# Universal hybrid quantum computing in trapped ions

R. T. Sutherland<EMAIL_ADDRESS>Department of Electrical and
Computer Engineering, Department of Physics and Astronomy, University of Texas
at San Antonio, San Antonio, TX 78249, USA  R. Srinivas  Department of
Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1
3PU, U.K.

###### Abstract

Using discrete and continuous variable subsystems, hybrid approaches to
quantum information could enable more quantum computational power for the same
physical resources. Here, we propose a hybrid scheme that can be used to
generate the necessary Gaussian and non-Gaussian operations for universal
continuous variable quantum computing in trapped ions. This scheme utilizes
two linear spin-motion interactions to generate a broad set of non-linear
effective spin-motion interactions including one and two mode squeezing, beam
splitter, and trisqueezing operations in trapped ion systems. We discuss
possible experimental implementations using laser-based and laser-free
approaches.

##  I Introduction

Quantum information is typically encoded in a discrete set of eigenvalues,
such as two-level qubit systems. Practical computation using such systems will
require error correction, which may need up to millions of physical qubits,
potentially limiting near-term applications Fowler  _et al._ (2012).
Continuous variable quantum computation (CVQC) Lloyd and Braunstein (1999);
Gottesman  _et al._ (2001); Braunstein and Van Loock (2005); Lau and Plenio
(2016) offers an alternative approach by storing information in states with a
continuum of eigenvalues, enabling higher density encoding for the same
physical resources. This approach, however, requires Gaussian operations
(displacement, squeezing, etc) and at least one non-Gaussian operation
(trisqueezing) Lloyd and Braunstein (1999). Experimentally, it has thus far
been challenging to implement a scheme with the flexibility to perform both
types of operations unilaterally. This challenge could be addressed with
hybrid platforms that incorporate both discrete and continuous variables,
utilizing the advantages of each system Andersen  _et al._ (2015); Lau and
Plenio (2016).

Trapped ions—a leading platform for quantum information Cirac and Zoller
(1995); Monroe  _et al._ (1995); Nielsen and Chuang (2010); Häffner  _et al._
(2008); Blatt and Wineland (2008); Harty  _et al._ (2014); Ballance  _et al._
(2016); Gaebler  _et al._ (2016); Srinivas  _et al._ (2021), quantum
simulation Porras and Cirac (2004); Jurcevic  _et al._ (2017); Sutherland
(2019), and quantum metrology Heinzen and Wineland (1990); Wineland  _et al._
(1994); Burd  _et al._ (2019); Brewer  _et al._ (2019); McCormick  _et al._
(2019) —are a prime candidate for CVQC Lau and James (2012) and hybrid quantum
computing Andersen  _et al._ (2015); Lau and Plenio (2016); Flühmann  _et al._
(2019). The Jaynes-Cummings-type interaction between the (discrete) internal
states of ions coupled to their (continuous) motional states can generate
Gaussian Meekhof  _et al._ (1996); Zeng  _et al._ (2002); Kienzler  _et al._
(2015) and non-Gaussian Leibfried  _et al._ (2002) operations. The strength of
this spin-motion coupling is typically determined by the Lamb-Dicke factor
$\eta\equiv\Delta kx_{0}$, where $\Delta k$ is the effective wave number of
the applied field and $x_{0}$ is the extent of the ground state motional wave
packet; typically, $\eta$ is $\sim$ 0.1 for laser-based interactions and
$\sim\,0.0001$ for laser-free interactions. Existing schemes that generate
non-linear interactions with respect to their motional coupling are
$\propto\eta^{2}$ or higher-order, making them significantly weaker than
$\propto\eta$ interactions. These higher-order interactions include the state
dependent beam splitter and one or two mode squeezing operations, and can also
be used to generate non-Gaussian states Gan  _et al._ (2020); Drechsler  _et
al._ (2020); Cardoso  _et al._ (2021). Here, we introduce a new scheme that
scales linearly with respect to $\eta$, enabling stronger, higher-order
interactions, and can also be used to implement multiple types of operations
using the same control fields, reducing experimental overhead. We show that we
can use two, simultaneously applied, spin-dependent displacement interactions
Mølmer and Sørensen (1999); Sørensen and Mølmer (2000); Leibfried  _et al._
(2003), typically used in geometric phase gates, to selectively generate
effective couplings that are non-linear in their motional component; we will
thus refer to this as a geometric phase gate interaction.

This paper is organized as follows: In Sec. II.1, we show that one can
selectively implement a broad set of effective Hamiltonians, representing
Gaussian and non-Gaussian interactions, by simultaneously applying two
geometric phase gate interactions to one ion, where the spin-components of the
interactions do not commute. We can select which effective Hamiltonian to
implement by simply adjusting the frequencies of the phase-gate interactions.
In Sec. II.2, we present a method for implementing our scheme in a laser-free
system with a single spin-motion coupling interaction. In Sec. III, we compare
numerical simulations of the (exact) geometric phase gate Hamiltonians with
the (approximate) effective Hamiltonians, which converge for long enough gate
durations. In Sec. IV, we give conclusions and prospects for future work.

##  II Theory

In this section, we describe how two non-commuting phase-gate interactions can
create Gaussian and non-Gaussian operations, sufficient for universal CVQC
Lloyd and Braunstein (1999); sup . Such phase-gate interactions are typically
used in trapped-ion experiments to generate entanglement with or without
lasers. We also show that this scheme can be implemented in a laser-free
experiment with a single radiofrequency gradient Sutherland  _et al._ (2019);
Srinivas  _et al._ (2019); Sutherland (2019). We consider a single spin, with
states $\ket{\downarrow}$ and $\ket{\uparrow}$, whose interaction is
represented by Pauli operators $\hat{\sigma}_{\alpha}$, where
$\alpha\in\\{x,y,z\\}$. The spin is coupled to one or two harmonic oscillator
modes $j$, whose interactions are represented with an annihilation(creation)
operator $\hat{a}_{j}(\hat{a}_{j}^{\dagger})$.

