# Variational quantum state preparation via quantum data buses

Viacheslav V. Kuzmin Center for Quantum Physics, Faculty of Mathematics,
Computer Science and Physics, University of Innsbruck, A-6020, Innsbruck,
Austria  Institute for Quantum Optics and Quantum Information of the Austrian
Academy of Sciences, A-6020 Innsbruck, Austria  Pietro Silvi Center for
Quantum Physics, Faculty of Mathematics, Computer Science and Physics,
University of Innsbruck, A-6020, Innsbruck, Austria  Institute for Quantum
Optics and Quantum Information of the Austrian Academy of Sciences, A-6020
Innsbruck, Austria

###### Abstract

We propose a variational quantum algorithm to prepare ground states of 1D
lattice quantum Hamiltonians specifically tailored for programmable quantum
devices where interactions among qubits are mediated by Quantum Data Buses
(QDB). For trapped ions with the axial Center-Of-Mass (COM) vibrational mode
as single QDB, our scheme uses resonant sideband optical pulses as resource
operations, which are potentially faster than off-resonant couplings and thus
less prone to decoherence. The disentangling of the QDB from the qubits by the
end of the state preparation comes as a byproduct of the variational
optimization. We numerically simulate the ground state preparation for the Su-
Schrieffer-Heeger model in ions and show that our strategy is scalable while
being tolerant to finite temperatures of the COM mode.

##  1 Introduction

Realizing many-body quantum states on quantum devices offers an experimental
pathway for studying the equilibrium properties of interacting lattice models
[1, 2, 3], quench dynamics [4, 5, 6, 7], or it can be viewed as a quantum
resource, e.g., for sensing [8, 9, 10, 11]. Adiabatic state preparation has
been experimentally demonstrated, e.g., on trapped ions [12] and atoms in
optical tweezers [13]. However, this strategy is currently limited by the
finite coherence times of quantum devices, incompatible with the long
annealing times required by the adiabatic criterion [14]. An alternative,
promising pathway towards quantum ground state preparation is given by
feedback-loop quantum algorithms, and especially by the Variational Quantum
Eigensolver [15, 16, 17, 18, 19, 20, 21] (VQE). In VQE, parametrized $-$
generally non-universal $-$ quantum resource operations, available on a given
quantum device, are arranged in a variational sequence, or circuit ansatz, to
generate entangled trial states. The trial states approximate a target ground
state as a result of a closed-loop optimization of the variational parameters,
where an energy cost function is minimized.

For trapped ion qubits, VQE was recently demonstrated using as entangling
resource either the Mølmer-Sørensen gate [22], or a programmable analog
quantum simulator of the long-range XY spin model [23]. Both these entangling
resources generate effective interactions between ion qubits by coupling them
to vibrational modes (phonons). Implementing the optical couplings off-
resonantly, one effectively achieves robustness of these resource operations
to the temperatures of the phonon modes. However, this strategy yields
relatively slow rates [24, 25], limiting the performance of VQE, as the
quantum processing suffers from decoherence.

Figure 1: (a) Blue-detuned optical pulse acting on ion $j$ realizes
interaction $H_{j}$, Eq. (1), which couples the internal ion levels to levels
of the COM phonon mode. The coupling rates depend on the state of the system.
(b) QDB-MPS circuit in an ion trap generating trial states as in Eq. (6). The
COM mode (red line), as single QDB, is initially cooled to a low-temperature
state $\rho_{0}$, and ions (black lines) are initialized in pure states.
Variational operations $\text{exp}(-i\theta_{k}H_{j_{k}})$ (blue links) are
virtually grouped into boxes, parametrized by corresponding vectors
$\boldsymbol{\theta}_{i}=\\{\theta_{k}\\}$, as explicitly illustrated for the
first two boxes. The out-coming ions are measured, while the state of the QDB
is discarded. (c) Mapping of circuit (b) with the optimized parameters,
$\\{\boldsymbol{\theta}_{i}^{\text{(opt)}}\\}$, to an MPS diagram with the
qubits state approximating $\left|\psi_{\text{targ}}\right\rangle$.

Here we propose a strategy to improve the efficiency of VQE on programmable
quantum devices where interaction between qubits is mediated by auxiliary
degrees of freedoms: Quantum Data Buses (QDBs). This strategy employs
interactions between qubits and QDBs, naturally available in these devices, as
variational resource operations in the quantum algorithm. We illustrate our
approach on trapped ions, using the acoustic Center-Of-Mass (COM) phonon mode
$-$ the axial vibrational mode where all ions oscillate in phase $-$ acting as
a single QDB. We employ resonant optical coupling of ion qubits with the COM
mode, yielding faster rates than off-resonant strategies, and thus being less
prone to decoherence. Unlike logical gates designed with the resonant
coupling, such as the C-phase gate [26], our variational approach is tolerant
to finite temperatures of the COM mode, which do not limit the state
preparation fidelity. Moreover, our strategy does not require to disentangle
the QDB after each resource operation, but only at the end of the state
preparation, ultimately yielding faster processing. The task of disentangling
the QDB from the qubits is carried out by the optimization algorithm itself:
As the output qubit state approaches the unique ground state of a target non-
degenerate Hamiltonian, the QDB becomes disentangled without ever being
measured.

