84\mu m$. b) In order to measure cross-talk even
for low values, we resonantly address the ions using the individual addressing
path. Cross-talk Rabi frequency ratio is approximately $10^{-1}$ for $3.71\mu
m$ inter-ion distance. c) An example of single qubit Rabi oscillations using
individual addressing in the light shift configuration. Here we use a five-ion
chain, with a minimal inter-ion distance of $2.83\mu m$.

The individual addressing system is used to address the ions off-resonantly,
generating a light shift with effective Rabi frequency
$\Omega_{ls}=\frac{\Omega^{2}_{res}}{\Delta}$, where $\Omega_{res}$ is the
resonant Rabi frequency and $\Delta$ is the detuning from resonance. Typical
parameters are $\Omega_{res}=500$ kHz and $\Delta=5$ MHz, implying an
effective light shift of $\Omega_{ls}=50$ kHz. Limiting the individual path to
$\sigma_{z}$ rotations via off resonant illumination has significant
advantages as compared with resonant coupling. Coherent resonant coupling from
both global and individual paths require maintaining phase coherence between
the two, which is challenging due to the very different optical trajectories.
Furthermore, off-resonant addressing minimizes crosstalk by effectively
decreasing the individual path spot size. While resonant coupling strength is
determined by the Rabi frequency $\Omega_{res}$, off-resonant coupling is
proportional to $\Omega_{res}^{2}$; hence the effective beam cross section
depends on the amplitude squared rather than the amplitude and leads to a
$\sqrt{2}$ narrower cross-section. $\sigma_{z}$ rotations in conjunction with
$\sigma^{\phi}$ global rotations can generate any single-qubit unitary.

We characterize the beam profile by measuring the lights-hift on the ion,
which is proportional to the laser intensity, as a function of AOD frequency,
or correspondingly, beam position. The results are shown in Fig. 2. A useful
measure of the spot size is the beam waist diameter $2w_{0}$, which is the
$1/e^{2}$ intensity diameter of a Gaussian beam. Making use of the full
aperture of our 0.34 NA objective, for a diffraction limited beam, one would
expect a beam waist diameter $2w_{0}=\frac{2\lambda}{\pi NA}\approx 1.3$ $\mu
m$. However, we measure a significantly larger beam waist diameter of
$2w_{0}=3.84$ $\mu m$. The discrepancy is a result of an effectively smaller
NA and aberrations. Use of the entire aperture of the objective would clip the
outer tail of the Gaussian profile; this would imply a focus on the ions that
is a convolution of a Gaussian and an Airy beam profile. The latter introduces
intensity oscillations that decay with distance from the beam center $r$ as
$1/r^{2}$, which may then generate non-negligible cross-talk between ions. To
avoid significant Airy-like features, the beam waist diameter on the objective
is set to be smaller than the objective aperture, implying an effective NA of
approximately $NA\approx 0.19$ which, for a diffraction-limited spot, would
result in a waist diameter $2.3\mu m$. In practice, we achieve a diameter that
is $\sim 1.7$ times larger due to aberrations, which are captured by a beam
quality parameter $M^{2}\approx 1.7$.

As long as the distance between ions is larger than the beam waist radius of
$w_{0}=1.92\mu m$, which is always true for our experiments, the cross-talk is
determined by the beam tail. Perfectly Gaussian beams decay strongly, so in
practice tails are dominated by small aberrations, such as weak Airy
contributions from the aperture cutoff, or minor astigmatism and coma from
imperfect alignment. Hence, the beam waist is not necessarily a reliable
parameter for evaluating cross-talk. The cross-talk parameter of interest is
$R_{ls}=\max_{i,j}{\frac{\Omega^{i,j}_{ls}}{\Omega^{i,i}_{ls}}}$, where
$\Omega_{ls}^{i,j}$ is the light-shift induced on ion $j$ when attempting to
address ion $i$. $R_{ls}$ can be used to give an upper bound on errors
generated by cross-talk per single-qubit operation: $\epsilon_{crosstalk}\leq
2R_{ls}^{2}$ (under the assumption that nearest-neighbour cross-talk is the
main source of error). As the $R_{ls}$ is quite low, it is difficult to
properly measure directly. Instead, we measure the resonant cross-talk
parameter $R_{res}=\max_{i,j}{\frac{\Omega^{i,j}_{res}}{\Omega^{i,i}_{res}}}$,
from which one can straightforwardly derive $R_{ls}=R_{res}^{2}$. To do so, we
trap a 3-ion chain with inter ion distances $3.7\mu m$ (a typical inter-ion
distance for small chains in our trap), drive resonant Rabi oscillations while
attempting to address each ion, and measure the Rabi frequency on all ions
simultaneously. The results are shown in Fig. 2. For an inter-ion distance of
3.7 $\mu m$s, we measure $R_{res}=0.12(1)$, implying $R_{ls}=0.014(2)$ and
$\epsilon_{crosstalk}\leq 3.9(5)\times 10^{-4}$. Hence, for such inter-ion
distances, the error induced by cross-talk is significantly lower than other
error sources. An example of single qubit Rabi rotations in a five-qubit
chain, induced by light-shift individual addressing, is shown in Fig. 2 (c),
with no discernible cross talk.