###  II.1 Spin-dependent non-linear spin-motion coupling

In this section, we discuss applying interactions that are linear in
$\hat{a}_{j}$, to generate effective spin-motion interactions that are non-
linear in $\hat{a}_{j}$. We analyze our effective Hamiltonians using the
Magnus expansion for a unitary time-propagator acting under Schrödinger’s
equation Magnus (1954):

$\displaystyle\hat{U}(t)$ $\displaystyle\simeq$
$\displaystyle\exp\Big{(}\frac{-i}{\hbar}\int^{t}_{0}dt_{1}\hat{H}_{1}-\frac{1}{2\hbar^{2}}\int^{t}_{0}\int^{t_{1}}_{0}dt_{1}dt_{2}\big{[}\hat{H}_{1},\hat{H}_{2}\big{]}$
(1) $\displaystyle+$
$\displaystyle\frac{i}{6\hbar^{3}}\int^{t}_{0}\int^{t_{1}}_{0}\int^{t_{2}}_{0}dt_{1}dt_{2}dt_{3}\Big{\\{}\Big{[}\hat{H}_{1},\big{[}\hat{H}_{2},\hat{H}_{3}\big{]}\Big{]}$
$\displaystyle+$
$\displaystyle\Big{[}\hat{H}_{3},\big{[}\hat{H}_{2},\hat{H}_{1}\big{]}\Big{]}\Big{\\}}\Big{)},$

where $\hat{H}_{k}\equiv\hat{H}(t_{k})$ represents the Hamiltonian describing
the system at time $t_{k}$. We have truncated the expansion after the
$3^{rd}$-order, leaving analysis of higher-order interactions to future work.
In the following, we will discuss effective Hamiltonians $\hat{H}_{eff}$,
where $\hat{U}(t)\simeq\exp({-it\hat{H}_{eff}/\hbar})$.

We begin with a Hamiltonian that represents two geometric phase gate
interactions acting on a single ion in the interaction picture with respect to
the ion’s qubit and motional energies:

$\displaystyle\hat{H}(t)=\hbar\Omega_{\alpha}\hat{\sigma}_{\alpha}\hat{a}_{j}e^{-i\Delta
t}+\hbar\Omega_{\alpha^{\prime}}\hat{\sigma}_{\alpha^{\prime}}\hat{a}_{j^{\prime}}e^{-i(n\Delta
t+\phi)}+c.c.,$

where $\Omega_{\alpha,\alpha^{\prime}}$ are the Rabi frequencies of the
corresponding spin-motion coupling terms, $\Delta=2\pi/t_{i}$ is the detuning,
$n$ is an integer, and $\phi$ is an arbitrary phase. Note that
$\Omega_{\alpha,\alpha^{\prime}}$ both scale linearly with the Lamb-Dicke
factor $\eta$. We assume $\alpha\neq\alpha^{\prime}$. The first-order term in
the Magnus expansion is 0 for integer multiples of $t_{i}$. The total
interaction duration is $t_{f}=Kt_{i}$, where $K$ is an integer.

Plugging Eq. (II.1) into Eq. (1), setting $n=\pm 1$, and solving for a
duration $t_{f}$ gives:

$\displaystyle\hat{U}_{2}(t_{f})$ $\displaystyle=$
$\displaystyle\exp\Big{(}\Omega_{2}t_{f}\varepsilon_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}\hat{\sigma}_{\alpha^{\prime\prime}}\Big{\\{}\delta_{n,-1}(\hat{a}^{\dagger}_{j}\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi}-\hat{a}_{j}\hat{a}_{j^{\prime}}e^{-i\phi})$
(3)
$\displaystyle~{}~{}~{}~{}~{}~{}+\delta_{n,1}(\hat{a}^{\dagger}_{j}\hat{a}_{j^{\prime}}e^{-i\phi}-\hat{a}_{j}\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi})$
$\displaystyle~{}~{}~{}~{}~{}~{}+\delta_{j,j^{\prime}}\delta_{n,1}\cos(\phi)\Big{\\}}\Big{)}+\mathcal{O}([\Omega_{\alpha}/\Delta]^{3}),$

where $\Omega_{2}\equiv 2\Omega_{\alpha}\Omega_{\alpha^{\prime}}/\Delta$. For
the error term, we assume $\Omega_{\alpha}$ and $\Omega_{\alpha^{\prime}}$ are
the same order of magnitude. Here, the $2^{\text{nd}}$-order terms in the
Magnus expansion are the largest contributions to the overall dynamics of the
system. By appropriate choice of $n$, $\phi$, $j$, and $j^{\prime}$ we can
create a set of interactions $\hat{H}_{eff}$ that are spin-dependent and
$2^{\text{nd}}$-order in $\hat{a}_{j},\hat{a}_{j^{\prime}}$ (see Table 1),
including one- (Sec. III.1) and two-mode (Sec. III.2) squeezing, as well as
beam splitter (Sec. III.3) interactions.

Similarly, for $n=\pm 2$, the $3^{\text{rd}}$-order terms in the Magnus
expansion are the largest contribution to the dynamics; the $1^{\text{st}}$\-
and $2^{\text{nd}}$-order terms integrate to zero. Plugging Eq. (II.1) into
Eq. (1) again, we get:

$\displaystyle\hat{U}_{3}(t_{f})$ $\displaystyle=$
$\displaystyle\exp\Big{(}-i\Omega_{3}t_{f}\hat{\sigma}_{\alpha^{\prime}}\Big{\\{}\delta_{n,2}(\hat{a}_{j}\hat{a}^{\dagger}_{j^{\prime}}\hat{a}_{j}e^{i\phi}+\hat{a}^{\dagger}_{j}\hat{a}_{j^{\prime}}\hat{a}^{\dagger}_{j}e^{-i\phi})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\delta_{n,-2}(\hat{a}_{j}^{\dagger
2}\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi}+\hat{a}^{2}_{j}\hat{a}_{j^{\prime}}e^{-i\phi})\Big{\\}}\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\mathcal{O}([\Omega_{\alpha}/\Delta]^{4}),$

where $\Omega_{3}\equiv
2\Omega_{\alpha^{\prime}}\Omega_{\alpha}^{2}/\Delta^{2}$. This time-evolution
operator corresponds to spin-dependent effective Hamiltonians that are
$3^{\text{rd}}$-order in $\hat{a}_{j},\hat{a}_{j^{\prime}}$ (see Table. 1),
which includes trisqueezing (Sec. III.4).