Via numerical simulations, we investigate the scaling and robustness
properties of our strategy for 1D lattice models. As a benchmark goal, we aim
to variationally prepare the ground state, $|\psi_{\text{targ}}\rangle$, of
the Su-Schrieffer-Heeger (SSH) Hamiltonian [27, 28, 29] on ion qubits in a
linear trap. Using only blue-detuned sideband optical pulses [30] as resource
operations, we design the variational circuit ansatz shown in Fig. 1, which
can efficiently realize Matrix Product States (MPS), a class of tailored
variational wavefunctions capable of accurately capturing the equilibrium
physics of many-body quantum systems in 1D [31, 32]. This ‘QDB-MPS circuit’
can incorporate various symmetries of the target model for enhanced
performance, including approximate translational invariance in the bulk. In
this paper, we will show that our approach is scalable, as we can approximate
1D ground states at saturating precision in the system size $N_{\text{ions}}$,
without increasing the number of variational parameters $N_{p}$ [see Fig.
2(a)]. Additionally, we compare the results from the QDB-MPS circuit with
other VQE strategies, still designed for trapped ion hardware, but using
different sets of (coherent) entangling resources: (i) site-filtered Mølmer-
Sørensen gates (ii) an analog quantum simulator of a long-range XXZ model
[23]. We evaluate the respective accuracies in terms of various figures of
merit, including excitation energy, fidelity, and two-point correlations. We
demonstrate that our strategy can realize highly-accurate ground states even
for QDB initialized at finite temperatures. Finally, we show that the QDB-MPS
circuit can be up-scaled beyond the single-trap limit by being implemented in
modular ion traps [33, 34]. Overall, we consider our strategy a viable,
efficient route towards quantum state preparation in ion traps as well as in
other quantum platforms that rely on QDBs, such as atom qubits coupled via
photons in waveguides [35, 36] or cavities [37, 38, 39, 40], or
superconducting qubits coupled by microwave resonators [41, 42, 43, 44].

The paper is organized as follows. In section 2, we describe the ion trap
resource operations which we employ in the QDB-MPS circuit. We also review the
resources used by other VQE strategies in ion traps, which we consider for
comparison. We briefly introduce the target SSH model in section 3, and we
discuss in detail the VQE strategies. We exhibit the results of our numerical
simulation in section 4. Finally, in section 5, we summarize our conclusions
and argue future perspectives for our approach.

##  2 Quantum resources in trapped ions

Figure 2: (a) Error function $F_{\text{err}}$ (7) versus size
$N_{\text{ions}}$ of the states prepared in the circuits QDB-MPS, CSA shown in
(b), and CSD-MPS shown in (c) for the fixed number of variational parameters
$N_{p}=18$. The insets show correlation matrix $C_{ij}^{\text{DMRG}}$ (8)
(right) with $i<j$ for $\left|\psi_{\text{targ}}\right\rangle$ with
$N_{\text{ions}}=12$ obtained by the numerical DMRG method, and deviations
$\Delta C_{ij}=C_{ij}-C_{ij}^{\text{DMRG}}$ for $C_{ij}$ obtained for states
prepared by the circuits CSA (left) and QDB-MPS (middle); the arrows indicate
the corresponding $F_{\text{err}}$ data-points. The dashed pixels represent
arias with negligible absolute value $<0.01$. (b) CSA and (c) CSD-MPS circuits
generating trial states (5). CSA circuit uses global all-to-all qubits
operations (3) and singe-qubit rotations $\sigma_{i}^{z}$. CSD-MPS circuit
uses local entangling MS gates (2) and singe-qubit rotations $\sigma_{i}^{z}$.

We consider a setup of $N_{\text{ions}}$ ions in a linear Paul trap as a
relevant quantum platform. The system qubit, or spin 1/2, at site $j$ is
encoded by two internal electronic levels of ion $j$ [30], labeled
$\left|\downarrow\right\rangle_{j}$ and $\left|\uparrow\right\rangle_{j}$.
Each ion, driven by laser fields in the transition between levels
$\left|\downarrow\right\rangle_{j}\leftrightarrow\left|\uparrow\right\rangle_{j}$,
experiences a recoil associated with absorption or emission of photons. This
mechanism couples the internal ion levels with vibrational (phonon) modes,
either axial or radial, depending on the optical setup [30], and allows
phonons to be used as bosonic QDBs for the ion qubits. In ion traps, as well
as in other quantum hardware using QDBs [45, 35, 36, 37, 38, 39, 40, 41, 42,
43, 46], various strategies have been developed to construct entangling
quantum operations between qubits, while ensuring that the QDBs are
disentangled from the qubits by the end of each operation. In what follows, we
review some of these strategies.

