# Error Correcting States in Ultracold Atoms

Harry C. P. Kendell  Quantum Engineering Centre for Doctoral Training,
University of Bristol, Bristol BS8 1FD, UK  Quantum Engineering Technology
Laboratories, H. H. Wills Physics Laboratory and Department of Electrical,
Electronic, and Mechanical Engineering, University of Bristol, Bristol BS8
1FD, UK  Giacomo Ferranti  Quantum Engineering Technology Laboratories, H. H.
Wills Physics Laboratory and Department of Electrical, Electronic, and
Mechanical Engineering, University of Bristol, Bristol BS8 1FD, UK  Carrie A.
Weidner<EMAIL_ADDRESS>Quantum Engineering Technology Laboratories,
H. H. Wills Physics Laboratory and Department of Electrical, Electronic, and
Mechanical Engineering, University of Bristol, Bristol BS8 1FD, UK

# Error Correcting States in Ultracold Atoms

Harry C. P. Kendell  Quantum Engineering Centre for Doctoral Training,
University of Bristol, Bristol BS8 1FD, UK  Quantum Engineering Technology
Laboratories, H. H. Wills Physics Laboratory and Department of Electrical,
Electronic, and Mechanical Engineering, University of Bristol, Bristol BS8
1FD, UK  Giacomo Ferranti  Quantum Engineering Technology Laboratories, H. H.
Wills Physics Laboratory and Department of Electrical, Electronic, and
Mechanical Engineering, University of Bristol, Bristol BS8 1FD, UK  Carrie A.
Weidner<EMAIL_ADDRESS>Quantum Engineering Technology Laboratories,
H. H. Wills Physics Laboratory and Department of Electrical, Electronic, and
Mechanical Engineering, University of Bristol, Bristol BS8 1FD, UK

###### Abstract

We demonstrate a method for encoding Gottesman-Kitaev-Preskill (GKP) error-
correcting qubits with single ultracold atoms trapped in individual sites of a
deep optical lattice. Using quantum optimal control protocols, we demonstrate
the generation of GKP qubit states with $10\text{\,}\mathrm{dB}$ squeezing,
which is the current minimum allowable squeezing level for use in surface code
error correction. States are encoded in the vibrational levels of the
individual lattice sites and generated via phase modulation of the lattice
potential. Finally, we provide a feasible experimental protocol for the
realization of these states. Our protocol opens up possibilities for
generating large arrays of atomic GKP states for continuous-variable quantum
information.

††preprint: APS/123-QED

##  I Introduction

Fault-tolerant quantum information processing will require fast low error-rate
operations on qubits [1]. This ensures large algorithms can be fully
implemented without the exponential build up of errors rendering the final
result useless. However, current systems suffer from relatively high physical
error rates [2]. This must, at least in the near term, be offset through the
use of error-correcting codes [3]. These rely upon a larger state space upon
which a smaller subspace of information is encoded. This proceeds either
through the representation of a single logical qubit with multiple physical
qubits or through the use of a bosonic oscillator, which itself describes a
larger state space.

The Gottesman-Kitaev-Preskill (GKP) states [4] are the canonical example of
this latter case. These states have been shown to have the highest error-
correction performance compared to similar encodings [5], and they have been
shown to have utility in cluster-state quantum computing [6]. GKP states
consist of periodic delta functions evenly spaced in position space. The two
encoded qubit states are identical, except that one is displaced by half the
spacing period. This nominally allows for perfect error correction up to a
displacement by a quarter of the inter-delta spacing. Finite approximations to
these ideal infinitely-squeezed states provide consequently approximate error
correction.

The difficulty in preparing these states lies in the challenge of generating
adequate squeezing to offset the overhead of the encoding and reduce the
logical error rate. Furthermore, the control of these states is very sensitive
to noise, for more accurate approximations, due to the large amplitude in
phase space [7]. These states must therefore be considered not only in their
error-correction ability but also the difficulty with which they may be
implemented and controlled. Nevertheless, approximate GKP states have been
experimentally demonstrated in ions [8], transmon qubits [9], and photonic
systems [10].

Here, we present a proposal for the realization of GKP states in ultracold
atoms, specifically, arrays of single atoms trapped in the individual sites of
a deep optical lattice potential. Atoms are excellent candidates for qubits,
and a reprogrammable tweezer-based quantum information processor with error
correction has recently been demonstrated [11], where quantum information is
processed via atom-atom interactions mediated by selective excitation of atoms
into high-lying Rydberg states [12]. Our protocol can be straightforwardly
adapted to atoms in optical tweezers [13, 14, 15], but we focus here on atoms
in optical lattices. This is due to their straightforward scalability and
controllability, even at the single-atom level [16, 17, 18]. Additionally,
optical-lattice-based systems have found use in an array of high-precision
quantum technologies, including atomic clocks [19, 20], quantum simulators
[21, 22], and inertial sensors [23, 24, 25, 26].