The error incurred by an individual addressing operation on one of the ions is
not limited to cross-talk; indeed, we find that it is dominated by other
terms. We measured the fidelity of an individual addressing $\sigma^{z}$ $\pi$
angle rotation. This was achieved by initializing the 3 ions in a
$|111\rangle$ state, applying a global $\sigma^{x}$ $\pi/2$ pulse, followed by
an individual addressing $\sigma^{z}$ $\pi$ pulse on one of the ions, and then
a global $\sigma^{x}$ $-\pi/2$ pulse (applied by varying the phase of the
global 674 nm laser by $\pi$). The desired result is a chain in which only one
of the qubits is in the excited state, e.g. the $|101\rangle$ state. After
correcting for state preparation and measurement (SPAM) errors due to the
global shelving pulses, optical pumping initialization, and measurement
fidelity, we arrive at a fidelity of $0.9964(10)$ for a $\sigma^{z}$ $\pi$
pulse. We also find that while the target ion undergoes an operation, other
ions suffer an error of $0.001-0.002$ per ion; this may be due to background
$\sigma^{z}$ noise.

###  III.2 Detection

Qubit state detection is performed through state-selective fluorescence
scattering of the 422 nm laser on the $5S_{\frac{1}{2}}\leftrightarrow
5P_{\frac{1}{2}}$ transition: fluorescence is observed only if the electron
collapses to the $5S_{\frac{1}{2}}$ state. The scattered fluorescence is
picked up by the LENS-Optics 0.34 NA objective, which focuses the images of
the ions directly onto an Andor iXon Ultra 897 electron multiplying charge
coupled-device (EMCCD) camera, placed approximately 1230 mm away from the
objective. This results in a magnification of about $\times 41$, measured by
comparing distance between ions on camera to calculated values James (2000).
Apart from the objective itself and aligning mirrors, the only optical element
in the path of fluorescence light is a dove prism; the prism is used to rotate
the chain image such that it is aligned horizontally on the EMCCD, and is
parallel with the pixel axis, allowing for a smaller pixel region of interest
(ROI) and hence faster readout. The camera is placed on a custom made holder,
physically separated from any other element in the system, in order to
suppress detrimental acoustic vibrations caused by the camera cooling fans.

The main advantage of camera detection is the spatial resolution that it
provides, allowing for simple differentiation between ions. Beyond the optical
resolution, the iXon Ultra has high quantum efficiency; high readout speeds
for a CCD camera; thermoelectrically cools the CCD array to $-90^{\circ}$ C,
substantially mitigating thermal detection noise; and possesses an electron-
multiplying capability that can increase gain to $\times 500$. These
attributes are important advantages for ion detection, giving good signal-to-
noise ratios (S/N) for short detection times. However, the iXon Ultra uses a
masked CCD storage area (designed to improve duty cycle); transferring the
signal through the additional CCD frame storage area implies a readout time
overhead of approximately $300-400\mu s$.

Typically, we define a rectangular region of interest (ROI) on the camera of
approximately $6\times 20-40$ pixels, where the latter depends on the number
of ions in the chain. Vertically smaller ROIs result in lower readout times
for the camera, and are thus beneficial. Readout times can be further
minimized by placing the ion chain image in the corner of the pixel array that
is closest to the analog-to-digital-converter (ADC). We achieve typical
readout times (not including the detection exposure time itself) of 600-700
$\mu s$ including the frame storage overhead.

During a standard calibration procedure, a set of 1500 images of the ions
fluorescing, and 1500 images of the ions not fluorescing, are acquired. Then,
using these images, a set of pixels are allocated to each ion. For each ion,
the distribution of the signal sum on the corresponding pixels are compared
for the fluorescing and non fluorescing case, and an intermediate value is
chosen as a threshold discriminator. After calibration, for every measurement
and for every ion, the pixel sum is compared to this threshold in order to
differentiate the fluorescing state (implying $|1\rangle=5S_{\frac{1}{2}}$)
from a non fluorescing state (implying $|0\rangle=4D_{\frac{5}{2}}$).