Resonant Sideband Resources $-$ One possible approach to exploit QDBs in
quantum hardware is to transfer information from one qubit to another by
populating QDBs with real excitations [26, 46, 45, 43, 42]. In trapped-ion
quantum computers, a typical choice is to elect the axial COM mode as single
QDB, thanks to notable advantages: The mode frequency is well separated from
the other phonon modes thanks to the linear dispersion at $k=0$ of the
acoustic band, and additionally the qubit-phonon coupling is (roughly)
homogeneous along the chain. Using focused beams, each individual ion can be
coupled to the COM mode, usually via sideband optical pulses [26] implemented
in the Lamb-Dicke regime [30] (where the Lamb-Dicke parameter $\eta$ is small,
$\eta\ll 1$). These pulses have frequencies detuned from the atomic transition
exactly by the frequency of the COM mode $\nu$. For the case considered
further in this paper, it is sufficient to consider a single resonant
sideband, either the $-\nu$ negative detuned (red sideband) or the $+\nu$
positive detuned (blue sideband). As the two choices lead to equivalent
results, here we only illustrate the blue sideband case for simplicity, which
is described by the Anti-Jaynes-Cummings Hamiltonian

$H_{j}=i\eta\Omega\left(a\sigma_{j}^{-}-a^{\dagger}\sigma_{j}^{+}\right)$ (1)

addressing ion $j$, where $\Omega$ is the Rabi frequency, and achievable rates
are of the order $\eta\Omega\propto 25\,\text{kHz}$ [47, 48]. In this picture,
$a$ is the destruction operator for the COM mode, and
$\sigma_{j}^{-}=(\sigma_{j}^{+})^{\dagger}=\left|\downarrow\right\rangle\\!\left\langle\uparrow\right|_{j}$
is the lowering operator for ion $j$. In Eq. (1), we used a gauge
transformation $a\to e^{i\varphi}a$ to highlight the anti-unitary symmetry
$(\sigma^{\pm}\to\sigma^{\mp},a\to a^{\dagger},t\to-t)$ in the interaction
Hamiltonian.

In Ref. [26], it was proposed to use resonant sideband pulses to realize the
C-phase gate in ion traps. However, the main hindrance in using gates built
out of resonant interactions with a bosonic QDB, as Eq. (1), is the
requirement for the QDB to be perfectly initialized in a pure state, typically
the vacuum $\left|0\right\rangle_{a}$, where index $a$ refers to the QDB mode.
This requirement stems from the incommensurate rates $\sqrt{l+1}\eta\Omega$ of
the transitions
$\left|\downarrow\right\rangle_{j}\left|l\right\rangle_{a}\leftrightarrow\left|\uparrow\right\rangle_{j}\left|l+1\right\rangle_{a}$,
depending on the number of excitations $l$ stored in the QDB, as highlighted
in Fig. 1(a). As a result, the gate fidelity of resonant sideband gates in ion
traps decreases roughly linearly with temperature $T$ of the initial thermal
state $\rho_{0}(T)=(1-e^{-1/T})\exp(-a^{\dagger}a/T)$ (in units of
$k_{B}=\hbar=1$) of the COM mode.

Off-Resonant Sideband Resources $-$ To address this hindrance, alternative
strategies to realize entangling operations were developed, in which couplings
of the qubits to QDBs are implemented off-resonantly, and the QDBs are only
virtually populated [49, 50, 51, 52, 39]. A prototypical example in ion traps,
the Mølmer-Sørensen (MS) gate [49], can be implemented by coupling a subset
$\mathcal{S}$ of ions to the COM mode by a bichromatic laser beam with two
frequencies, detuned from the qubit frequency by $\pm\left(\nu-\delta\right)$
with $\delta\ll\nu$. The resulting Hamiltonian

$H_{\text{MS}}=\frac{(\eta\Omega)^{2}}{\delta}\sum_{i<j}^{\mathcal{S}}\sigma_{i}^{x}\sigma_{j}^{x},$
(2)

is an all-to-all interaction among qubits in $\mathcal{S}$, with
$\sigma_{j}^{x}=\sigma_{j}^{+}+\sigma_{j}^{-}$. The condition
$\delta\gg\eta\Omega$ [53] ensures that the qubits interact with each other
via second-order processes, populating the COM mode only virtually and keeping
it disentangled from the qubits. This approach is insensitive to the
temperature of the COM mode. However, its rates are intrinsically lower than
the rates of the resonant sideband resources (1),
$(\eta\Omega)^{2}/\delta\ll\eta\Omega$.