Effective Hamiltonian | Gaussian | n | $\mathbf{j}=\textbf{j}^{\mathbf{\prime}}$
---|---|---|---
$\hat{H}_{eff}\simeq i\hbar\Omega_{2}\varepsilon_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}\hat{\sigma}_{\alpha^{\prime\prime}}\Big{(}\hat{a}_{j}^{\dagger 2}e^{i\phi}-\hat{a}^{2}_{j}e^{-i\phi}\Big{)}$ | yes | -1 | yes
$\hat{H}_{eff}\simeq i\hbar\Omega_{2}\varepsilon_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}\hat{\sigma}_{\alpha^{\prime\prime}}\Big{(}\hat{a}^{\dagger}_{j}\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi}-\hat{a}_{j}\hat{a}_{j^{\prime}}e^{-i\phi}\Big{)}$ | yes | -1 | no
$\hat{H}_{eff}\simeq i\hbar\Omega_{2}\varepsilon_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}\hat{\sigma}_{\alpha^{\prime\prime}}\Big{(}\hat{a}_{j}^{\dagger}\hat{a}_{j}e^{-i\phi}-\hat{a}_{j}\hat{a}^{\dagger}_{j}e^{i\phi}+\cos{\phi}\Big{)}$ | yes | 1 | yes
$\hat{H}_{eff}\simeq i\hbar\Omega_{2}\varepsilon_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}\hat{\sigma}_{\alpha^{\prime\prime}}\Big{(}\hat{a}_{j}^{\dagger}\hat{a}_{j^{\prime}}e^{-i\phi}-\hat{a}_{j}\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi}\Big{)}$ | yes | 1 | no
$\hat{H}_{eff}\simeq\hbar\Omega_{3}\hat{\sigma}_{\alpha^{\prime}}\Big{(}\hat{a}_{j}^{\dagger 3}e^{i\phi}+\hat{a}^{3}_{j}e^{-i\phi}\Big{)}$ | no | -2 | yes
$\hat{H}_{eff}\simeq\hbar\Omega_{3}\hat{\sigma}_{\alpha^{\prime}}\Big{(}\hat{a}_{j}^{\dagger 2}\hat{a}_{j^{\prime}}^{\dagger}e^{i\phi}+\hat{a}^{2}_{j}\hat{a}_{j^{\prime}}e^{-i\phi}\Big{)}$ | no | -2 | no
$\hat{H}_{eff}\simeq\hbar\Omega_{3}\hat{\sigma}_{\alpha^{\prime}}\Big{(}\hat{a}_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}e^{i\phi}+\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{a}^{\dagger}_{j}e^{-i\phi}\Big{)}$ | no | 2 | yes
$\hat{H}_{eff}\simeq\hbar\Omega_{3}\hat{\sigma}_{\alpha^{\prime}}\Big{(}\hat{a}_{j}^{2}\hat{a}_{j^{\prime}}^{\dagger}e^{i\phi}+\hat{a}_{j}^{2\dagger}\hat{a}_{j^{\prime}}e^{-i\phi}\Big{)}$ | no | 2 | no
Table 1: List of effective Hamiltonians $\hat{H}_{eff}$ that can be generated
by choosing the values of $n$, $\phi$, $j$, and $j^{\prime}$ in Eq. (II.1).
Setting $n=\pm 1$ generates forms of $\hat{H}_{eff}$ that correspond to the
$2^{\text{nd}}$-order term in the Magnus expansion, producing Gaussian
operations on $1$ or $2$ phonon modes, with a Rabi frequency $\Omega_{2}\equiv
2\Omega_{\alpha}\Omega_{\alpha^{\prime}}/\Delta$. Setting $n=\pm 2$ generates
forms of $\hat{H}_{eff}$ that correspond to the $3^{\text{rd}}$-order terms in
the Magnus expansion, producing non-Gaussian operations on $1$ or $2$ modes,
with Rabi frequency $\Omega_{3}\equiv
2\Omega_{\alpha^{\prime}}\Omega_{\alpha}^{2}/\Delta^{2}$.

In both Eq. (3) and Eq. (II.1), the time evolution primarily corresponds to a
selected $\hat{H}_{eff}$. The undesired, higher-order terms in the Magnus
expansion decrease faster with $\Delta$ than $\hat{H}_{eff}$. Therefore, the
time evolution converges to that of $\hat{H}_{eff}$ as the iterations $K$ of
the interaction and $\Delta$ increase, such that $\Omega_{2}t_{f}$ or
$\Omega_{3}t_{f}$ is fixed. Thus,
$\hat{U}(t_{f})\rightarrow\exp({-\frac{i}{\hbar}\hat{H}_{eff}t_{f}})$ for
large values of $K$. As $t_{f}=Kt_{i}=2\pi K/\Delta$, for
$2^{\text{nd}}$-order interactions this results in

$\frac{4\pi\Omega_{\alpha}\Omega_{\alpha^{\prime}}K}{\Delta^{2}}=\Omega_{2}t_{f},$
(5)

which must be held constant. Thus, as $K$ increases from $1$,
$\Delta\rightarrow\Delta K^{1/2}$ and $t_{f}\rightarrow t_{f}K^{1/2}$.
Similarly, for $3^{\text{rd}}$-order interactions, we see that

$\frac{4\pi\Omega^{2}_{\alpha^{\prime}}\Omega_{\alpha}K}{\Delta^{3}}=\Omega_{3}t_{f},$
(6)

must be held constant. Therefore, increasing the value of $K$ from $1$ gives
$\Delta\rightarrow\Delta K^{1/3}$ and $t_{f}\rightarrow t_{f}K^{2/3}$.
Therefore, for large values of $K$, the evolution of the interactions reported
in Table 1 can be made accurate to an arbitrary degree. In Sec. III, we verify
this behaviour through direct numerical integration of Eq. (II.1).