For exposure times of 1 millisecond, we find that detection error is dominated
by spontaneous emission of the $4D_{5/2}$ orbital, giving a detection error of
$1-e^{-1/390}\approx 2.6\times 10^{-3}$. We perform $2\times 10^{4}$ detection
measurements, where the ion state is predetermined by the application; or lack
thereof; of the 1092 nm repump laser. When this laser is turned off,
spontaneous emission errors are practically undetectable; if an emission event
occurs, the electron is quickly pumped back into the non-fluorescing
$4D_{\frac{3}{2}}$ state and remains there until another emission event or the
end of the measurement. With spontaneous emission suppressed, we find no
errors for these $2\times 10^{4}$ measurements, implying with high confidence
an error rate lower than $<10^{-3}$ from other sources. Thus we conclude the
spontaneous emission error is dominant.

Figure 3: An image of an 8-ion chain on the iXon Ultra camera. Here the
minimal inter-ion distance is about $2.3\mu m$. The ion chain is aligned along
the pixel axis using a dove prism set before the camera.

For a series of acquisitions, each image is read via ADC and saved in the
camera’s internal buffer. The buffer is read out at the end of the series by a
USB, where the measurement results are determined retroactively through the
simple image analysis described above. In certain scenarios (e.g. QEC)
determining the measurement result retroactively is not good enough, and a
real-time decision must be made. We achieve this as well with the Andor iXon
camera, as described in a succeeding section detailing our real time quantum
coherent feedback module.

##  IV Robust Entangling Gates

Entangling gates are a key component in any realization of a quantum computer.
They are typically the most challenging gates of the universal gate set and in
ion quantum processors are often substantially slower and less robust than
single qubit operations. The most prevalent method to entangle qubits in an
ion trap quantum processor is by using the Mølmer-Sørensen (MS) technique
Sørensen and Mølmer (1999, 2000) in which a spin-dependent force is exerted on
a collective normal-mode of motion of the ion chain, resulting in an effective
qubit-qubit interaction.

MS gates are very successful in two-ion chains Gaebler  _et al._ (2016);
Ballance  _et al._ (2016); Srinivas  _et al._ (2021); Clark  _et al._ (2021).
However, in long ion-chains, the gates have an increased sensitivity to even
small implementation imperfections. Here we utilize an entangling gate
technique that employs coherent control in order to decrease the sensitivity
of the gate operation to experimental imperfections Shapira  _et al._ (2018);
Webb  _et al._ (2018). This technique is an extension of the MS gate and is
beneficial for scaling up the ion qubit register. We demonstrate our
entangling scheme on pairs of ions in a five ion chain, and in addition as an
all-to-all gate in a four ion chain.

Our method is based on driving the gate with a multi-tone laser that is tuned
on resonance with the optical qubit transition along with a sequence of
amplitude modulations at frequencies $\omega_{j}=\nu+n_{j}\xi$, where $\nu$ is
the frequency of the normal-mode of motion used to mediate the interaction,
$n_{j}$ is an integer and $\xi$ is a detuning, such that the gate time is
$T=2\pi/\xi$. One may write the interaction Hamiltonian due to this driving
field as,

$V=\frac{1}{2}\Omega\sum_{i}\sigma^{x}_{i}\cos(kx_{i}-\omega_{eg}t)\sum_{j}r_{j}\cos\left(\omega_{j}t\right).$
(4)

Here $\Omega$ is the resonant Rabi frequency, $\sigma^{x}_{i}$ is the Pauli
operator on the optical qubit subspace of ion $i$, $k$ is the optical
wavenumber, $x_{i}$ is the position of ion $i$, $\omega_{eg}$ is the resonant
frequency for the optical qubit transition,and $r_{j}$ represents the
magnitude of the amplitude modulation at frequency $\omega_{j}$. The regular
MS gate is formed by the special choice of a single modulation frequency with
$n=1,r=1$, $\xi=2\eta\Omega$ and $\eta=kx_{0}$, where $x_{0}$ is the ground
state extent of the relevant motional mode.

Figure 4: (a) Fidelities for generating a two-qubit maximally entangled state
by applying an entangling gate for every pair of qubits on a 5-ion chain. The
values are SPAM-corrected, with an average fidelity of 0.973(2). The
entangling gate is a cardioid (1,3) Mølmer-Sørensen, applied with an echo
pulse midway through the gate. (b) An example parity measurement on a pair of
qubits in the chain.

