# Variational protocols for emulating digital gates
using analog control with always-on interactions

Claire Chevallier  PASQAL SAS, 7 rue Léonard de Vinci - 91300 Massy, Paris,
France  Joseph Vovrosh<EMAIL_ADDRESS>PASQAL SAS, 7 rue Léonard de
Vinci - 91300 Massy, Paris, France  Julius de Hond  PASQAL SAS, 7 rue Léonard
de Vinci - 91300 Massy, Paris, France  Mario Dagrada  PASQAL SAS, 7 rue
Léonard de Vinci - 91300 Massy, Paris, France  Alexandre Dauphin  PASQAL SAS,
7 rue Léonard de Vinci - 91300 Massy, Paris, France  Vincent E. Elfving
PASQAL SAS, 7 rue Léonard de Vinci - 91300 Massy, Paris, France

###### Abstract

We design variational pulse sequences tailored for neutral atom quantum
simulators and show that we can engineer layers of single-qubit and multi-
qubit gates. As an application, we discuss how the proposed method can be used
to perform refocusing algorithms, SWAP networks, and ultimately quantum
chemistry simulations. While the theoretical protocol we develop still has
experimental limitations, it paves the way, with some further optimisation,
for the use of analog quantum processors for variational quantum algorithms,
including those not previously considered compatible with analog mode.

##  I Introduction

Quantum computers present an exciting alternative approach to their classical
analogues. By leveraging special-purpose compute elements, which exhibit
quantum properties, such as qubits, it has the potential to give exponential
speed ups to very specific problems such as the simulation of quantum systems
Lloyd (1996) and breaking cryptographic codes Shor (1994). The most common
approach to quantum computing is digital, in which a set of one-qubit and two-
qubits gates are used to build a theoretically universal quantum computer.
There have been attempts to build such a universal device with various
technologies, for example, superconducting circuits Koch  _et al._ (2007),
trapped ions Kielpinski  _et al._ (2002); Bruzewicz  _et al._ (2019), neutral
atoms Henriet  _et al._ (2020); Bluvstein  _et al._ (2023) and photonic
devices Slussarenko and Pryde (2019).

Currently, a fully fault-tolerant quantum computer remains out of reach but
the last decade has seen much progress towards this goal. Quantum error
correction protocols have been proposed to remove these errors but, before
they can be implemented at a meaningful scale, the number of qubits in Noisy
Intermediate-Scale Quantum (NISQ) devices and the noise level must be improved
by many orders of magnitude Ryan-Anderson  _et al._ (2021). As the quantum
circuits that are promised to solve real world problems require large numbers
of qubits and deep circuits, for example due to trotterisation Suzuki (1976),
these devices are not yet practically useful. Despite these issues, there have
been impressive results in recent years, for example, solving classically
intractable sampling problems Arute  _et al._ (2019); Zhong  _et al._ (2020),
and accurate quantum simulation of a spin system using more than 100 qubits
Kim  _et al._ (2023), albeit for a classically tractable problem Patra  _et
al._ (2023); Tindall  _et al._ (2023); Liao  _et al._ (2023). As a result of
these successes, quantum computers have gathered much interest from
researchers in fields such as quantum chemistry Elfving  _et al._ (2020);
Daley  _et al._ (2022), the financial sector Orús  _et al._ (2019),
pharmaceuticals Cao  _et al._ (2018), and artificial intelligence Schuld and
Petruccione (2021); Bharti  _et al._ (2022); Cerezo  _et al._ (2021); Dawid
_et al._ (2023).

Figure 1: A schematic showing how to use an analog pulse sequence to
approximate a variety of one- and two-qubit gates on neutral atom hardware.
(a) We variationally prepare pulse sequences to achieve global $\pi/2$
rotations through tuning global Rabi frequencies, detunings and phases of
external lasers. (b) These global rotations along side single-qubit Rz gates
and analog pulses of Eq. (1) lead to a general set of gates available in a
purely analog fashion.

Analog quantum processors are alternative avenue for solving complex problems.
By carefully isolating a given register of particles and understanding the
underlying physics that governs the interactions between these particles we
have access to and control over a resource Hamiltonian which we can use to
encode and solve problems of interest. In an attempt to build analog devices,
technologies such as trapped ions Zhang  _et al._ (2017), cold atoms yoon Choi
_et al._ (2016) and photonics Tillmann  _et al._ (2013) have been used.
However, in this work we focus our attention on neutral atoms as the particles
in use Henriet  _et al._ (2020); Briegel  _et al._ (2000). While analog
devices are not universal, and are thus more limited in their potential use
cases, they have lower error rates than current digital quantum computers and
therefore are likely to show the first industry-relevant quantum advantage
Bloch  _et al._ (2012); Preskill (2018).

Naturally, as analog devices are closely related to quantum experimental
setups, applications for quantum simulation are the most promising contender
for early quantum advantage. In fact, there have been experimental results
published that challenge state-of-the-art classical simulation methods already
yoon Choi  _et al._ (2016); Scholl  _et al._ (2021); Shaw  _et al._ (2023).
Generalising the use of an analog system to problems outside the realm of
quantum simulation is tough. One issue is that of always-on interactions;
typically in physical systems, interactions between particles scale with their
separation. While the exact scaling is defined by the type of particles in use
the consequence is the same, for a fixed register of atoms the interactions
are fixed. As a result, performing quantum operations that act on an isolated
part of the quantum system is difficult. This greatly limits the potential
applications for analog quantum devices.