Since the convergence criteria for both the $2^{\text{nd}}$\- and
$3^{\text{rd}}$-order interactions depend only on $\Omega_{\alpha}/\Delta$,
and $\Omega_{\alpha}$ is linear with respect to $\eta$, $\Delta$ is also
linear with respect to $\eta$. Thus, $\Omega_{2}$ and $\Omega_{3}$ are linear
with respect to $\eta$ as well; this makes our scheme, to the best of our
knowledge, the first method for generating spin-motion coupling that is non-
linear in $\hat{a}_{j}$ with field interactions that are only linear in
$\eta$. This linearity is crucial for laser-free based approaches, where
$\eta$ is small and non-linear interactions are hard to generate.

#### Physical Interpretation

In the above discussion, a clear pattern emerged: the largest non-zero
contribution to the overall dynamics of the system comes from terms in the
Magnus expansion that oscillate at opposite frequencies to one another. When
$n=1$ in Eq. (II.1), for example, the $\hat{a}_{j,j^{\prime}}$ and
$\hat{a}^{\dagger}_{j,j^{\prime}}$ terms oscillate with frequencies $-\Delta$
and $\Delta$, respectively, leading to
$\hat{H}_{eff}\propto\hat{a}^{\dagger}_{j^{\prime}}\hat{a}_{j}+c.c.$. If
$n=-1$, the opposite is true, leading to
$\hat{H}_{eff}\propto\hat{a}_{j^{\prime}}\hat{a}_{j}+c.c.$. We also see the
same behaviour for $3^{\text{rd}}$-order terms in the Magnus expansion when
$n=\pm 2$. If we consider that the values of $\Delta$ and $n\Delta$ represent
the detuning of the gate fields from their respective transitions, we can
interpret this apparent pattern as a result of the conservation of energy.
This means that $\hat{H}_{eff}$ describes the leading-order energy conserving
transition in the Magnus expansion. From this perspective, the physics we
describe here parallels that of a Raman transition, where the spin-dependent
displacement acts as an auxiliary state. In Sec. III we show that, similar to
a Raman transition, the larger the detuning from the auxiliary state, the more
the time evolution corresponds to $\hat{H}_{eff}$, due to the reduced effect
off-resonant processes. This leads to the question of whether a more general
pattern exists, where, for example, larger values of $n$ in Eq. (II.1)
generate forms of $\hat{H}_{eff}$ that are higher-order in
$\hat{a}_{j,j^{\prime}}$ beyond the $3^{\text{rd}}$-order interactions shown
here. In this work, we only postulate that such a pattern exists, leaving a
rigorous proof to future work.

###  II.2 Laser-free implementation with radiofrequency gradient

Equation (II.1) may be implemented with multiple pairs of lasers,
symmetrically detuned around the transition frequency of the qubit. The same
interaction may also be implemented without lasers, using multiple pairs of
gradients oscillating close to the qubit frequency Ospelkaus  _et al._ (2008,
2011). In this subsection, we present a method of implementing Eq. (II.1) in a
laser-free system using a single radiofrequency gradient in addition to a pair
of weak, symmetrically detuned microwave fields Sutherland  _et al._ (2019);
Srinivas  _et al._ (2019, 2021). Such a scheme would simplify the
experimental, overhead as multiple gradients are typically hard to generate.

In the interaction picture with respect to the motion and qubit frequencies,
the Hamiltonian for such a microwave-driven system, after the rotating wave
approximation, takes the form Sutherland  _et al._ (2019):

$\displaystyle\hat{H}_{r}(t)$ $\displaystyle=$ $\displaystyle
2\hbar\Omega_{\mu}\hat{\sigma}_{x}\cos(\delta t)$
$\displaystyle+2\hat{\sigma}_{z}\cos(\omega_{g}t)\sum_{j}\hbar\Omega_{g,j}\Big{\\{}\hat{a}_{j}e^{-i\omega_{j}t}+\hat{a}_{j}^{\dagger}e^{i\omega_{j}t}\Big{\\}},$

where $\Omega_{g,j}$ is the Rabi frequency of the radiofrequency gradient
coupled to one or two motional modes $j$, $\Omega_{\mu}$ is the Rabi frequency
of the symmetrically detuned microwave pair, and $\delta$ is the magnitude of
the detuning of each microwave pair from the qubit frequency. We have assumed
the microwave pair is polarized in the $x$ direction.

We analyze the dynamics of Eq. (II.2) by transforming into the interaction
picture with respect to the bichromatic microwave pair Roos (2008); Jonathan
_et al._ (2000); Sutherland  _et al._ (2019):

$\displaystyle\hat{H}_{I}(t)=\hat{U}_{I}^{\dagger}(t)\hat{H}_{r}(t)\hat{U}_{I}(t)+i\hbar\dot{\hat{U}}_{I}^{\dagger}(t)\hat{U}_{I}(t),$
(8)

where the frame transformation $\hat{U}_{I}(t)$ is given by:

$\displaystyle\hat{U}_{I}(t)=\exp\Big{\\{}-\frac{2i\Omega_{\mu}\sin(\delta
t)}{\delta}\hat{\sigma}_{x}\Big{\\}}.$ (9)

Here, $\hat{H}_{I}(t)$ takes the form:

$\displaystyle\hat{H}_{I}(t)$ $\displaystyle=$ $\displaystyle
2\hbar\cos(\omega_{g}t)\Big{\\{}\sum_{j}\Omega_{g,j}(\hat{a}_{j}e^{-i\omega_{j}t}+\hat{a}_{j}^{\dagger}e^{i\omega_{j}t})\Big{\\}}\times$
(10)
$\displaystyle\Big{\\{}\hat{\sigma}_{z}\Big{[}J_{0}\Big{(}\frac{4\Omega_{\mu}}{\delta}\Big{)}+~{}2\sum_{m=1}^{\infty}J_{2m}\Big{(}\frac{4\Omega_{\mu}}{\delta}\Big{)}\cos(2m\delta
t)\Big{]}$
$\displaystyle+~{}2\hat{\sigma}_{y}\sum^{\infty}_{m=1}J_{2m-1}\Big{(}\frac{4\Omega_{\mu}}{\delta}\Big{)}\sin([2m-1]\delta
t)\Big{\\}},$