A derivation similar to that found in Shapira  _et al._ (2018) shows that the
fidelity of a two ion entangling gate in an $N$ ion chain is given by,

$\begin{split}F_{2,N}=&\frac{1}{8}\bigg{[}4e^{-\frac{F^{2}+G^{2}}{2}\left(\bar{n}+\frac{1}{2}\right)}\cos\left(N\left(A+\frac{FG}{2}-\frac{\pi}{2}\right)\right)\\\
&\cdot\cos\left(A+\frac{FG}{2}-\frac{\pi}{2}\right)+e^{-2\left(F^{2}+G^{2}\right)\left(\bar{n}+\frac{1}{2}\right)}\\\
&\cdot\cos\left(2N\left(A+\frac{FG}{2}-\frac{\pi}{2}\right)\right)+3\bigg{]}\end{split}$
(5)

where $G+iF=\int\limits_{0}^{T}dt\sum_{j}r_{j}e^{in_{j}\xi t}$,
$A=-\int\limits_{0}^{T}dtgF$, and $\bar{n}$ is the mean occupation of the
normal mode, assuming an initial thermal state. Expanding Eq. (5) in powers of
$N$ we find that for a MS gate, the infidelities due to an inaccurate gate
time, $T$, or normal-mode frequency, $\nu$, scale quadratically with $N$.
Thus, using a gate that is robust to these type of errors is necessary for
scaling up the ion-trap quantum processor. Similar scaling laws occur for
$N$-qubit all-to-all entangling gates. Furthermore, the gate time typically
scales as $\sqrt{N}$ resulting in an even increased sensitivity to errors,
which can be mitigated only by coupling to additional normal-modes of motion
of the ion-chain Shapira  _et al._ (2020).

By choosing $\\{n_{j},r_{j}\\}$ correctly, one can use the additional degrees
of freedom to suppress unwanted sources of infidelity, in an order-by-order
manner. In our quantum information processor, we use two modulation terms
rather than one which leads to a Cardioid-shaped trajectory of the normal-mode
in phase space. A Cardioid gate eliminates the leading order contribution of
carrier coupling and errors due to timing inaccuracies, and reduces errors due
to normal-mode frequency inaccuracies and normal-mode heating.

While the most efficient implementation uses harmonics $n_{j}=1,2$, we
realized the technique with the harmonics $n_{j}=2,3$ or $n_{j}=1,3$, in order
to avoid errors caused by spurious frequencies due to third-order
nonlinearities in the AOM or the RF amplifier. The double amplitude modulation
is equivalent to a four-frequency field which is generated by driving an AOM
with RF generated by a Keysight Trueform 33622a arbitrary waveform generator
(AWG).

In addition to the Cardioid modulation, we implement a Hahn echo pulse midway
through the gate operation. For the MS gate, this implies a prolongation of
the gate time by a factor of $\sqrt{2}$, but increases the gate robustness to
$\sigma^{z}$ noise, which is our main source of decoherence.

Figure 5: Population of the ground state, $|1111\rangle$ (blue), and the fully
excited state, $|0000\rangle$ (red) are shown. The system is initialized at
$t=0$ to the ground state. Ideally, the population of these states at gate
time, $t=T$, is $0.5$, implying a maximally entangled qubit state. In
addition, the population of the 14 remaining states is shown (yellow), Ideally
the population of these states at gate time is 0, indicating that the qubit
state is disentangled from the motion. Error bars reflect shot noise due to
400 repetitions of each evolution time. In the MS gate (top), the population
changes sharply at the gate time, indicating an increased sensitivity to pulse
timing. While the Cardioid gate (bottom) approaches gate time smoothly,
indicating a robust gate. Furthermore the Cardioid generates populations that
are closer to the ideal case, due to reduced carrier coupling.

In order to characterize the fidelity of our entangling gates we implemented
two-qubit Cardioid gates on all pairs of qubits in a 5-qubit chain. We encoded
qubits on the $|1\rangle=5S_{\frac{1}{2},\frac{1}{2}}$ and
$|0\rangle=4D_{\frac{5}{2},\frac{3}{2}}$ states, and used the
$|\tilde{1}\rangle=S_{\frac{1}{2},-\frac{1}{2}}$ and
$|\tilde{0}\rangle=4D_{\frac{5}{2},-\frac{3}{2}}$ states as a “hidden
subspace”; qubits encoded on the latter are spectrally decoupled from the
entangling interaction. For each qubit pair, we first prepared the register in
the $|11111\rangle$ state using optical pumping. We used individual addressing
and global 674 nm pulses, as well as RF pulses on the Zeeman qubit, to hide
all qubits but the targeted pair onto their corresponding hidden subspace,
producing the state $|\tilde{1}\tilde{1}\tilde{1}\rangle\otimes|11\rangle$,
where the latter two represent the target qubits. We then carried out a global
Cardioid gate with an echo pulse, ideally producing the maximally entangled
Bell state $|\psi_{Bell}\rangle=\frac{1}{\sqrt{2}}(|11\rangle+i|00\rangle)$
for these two qubits. We measured the state fidelity resulting from this
procedure by measuring the populations and coherence of this two-qubit state
through a parity scan.