A third regime of interest is Digital-Analog Quantum Computing (DAQC). Here, a
quantum device has access to both analog pulses along-side single-qubit gates.
NMR setups are known to work within this regime Jones (2011), and proposals
have considered realising a DAQC device in superconducting qubits Lamata  _et
al._ (2018); Gonzalez-Raya  _et al._ (2021); Galicia  _et al._ (2020) with
some early stage experimental results Greenaway  _et al._ (2023). In fact,
under the right conditions, this type of quantum computing can be universal
Parra-Rodriguez  _et al._ (2020); de Andoin  _et al._ (2023). There are many
proposed applications for such a device, for example, quantum chemistry Kumar
_et al._ (2023), implementing the quantum approximate optimization and Harrow-
Hassidim-Lloyd algorithms Headley  _et al._ (2022); Martin  _et al._ (2023),
and computing the quantum Fourier transformation Martin  _et al._ (2020).
However, in order to realise single-qubit gates previous work has assumed
either that interactions can be turned off, external laser strengths much
larger than these interactions - regimes that are difficult to realise for
many technologies, or the availability of analog pulses with lengths that are
below experimental feasibility.

In this work, we show how to realise a general set of single-qubit gates for a
device that has always-on interactions in the regime where the external laser
fields and pulse lengths are on the same order as the interactions between
qubits. In particular, we show how a surprisingly simple variational protocol
requiring just access to single-qubit addressability in a given field of the
governing Hamiltonian highlights a potential solution of this long-known
problem. Furthermore, we present how this method along side a refocusing
algorithm allows for approximating a wide variety of two-qubit gates in a
purely analog fashion. Figure 1 shows a schematic of the process outlined in
this article. This work acts as the first step to bridge the gap between
analog and digital quantum computing and potentially allows purely digital
algorithms to be implemented in an analog quantum device.

This paper is structured as follows: In Section II we introduce the
Hamiltonian that governs an analog neutral atom device. In Section III we
outline the main result of this work, i.e. a method to approximate layers of
single-qubit gates using only global pulses and single qubit addressibility in
a single field. In Section IV we show, given the access to layers of single-
qubit gates, how to perform a refocusing algorithm and thus approximate two-
qubit gates. In Sections V and VI we exemplify the use of these methods
through preparing an approximate SWAP gate network and solving simple
chemistry problems. In Section VII we detail further the experimental
considerations that need to be taken into account to realise this protocol.
Lastly, in Section VIII we summarise the work presented in this report and
outline open questions for future work.

##  II  Rydberg atoms in tweezers

In this article we concentrate on the analog capabilities of neutral atom
devices. Such systems have attracted much interest for research purposes in
many-body physics and more recently for potential applications in industrially
relevant use cases. It has been shown that the Maximum Independent Set problem
can be solved directly through a mapping to neutral atoms Pichler  _et al._
(2018); Wurtz  _et al._ (2022); Many machine learning schemes have been
proposed that utilise neutral atoms Bravo  _et al._ (2022); Henry  _et al._
(2021); Leclerc  _et al._ (2022); Albrecht  _et al._ (2023); A blueprint for
VQE protocols taking advantage of the freedom of lattice geometry in registers
of neural atoms was recently published Michel  _et al._ (2023); Recently, it
was also shown how neutral atoms can be used for drug discovery D’Arcangelo
_et al._ (2023).

In neutral atom devices a qubit is encoded into two electronic states of an
atom. Typical choices for atomic species are rubidium or strontium atoms, in
which two energy levels can be chosen to represent the two level system of a
qubit. Since the number of electronic states for a given atom is large, the
various possible choices of $\ket{0}$ and $\ket{1}$ can lead to different
interaction landscapes. Throughout this work we will consider encoding
$\ket{0}$ in a low-lying energy level and encoding $\ket{1}$ in a Rydberg
state; the resultant Hamiltonian of the system with this choice is of Ising
type Henriet  _et al._ (2020)

$\mathcal{H}(t)=\hbar\sum\limits_{j}\frac{\Omega(t)}{2}\left\\{\cos\left[\Phi\left(t\right)\right]\hat{\sigma}_{j}^{x}-\sin\left[\Phi\left(t\right)\right]\hat{\sigma}_{j}^{y}\right\\}\\\
-\hbar\sum\limits_{j}\frac{\delta_{j}(t)}{2}\hat{\sigma}_{j}^{z}+\sum\limits_{i>j}\frac{C_{6}}{r_{ij}^{6}}\hat{n}_{i}\hat{n}_{j},$
(1)

where $\hat{\sigma}^{x,y,z}_{i}$ are the Pauli $x$, $y$ and $z$ matrices
acting on the $i^{\text{th}}$ qubit,
$\hat{n}_{i}=\frac{1-\hat{\sigma}_{i}^{z}}{2}$ is the number operator,
$r_{ij}$ is the distance between the qubits $i$ and $j$, $\phi$ is the phase,
$\Omega$ is the Rabi frequency and $\delta$ the detuning of the external laser
that couples the qubit ground and Rydberg states. These can be seen as an
effective magnetic field, with transverse and longitudinal components
$\propto\Omega(t)$ and $\propto\delta(t)$ respectively. Note, here we allow
for the detuning to be site dependant. Each field can be varied by changing
the intensity and frequency of the laser field. The third term in Eq. (1)
relates to the interactions between individual atoms. More specifically, it
corresponds to an energy penalty that two qubits experience if they are both
in the Rydberg state at the same time - this leads to the well known Rydberg
blockade Urban  _et al._ (2009). This coupling between two atoms depends on
the coefficient $C_{6}$ which is defined by the choice of Rydberg state, $n$.