Analog Spin-model Resource $-$ Recent experiment [23] showed that VQE in ion
traps can also employ a programmable analog quantum simulator as entangling
resource operation. Such analog quantum simulator is realized by using a
bichromatic laser beam that off-resonantly couples all the ions to all the
transverse phonon modes, ultimately implementing the XY Hamiltonian [50, 54]

$H_{XY}=\sum_{i=1}^{N_{\text{ions}}-1}\sum_{j=i+1}^{N_{\text{ions}}}\frac{J_{ij}}{2}\left(\sigma_{i}^{x}\sigma_{j}^{x}+\sigma_{i}^{y}\sigma_{j}^{y}\right)+B\sum_{i=1}^{N_{\text{ions}}}\sigma_{i}^{z}$
(3)

with power-law $J_{ij}\sim J_{0}\left|i-j\right|^{-\alpha}$ long-range
couplings, where $\sigma_{j}^{\\{x,y,z\\}}$ are the Pauli operators acting on
qubit $j$ and $B$ is a uniform magnetic field. Thanks to the off-resonant
couplings, the interaction (3) is also insensitive to the temperature of the
phonon modes. However, typical rates $J_{0}\sim 0.05-0.5\,\text{kHz}$ [24, 25]
are, again, much slower than the rates of resonant sideband interactions (1).

##  3 Variational Quantum Eigensolvers

VQE enables the experimental realization of ground states of arbitrary
Hamiltonians, providing the best approximation for the given resource
operations. Once the optimization is converged, the optimal quantum state can
then be re-prepared on demand. For example, this state can be used as a
resource to initialize the system for digital quantum simulation [55], which
was recently demonstrated on ion traps in the context of lattice gauge
theories [6].

Target model: Su-Schrieffer-Heeger $-$ As an example, in this work, we
consider VQE for the ground state preparation of the SSH model on ion qubits.
The SSH is an exactly solvable Hamiltonian [27, 28, 29], and a free-fermion
prototype for symmetry-protected topological order in 1D, which reads

$H_{\text{SSH}}=\sum_{j=1}^{N_{\text{ions}}-1}\left(1-\left(-1\right)^{j}t\right)\left(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}\right)\\\
+\tilde{B}\left(\sigma_{1}^{z}-\sigma_{N_{\text{ions}}}^{z}\right)$ (4)

in open boundary conditions. $H_{\text{SSH}}$ is real in the canonical basis,
exhibits $z$-magnetization symmetry
$[H_{\text{SSH}},\sum_{j}\sigma_{j}^{z}]=0$, and has zero-magnetization ground
states with real amplitudes. We added a small staggered magnetic field
$\tilde{B}=0.1$ at the boundaries, to lift the four-fold degeneracy arising in
the topological phase ($t<0$). With this prescription, $H_{\text{SSH}}$ is
also CP symmetric (invariant under global spin-flip plus spatial reflection),
translationally invariant in the bulk with a period of two sites, and has a
critical point at $t=0$.

VQE with Closed System Analog resources (CSA) $-$ In Ref. [23], VQE was
performed on trapped ions using Closed-System programmable Analog resources as
in Eq. (3). We report the variational circuit ansatz considered in that work
in Fig 2(b). This circuit alternates global resource operations, generated by
$H_{XY}$, and single-qubit rotations $\exp(-i\theta\sigma_{j}^{z})$. It was
shown that this strategy is potentially scalable and capable of approximating
ground states of 1D lattice models, including lattice gauge theories, with
high fidelity. Here, we consider the CSA circuit as a comparison benchmark for
the QDB-MPS circuit in preparing of the ground state of $H_{\text{SSH}}$. We
briefly review some details of the CSA strategy to highlight the differences
with respect to the QDB-MPS.

The variational CSA circuit built with closed-system resource interactions
$H_{k}^{(r)}\in\\{H_{XY},\sigma_{j}^{z}\\}$ generates (ideally pure) trial
states

$\displaystyle\left|\psi_{\text{out}}(\boldsymbol{\theta})\right\rangle=\prod_{k}\text{exp}(-i\theta_{k}H_{k}^{(r)})\left|\psi_{\text{in}}\right\rangle,$
(5)

where the product of operators is ordered right-to-left for increasing $k$.
The system qubits are initialized in a Néel product state
$\left|\psi_{\text{in}}\right\rangle=\left|\downarrow\right\rangle_{1}\left|\uparrow\right\rangle_{2}\left|\downarrow\right\rangle_{3}...\left|\uparrow\right\rangle_{N_{\text{ions}}}$
which can be prepared with high fidelity. For each given set of trial
parameters $\boldsymbol{\theta}$, multiple instances of
$\left|\psi_{\text{out}}(\boldsymbol{\theta})\right\rangle$ are prepared, and
projectively measured in various local bases to reconstruct the energy cost
function $\left\langle
H_{\text{SSH}}\right\rangle_{\boldsymbol{\theta}}=\left\langle\psi_{\text{out}}(\boldsymbol{\theta})\right|H_{\text{SSH}}\left|\psi_{\text{out}}(\boldsymbol{\theta})\right\rangle$
from one- and two-point correlation functions $\langle\sigma_{i}^{a}\rangle$,
$\langle\sigma_{i}^{a}\sigma_{j}^{a}\rangle$. The cost function $\left\langle
H_{\text{SSH}}\right\rangle_{\boldsymbol{\theta}}$, function of
$\boldsymbol{\theta}$, is evaluated at finite precision and minimized with a
search algorithm capable of taking into account statistical errors. Upon
convergence, $\left\langle
H_{\text{SSH}}\right\rangle_{\boldsymbol{\theta}_{\text{opt}}}$ reaches its
global minimum, and the optimal set of parameters
$\boldsymbol{\theta}_{\text{opt}}$ can be used to re-prepare the (approximate)
non-degenerate ground state
$\left|\psi_{\text{out}}(\text{$\boldsymbol{\theta}_{\text{opt}}$})\right\rangle$
on demand.