The results of our measurements are summarized in Fig. 4. After correcting for
the preparation cost of hiding three out of five ions, measured independently,
as well as state measurement error, we reach an average fidelity of
$0.973(2)$. The gate duration, including the $\sqrt{2}$ prolongation implied
by the echo pulse addition, is approximately 200 $\mu s$. These fidelities are
comparable with those reported by state of the art leading trapped ion quantum
computers Figgatt  _et al._ (2019); Pogorelov  _et al._ (2021b). We believe
that our major source of error is due to magnetic $\sigma^{z}$ noise, which is
only partially mitigated by the echo pulse, as the optical qubit coherence
time is limited to approximately two milliseconds, only a a single order of
magnitude longer than the gate time. Hence, we expect significant improvements
in fidelity with straightforward improvements to our experimental setup, such
as the addition of a $\mu$-metal shield.

Using the same methods described above, we also implemented a Cardioid all-to-
all gate on a $N=4$ ion chain, and compared it with a conventional MS gate.
Figure 5 shows the dynamics of four ions under the MS (top) and Cardioid
(bottom) all-to-all entanglement gates. Robustness of the Cardioid to pulse
time errors is apparent as the evolution in the vicinity of $t=T$ is much
smoother than that of the MS gate. Furthermore, the dynamics generated by the
Cardioid gate is closer to the ideal case compared to MS.

To further demonstrate the robustness of the Cardioid to pulse time errors we
measured the gate fidelity at different evolution times. Figure 6 compares the
4-qubit gate fidelity of the Cardioid (blue) and MS (red) gates. Clearly the
Cardioid exhibits a higher fidelity which is also less sensitive to incorrect
gate times. The peak fidelties are $0.84(3)$ and $0.69(3)$ for Cardioid and MS
gates respectively.

Figure 6: The state fidelity for a 4-qubit GHZ state generated by the Cardioid
(blue) and MS (red) gates is measured for different gate times. The Cardioid
gate exhibits a higher peak fidelity and reduced sensitivity to pulse time
errors compared to MS.

##  V Coherent feedback

Neither complete isolation of a quantum computer from the environment nor
perfectly executed unitary operations are physically realizable goals, and
errors will inevitably decohere the quantum computer states. Fortunately, a
family of techniques known as quantum error correction (QEC) can enable fault-
tolerant quantum computation for arbitrarily long computations given low
enough error rates Aharonov and Ben-Or (1997); Knill and Laflamme (1997). QEC
protects quantum information by encoding it non-locally, using highly
entangled states. As the information is encoded over high order correlations,
the otherwise harmful local noise does not destroy information, and the errors
it inflicts can therefore be rectified.

QEC requires measuring parts of the qubit register while maintaining the
quantum coherence of the rest, and generating feedback operations conditioned
on the measurement results. Generally, this is a technically difficult task:
measurement is a highly dissipative process, strongly coupling parts of the
quantum computer to the environment; at the same time, other (often
neighboring) parts of the computer must remain completely decoupled from the
environment. Furthermore, the measurement must be read out, and a conditional
operation must be applied, all within the coherence time of the system.

Standard state selective fluorescence detection on an EMCCD camera is ill-
equipped to fulfill these demands. Photon scattering from a laser beam
overlapping the entire chain collapses the wavefunction of all qubits in the
register. Even if the measurement beam is focused onto single ions, since
photon collection is limited, typically thousands of photons have to be
scattered before the state of the ion can be inferred with sufficient
statistical certainty. Secondary photon scattering will lead to errors on
other ions with high probability.

Furthermore, the readout of EMCCD cameras is typically slow since the transfer
of charge across the camera pixel array takes time. Thus, in most camera-based
readout, the processing of the detection results is done retroactively.
Experiments are almost always repeated many times, and in each iteration, the
image acquired by the camera is saved onto a buffer. Only after all iterations
are complete, the entire set of images is transferred from the buffer to a
computer via USB, where image processing is performed to determine the
measurement results. This retroactive processing is not adequate for real-time
coherent feedback. Since the USB interface is slow and ultimately controlled
by the CPU, simply reading out each measurement individually is not a viable
solution to this problem. Furthermore, after readout, each image must be
processed quickly, and the result should be fed back into the system within
the coherence time of the qubits.