##  II Background

The GKP states have a rich history of approximation, given the full code
states are physically unrealizable. A number of these possess attractive
qualities, such as the representation of approximate GKP states as sums of
translated coherent states, which naturally admits an experimental
construction in ions [8]. However, these approximations largely agree up to a
redefinition of parameters [27]. We therefore choose, without loss of
generality, an approximation as a series of displaced squeezed vacuum states
with an overall Gaussian envelope. We additionally work with symmetric GKP
states in the two quadratures, motivated by the fact there is no _a priori_
reason to weight error correction performance to one quadrature. This
introduces an additional squeezing factor $\hat{S}_{0}$ [27].

We denote the GKP code states by $\lvert\mathrm{GKP} 0,1\rangle$ and Fock
states by $\lvert n\rangle$. The finite GKP code state is, therefore, up to
normalization,

$\begin{split}\lvert\mathrm{GKP} k\rangle\propto\hat{S}_{0}
\sum_{s\in\mathbb{Z}}e^{-\frac{1}{2}\sigma^{2}x_{t}^{2}} \hat{X}(x_{t})
\hat{S}(\Delta) \lvert 0\rangle.\end{split}$ (1)

The displacement value $x_{t}=\sqrt{\pi}(2s+k)$ for each squeezed vacuum
state, given the code state $k\in\\{0,1\\}$. The additional squeezing factor
is given by

$\hat{S}_{0}=\hat{S}\big{(}\ln(\sqrt{1+\sigma^{2}\Delta^{2}})\big{)},$ (2)

with $\Delta=-\ln\big{[}\sqrt{\sigma^{2}/(1-\sigma^{4})}\big{]}$. The finite
GKP state is then characterised solely by the level of squeezing as
$\zeta=-10\log_{10}(\sigma^{2}) $\text{\,}\mathrm{dB}$$.

Here, we find optimal controls that can be used to generate GKP states with
$\zeta=$10\text{\,}\mathrm{dB}$$ of squeezing with atoms in optical lattices.
This level of squeezing is generally recognized to be the point at which such
states are useful from a quantum information perspective [28, 29]. These
states may also be concatenated with a surface code [30]. This provides
enhanced error correction performance at the cost of additional overhead and
complexity. As a result, fault-tolerant quantum computing can be achieved at
as low as $8.1\text{\,}\mathrm{dB}$ of squeezing [31], assuming zero losses in
the system.

The Hamiltonian of the optical lattice system used to generate these states
takes the form

$H=\frac{p^{2}}{2m}+U\cos{\big{(}2k_{\mathrm{L}}[x+u(t)]\big{)}}$ (3)

where the lattice wavenumber is $k_{\mathrm{L}}=2\pi/\lambda$ for a lattice of
wavelength $\lambda$, the lattice depth is $U$ (typically quoted in units of
the photon recoil energy $E_{\mathrm{R}}=\hbar^{2}k_{\mathrm{L}}^{2}/2m$ for
an atom of mass $m$), and $u(t)$ is our control, with units of length. By
varying $u(t)$, the lattice is phase modulated, driving transitions between
different external atomic states while maintaining the atom’s internal state.

We assume the atom is trapped in a single site of a deep 1D lattice potential,
which is straightforward to experimentally prepare [32, 17]. We then define,
as in Ref. [33], our Fock states $\lvert n\rangle$ to correspond to the bound
vibrational levels within a given lattice site, i.e. $n=0,...,N$. Thus, the
total number $N$ of Fock states bound to a single lattice site directly
affects the maximum obtainable squeezing of our resulting GKP states. Figure 1
shows, for a given squeezing level, the required lattice depth and resulting
Fock basis size (i.e., the number of vibrational states bound to a given
lattice site). The purpose of our phase modulation is then to control
transfers between these Fock states to generate and manipulate the desired GKP
states.

Figure 1: Lattice depth (left axis) and required basis size (right axis) as a
function of desired squeezing level where the basis size is chosen such that
the GKP state can be reconstructed with a fidelity of $\mathcal{F}\geq 0.99$.
The large plateaus in depth and basis size arise due to the fact that, as more
accessible Fock states are introduced into the problem, there is a range of
available squeezing parameters realizable to within our target fidelity.