Convergence of the search algorithm can be enhanced by incorporating
symmetries of the target model in the trial states. For the CSA circuit, the
zero $z$-magnetization of the initial Néel state
$\left|\psi_{\text{in}}\right\rangle$ is protected both by the entangler
$H_{XY}$ and rotations $\text{exp}(-i\theta_{j}\sigma_{j}^{z})$. CP symmetry
is protected by $H_{XY}$, and can be enforced on single-qubit transformation
‘layers’ by using correlated rotations angles
$\theta_{j}=-\theta_{N_{\text{ions}}+1-j}$. can be approximated as well, by
repeating odd-site and even-site rotation angles, $\theta_{j}=\theta_{j+2}$,
away from the system edges [23].

While VQE provides the best approximate state preparation for a given set of
resources in the presence of actual imperfections and noise [10], it is still
affected by decoherence. By considering a different VQE circuit ansatz for
trapped ions, which uses the high-rate resonant sideband resources of Eq. (1),
we aim to tackle decoherence by speeding-up the preparation of trial states.

VQE with Quantum Data Buses (QDB-MPS)$-$ We design a variational circuit
ansatz $U(\boldsymbol{\theta})=\prod_{k}\text{exp}(-i\theta_{k}H_{j_{k}})$
built on ion-COM mode interaction pulses (1), according to Fig. 1(b). Since
now the COM mode becomes entangled to the ion qubits, the state of the qubits
is generally mixed, both during the processing and in the output trial state

$\displaystyle\rho_{\text{out}}(\boldsymbol{\theta})=\text{Tr}_{\text{COM}}[U(\boldsymbol{\theta})(\left|\psi_{\text{in}}\right\rangle\left\langle\psi_{\text{in}}\right|\otimes\rho_{0})\,U^{\dagger}(\boldsymbol{\theta})].$
(6)

Parameters $\\{\theta_{k}\\}$ can be controlled by the duration of the
sideband laser pulses. Similar to the closed-system case, the cost function
$\langle
H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}=\text{Tr}[\rho_{\text{out}}(\boldsymbol{\theta})H_{\text{SSH}}]$
is reconstructed from correlation functions evaluated on the trial qubit
state, while the output state of the COM mode is discarded. As $\langle
H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}$ is minimized to approach the
ground energy, $\rho_{\text{out}}(\boldsymbol{\theta})$ approximates the
unique, pure ground state
$|\psi_{\text{targ}}\rangle\langle\psi_{\text{targ}}|$, and the output state
of the COM mode becomes automatically disentangled.

In our design, resource operations are arranged such that the circuit
ultimately generates a variational Matrix Product State (MPS). In fact, MPS
efficiently approximate ground states of 1D quantum lattice Hamiltonians,
including the SSH model [56, 57]. In contrast to CSA resources, which are
inherently global, blue-sideband operations (1) offer single-site
addressability. This property allows us to progressively tailor the MPS from
one edge to the other, as shown by the Tensor Network diagram [56, 57] in Fig.
1(c), in analogy to quantum circuits developed in the context of quantum
machine learning [58]. Previous strategies for realizing MPS on quantum
devices were proposed, both for deterministic state preparation [59, 60], as
well as for variational state preparation [61] which employed long-range
interactions. In contrast, our variational circuit requires only local ion-
phonon interactions, and, therefore, can be up-scaled to modular ion trap
architectures (see next section). Additionally, we introduce an approximate
bulk-translational invariance to the ansatz, to reduce its variational
complexity: In our circuit, interactions in the bulk can be virtually grouped
in ‘boxes’ (drawn in Fig. 1(b) as grey rectangles at the edges, and orange and
green for the bulk). Each box is defined by a set of variational parameters,
which are repeated in the bulk boxes of the same color. At the same time, the
maximum achievable bond dimension $D\sim 2^{l-1}$ of the generated MPS can be
controlled by increasing the size of the bulk boxes $l$ (see Appendix A).
Figure 1 shows a specific design of the bulk boxes using five operations per
box. Numerical results suggest that the number of operations per box is
crucial towards the accuracy of the ansatz, while the ordering is marginal
(data not shown). Appendix E illustrates our basic strategy to arrange
operations inside a box.