Figure 7: The control scheme for the quantum coherent feedback module. The
user controls the experiment using the CPU, which through USB determines
parameters for the RF frequency sources and camera, and via TCP/IP uploads
detection parameters to the frame grabber FPGA, and an experimental sequence
onto the main FPGA. Once started, the main FPGA executes the sequence by
controlling a set of TTL digital outputs, which can turn on or off
electromagnetic fields addressing the ions, and can trigger camera detection.
Live feedback is enabled by the addition of a frame grabber FPGA. The
framegrabber receives real-time camera images through a Camera-Link
communication protocol, processes the images according to parameters uploaded
via TCP/IP, and outputs the results in SPI form to the main FPGA. The latter
is programmed to continue the sequence according to these results.

In order to address these challenges, we developed a module and a procedure
for live readout and real-time quantum-coherent feedback in our system. In
order to measure only pre-selected qubits, we use individual addressing pulses
to “hide” all other qubits by encoding them in levels which are completely
decoupled from 422 nm scattering and therefore the measuring process. This is
achieved by making use of another state in the six-dimensional Zeeman manifold
of the $4D_{\frac{5}{2}}$ orbital. Specifically, while qubits to be measured
remain encoded on the optical qubit states
$|1\rangle=|5S_{\frac{1}{2},\frac{1}{2}}\rangle$ and
$|0\rangle=|4D_{\frac{5}{2},\frac{3}{2}}\rangle$, hidden qubits are encoded
into the $|0\rangle=|4D_{\frac{5}{2},\frac{3}{2}}\rangle$ and
$|\tilde{0}\rangle=|4D_{\frac{5}{2},-\frac{3}{2}}\rangle$ states (see Fig. 8
(a)). The hiding process is realized using a series of RF and optical pulses,
as shown in Fig. 8 (b). After the hiding protocol, we measure the remaining
qubits by turning on the 422 and 1092 nm lasers, performing our standard
measurement protocol, but reading out and processing the results in real time.
Then, the hiding process is reversed, and an operation conditioned on the
measurement result can be applied.

To implement real-time readout, processing and feedback, we realize a scheme
which completely bypasses the camera buffer, the USB readout, the retroactive
image processing, and in fact, any PC involvement. Fast readout is enabled by
the Camera-Link port of the iXon Ultra camera. Camera-Link is a communication
protocol that is designed for fast reading of images, and in the iXon Ultra
outputs data before they reach the camera buffer. The acquired image is read
out and processed in real-time by a dedicated framegrabber FPGA (National
Instruments NI-PCIe-1473R), which comes with a built-in Camera-Link port. The
framegrabber can perform image processing on the Camera-Link data on the fly,
while information is still arriving. In this way, decisions on the outcome of
the measurement, expressed as a bit state for each qubit, are made very
quickly, and for some cases, qubit states are inferred before the complete
image has even been read out by the analog-to-digital converter (ADC). The
processing algorithm is pre-configured on the framegrabber FPGA in advance,
and is identical to the algorithm realized in PC image processing: a set of
pixels and a threshold value are allocated to each ion; for every image, the
framegrabber sums over the values on pixels ascribed to each ion and
determines if the result is larger than the corresponding threshold value; a
binary string with the length of the ion chain is given as output.

In order for this fast determination of the ion state to be employed for
coherent feedback, it must be available for use in real time by the main FPGA
controlling all other parts of the experiment. Therefore, some communication
between the two is necessary. We realized one-way synchronous (clock-
controlled) serial communication using Serial Peripheral Interface (SPI). The
framegrabber output is transferred to the main FPGA through SPI, while the
main FPGA is configured to wait for the measurement results and to incorporate
conditioned operations based on these results. In the opposite direction, the
main FPGA sends a trigger to both the camera and the framegrabber at the
beginning of every experiment, preparing them for an upcoming measurement
Gazit (2020). Our control scheme is shown diagrammatically in Fig. 7.

The speed in which the image is digitally read out by the camera, sent through
Camera-Link to the framegrabber, processed, the results sent to the main FPGA,
and the conditioned operation is triggered, is orders of magnitude shorter
than either the acquisition or readout time (over 500 $\mu s$ each). Hence,
this process introduces negligible time overhead.