where $J_{m}$ represents the $m^{\text{th}}$ Bessel function of the
$1^{\text{st}}$ kind, which we assume has an argument of
$4\Omega_{\mu}/\delta$ from here on. As $\delta\gg\Omega_{g,j}$ in typical
laser-free experiments, Eq. (10) shows that there is an infinite set of
potential spin-motion interactions that can be generated by tuning an integer
multiple of $\delta$ to be near any of the $\omega_{j}\pm\omega_{g}$
sidebands. Choosing $\delta$ such that it is an even multiple of
$\omega_{j}\pm\omega_{g}$ produces a $\hat{\sigma}_{z}$ coupling to the spin,
while an odd multiple of $\delta$ produces a $\hat{\sigma}_{y}$ coupling
Sutherland  _et al._ (2019, 2020). Finally, we must ensure that the dynamics
one expects from $\hat{H}_{I}(t)$ correspond to that of $\hat{H}_{r}(t)$. If
the bichromatic microwave pair is ramped on and off slowly compared to
$1/\delta$, $\hat{U}_{I}(t_{f})\rightarrow\hat{I}$ and the time-propagator
that results from $\hat{H}_{r}(t)$ converges to that of $\hat{H}_{I}(t)$. We
can, therefore, we can consider $\hat{H}_{I}(t)$ exclusively when discussing
the dynamics of $\hat{H}_{r}(t)$ Sutherland  _et al._ (2019, 2020).

Reference Sutherland  _et al._ (2020) discusses methods to set the values of
$\delta$ and $\omega_{g}$ to generate an entangling interaction that is
insensitive to decoherence of the qubit and motion. We implement a similar
idea here to produce a Hamiltonian that takes the form of Eq. (II.1). Assuming
that the values of $\omega_{j}$ and $\omega_{j^{\prime}}$ are fixed, we
generate Eq. (II.1) by setting even and odd multiples of $\delta$ to be near
the $\omega_{j}-\omega_{g}$ and $\omega_{j^{\prime}}+\omega_{g}$ sidebands,
respectively. As an example, we choose $\delta$ and $\omega_{g}$ to satisfy:

$\displaystyle 2\delta$ $\displaystyle=$
$\displaystyle(\omega_{j}-\omega_{g})-\Delta,$ $\displaystyle 3\delta$
$\displaystyle=$ $\displaystyle(\omega_{j^{\prime}}+\omega_{g})-n\Delta,$ (11)

where in the above equation $j^{\prime}$ may or may not refer to the same mode
as $j$. Upon making the rotating wave approximation, this gives:

$\displaystyle\hat{H}_{I}(t)\simeq\hbar\Big{\\{}\Omega_{g,j}J_{2}\hat{\sigma}_{z}\hat{a}_{j}e^{-i\Delta
t}-i\Omega_{g,j^{\prime}}J_{3}\hat{\sigma}_{y}\hat{a}_{j^{\prime}}e^{-in\Delta
t}\Big{\\}}+c.c.,$

taking the form of Eq. (II.1) wherein $\Omega_{\alpha}=\Omega_{g,j}J_{2}$,
$\Omega_{\alpha^{\prime}}=\Omega_{g,j^{\prime}}J_{3}$, and $\phi=\pi/2$. Thus,
our scheme enables non-linear spin-motion coupling, without lasers, using a
single gradient.

##  III Numerical simulations

Here, we show that, for a selection of the $\hat{H}_{eff}$ shown in Table 1,
the time-dynamics produced by $\hat{H}_{eff}$ converge to those of Eq. (II.1)
when $\Delta$ is large compared to $\Omega_{\alpha,\alpha^{\prime}}$. Here, we
assume that $\Omega_{\alpha}$ and $\Omega_{\alpha^{\prime}}$ have the same
value $\Omega$. We also assume that $\alpha=y$ and $\alpha^{\prime}=x$. We
show the convergence to $\hat{H}_{eff}$ in Figs. 1-4 with the ‘fidelity’
$\mathcal{F}$ of each operation, defined as
$\mathcal{F}\equiv|\braket{T}{\psi(t)}|^{2}$, where $\ket{T}$ is the target
state obtained through direct integration of $\hat{H}_{eff}$, and
$\ket{\psi(t)}$ is the state obtained through direct integration of Eq.
(II.1). After the simulation, we can approximate experimental run-times for
both laser-based and laser-free experiments, using Rabi frequencies of
$\Omega/2\pi=10~{}\mathrm{kHz}$ and $\Omega/2\pi=1~{}\mathrm{kHz}$,
respectively, which are within the parameter regime of current experiments
Ballance  _et al._ (2016); Gaebler  _et al._ (2016); Srinivas  _et al._
(2019).

###  III.1 One-mode squeezing

Figure 1: One-mode squeezing operation generated by Eq. (II.1). We plot the
infidelity $1-\mathcal{F}$ versus normalized gate duration $t_{f}\Omega$,
while keeping $\Omega_{2}t_{f}$ constant. This dependence is shown for
squeezing parameters $2\Omega_{2}t_{f}\equiv r_{s}$ of 1.5 (blue dashed), 1.0
(purple dotted), and $0.5$ (solid light blue). For large values of $t_{f}$,
$1-\mathcal{F}\rightarrow 0$, as described in the text. The inset shows the
probability $\mathcal{P}$ of phonon state $\ket{n_{j}}$ for a squeezed state
with $r_{s}=1.5$, generated by integrating Eq. (II.1) for
$t_{f}\Omega/2\pi\simeq 2.2$.

One mode squeezing has been shown to enhance quantum metrology Caves (1981);
Heinzen and Wineland (1990); Aasi  _et al._ (2013); Burd  _et al._ (2019), the
speed of entangling gates Ge  _et al._ (2019); Burd  _et al._ (2020), and is
an integral component of CVQC Lloyd and Braunstein (1999). In trapped ions,
squeezing of the motion can be generated through modulation of the trapping
potential at twice the motional frequency Heinzen and Wineland (1990); Burd
_et al._ (2019), a diabatic change of the trapping frequency Heinzen and
Wineland (1990); Wittemer  _et al._ (2019), a pair of laser beams with a
difference frequency that is twice the motional frequency Meekhof  _et al._
(1996); Dupays and Chenu (2021), or reservoir engineering Kienzler  _et al._
(2015). Here, we present an alternative scheme, where each geometric phase
gate interaction in Eq. (II.1) operates on the same mode ($j=j^{\prime}$) with
opposite detunings $\Delta$ from one another ($n=-1$). This interaction
produces a spin-dependent squeezing operation:

$\displaystyle\hat{H}_{eff}\simeq
i\hbar\Omega_{2}\hat{\sigma}_{z}\Big{(}\hat{a}_{j}^{\dagger
2}e^{i\phi}-\hat{a}^{2}_{j}e^{-i\phi}\Big{)}.$ (13)

This spin-dependent squeezing operation can be combined with single qubit
rotations and spin-measurements to create non-Gaussian states Drechsler  _et
al._ (2020).