As in the CSA case, we again incorporate symmetries of the target model in the
trial states. First of all, since $H_{j}$ is fully imaginary, the operation
$U(\boldsymbol{\theta})$ is real as well as
$\left|\psi_{\text{in}}\right\rangle$ and $\rho_{0}(T)$, and, thus, the
generated states $\rho_{\text{out}}(\boldsymbol{\theta})$ have real
amplitudes. Moreover, operations (1) protect an ‘extended’ magnetization
symmetry $1/2\sum_{j}\sigma_{j}^{z}-a^{\dagger}a$, forcing the output qubits
state to have well-defined $z$-magnetization upon qubits and QDB becoming
variationally disentangled (see Appendix B for details). We have analytically
verified that, in the theoretical limit of infinite-depth circuits, the
resources given by Eq. (1) are sufficient to generate ground states of _any_
real, magnetization conserving Hamiltonian (proof in Appendix C).
Controllability for an arbitrary lattice model (e.g., without symmetries) can
be achieved by adding to the resources a resonant carrier operation
$\sigma_{j}^{z}$ for at least one ion.

We stress that resources of the form (1) are typical in various experimental
platforms with bosonic QDBs, such as superconducting qubits connected via a
resonator [46, 42, 43]. As such, the numerical analysis presented below can be
generalized to these platforms.

CSD-MPS Circuit Ansatz $-$ We also compare our QDB-based approach with a
variational MPS circuit ansatz realized with Closed System Digital resource
operations (CSD-MPS circuit), as presented in Fig 2(c). This circuit is built
with single-qubit $\sigma_{i}^{z}$ rotations and site-filtered MS gates (2)
applied to a local set of ions. Such resource operations protect
z-magnetization in the trial state.

##  4 _Results_

Performance$-$ We compare the performance of the variational circuits by
numerically simulating VQE for preparation of the ground state
$|\psi_{\text{targ}}\rangle$ of Hamiltonian (4) in the gapped phase with
$t=0.5$. Trial states are obtained by evolving the initial quantum state in
terms of density matrixes. For the QDB-MPS circuit, we consider the COM mode
to be prepared in a thermal state $\rho_{0}(T)$ with realistic average number
of excitations
$n_{0}(T)=\text{Tr}[a^{\dagger}a\,\rho_{0}(T)]=1/(e^{1/T}-1)=0.01$ [62, 63].
Other sources of imperfections are not taken into account, and, therefore, in
contrast to the QDB-MPS circuit, simulations of the CSA and CSD-MPS circuits
are free of imperfections. We consider the theoretical limit of unlimited
measurement budget [10, 23], i.e., the cost function is the exact expectation
value $\langle H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}$. To optimize
parameters $\boldsymbol{\theta}$, we use a gradient-based optimization
algorithm (see Appendix D), which can be adapted for noisy cost functions [64,
65, 66]. We emphasize that we consider a low number of free variational
parameters, $N_{p}\sim 10-25$, which enables global search methods, such as
the dividing rectangles algorithm [67, 68], already adopted in VQE [23]. For
the analog resource entangler $H_{XY}$, realistic values $\alpha=1.34$ and
$B=20$ [23] are used. Further details on the circuit ansätze and their
simulation are given in Appendix E.

Figure 3: Energy error related to the excitation gap, $\varepsilon$, (a)
versus size of the prepared states $N_{\text{ions}}$ for fixed number
variational parameters $N_{p}=18$ and (b) versus $N_{p}$ for
$N_{\text{ions}}=12$ in circuits QDB-MPS, CSD-MPS, and CSA, as indicated in
(a). Gray small rounds in (b) give values of $\varepsilon_{q}$ obtained in the
same QDB-MPS circuit and parameters as the black plot but with different
initial pure states of the QDB $\left|q\right\rangle_{a}$ for $q\in\\{0,1\\}$.

To assess the accuracy of the ground state preparation, we employ various
figures of merit. Besides the fidelity and the excitation energy (defined
later on), we also consider a correlation-based error function

$\displaystyle F_{\text{err}}$
$\displaystyle=\sum_{i=1,\,j>i}^{N_{\text{ions}}}\left|C_{ij}^{\text{DMRG}}-C_{ij}\right|/\sum_{i=1,\,j>i}^{N_{\text{ions}}}\left|C_{ij}^{\text{DMRG}}\right|,$
(7)

calculated using (parallel) two-point correlators

$\displaystyle C_{ij}$
$\displaystyle=\langle\sigma_{i}^{x}\sigma_{j}^{x}\rangle-\langle\sigma_{i}^{x}\rangle\langle\sigma_{j}^{x}\rangle.$
(8)