Figure 8: (a) For qubits that should remain hidden from measurement, the
$|1\rangle$ state is mapped onto $|\tilde{0}\rangle$; after the mapping
procedure, these qubits are entirely encoded in the $4D_{\frac{5}{2}}$
manifold, and are decoupled from the measurement process. (b) Mapping is
carried out by a series of operations utilizing optical control of the
standard optical qubit $(|0\rangle,|1\rangle)$, a “shadow” optical qubit
comprised of $(|\tilde{0}\rangle,|\tilde{1}\rangle)$, and RF control of the
$5S_{\frac{1}{2}}$ manifold qubit. Individual addressing pulses are used to
single out the hidden ions. (c) After hiding, the remaining qubits are
measured. During measurement the hidden qubits must maintain coherence in face
of dephasing noise. We use RF pulses resonant with the $D$ manifold Zeeman
splitting to flip between $|0\rangle$ and $|\tilde{0}\rangle$, echoing out the
noise. We apply four $\pi$ pulses in a double CPMG configuration, such that
noise during exposure and readout are decoupled separately. Typical times are
$\tau_{e}\approx 700\mu s$ and $\tau_{r}\approx 600\mu s$. (d) In order to
test coherent feedback, two qubits are initialized in the $|1\rangle$ state.
We first apply a $\pi/2$ pulse on both qubits. Then, we apply a readout
procedure, where one ion is hidden, the other is measured in parallel with the
echo sequence shown in (c), and the hiding process is then reversed. Finally,
a $\pi/2$ pulse is applied on the target qubit, with a phase that is
controlled by the measurement result, and the two qubits are measured. (e)
Parity measurement results are plotted as a function of $\phi$. For a non-
conditioned pulse, or for total loss of coherence, we would expect a flat
parity signal that is independent of phase. Instead, we measure an 0.83
success probability for the feedback operation, determined by the amplitude of
the parity oscillations.

The image acquisition time, including both EMCCD exposure and readout time, is
typically longer than 1 ms and can be as high as 2 ms, during which require
that the hidden ions maintain quantum coherence. However, the encoded states
$(|0\rangle,|\tilde{0}\rangle)$ are highly susceptible to magnetic fields, and
therefore magnetic noise in our lab limits the coherence times of the hidden
qubits to about $500\mu s$. This is strongly prohibitive for performing
coherent feedback. In order to suppress noise, we apply dynamical decoupling
(DD) pulses on the $4D_{\frac{5}{2}}$ manifold _during_ the measurement of the
rest of the qubits. The DD pulses are realized by driving an RF magnetic field
resonant with the $4D_{\frac{5}{2}}$ Zeeman splitting. This produces a $J_{x}$
Hamiltonian on the spin 5/2 Zeeman manifold, and for a correct time exactly
realizes an effective $\pi$ pulse, translating the states
$4D_{\frac{5}{2},m}\leftrightarrow 4D_{\frac{5}{2},-m}$. Thus, this operation
flips magnetic susceptibility and can be used as an echo pulse to suppress
noise. Moreover, this driving has no bearing on the 422 nm measurement, as
either optical qubit states remain within their subspace and appropriately
maintain an identical measurement output. We realize the dynamical decoupling
as a series of four echo pulses in a Carr-Purcell-Meiboom-Gill (CPMG) protocol
Carr and Purcell (1954); Meiboom and Gill (1958).

###  V.1 Demonstrating coherent feedback

In order to verify and characterize the module’s performance, we performed two
tests for which real-time, coherent, quantum feedback is necessary. Both
experiments involve two qubits encoded on two ions: a control qubit, which is
measured; and a target qubit, which is subject to a feedback operation. For
both experiments, the operation on the target qubit is successful only if it
is correctly conditioned on the result of the measured qubit, and if the
target qubit remains coherent. For the first experiment the two qubits are in
a separable state, while in the second experiment the ions are entangled prior
to measurement.

In the first experiment, we prepared both qubits in the $|1\rangle$ state and
applied a $\pi/2$ pulse, generating the separable state

$|\psi_{1}\rangle=\frac{1}{2}\left(|11\rangle+i|10\rangle+i|01\rangle-|00\rangle\right).$
(6)

We then hid the target qubit, measured the control qubit, and reversed the
hiding process, according to the protocol described earlier. Finally, another
$\pi/2$ pulse was applied on the target qubit, with a phase that was
conditioned on the measurement result: $\phi-\pi/2$ if $|1\rangle$ was
measured, and $\phi+\pi/2$ if $|0\rangle$ was measured. As examples of the
consequence of this conditioned operation, consider that in the case
$\phi=\pi/2$, the target qubit will always be found in the opposite state to
that of the control, whereas in the case of $\phi=-\pi/2$ the target and
control qubit will be measured in the same state. Notice that if the operation
is not conditioned on the measurement result, which is completely random, _or_
if it is conditioned, and the target qubit is no longer coherent, then the
qubits will exhibit no correlations. More generally, we measured the
correlation/anti-correlation of the control and target qubit measurement
outcomes as a function of $\phi$. We successfully realized our desired
correlation with a probability of 0.832(24), compared to random 0.5
probability, as shown in Fig. 8e. The main source of error is likely the
remainder magnetic field noise, along with imperfect implementation of DD RF
pulses leading to population remaining in the $4D_{\frac{5}{2}}$ manifold.