In Fig. 1, we numerically integrate both Eq. (II.1) and Eq. (13), showing
$1-\mathcal{F}$ versus $t_{f}$ for squeezing parameters $r_{s}\equiv
2\Omega_{2}t_{f}$ of $0.5$, $1.0$, and $1.5$. For all three calculations, the
results of Eq. (II.1) and Eq. (13) converge to one another, showing that our
scheme can produce an effective squeezing operation to arbitrary accuracy for
large enough values of $t_{f}$. We can use the results shown in Fig. 1 to
approximate the experimental run-time needed to induce a given value of
$r_{s}$ and $\mathcal{F}$. For example, a squeezing parameter of $r_{s}=1.5$
with fidelity $\mathcal{F}\simeq 0.99$ has a normalized gate duration of
$t_{f}\Omega/2\pi=1.2$. This corresponds to 120 $\mu$s for laser-based
($\Omega/2\pi=10~{}\mathrm{kHz}$), and $1.2\,$ms for laser-free
($\Omega/2\pi=1~{}\mathrm{kHz}$) implementations.

###  III.2 Two-mode squeezing

Figure 2: Two-mode squeezing operation generated by Eq. (II.1). Here we show
the infidelity $1-\mathcal{F}$ versus the normalized gate duration
$t_{f}\Omega/2\pi$. This dependence is shown for squeezing parameters $r_{2s}$
of 0.75 (red dashed), 0.5 (pink dotted), and 0.25 (orange solid). For larger
values of $t_{f}$, all three calculations converge to that of $\hat{H}_{eff}$.
The inset shows the probabilities $\mathcal{P}$ of occupying a state with
$n_{j}$ phonons in the $j$ mode and $n_{j^{\prime}}$ phonons in the
$j^{\prime}$ mode for $r_{2s}=0.75$ and $t_{f}\Omega/2\pi\simeq 1.7$.

Two-mode squeezing generates an entangled state between two bosonic modes
Agarwal (2012). It has been used as a resource for CVQC, quantum cryptography,
as well as quantum teleportation Braunstein and Van Loock (2005); Weedbrook
_et al._ (2012); Grosshans  _et al._ (2003); Furusawa  _et al._ (1998), and is
potentially useful for quantum metrology Cardoso  _et al._ (2021). Experiments
have been demonstrated with optical Furusawa  _et al._ (1998); Grosshans  _et
al._ (2003) and microwave Eichler  _et al._ (2011) photons, but not the phonon
modes of a trapped ion crystal. Two-mode squeezing in trapped ions has been
proposed through the non-linear motional coupling of the ions to laser fields
oscillating at $\omega_{j}+\omega_{j^{\prime}}$ Zeng  _et al._ (2002); Cardoso
_et al._ (2021); our approach offers an alternative that uses only linear
motional coupling. Implementing Eq. (II.1), where each geometric phase gate
interaction operates on different modes ($j\neq j^{\prime}$) and each of the
interactions have equal and opposite detunings ($n=-1$), the effective
interaction is:

$\displaystyle\hat{H}_{eff}\simeq
i\hbar\Omega_{2}\hat{\sigma}_{z}\Big{(}\hat{a}^{\dagger}_{j}\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi}-\hat{a}_{j}\hat{a}_{j^{\prime}}e^{-i\phi}\Big{)},$
(14)

representing spin-dependent two-mode squeezing. Since this Hamiltonian is
spin-dependent, it may be combined with single-qubit operations and
observations of the spin to create the superpositions of multiple two-mode
squeezed states described in Ref. Cardoso  _et al._ (2021).

In Fig. 2, we numerically integrate Eq. (II.1) and Eq. (14) for $n=-1$ and
$j\neq j^{\prime}$, showing $1-\mathcal{F}$ versus the normalized gate
duration $t_{f}\Omega/2\pi$ for squeezing parameters
$r_{2s}\equiv\Omega_{2}t_{f}$ Agarwal (2012) of $0.25$, $0.5$, and $0.75$;
here, there is a factor of two difference in the definitions of $r_{2s}$ and
$r_{s}$ Agarwal (2012). For all three calculations, the results of Eq. (II.1)
and Eq. (14) converge, showing that our scheme is capable of producing an
effective two-mode squeezing operation to arbitrary accuracy for large enough
values of $t_{f}$. For $r_{2s}=0.75$ and $\mathcal{F}\simeq 0.99$,
$t_{f}\Omega/2\pi=1.4$, giving experimental run-times of $t_{f}\simeq
140~{}\mu\mathrm{s}$ and 1.4 ms for laser-based
($\Omega/2\pi=10~{}\mathrm{kHz}$) and laser-free parameters
($\Omega/2\pi=1~{}\mathrm{kHz}$), respectively.

###  III.3 Beam splitter

Figure 3: Beam-splitter operation generated by Eq. (II.1). Here, we show the
infidelity $1-\mathcal{F}$ versus normalized gate duration $t_{f}\Omega/2\pi$.
This dependence is shown for two squeezing parameters $r_{bs}=\pi/4$ (solid
turquoise) and $r_{bs}=\pi/2$ (dashed green), for a system initialized to the
state $\ket{\downarrow}\ket{n_{j}=1,n_{j^{\prime}}=0}$. The inset shows the
probabilities of the phonon subspace being in the state $\ket{10}$ and
$\ket{01}$, versus time, for a $r_{bs}=\pi/2$ interaction such that
$t_{f}\Omega/2\pi\simeq 2.5$.