Here $C_{ij}$ is evaluated on the optimized state
$\rho_{\text{out}}\left(\boldsymbol{\theta}_{\text{opt}}\right)$ and
$C_{ij}^{\text{DMRG}}$ is the exact correlator of $|\psi_{\text{targ}}\rangle$
obtained by DMRG method [69]. We consider this error function in Fig. 2(a), to
show that the proposed QDB-MPS circuit is able to prepare states with
$F_{\text{err}}$ saturating in $N_{\text{ions}}$ at fixed number of
variational parameters $N_{p}$. In particular, the insets show that these
states have correlations $C_{ij}$ accurately capturing $C_{ij}^{\text{DMRG}}$.
By contrast, $F_{\text{err}}$ arising from the CSA circuit grows rapidly for
this specific task, and the correlators $C_{ij}$ decay slower than in
$|\psi_{\text{targ}}\rangle$. We argue that this follows from the necessity to
variationally remove the strong (power-law) correlations established by the
long-range interactions in $H_{XY}$ (3) to recover the exponential decay of
correlations in the ground state. Moreover, the SSH ground state exhibits
considerable (short-range) entanglement dimerization, which in turn seems to
favor quasi-local resource circuits over global resource circuits. While the
results for CSD-MPS and QDB-MPS circuits given in Fig 2(a) are compatible, the
QDB-MPS is realized with higher-rate operations, and, therefore, is expected
to be less prone to decoherence.

_S_ _calability_ $-$ We study the scalability of the circuits in terms of the
excitation energy in units of the energy gap, which bounds infidelity of the
prepared states $1-F\leq\varepsilon=(\langle
H_{\text{SSH}}\rangle_{\boldsymbol{\theta}_{\text{opt}}}-E_{0})/\Delta$
(Appendix F), and is thus a figure of accuracy. Here,
$F=\langle\psi_{\text{targ}}|\rho_{\text{out}}\left(\boldsymbol{\theta}_{\text{opt}}\right)|\psi_{\text{targ}}\rangle$
and $\Delta=E_{1}-E_{0}$ is the energy gap obtained by DMRG, which converges
to a finite non-zero value in the thermodynamical limit
$N_{\text{ions}}\to\infty$ since the target model is non-critical. Figure 3(a)
shows that, for the QDB-MPS and CSD-MPS circuits, $\varepsilon$ grows linearly
with $N_{\text{ions}}$, in contrast to the CSA circuit. This is compatible
with the previous observation that the two-point correlation functions carry,
on average, an error saturating in $N_{\text{ions}}$, and the energy
functional contains a linear amount of such correlators.

Scaling of the accuracy $\varepsilon$ with $N_{p}$, for the various methods
and at fixed $N_{\text{ions}}$, is plotted in Fig. 3(b). Again, we observe
that MPS circuits offer asymptotically better accuracy (or better parametric
efficiency) than the CSA global resources. While increasing the number of
parameters $N_{p}$, we observe the CSD-MPS circuit reaches slightly better
accuracies than the QDB-MPS circuit; we attribute this effect to the finite
temperature of the COM mode we employed for the simulations. This argument is
supported by the gray curves in Fig. 3(b), showing relative excitation
energies $\varepsilon_{q}$ obtained with the same QDB-MPS circuit and the same
parameter values as the black plot but with different initial pure states of
the COM mode $\left|q\right\rangle_{a}$ for $q\in\\{0,1\\}$. For $n_{0}=0.01$,
we have $\varepsilon=\varepsilon_{0}p_{0}+\varepsilon_{1}p_{1}+O(n_{0}^{2})$,
with $p_{q}=\langle q|_{a}\rho_{0}(T)|q\rangle_{a}\sim n_{0}^{q}$. The plot
shows that the two quantum trajectories are both variationally optimized in
such a way that their error contributions are commensurate,
$\varepsilon_{0}p_{0}\sim\varepsilon_{1}p_{1}$.

Figure 4: (a) Fidelity and (b) infidelity of the $6$ qubit state prepared in
the QDB-MPS circuit as functions of number of excitations $n_{0}$ in
$\rho_{0}(T)$ and number of variational parameters $N_{p}$ in the circuit. In
(b), the red line indicates the lowest number of $N_{p}$ required to preserve
infidelity threshold $1-F\leq 0.002$ at given $n_{0}$.