##  III  Realising analog single-qubit gates

In this section we propose a method to engineer layers of single-qubit gates
on a purely analog neutral atom device, i.e., in the presence of always-on
interactions described by Eq. (1). Given the local addressibility of detunings
in Eq. (1) coupled with the fact that the effect of detuning in the
Hamiltonian commutes with the qubit interactions, the implementation of a
single qubit Rz gate is quite natural. We first show how these can be
routinely realised.

In order to create pulse sequences (sequences of time-evolution under the
Rydberg Hamiltonian with different values of $\Omega$ and $\delta$) that are
equivalent to single qubit Rx and Ry gates, we turn to global $\pi/2$
rotations that allow us to change freely between the $X$, $Y$ and $Z$ basis.
Given that the strength of the Rabi frequency and detuning are on the same
order as the interaction strength between atoms, the protocols that will
achieve this are non-trivial. However, in the following we show how a
surprisingly simple setup can be used to variationally prepare experimental
pulse sequences that approximate these rotations with high accuracy.

Once these global rotations are obtained, they can be combined with local Rz
gates to implement layers of Rx, Ry gates, see Fig. 2(a). For example,

$\text{Rx}\left(\vec{\theta}\right)\equiv\text{RY}\left(\frac{\pi}{2}\right)\otimes\text{Rz}\left(\vec{\theta}\right)\otimes\text{RY}\left(-\frac{\pi}{2}\right),$
(2)
$\text{Ry}\left(\vec{\theta}\right)\equiv\text{RX}\left(\frac{\pi}{2}\right)\otimes\text{Rz}\left(\vec{\theta}\right)\otimes\text{RX}\left(-\frac{\pi}{2}\right).$
(3)

where $\vec{\theta}$ is a vector of angles in which each qubit will rotate.
Thus, the combination of these rotations and Rz gates allows one to then
perform very general single-qubit gates. Note, in the above and throughout
this work we will use the notation RX for a global rotation and Rx for a local
rotation (and equivalently for $y$ and $z$ rotations).

###  III.1 Choice of the geometry

While the procedure we outline in the following section method can be adapted
for more general lattice geometries, as a first example we consider a linear
chain of $N$ qubits with nearest neighbour (NN) separation $r$ and periodic
boundary conditions (PBC). Furthermore, we neglect next-nearest interactions
as they are 64 times smaller than the nearest neighbour interactions. As a
result, the interaction term in our Hamiltonian simplifies to

$J\sum\limits_{i}\hat{n}_{i}\hat{n}_{i+1},$ (4)

where $J=\frac{C_{6}}{r^{6}}$. However, in Section. VII we consider the effect
of the full long-range tail of the qubit interactions on the fidelities of the
pulse sequences we variationally prepare in the NN setting.

###  III.2 Local Rz gates

A direct consequence of the NN approximation is that we can easily realise any
Rz rotation through the single-qubit addressibility in detuning. For
$\Omega=0$, the evolution operator is given by

$U(t,\vec{\delta})\equiv e^{-i\mathcal{H}(\vec{\delta})t},$ (5)

where

$\mathcal{H}(\vec{\delta})=J\sum\limits_{i}\hat{n}_{i}\hat{n}_{i+1}-\hbar\sum\limits_{i}\frac{\delta_{i}}{2}\hat{\sigma}_{j}^{z}.$
(6)

Given that the longitudinal field induced by the laser detuning commutes with
the qubit interactions, the effect of these interactions in the Hamiltonian
evolution can be removed by carefully choosing the evolution time $t$ such
that $tJ=2\pi$ and thus the effect of
$\exp(-itJ\sum_{i}\hat{n}_{i}\hat{n}_{i+1})\equiv\exp(-i2\pi\sum_{i}\hat{n}_{i}\hat{n}_{i+1})$
is equivalent to the identity. Thus, we are left with
$U(2\pi/J,\vec{\delta})=\exp\left(i\frac{2\pi}{J}\hbar\sum\limits_{i}\frac{\delta_{i}}{2}\hat{\sigma}_{i}^{z}\right)$
and, by choosing $\delta_{i}=\frac{\theta_{i}}{t}$, this can be tuned to act
as Rz gates acting on each qubit with angles $\theta_{i}$.

###  III.3 Global RX/RY gates

As shown in Eqs. (2) and (3), the combination of local Rz rotations with
global $\pi/2$ rotations allows for the realisation of single qubit Rx and Ry
rotations. In the following we present the procedure to variationally prepare
these global rotation gates,
$\left\\{\text{RX}\left(\frac{\pi}{2}\right),\text{RX}\left(-\frac{\pi}{2}\right),\text{RY}\left(\frac{\pi}{2}\right),\text{RY}\left(-\frac{\pi}{2}\right)\right\\}$,
in an analog setting.

Problem setup – We would like to find a pulse sequence that approximates a
global $\text{R}\left(\frac{\pi}{2}\right)$ by optimizing a global Rabi
frequency, global phase and global detuning of each pulse, within the
constraints $\hbar\Omega<J$ and $\hbar|\delta|<2J$ to ensure experimental
feasibility Wurtz  _et al._ (2023); Henriet  _et al._ (2020), to minimise a
well-chosen loss function. In the following we consider a 4-qubit periodic
system with NN interactions. We consider the loss function as the infidelity,
$L=1-F$, such that

$F=\frac{1}{\mathrm{dim}(G)}\left|\mathrm{tr}\left(G^{\dagger}\prod\limits_{i=1}^{p}\exp\left(-i\frac{\Delta
t}{p\hbar}H(\bm{\Theta})\right)\right)\right|,$ (7)

where $G$ is the desired unitary operation acting on $N$ qubits, in our case
$G=\text{R}\left(\frac{\pi}{2}\right)$, $p$ is the number of pulses
considered,
$\bm{\Theta}=(\Omega_{1},\Phi_{1},\delta_{1},\cdots,\Omega_{p},\Phi_{p},\delta_{p})$
and $H$ is the Ising Hamiltonian described by Eq. (1) restricted to NN
interactions and strictly global pulses. Note that $L=0$ implies that the
pulse sequence found is correct up to a global phase shift.