In our simulations, we initialize our atomic state in a given initial state
$\lvert\psi(t=0)\rangle$ (here, $\lvert 0\rangle$, considering the generation
of GKP states from the ground state of the lattice potential). We then
generate an initial guess (seed) for our shaking function $u(t)$ where $0\leq
t\leq T$ and use the symmetrized split-step method [34] to evolve the state
using the time-dependent Schrödinger equation. This results in a final state
$\lvert\psi(t=T)\rangle=\lvert\psi_{\mathrm{F}}\rangle$ where the time $T$ is
varied; higher values of $T$ typically lead to higher fidelity protocols. The
optimization protocol then adjusts $u(t)$ in an attempt to maximize fidelity.

We define our state transfer fidelity as
$\mathcal{F}=|\langle\psi_{\mathrm{D}}|\psi_{\mathrm{F}}\rangle|^{2}$, where
$\lvert\psi_{\mathrm{D,F}}\rangle$ are the desired state and final state after
a given shaking protocol $u(t)$, respectively. In Fig. 1 and this work in
general, our target fidelity is $\mathcal{F}\geq 0.99$; extensions beyond this
will require more bound states and a deeper lattice, posing increasing
experimental challenges. It remains to be seen what fidelities are required
for atomic GKP states to be useful in quantum information protocols; finite
fidelity likely manifests similarly to loss in photonic systems.

To find our controls, we use the GRAPE method [35] with BFGS built into the
QEngine C++ library [36]. Furthermore, bandwidth and amplitude limits for
$u(t)$ are implemented via regularization and soft boundary terms in our
optimization functional. Practically, we used a smooth low pass filter
centered at $0.5\text{\,}\mathrm{MHz}$ to ensure the experimental viability of
the controls [37], in addition to limiting the shaking amplitude to no more
than half a lattice site.

##  III Generating GKP states in atoms

Figure 2: Contour plot of lattice trap lifetime $T$ (black lines and shading)
and lattice depth $U$ (white lines) as a function of laser power and
wavelength between the D1 and D2 lines for (a) rubidium and (b) cesium,
assuming a 1D lattice with a beam waist of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$. For the lattice depths of
$U\approx 1500E_{\mathrm{R}}$ that give rise to useful squeezing levels of
$\approx$10\text{\,}\mathrm{dB}$$, the cesium atom lifetime is roughly a
factor of six longer than that of rubidium.

The experimental proposal requires atoms to be loaded into the ground state of
an optical lattice generated by two lasers with identical wavelength but
varying phase. Ground-state loading can be achieved by either loading from a
Bose-Einstein condensate or sideband cooling [38, 39] atoms loaded into the
lattice directly after laser cooling. Phase modulation can be achieved by
changing the phase of one lattice beam relative to the other, e.g., by
controlling the RF tone sent to an acousto-optic modulator (AOM). In this way,
we can implement the shaking protocols required for GKP state generation and
manipulation.

State verification uses a recent proposal for direct measurement of the Wigner
function of atoms in an optical lattice potential [33]. This method, which was
shown to work even for anharmonic traps, requires one to take advantage of the
differential light shift between two hyperfine ground states of the atoms
(e.g., the $\lvert F,m_{F}\rangle=\lvert 2,-2\rangle$ and $\lvert 1,-1\rangle$
hyperfine states of rubidium [18]) due to their differing vector
polarizabilities. Similar methods of differential lattice manipulation have
been used, e.g., to perform quantum walk experiments [40], but the drawback is
the lattice wavelength must be between the D1 and D2 lines of the alkali atom
used, limiting atom lifetime in the lattice.

Figure 3: Wigner functions of the $10\text{\,}\mathrm{dB}$ squeezed
$\lvert\mathrm{GKP} 0,1\rangle$ states. (a) and (c) show the ideal target
states (assuming $24$ bound states in the lattice) for the $\lvert\mathrm{GKP}
0,1\rangle$ states, respectively. These are compared with the time-optimal
$\mathcal{F}\geq 0.99$ results for $\lvert
0\rangle\rightarrow\lvert\mathrm{GKP} 0,1\rangle$ states for rubidium in (b)
and (d), respectively. The Wigner functions are given in units of the vacuum
state variance in $x$ ($\Delta X_{0}$) and $p$ ($\Delta P_{0}$), which, for a
very deep lattice, is, to an excellent approximation, the harmonic oscillator
width.