The beam splitter interaction swaps the states of two boson modes Agarwal
(2012). This interaction is useful for CVQC Lloyd and Braunstein (1999);
Braunstein and Van Loock (2005); Weedbrook  _et al._ (2012), and can also be
used to cool multiple motional modes of trapped ion crystals Gorman  _et al._
(2014); Lau (2014). This interaction has been experimentally demonstrated in
trapped ions by modulating the voltages of the trap electrodes Gorman  _et
al._ (2014), and with a running optical lattice Gan  _et al._ (2020). Instead,
we here implement Eq. (II.1), where each geometric phase gate interaction
operates on different modes ($j\neq j^{\prime}$) and each interaction has the
same detuning ($n=1$). The resulting interaction is

$\hat{H}_{eff}\simeq
i\hbar\Omega_{2}\hat{\sigma}_{z}\Big{(}\hat{a}_{j}\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi}-\hat{a}_{j}^{\dagger}\hat{a}_{j^{\prime}}e^{-i\phi}\Big{)},$
(15)

representing a spin-dependent beam splitter.

In Fig. 3, we numerically integrate Eq. (II.1), for $n=1$ and $j\neq
j^{\prime}$, and Eq. (15), showing $1-\mathcal{F}$ versus $t_{f}\Omega/2\pi$.
The figure compares two calculations where the system is initialized to the
state $\ket{\downarrow}\ket{n_{j}=1,n_{j^{\prime}}=0}$, for
${r_{bs}\equiv\Omega_{2}t_{f}}$ of $\pi/4$ and $\pi/2$. The inset of the
figure shows the probabilities of finding the motion in the states $\ket{10}$
and $\ket{01}$, traced over the spin degree-of-freedom, for
$t_{f}\Omega/2\pi\simeq 2.5$ and $r_{bs}=\pi/2$. For a value of
$r_{bs}=\pi/4$, a value of $t_{f}\Omega/2\pi\simeq 1.8$ is needed to achieve
$\mathcal{F}\simeq 0.99$. This duration is $t_{f}\simeq 180~{}\mu\mathrm{s}$
for laser-based parameters ($\Omega/2\pi=10~{}\mathrm{kHz}$), and $t_{f}\simeq
1.8~{}\mathrm{ms}$ for laser-free parameters ($\Omega/2\pi\simeq
1~{}\mathrm{kHz}$).

###  III.4 Trisqueezing

Figure 4: Trisqueezing operation generated by Eq. (II.1) . We show the
infidelity $1-\mathcal{F}$ versus normalized gate duration $t_{f}\Omega/2\pi$
for three squeezing parameters $r_{3s}=0.12$ (brown dashed), $r_{3s}=0.09$
(light brown dotted), and $r_{3s}=0.06$ (tan solid). The inset shows the
probabilities $\mathcal{P}$ of each phonon state $n_{j}$ for $r_{3s}=0.12$ and
$t_{f}\Omega/2\pi\simeq 2.9$ generated by Eq. (II.1), showing that only states
with phonon numbers that are multiples of $3$ are populated.

Universal CVQC requires at least one non-Gaussian operation Lloyd and
Braunstein (1999). For $n=-2$ and $j=j^{\prime}$ in Eq. (II.1), we generate
the effective interaction:

$\displaystyle\hat{H}_{eff}\simeq\hbar\Omega_{3}\hat{\sigma}_{x}\Big{(}\hat{a}_{j}^{\dagger
3}e^{i\phi}+\hat{a}^{3}_{j}e^{i\phi}\Big{)},$ (16)

which is the trisqueezing Hamiltonian Fisher  _et al._ (1984); Hillery  _et
al._ (1984); Braunstein and McLachlan (1987); Hillery (1990), which has
recently been demonstrated using microwave photons Chang  _et al._ (2020).
Here, we stay in a regime where the squeezing parameter is small
$t_{f}\Omega_{3}\equiv r_{3s}\leq 0.12$. Importantly, implementing the non-
Gaussian trisqueezing interaction using our scheme has no additional
experimental overhead compared to the Gaussian interactions discussed above.

In Fig. 4, we numerically integrate Eq. (II.1) starting in the motional ground
state, and compare the results to Eq. (16) by showing $1-\mathcal{F}$ versus
the normalized time $t_{f}\Omega/2\pi$. The inset of the figure shows the
probability of the motional mode having a given number of phonons $n_{j}$,
tracing over the spin degree-of-freedom for an implementation of Eq. (II.1)
with and $t_{f}\Omega/2\pi\simeq 2.9$; this shows that, as expected, only
states with phonon numbers that are multiples of three have non-zero
populations. For $r_{3s}=0.12$, a gate duration of $t_{f}\Omega/2\pi\simeq
1.4$ is needed to achieve $\mathcal{F}\simeq 0.99$ which corresponds to
$t_{f}\simeq 140~{}\mu\mathrm{s}$ and $t_{f}\simeq 1.4~{}\mathrm{ms}$ using
laser-based ($\Omega/2\pi=10~{}\mathrm{kHz}$), and laser-free parameters
($\Omega/2\pi=1~{}\mathrm{kHz}$), respectively.

##  IV Conclusion

In conclusion, we have introduced a new hybrid scheme for implementing spin-
motion interactions that are non-linear in their motional component, using
only spin-motion interactions that are linear in their motional component.
Through the application two geometric phase gate interactions acting on a
single ion, we showed that it is possible to create a broad set of Gaussian
and non-Gaussian operations, sufficient for universal quantum computation over
continuous variables. The type of operation can be chosen by simply changing
the frequencies of the individual phase gate interactions. We also proposed an
implementation of our scheme in a laser-free setup using a single
radiofrequency gradient. Finally, we verified the validity of our scheme
numerically, and provided approximate experimental run-times. While most
experimental demonstrations of non-linear coupling to a boson mode have only
focused on one type of interaction, all the interactions discussed here can be
implemented in the same experimental setup.

## Acknowledgements

We would like to thank D. T. C. Allcock, C. J. Ballance, O. Bazavan, M.
Minder, S. Saner, and D. J. Wineland for thoughtful discussions.