Optimization – We begin by minimizing the loss function $L$ for a single
pulse, i.e. $p=1$, of length $T$ using an L-BFGS-B optimizer. We then use the
parameters found with this first optimisation process as an input to a second
round of optimisation with two pulses, both of length $T/2$. The effect of
this is to increase the expressibility of the pulse sequence. We repeat this,
effectively halving each pulse in the sequence split into two individual
pulses at each step, until we reach a pulse length $dt$ for each pulse; a
predetermined minimum pulse length set by experimental considerations. The
full algorithm used is given in Algorithm III.3.

Note, given each global rotation can be transformed into any other via a
constant shift in the values found for $\phi$, this procedure only needs to be
performed once.

Global rotation pulse

$p\mathrm{{}_{max}}$, $dt\leftarrow$ initialized by user

$T\leftarrow 2^{p\mathrm{{}_{max}}}dt$

thresh $\leftarrow$ threshold initialized by user

$L$ $\leftarrow$ initialized to a value $>$ thresh

while $L>\text{thresh}$ do

$\bm{\Theta}_{0}\leftarrow$ initialized to random values

$t\leftarrow T$

for $p$ in $1:p\mathrm{{}_{max}}$ do

$L,\bm{\Theta}_{p+1}\leftarrow\min\limits_{\bm{\Theta_{p}(t)}}(L)$

$t\leftarrow T/(2^{p})$

end for

end while

Results – We show the results of the procedure outlined above in Fig. 2(b),
here we present the exact pulse sequences found through this procedure that
realise all $\pm\pi/2$ rotations with $>99.9\%$ fidelity for a chain with PBC.
The fidelities found for the implementation of these pulses on different qubit
systems are presented in Table 1. While this optimisation was performed on a
small system, as we consider global pulses and work in the limit of negligible
next-nearest neighbour interactions, these pulses generalise well to larger
chains. We clearly see that using this measure of fidelity we obtain very
accurate results even when doubling the number of qubits in the chain with
respect to the number in which the optimisation procedure was implemented on.
Counterintuitively, even on such small systems the fidelities of these pulses
are even higher when we switch from PBC to open boundary conditions (OBC).

Figure 2: Results of procedure to find a global $\pi/2$ rotation using a neutral atom setup. (a) A schematic showing how these global pulses can be combined with local $Z$ gates to perform local $X$ or $Y$ gates. (b) The precise pulse sequences found to implement global rotations, here $dt\sim 1.6J$ . $\mathrm{n_{qubits}}$ | NN PBC |  NN OBC
---|---|---
4 | $99.92\%$ |  $99.97\%$
5 | $99.90\%$ |  $99.94\%$
6 | $99.87\%$ |  $99.92\%$
7 | $99.85\%$ |  $99.90\%$
8 | $99.83\%$ |  $99.88\%$
Table 1: Fidelities of the global analog qubit rotations for a NN interacting
linear chain with PBC and OBC. Figure 3: The pulse sequence required to
perform a $CZ(0,1)CZ(2,3)$ gate (a) and $CZ(0,1)CZ(1,2)$ gate (b) on a 4 qubit
nearest neighbor linear chain with PBC. The pulse sequence required to perform
a $CZ(0,1)$ gate (c) and a $CZ(1,2)$ gate (d) on a 4 qubit nearest neighbor
linear chain with OBC.

##  IV Realizing Analog two-qubit gates

Building on the previous section, we now discuss how to realize two-qubit
gates with the help of analog rotations. To this end, we first discuss how to
‘turn off’ interactions between qubits with a refocusing algorithm to realise
some two-qubit gates Jones and Knill (1999); Leung  _et al._ (2000); Dodd  _et
al._ (2002); Guseynov and Pogosov (2022). The fundamental idea behind the
refocusing algorithm is: by evolving a system under a given pulse, followed by
a second pulse of the same length in which the sign of the unwanted
interactions is reversed, the resultant effect of this entire pulse sequence
is only that of the desired interactions throughout the whole evolution
Vandersypen and Chuang (2005).

In general reversing the interaction strength of a given quantum system is not
physical but this can be effectively achieved for an Ising-type Hamiltonian by
using single qubit $X$ gates,

$X_{i}e^{t\sigma^{z}_{i}\sigma^{z}_{j}}X_{i}=e^{-t(\sigma^{z}_{i}\sigma^{z}_{j})},$
(8)

which relies on temporarily flipping the computational basis from $\\{0,1\\}$
to $\\{1,0\\}$: in that basis the Pauli-Z operator acquires a sign flip Parra-
Rodriguez  _et al._ (2020).

Given the close relationship between Rydberg interactions and Ising
interactions, and given that layers of $X$ gates can be engineered through the
pulse sequences outlined in the previous sections, we are able to realise an
analog implementation of the refocusing algorithm. Effectively, this idea will
allow us to entangle subsections of the qubit register in a controllable
manner.