It is important to note that another method for Wigner function determination
via density matrix reconstruction has been demonstrated experimentally [41].
This method relies on time-of-flight data, and thus the stringent limits on
lattice wavelength are relaxed. Moving to a far-off-resonant trap (e.g., at
the common lattice wavelength of $1064\text{\,}\mathrm{nm}$) will require more
lattice power to achieve a given depth (as depth goes as $I/\tilde{\Delta}$,
where $I$ is the intensity of the lattice light and $\tilde{\Delta}$ is the
detuning from resonance) [42]. However, the associated atom lifetime will be
longer, as the photon scattering rate $\Gamma\propto I/\tilde{\Delta}^{2}$. We
will, however, consider a lattice between the D1 and D2 lines of an alkali
atom as a worst-case scenario for our proposal.

As a first step, we need to determine the lattice depth that gives us the
required squeezing. From Fig. 1, we see that to obtain
$10\text{\,}\mathrm{dB}$ of squeezing, we require $24$ basis states and a
lattice depth of $U\approx 1500E_{R}$. Figure 2 shows a contour plot of the
lattice depth and scattering-limited lifetimes for Rb and Cs as a function of
laser power and wavelength, restricted between the D1 and D2 lines. From the
figure, we see for $U=1500E_{\mathrm{R}}$, the best rubidium lifetime is
between $5$ and $10\text{\,}\mathrm{ms}$, but for cesium, this is about six
times larger, assuming the laser power is below $100\text{\,}\mathrm{mW}$,
which is readily achievable in the laboratory. There is a minor tradeoff,
however, due to the increased wavelength of the cesium lattice that causes, in
the conversion from scaled to unscaled units, the cesium protocols to be
roughly a factor of two longer than the rubidium protocols, so the overall
gain of a cesium system is only about three times that of rubidium. Note that
for lighter alkali atoms, the achievable lattice lifetime will be lower, as
the D1/D2 splitting increases with atomic mass. As such, cesium, the heaviest
easily trappable alkali (although there are efforts focusing on the heavier,
radioactive francium [43]), is the best alkali atom for these experiments.

The modulation times for the time-optimal (i.e., lowest $T$ where
$\mathcal{F}\geq 0.99$) protocols considered here are
$$141\text{\,}\mathrm{\SIUnitSymbolMicro
s}$($239\text{\,}\mathrm{\SIUnitSymbolMicro s}$)$ for the $\lvert
0\rangle\rightarrow\lvert\mathrm{GKP} 0\rangle$ transitions in Rb (Cs) and
$$158\text{\,}\mathrm{\SIUnitSymbolMicro
s}$($270\text{\,}\mathrm{\SIUnitSymbolMicro s}$)$ for the $\lvert
0\rangle\rightarrow\lvert\mathrm{GKP} 1\rangle$ transitions. The resulting
rubidium Wigner functions are compared with the ideal in Fig. 3; Wigner
results for cesium can be found in the Appendix, where we also present example
controls and their spectra. Our protocols reproduce the ideal states with good
fidelity; in particular, the grid-like signature of the GKP state is readily
apparent.

When considering the viability of our protocols with respect to atom lifetime
in the lattice, one must also consider the time required for the Wigner
measurement protocol; in Ref. [33] the protocol duration was on the order of
hundreds of $\text{\,}\mathrm{\SIUnitSymbolMicro s}$; the protocol time
depends on the differential trap depth, and thus, with a sufficiently large
variation in the power between the different circular polarizations of the
lattice light, we expect this protocol time should be maintained even at the
high lattice depths required for these experiments. As such, we find, without
accounting for measurement time, our modulation protocols take
$\approx$3\text{\,}\mathrm{\char 37\relax}$$ of the atom lifetime in the
lattice, assuming a very conservative Rb lattice lifetime of
$5\text{\,}\mathrm{ms}$. Cesium protocols require a fraction of a percent of
the atom lifetime. Including an (again, conservative)
$0.5\text{\,}\mathrm{ms}$ Wigner measurement time still keeps us within
$\leq$15\text{\,}\mathrm{\char 37\relax}$$ of the atom lifetime in the
lattice.

Finally, we would like to remark on the scalability of this method.
Empirically, we find our protocols require the lattice depth to be stable to
within $0.1\text{\,}\mathrm{\char 37\relax}$ before substantial fidelity loss
incurs. If we assume our atoms are trapped in a three-dimensional lattice with
sufficiently large lattice beam waists in each dimension, the lattice depth
variation across an atom cloud can be minimized. Even for modest beam waists
of $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$, one can populate $\sim 20$
lattice sites in each direction before the fidelity drops appreciably,
allowing for $8000$ atoms to be simultaneously interrogated so that high-
fidelity GKP states can be created along one Cartesian dimension.
Additionally, state stabilization protocols can be applied along the
orthogonal dimensions during state preparation so the population remains in
the ground state in these dimensions; this is considered in Ref. [44] and is
straightforward to include in our system as well. Thus, we expect our
protocols are readily scalable to considerable atom numbers and feasible to
implement with current experimental technology.