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Supplemental Materials

## Universal continuous variable quantum computation with linear spin-motion
coupling

Reference Lau and Plenio (2016) showed that a spin’s non-linear interaction
with a harmonic oscillator may be used to generate non-Gaussian motional
states. We here show that, when combined with single qubit rotations on the
Bloch sphere, the interaction used in geometric phase gates can achieve
universal CVQC. This Hamiltonian takes the form:

$\displaystyle\hat{H}(\phi,\alpha,j)=\hbar\Omega\hat{\sigma}_{\alpha}(\hat{a}^{\dagger}_{j}e^{i\phi}+\hat{a}_{j}e^{-i\phi}),$
(S1)

where $\Omega$ is the interaction Rabi frequency, $\phi$ is a phase,
$\alpha\in\\{x,y,z\\}$, and $j$ corresponds to the motional mode. Following
Ref. Lloyd and Braunstein (1999), we can generate higher-order interactions
from Eq. (S1) using the sequence:

$\displaystyle e^{-(i/\hbar)\hat{H}\delta
t}e^{-(i/\hbar)\hat{H}^{\prime}\delta t}e^{(i/\hbar)\hat{H}\delta
t}e^{(i/\hbar)\hat{H}^{\prime}\delta t}$ $\displaystyle=$ $\displaystyle
e^{(\delta t/\hbar)^{2}[\hat{H}^{\prime},\hat{H}]}+\mathcal{O}([\delta
t\Omega]^{3}).$

This sequence has been used in trapped ions to simulate a spin-1/2 particle in
an external potential Leibfried  _et al._ (2002), but relied on interactions
that were non-linear in $\hat{a}_{j}$ to achieve a universal continuous
variable gate set. Here, we use interactions that are only linear in
$\hat{a}_{j}$, relaxing the experimental requirements.

Choosing $\hat{H}(\phi,\alpha,j)$ and
$\hat{H}^{\prime}=\hat{H}(\phi^{\prime},\alpha^{\prime},j^{\prime})$, we can
obtain a general $2^{nd}$ order gate interaction:

$\displaystyle\hat{U}_{2g}$ $\displaystyle=$
$\displaystyle\exp\Big{(}-2i\delta
t^{2}\Omega^{2}\varepsilon_{\alpha,\alpha^{\prime},\alpha^{\prime\prime}}\hat{\sigma}_{\alpha^{\prime\prime}}\Big{\\{}i\delta_{jj^{\prime}}\sin(\phi-\phi^{\prime})+(\hat{a}^{\dagger}_{j}e^{i\phi}+\hat{a}_{j}e^{-i\phi})(\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi^{\prime}}+\hat{a}_{j^{\prime}}e^{-i\phi^{\prime}})\Big{\\}}\Big{)}+\mathcal{O}([\delta
t\Omega]^{3}),$ (S3)

where $\delta_{jj^{\prime}}$ is the Kronecker delta function, and
$\varepsilon_{\alpha,\alpha^{\prime},\alpha^{\prime\prime}}$ is the Levi-
Civita symbol. If we assume $\alpha\neq\alpha^{\prime}$, and apply a single-
qubit rotation to transform the spin to a $\mp 1$ eigenstate of
$\hat{\sigma}_{\alpha^{\prime\prime}}$, and ignore a global phase, the gate
operator becomes:

$\displaystyle\hat{U}_{2g}$ $\displaystyle=\exp\Big{(}\pm 2i\delta
t^{2}\Omega^{2}\Big{\\{}$
$\displaystyle(\hat{a}^{\dagger}_{j}e^{i\phi}+\hat{a}_{j}e^{-i\phi})(\hat{a}^{\dagger}_{j^{\prime}}e^{i\phi^{\prime}}+\hat{a}_{j^{\prime}}e^{-i\phi^{\prime}})\Big{\\}}\Big{)}+\mathcal{O}(\delta
t^{3}).$ (S4)

In combination with coherent displacements , which can be generated by Eq.
(S1) alone, this represents the Gaussian operations needed for universal CVQC
Weedbrook  _et al._ (2012). Gaussian operations alone are not sufficient for
universal quantum computation Lloyd and Braunstein (1999), however, and can be
classically simulated efficiently if the initial state is Gaussian Gottesman
(1998); Bartlett and Sanders (2002). To achieve higher-order motional
couplings, we can apply Eq. (Universal continuous variable quantum computation
with linear spin-motion coupling) again, substituting $\hat{U}_{2g}$ for
$e^{\pm(\delta t/\hbar)^{2}\hat{H}^{\prime}}$, where the required
$\hat{H}^{\prime}\rightarrow-\hat{H}^{\prime}$ may be achieved by applying
$\pi$ pulses to the spin Jost  _et al._ (2009). This interaction will generate
the degree-$3$ motional coupling needed to satisfy the Lloyd-Braunstein
criterion for universal CVQC Lloyd and Braunstein (1999). For example, if we
generate $\hat{U}_{2g}$ with $\hat{H}(0,x,j)$ and $\hat{H}^{\prime}(0,y,j)$
and, again, apply Eq. (Universal continuous variable quantum computation with
linear spin-motion coupling) using $\hat{H}=\hat{H}(0,x,j)$ we get:

$\hat{U}_{3g}=\exp\Big{(}4i\delta
t^{3}\Omega^{3}[\hat{a}^{\dagger}_{j}+\hat{a}_{j}]^{3}\hat{\sigma}_{y}\Big{)}.$
(S5)

If this operator acts on a $\pm 1$ eigenstate of $\hat{\sigma}_{y}$, we get:

$\hat{U}_{3g}\rightarrow\exp\Big{(}\pm 4i\delta
t^{3}\Omega^{3}[\hat{a}^{\dagger}_{j}+\hat{a}_{j}]^{3}\Big{)}.$ (S6)

This time evolution operator corresponds to a cubic phase gate Weedbrook  _et
al._ (2012), and, taken in combination with Gaussian operations, satisfies the
Lloyd-Braunstein criterion. Note that the cubic phase gate was chosen for
simplicity, and a similar prescription could be chosen to generate a broad set
of degree-$3$ gates through different choices of $\phi$ and $j$ at each step.

This section shows that our hybrid approach to universal CVQC has the same
physical gate requirements as discrete quantum computing. There are, however,
many situations where a digitized series of pulses is not desirable, for
example if Gaussian interactions are needed for reasons other than CVQC
Heinzen and Wineland (1990); Burd  _et al._ (2019, 2020). Therefore, in the
main manuscript we discuss a technique for generating higher-order spin-motion
coupling that uses continuously applied fields.