###  IV.1 From NN interaction to $CZ$ gates

As the most natural example for an analog neutral atom device we will present
the procedure to perform different layers of effective $CZ$ gates in an analog
operation. A $CZ$ gate acting on two qubits, $i$ and $j$, may be defined as

$CZ=\exp\left(i\pi\hat{n}_{i}\hat{n}_{j}\right).$ (9)

From this definition we can directly recognise the evolution of the nearest-
neighbour neutral atom device on an $N$ qubit linear chain with PBC and no
external laser fields as

$U(t,\vec{0})=CZ(0,1)CZ(1,2)\dotsb CZ(N,0),$ (10)

providing $t=\pi/J$. Once this equivalence has been established we can
consider using the refocusing algorithm to remove the unwanted $CZ$ gates from
Eq. (10). More explicitly, given that the effect of $X$ gates surrounding such
a pulse is

$\begin{split}&X_{j}e^{t\sum_{i}\hat{n}_{i}\hat{n}_{i+1}}X_{j}\\\
&=e^{t\left(\sum_{i\neq
j,j-1}\hat{n}_{i}\hat{n}_{i+1}-\hat{n}_{j}\hat{n}_{j+1}-\hat{n}_{j-1}\hat{n}_{j}+\hat{n}_{j-1}+\hat{n}_{j+1}\right)}.\end{split}$
(11)

By combining these two results and utilising single qubit detuning to
compensate for the linear term in the exponent of Eq. (11), we are able to
produce layers of $CZ$ gates between desired qubits. Note, to ‘turn off’ the
interaction between qubits $i$ and $j$, we surround the second pulse in the
sequence by $X$ gates acting on one of the two qubits involved in the
interaction we wish to turn off. When no $X$ gates act on the qubit pair or an
$X$ gate acts on both qubits, the interaction will not be affected. For
example, to remove the $CZ(0,1)$ and $CZ(2,3)$ from Eq.(10) we can perform the
pulse sequence

$\begin{split}&CZ(1,2)CZ(3,4)CZ(4,5)\dotsb CZ(N,0)=\\\ &X_{1}X_{2}\otimes
U\left(\frac{t}{2},\vec{0}\right)\otimes X_{1}X_{2}\otimes
U\left(\frac{t}{2},\vec{\delta}^{*}\right),\end{split}$ (12)

where $\vec{\delta}^{*}$ is chosen to account for the unwanted linear terms
accumulated by the action of each $X$ gate.

The types of layers of CZ gates realisable through this protocol of removing
given interactions depends on the geometry and boundary conditions of the
system. In the following we will explain these explicitly for a linear chain
with both PBC and OBC.

###  IV.2 Periodic Boundary Conditions: Even and odd qubit chains

Given that the action of each $X$ gate is to remove the effect of two distinct
interactions in a system with PBC, it becomes clear that the number of qubits
in the chain, even or odd, defines the subset of $CZ$ gates realisable.
Explicitly, in an even-numbered qubit chain we are able to realise layers of
an even number of $CZ$ gates, and in an odd-numbered qubit chain we are able
to realise layers of an odd number of $CZ$ gates. As a result, in order to
realise a single $CZ$ gate, for example $CZ(0,1)$, we are required to work on
a system containing an odd number of qubits. However, as many applications
benefit from parallelisation of two qubit gates, this does not greatly
restrict potential use cases of our method.

As an explicit example we consider a 4 qubit chain with PBC. The possible two
types of layers of $CZ$ gates realisable (up to transnational symmetry) are
$CZ(0,1)CZ(2,3)$, by, for example, using $X$ gates on qubits 0 and 1 and
$CZ(0,1)CZ(1,2)$ by, for example, using an $X$ gate on qubit 3. Schematics of
these pulse sequences are given in Figs. 3(a) and (b), each requiring pulse
sequences of length $4T+5\pi/J$. These achieve a fidelity of $99.6\%$ and
$99.7\%$ respectively for a system with NN interaction.

###  IV.3 Open Boundary Conditions

Given a linear chain with OBC we can readily prepare a given layer of CZ gates
with no restrictions - this is a consequence of the end qubits interacting
with only one other qubit. For example, we an prepare a $CZ(0,1)$ gate on
through the use of an $X$ gate on qubit 2 or a $CZ(1,2)$ through the use of
$X$ gates on qubits 1 and 2. Schematics of these pulse sequences are given in
Figs. 3(c) and (d), again each require pulse sequences of length $4T+5\pi/J$.
These achieve a fidelity of $99.7\%$ and $99.6\%$ respectively for a system
with NN interaction.

Given that the variationally generated pulse sequences for global rotations
have higher fidelity on linear chains with OBC as well as the additional
flexibilty to realise layers of CZ gates this setting gives, we will consider
such systems for the rest of this article. In the following sections, we show
how, in theory, analog single qubit and $CZ$ gates can be used to build a full
SWAP network to produce all-to-all connectivity in an analog neutral atom
device. We then use this result to perform some simple analog VQE simulations
of small molecules.

##  V  Generating a SWAP network

While some attempts have been made to build devices with all-to-all
connectivity (for instance using trapped ions Landsman  _et al._ (2019),
photons Bartolucci  _et al._ (2021) as well as some preliminary results in
neutral atoms Bluvstein  _et al._ (2022, 2023)) in currently available NISQ
devices based on superconducting qubits Patra  _et al._ (2023); Pan and Zhang
(2021) and neutral atoms Henriet  _et al._ (2020) connectivity is limited. An
example in which this low connectivity can be an issue is the simulation of
fermionic systems. When simulating the physics of such systems, the occupation
number of a spin orbital is encoded in the qubit state. Therefore, in order to
allow all orbitals to interact with each other we must apply two qubit gates
between all pairs of qubits. Mathematically, this can be seen through the loss
of locality when applying the Jordan-Wigner transformation on simple molecular
Hamiltonians to convert between the fermionic basis and qubit basis. Using
quantum annealing to solve certain difficult quadratic unconstrained binary
optimization (QUBO) problems is another example in which all-to-all
connectivity is required Dí ez-Valle  _et al._ (2023). Such connectivity
allows many NP-hard optimisation problems to be encoded in the ground state of
an Ising model. This is thought to be a promising direction for useful quantum
computation Lucas (2014).