Figure 9:  Coherent feedback on an entangled state. (a) A two qubit maximally
entangled Bell state is prepared using an MS gate, followed by a $\pi/2$ pulse
on the control qubit. Then, the target qubit is hidden, a measurement is
performed along with echo pulses, and the target qubit is mapped back onto the
optical qubit. A $\pi/2$ pulse with a phase $\phi-\pi/2$ or $\phi+\pi/2$ is
applied conditioned on the measurement result. (b) Qubit population as a
function of $\phi$. Due to the initial entanglement, the measurement collapses
the target qubit onto a superposition state defined by the control qubit
measurement result. This phase is effectively measured by applying the $\pi/2$
pulse. We measure an 0.862(18) success probability.

In a second experiment, we prepared the two qubits in the maximally entangled
Bell state
$|\psi_{Bell}\rangle=\frac{1}{\sqrt{2}}\left(|11\rangle+i|00\rangle\right)$
using the MS gate. This was followed by a $\pi/2$ pulse on the control qubit,
leading to the state,

$|\psi^{\prime}_{Bell}\rangle=\frac{1}{\sqrt{2}}\left(|\tilde{+}1\rangle+i|\tilde{-}0\rangle\right),$
(7)

where $|\tilde{\pm}\rangle=\frac{1}{\sqrt{2}}(|1\rangle\pm i|0\rangle)$ are
Pauli $Y$ eigenstates. We then hid the target qubit, measured the control
qubit, and reversed the hiding of the target qubit, with a sequence identical
to that described above.

The measurement projects the two-qubit state to either
$|1\rangle\otimes\left(\frac{1}{\sqrt{2}}(|1\rangle+i|0\rangle)\right)=|1\tilde{+}\rangle$
or
$|0\rangle\otimes\left(\frac{1}{\sqrt{2}}(|1\rangle-i|0\rangle)\right)=|0\tilde{-}\rangle$.
The target qubit is therefore projected onto a superposition state with some
well-defined phase, which is exactly opposite for the two cases. Finally, as
in the previous experiment, we apply a $\pi/2$ pulse with phase $\phi-\pi/2$
if the control is measured as $|1\rangle$ and $\phi+\pi/2$ if the control is
measured as $|0\rangle$. For conditioned coherent experiments, $\phi=\pi/2$
should always generate $|0\rangle$ on the target qubit, while $\phi=3\pi/2$
should always generate $|1\rangle$. The measurement result is completely
random, but because we exploit the correlation between the entangled qubits,
the conditioned operation results in a deterministic outcome. Moreover,
correct feedback relies on the phase coherence of the target qubit. For this
measurement we report a success probability of 0.863(15), on par with the
previous experiment, as can be seen in Fig.9.

While we demonstrated all necessary ingredient for performing quantum error
correction with our system, our coherent feedback fidelities need to be
improved. The main source of error is decoherence due to magnetic noise during
the long readout time overhead $(\sim 300\mu s)$ of the EMCCD camera. Future
systems, incorporating better-suited EMCCD cameras or PMT arrays and magnetic
shielding and stabilization, will potentially increase fidelity.

##  VI Conclusion

Trapped ion are among the leading candidate technologies for the realization
of a large scale quantum computer. Such a large scale QC is likely to be
composed out of an interconnected network of small quantum registers with high
performance of universal quantum computing .

Here we have demonstrated such a small scale, five qubit, quantum register. In
this register we have shown two important features. The first is the
implementation of two-qubits gates that are robust against gate timing errors.
Gates executed on pairs of ion-qubits out of a longer chain are more prone to
errors than when executed on two-ion crystals. The use of robust gates is
therefore important in order to reach high fidelity gates in longer crystals.
The second feature is our demonstration of mid-circuit measurement and
coherent feedback. The ability to perform mid-circuit measurement and feedback
is at the heart of most quantum error-correction schemes and other protocols
such as entanglement distillation or quantum teleprotation, all of which are
inherent ingredients in large scale quantum computing.

The improvement in performance of small scale quantum registers, as shown in
this work, is an important step en-route the development of large scale
quantum computing.

This work was performed with the support of the Israeli-Science Foundation.
The Willner Family Leadership Institute for the Weizmann Institute of Science,
The Crown Photonics Center and the Rosa and Emilio Segre Research Award.

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