##  IV Conclusion

In this work, we demonstrate optimal control protocols that drive an atom from
the ground state of an optical lattice potential into the $\lvert\mathrm{GKP}
0\rangle$ and $\lvert\mathrm{GKP} 1\rangle$ states. We additionally present an
experimental protocol for the generation, manipulation, and measurement of
these states in optical lattices that can be readily achieved with current
experimental technology. Our protocols require large lattice depths $U\approx
1500E_{R}$, which poses challenges for atom lifetime and laser power, but we
show that even in our worst-case scenario of rubidium atoms and lattice light
between the D1 and D2 lines, the proposed protocols can be implemented within
a few percent of the atom lifetime in the lattice with powers
$<$1\text{\,}\mathrm{W}$$. We expect that as error correcting protocols
improve, the required squeezing level for GKP state utility will also
decrease, allowing useful atomic GKP states to be generated with lower power
requirements. In addition, while this paper focuses on GKP state generation,
extensions to gate optimization are straightforward [44] and will be the focus
of future work.

The efficient experimental generation of these states will also require
finding modulation protocols that are robust, e.g., to variations in the
lattice depth or noise in the modulation; this can be done via adaptation of
the methods discussed in Ref. [45]. There is likely also utility in exploring
constrained basis methods [46], for these problems, as such methods have been
adapted to efficiently find state transfer protocols in other modulated-
lattice systems [47]. These will allow for controls that can mitigate off-
resonant heating effects [48], analogously to what was shown in [49] for
phase-modulated shallow lattices. This will also reduce the beam waist (and
thus power) requirements to achieve the required lattice depths. Cavity-
enhanced lattices [50] can also lower power requirements.

Furthermore, while our proposal centers on atoms trapped in optical lattice
potentials, similar protocols can also be realized in optical tweezer
potentials. Arrays of tweezer potentials do not pack atoms as tightly as
lattices can, but they have the advantage that each lattice site can be more
easily independently controlled, allowing for flexible continuous-variable
quantum information protocols (although lattice systems with high-resolution
imaging and potential projection capabilities can allow for single-site
manipulation [18]). Additionally, the proposed Wigner measurement protocol is
applicable to tweezer systems.

Finally, for useful quantum information protocols to be realized, one must
entangle atoms with one another. While entanglement protocols are a subject
for future work, we would like to highlight that proposals for the
implementation of $\sqrt{\mathrm{SWAP}}$ gates with atoms in optical lattices
has been proposed in Ref. [51]. Similar protocols could be used to generate
cluster states of atoms [52], leading to a new paradigm of continuous variable
quantum information in atomic systems.

In the preparation of this work, the authors became aware of similar work by
Grochowski et al. [44]. Reference [44] is generally applicable to any
anharmonic quantum system that admits similar driving terms to those
considered here and considers a wider array of interesting quantum states,
e.g., cat states [53]. Our work, in contrast, presents a feasible experimental
proposal for the generation of atomic GKP states with $10\text{\,}\mathrm{dB}$
of squeezing.

##  V Acknowledgments

This work was carried out using the computational facilities of the Advanced
Computing Research Centre, University of Bristol -
http://www.bristol.ac.uk/acrc/. H.C.P.K. was funded by the EPSRC CDT in
quantum engineering (EP/SO23607/1). The authors would like to thank G. Ferrini
and A. Alberti for interesting discussions that motivated this work and A.
Clark for valuable comments on the paper draft.

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## Appendix

###  .1 Lifetime and Wigner results for cesium

Here, we present plots like those of Fig. 3 for cesium.

Figure 4: Same as Fig. 3, but for cesium in (b) and (d) for
$\lvert\mathrm{GKP} 0,1\rangle$ respectively. The ideal GKP state plots (a)
and (c) are identical to those in Fig. 3 for ease of comparison.

###  .2 Controls and their spectra

Here we present our controls and their Fourier spectra for the transitions
considered in this work.

Figure 5: Control and Fourier spectrum for the time-optimal state transfer
protocol driving the $\lvert 0\rangle\rightarrow\lvert\mathrm{GKP} 0\rangle$
transition. The control (a), (b) and spectra (c), (d) are shown for rubidium
and cesium, respectively. For (a) and (b), the blue line shows the optimized
control as a function of time. In (c) and (d), the spectrum is shown in blue,
and the filter applied during optimization is denoted with the excluded
frequencies shaded in red. Figure 6: Same as Fig. 5 but for the $\lvert
0\rangle\rightarrow\lvert\mathrm{GKP} 1\rangle$ transition.