One popular method to circumvent the issue of low connectivity within a
digital device is a SWAP network. Such a network uses SWAP gates that switch
the quantum information between two given qubits. A SWAP network effectively
inverts the position of the qubits in the circuit and allows one to act the
required operations between the given qubits when they are adjacent to each
other in this cycle O’Gorman  _et al._ (2019). Clearly, performing such a SWAP
network is not native to an analog device. Typically, in analog systems long-
range entanglement is generated through quench dynamics and is limited by a
Lieb-Robinson bound of the resource Hamiltonian Michel  _et al._ (2023).
However, using the protocols outlined in the Sections III and IV, a SWAP
network can be directly imported into the analog setting. In particular, a
SWAP gate can be decomposed into three CNOT gates as shown in Fig. 4(a). As a
CNOT gate can be realised through

$CNOT(i,j)=\text{Ry}\left(\frac{\pi}{2}\right)\;CZ(i,j)\;\text{Ry}\left(-\frac{\pi}{2}\right),$
(13)

we conclude that with the pulse sequences given in this work, an analog SWAP
gate is, in principle, realisable.

As a simple benchmark, by using the gates found in the previous sections to
perform a SWAP network on 4 qubits in a NN linear chain, we realise a long-
range CNOT gate between qubits 0 and 3. This pulse sequence achieved a
fidelity of $~{}94\%$ with SWAP gates that each require pulse sequences of
length $3(6T+5\pi/J)$. In the following, we add one more layer of complexity
to this proposal by replacing SWAP gates with Givens SWAP gates that are known
to be an effective ansatz for VQE problems Elfving  _et al._ (2021).

Figure 4: (a) The decomposition of a SWAP gate as three CNOT gates. (b) The
decomposition of a Givens SWAP gate into $CZ$ gates and single-qubit Ry
rotations, note the angles of each rotation are shown in the schematic.

##  VI  Application: Variational Quantum Eigensolvers

As previously mentioned, one promising application of quantum computing that
strongly benefits from all-to-all connectivity is the simulation of molecules.
When simulating the physics of electronic systems the typical approach is to
use Fock basis in which each basis function represents a fermionic mode. A
standard method to simplify the task of simulating these systems as well as
half the required resources is the paired-electron approximation. Here, we
consider the simplification such that each molecular orbital is either
occupied by an electron singlet pair or unoccupied. Under this assumption the
Hamiltonian of a molecule is given by

$\hat{\cal{H}}=C+\sum_{p,q}h_{p,q}^{(r1)}\hat{b}_{p}^{\dagger}\hat{b}_{q}+\sum_{p\neq
q}h_{p,q}^{(r2)}\hat{b}_{p}^{\dagger}\hat{b}_{p}\hat{b}_{q}^{\dagger}\hat{b}_{q},$
(14)

where $\hat{b}_{q}$ are hardcore bosons representing the electron-pair
annihilation operator in mode $p$ that satisfy the relevant (anti-)commutation
relations and $h_{p,q}^{(r1)}$ and $h_{p,q}^{(r2)}$ can be calculated from the
one- and two-electron integrals Elfving  _et al._ (2021).

To map this problem to a form that is applicable for a quantum computer we can
use the transformation
$\hat{b}_{p}=\frac{1}{2}(\hat{\sigma}^{x}_{p}+i\hat{\sigma}^{y}_{p}),$
resulting in a Hamiltonian of the form

$\hat{\cal{H}}_{qb}\approx
C+\sum_{p}\frac{h_{p}^{(r1)}}{2}(\hat{I}_{p}-\hat{\sigma}_{p}^{z})+\sum_{p\neq
q}\frac{h_{p,q}^{(r1)}}{4}(\hat{\sigma}_{p}^{x}\hat{\sigma}_{q}^{x}+\hat{\sigma}_{p}^{y}\hat{\sigma}_{q}^{y})\\\
+\sum_{p\neq
q}\frac{h_{p,q}^{(r2)}}{4}(\hat{I}_{p}-\hat{\sigma}_{p}^{z}-\hat{\sigma}_{q}^{z}+\hat{\sigma}_{p}^{z}\hat{\sigma}_{q}^{z}).$
(15)

This form of the Hamiltonian is now in a basis that directly maps to qubits
and allows us to find the ground state energy through a Variational Quantum
Eigensolver (VQE) approach.

Figure 5: Analog VQE results for the molecules H2 in (a) and LiH in (b), both
run in Qadence Seitz  _et al._ (2024). Here, we show the results for multiple
ansatzes with varying depths of Given SWAP gates as a function of the number
of qubits, $n$. For H2 $n=4$ and LiH $n=6$. Here we manage to reduce the error
of the VQE protocol to be on the same order of magnitude of the chemical
accuracy (the grey dashed line).

Given the molecular Hamiltonian is number conserving, i.e.,
$\sum_{p=0}^{q}\hat{b}_{p}^{\dagger}\hat{b}_{q}=n$ where $n$ is the number of
electrons in the system, a good choice for initial state during a VQE protocol
is the Hartree-Fock state. This is the state in which the $n$ lowest energy
molecular orbitals are occupied with pairs of electrons, here $n$ is the total
number of pairs of electrons in the system. From this initial state a Givens
SWAP network has been shown to be a promising variational ansatz to find the
ground state within the paired-electron approximation Elfving  _et al._
(2021). In fact, a proof of principle experiment utilising this ansatz was
preformed on Google’s superconducting qubit quantum computer and successfully
found the ground state for some simple molecular Hamiltonians O’Brien  _et
al._ (2022). A Givens SWAP gate is the combination of a Givens rotation
followed by a SWAP gate, mathematically it is defined by

$GS(\theta)=\underbrace{\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{pmatrix}}_{SWAP}\underbrace{\begin{pmatrix}1&0&0&0\\\
0&\cos(\theta)&-\sin(\theta)&0\\\ 0&\sin(\theta)&\cos(\theta)&0\\\
0&0&0&1\end{pmatrix}}_{G(\theta)}$ (16)

in which $\theta$ is a variational parameter. A full digital Givens SWAP
network is equivalent to the SWAP network with each SWAP gate replaced by a
Givens SWAP gate.

We now turn our attention to the analog implementation of such an ansatz. As a
Givens SWAP network can be decomposed into three $CZ$ gates and layers of
single-qubit gates, see Fig. 4(b), we can again lean on the protocols outlined
in Sections III and IV to build an analog pulse sequence that approximates
this digital circuit.

We use such a variational architecture to find the ground state energy of H2
and LiH using an analog Givens SWAP network. For H2 we consider the 631g basis
in which 4 qubits are required in the paired-election approximation Jensen
(2017). For LiH we consider the sto-3g basis in which we require 6 qubits in
the paired-electron approximation Jensen (2017). In Fig. 5 we show the
evolution of errors throughout the VQE protocol. We define the error as the
difference between the measured energy and the lowest value achievable in the
paired electron approach. Theoretically, the number of layers of Givens SWAPs
required to reach the ground state is equal to the number of qubits. However,
since our gates do not have a $100\%$ fidelity, more layers are required. We
vary the depth of our network as a function of the number of paired electrons
$n$, i.e. $n=4$ for H2 and $n=6$ for LiH. Here, we used BFGS, a gradient based
optimizer and an a convergence criterion of $\epsilon=e-7$. In Fig. 5 see that
while chemical accuracy (an error of 1.6 Ha) was not achieved for either H2 of
LiH, we were to achieve an error on the same order of magnitude to this goal.

##  VII Experimental Consideration

In this section we consider realistic experimental constraints such as atomic
positions, interactions strengths, Rabi frequencies, detunings and coherence
times. From these we present the required coherence times to realise each gate
used throughout the results we present in this report. Furthermore, we show
the effect of the long-range nature of interactions in a realistic qubit
register on the accuracy of the pulses.

###  VII.1 Pulse sequence depths

We consider a rubidium-based neutral atom device with atoms coupled to the
$|60S_{1/2}\rangle$ Rydberg state, and include the full $1/r^{6}$ long-range
tail of the Van der Waals interactions. Furthermore, we consider $N$ atoms in
a circular geometry with inter-atomic spacing of $r=6.24~{}\mu$m. This results
in a nearest neighbour interaction of $J\sim 2\pi\times 2.3$ MHz. We consider
$dt=108$ ns such that the time to perform a local $RZ$ gate is $428$ ns with
$\left|\delta_{j}\right|=J$. Our variationally prepared pulse sequences for
the global $\pi/2$ rotation gates are such that $T=864$ ns, $\Omega/2\pi<1.2$
MHz and $\left|\delta/2\pi\right|<4.6$ MHz. From these values we can then
calculate the required pulse lengths of each of the gates we use in this work,
see Table. 2. While the analog SWAP and Givens SWAP gates may be beyond
current experimental limitations with the pulses presented here, given that
coherence times of up to $\sim\\!6\mu$s have been experimentally realised
Scholl  _et al._ (2021), it is not unreasonable to expect a layer of analog
CNOT gates to be feasible in the near future. Note, in order to run deep
quantum circuits with more than one unique layer of qubit gates, we would
require temporal control of the single qubit addressability in laser
detunings.

(layer of) Gate | Pulse Sequence Length ($\mu$s)
---|---
RZ | $0.4$
Global $\pi/2$ rotation | $0.9$
Local RX/RY | $2.1$
CZ | $4.5$
CNOT | $6.2$
SWAP | $18.7$
Givens SWAP | $22.1$
Table 2: The required pulse lengths of given gates using proposed analog
pulses.

###  VII.2 Effect of long-range interactions

In the following section we consider the effect of the full long-range
interaction tail on the fidelities of both our proposal for local Rz gates
proposed in Section. III as well as the pulses given in Fig. 2(b). To estimate
the effect of these long-range interactions in the limit $N\gg 1$ we consider
a linear spin chain with OBC.

While the protocol to produce a local Rz gate proposed in Section. III is
exact for NN interactions, it is only approximate in the presence of the full
long-range interaction tail defined in Eq. (1). As a result we find that the
fidelity drops to $~{}98.9\%$ for a system of 8 qubits. However, by deviating
the pulse length used away from the theoretically exact time for a NN system
we are able to retain a fidelity of $99.6\%$, see Fig. 6(a). Furthermore, in
this setting of 8 qubits, the fidelity of the global rotation pulse sequences
falls by about $2-3\%$. However, by again allowing the length of the pulse
sequence to vary we can obtain a fidelity of $98.7\%$, see Fig. 6(b).

While these fidelities are still high and may be acceptable for certain
applications, it highlights that, given access to more computational
resources, performing the optimisation protocol outlined above on a larger
system with the full long-range tail would generalise better to long chains.

###  VII.3 Generalisation to larger system sizes

In Fig. 6 we show how our protocol generalises to large qubit systems, $N=18$
which is more than four times the chain length the pulses were optimised over.
In these simulations we consider the full long-range tails of the Van der
Waals interactions, here we place the qubits in a linear chain with OBC. As
the fidelity given in Eq. (7) requires the calculation of the full unitary
time evolution operator for a given pulse sequence, it is an expensive metric
for chains of this length. Thus, we use a simplified metric for fidelity: the
overlap the state prepared by acting our proposed pulse sequence on the ground
state with the exact state that is prepared via the target gate, $G$, i.e.

$\tilde{F}=\bra{000\cdots
0}G^{\dagger}\prod\limits_{i=1}^{p}\exp\left(-i\frac{\Delta
t}{p\hbar}H(\bm{\Theta})\right)\ket{000\cdots 0}.$ (17)

We find that for the pulse sequences variationally found for global $\pi/2$
rotations retain an overlap of $\tilde{F}>97\%$.

Furthermore, we ran simulations for a local $X$ gate on a single qubit in this
experimentally inspired set up with a resulting overlap of $\tilde{F}\approx
85\%$. However, as mentioned previously, when introducing long-range
interactions the Rz gate implementation is no longer exact. Thus, as further
investigation we find that deviating the pulse length from the NN predicted
time to $412$ns, we are able to increase the fidelity of the single qubit Rx
gate to $\tilde{F}>93\%$. In Fig. 6(c) and (d) we show the resultant local
magnetisations of each qubit after the analog pulses for both a global
$\text{RY}\left(\frac{\pi}{2}\right)$ rotation and local $X$ gate in the bulk
of the chain. Both of these results show that our pulse sequences optimised on
only 4 qubits with NN interactions are still highly effective in large systems
with the full long-range tail, and clearly show (up to small errors) the
desired gate effect.

Figure 6: Full analog simulations of both the global rotations as well as a
single qubit $X$ gate using the full long-range tail of the Van der Waals
interactions. (a) and (b) Results for the fidelity of analog Rz and global
rotation gates respectively in the presence of long-range interactions showing
the effect of varying the pulse length from the analytically predicted value
using NN interactions (shown as a vertical dashed line). (c) and (d) The
resultant local magnetisations after the analog pulses for a global
$\text{RY}\left(\frac{\pi}{2}\right)$ and local $X_{8}$ gates respectively.

###  VII.4 Future directions

In order to both reduce the required coherence times to implement these analog
gates as well as increase the fidelities in a more realistic experimental set
up, there are many directions that can be considered. Firstly, by performing
the optimisation with more flexible pulse sequences, for example continuous
interpolated pulse sequences parametrized through a Fourier basis Geen and
Freeman (1991), or bang-bang pulses Yang  _et al._ (2017) and by making use of
the CRAB algorithm Caneva  _et al._ (2011)), we can aim to reduce $T$.
Secondly, varying both the Rydberg level in use and the inter-atomic spacing
as well as experimental improvements that allow both $dt$ to decrease and
coherence times to increase, would each allow deeper circuits to be run on
analog systems. Also, performing the optimisation over a larger number of
qubits in a circular register and taking into account the full long-range
nature of interactions would push up the fidelities observed in the
experimental setup used in this section.

Finally, in order to solve the control task more efficiently, this work could
be extended using several other optimization techniques. Classical simulation
algorithms that perform gradient-based optimization such as GRAPE Khaneja  _et
al._ (2005), for instance, would offer a drop-in replacement for the approach
taken here. In particular, GRAPE also discretizes the control pulses, but then
optimizes based on the (approximate) derivative of the cost function with
respect to the pulse train parameters. As a purely classical algorithm, it is
restricted to systems consisting of a small number of qubits. For larger
systems, hybrid approaches such as differentiable analog quantum computing
Leng  _et al._ (2022) offer an attractive alternative. Here, the (functional)
derivative of the cost function with respect to some control function is
calculated on a quantum system using a combination of parameter shift rules
and Monte-Carlo integration.

Alternatively, gradient-free techniques such as Bayesian optimization
Shahriari  _et al._ (2016); Frazier (2018) could be helpful. The downside of
such an approach, however, is its unfriendly scaling with the dimension of the
parameter space it has to search through. This is especially a problem with
fine-grained pulses that have a lot of degrees of freedom. Restricting oneself
to pulses generated by a small set of basis functions would make such methods
attractive.

##  VIII  Outlook

In this work we outlined a procedure to realise a general set of one- and two-
qubit gates in the presence of always-on interactions. In particular, we
considered simulations using an analog neutral atom device. In this setting,
we showed how local Rz rotations are realisable given local addressing on the
laser detunings. Furthermore, we presented a procedure to variationally
prepare global $\pi/2$ rotations to allow us to change basis between the $X$,
$Y$ and $Z$ axis with high fidelity. Combining these methods we are able to
perform layers of single-qubit gates. These in turn allowed us to build up a
repertoire of two-qubits gates taking advantage of a refocusing algorithm.

While currently using the methods in this work to import a fully digital
algorithm into an analog setting is still far beyond what is experimentally
feasible, there are many short term uses that are highly valuable. For
example, one hurdle for analog simulation on a neutral atom device is initial
state preparation. With the analog pulses outlined above, we are able to
prepare both $X$ and $Y$ polarised states as well as product states with high
fidelities. By generalising this protocol to higher dimensions, while non-
trivial, would open up the prospect of more interesting quantum simulations in
the near-term. Furthermore, this work opens the door to measurements in a
general basis. This would allow for measurements of new observables, for
example energy and general correlation functions. There has been some exciting
proposals on the use of randomised measurements in the neutral atom setting
Notarnicola  _et al._ (2023). A successful implementation of the above
protocols outlined in this work would open up the door for such methods to be
implemented in the near term.

###### Acknowledgements.

We thank P. Barkoutsos and D. Seitz for insightful discussions. AD and JV
acknowledge funding from the European Union under Grant Agreement 101080142
and the project EQUALITY.

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