Tolerance to temperature $-$ Here, we further investigate tolerance of the
QDB-MPS approach to finite temperatures $T$ of the COM mode. As $T$ increases,
more amplitudes $p_{q}$ are populated in $\rho_{0}(T)=\sum
p_{q}(T)\left|q\right\rangle_{a}\left\langle q\right|$, and, therefore, more
equations
$U\left(\boldsymbol{\theta}_{\text{opt}}\right)\left|\psi_{\text{in}}\right\rangle\left|q\right\rangle_{a}\sim\left|\psi_{\text{targ}}\right\rangle\left|q\right\rangle_{a}$
need to be simultaneously satisfied, requiring higher $N_{p}$ to achieve the
same accuracies. In Fig. 4, we consider the fidelity of the optimized output
state of $N_{\text{ions}}=6$ qubits, as a function of the average number of
excitations $n_{0}(T)$ and number of variational parameters $N_{p}$. Panel (a)
shows that, for fixed $N_{p}$, $F$ decreases linearly with $n_{0}$ (similarly
to the C-phase gate [26] built with sidebands), but with a slope that flattens
for increasing $N_{p}$. Therefore, to achieve a fidelity threshold, $N_{p}$
must be increased with $n_{0}$. Remarkably, for the case under consideration,
it seems that the $N_{p}$ required to obtain a static fidelity threshold
scales logarithmically with $n_{0}$. This behavior is illustrated in panel
(b), where the density plot shows the infidelity of the optimized state as a
function of $n_{0}$ and $N_{p}$. The red curve highlights a fixed fidelity
threshold $1-F\leq 0.002$. This property sets our QDB-based VQE apart from the
approaches using sideband interactions to construct logical gates [26], in
which imperfect cooling imposes a hard bound on the attainable accuracies.

Figure 5: QDB-MPS circuit in a modular ion trap architecture, which generates
a state on the ions chain with 4 ions per trap. The processing is implemented
consequently, starting from the left trap to the right, and from the bottom to
the top in a single trap as presented by the circuits. The red boxes with the
arrows indicate the transfer of the edge ions (or their internal states) to
the next trap after the implementation of all operations in a trap.

Scalability on a _modular ion trap architectures_ $-$ The proposed QDB-MPS
circuit 1(a) relies on interactions between individual ions and the COM mode.
However, the number of ions which can be coupled to the COM mode in a single
trap is limited by $\sim 10-100$ [34]. To further scale the ion trap quantum
processors, modular ion trap architectures were proposed [33, 34]. Following
these proposals, we describe ground state preparation with QDB-MPS circuit
across multiple traps, as shown in Fig. 5. We consider coherent shuttling of
few ions [33, 70] (or their internal states [34]) as a quantum communication
channel among the traps, and we assume it to be error-free. We consider each
trap containing a small number of ions, such that each ion is addressable via
blue sideband pulses, realizing interactions as Eq. (1) with the COM mode of
the corresponding trap. For simplicity, we assume that the initial thermal
states, $\rho_{0}$, of the COM modes are identical in each trap, as well as
and their effective coupling to the ions.

The variational circuit ansatz is constructed sequentially along the network
of traps, i.e., by applying operations in trap $k+1$ after all operations in
trap $k$ are completed. Since each trap has an independent COM mode as a
bosonic QDB, we introduce a variational ‘interface’ operation
$U_{I}(\boldsymbol{\theta}_{5})$, constructed with resource interactions (1),
which is applied to a set of $l-1$ ions (with $l$ the size of the bulk boxes)
as the last operation for trap $k$. This operation can be optimized in order
to disentangle the COM mode of trap $k$, effectively implementing
$U_{I}\rho_{\text{qubits},k}U_{I}^{\dagger}=\rho_{\text{qubits}}\otimes\rho_{0}$,
with $\rho_{\text{qubits},k}$ the common state of ions and the COM mode. After
transferring the set of $l-1$ ions to trap $k+1$, we apply the unitary
$U_{I}^{\dagger}$ to the same ions plus the COM mode of trap $k+1$ prepared in
state $\rho_{0}$, effectively reconstructing the state
$\rho_{\text{qubits},{k+1}}=U_{I}^{\dagger}(\rho{}_{\text{qubits}}\otimes\rho_{0})U_{I}=\rho_{\text{qubits},k}$.
Ultimately, this operation removed the ‘impurity’ introduced by the interface
between the two traps.

Figure 6: Energy error related to the excitation gap versus size
$N_{\text{ions}}$ of the optimized state in a single trap using $N_{p}=18$
variational parameters and in $N_{\text{traps}}$ traps with 6 ions per trap
using $N_{p}=18+2$ .

In Fig. 6, we compare the excitation energies, from optimizing the QDB-MPS
circuit, with a single trap and with a modular trap network. Both circuits
have the same number of variational operations in the boxes in the edges and
in the boxes of the bulk, and the interface operation $U_{I}$ contains $2$
variational sideband operations. The error emerging from the optimization of
the ion-trap network deviates only slightly from the error in the single trap
case, and we recover linear scaling of the error with the system size. This
result is an even stronger indicator of scalability of the QDB-MPS approach
for VQE, as it can treat hardware interfaces as impurities, and efficiently
compensate for them with small variations of the circuit ansatz.

##  5 Outlook

We proposed an approach for VQE which uses qubit-QDB interactions, available
in currently-existing programmable quantum devices, as variational resource
operations. In our strategy, the QDB is disentangled from the optimized states
only at the end of the state preparation, as a byproduct of the optimization.
For trapped ions, we discussed how to realize our strategy with resonant blue-
sideband pulses, potentially yielding faster state-preparation rates compared
to existing strategies, and thus helping to mitigate decoherence effects. The
variational ansatz circuit we designed is tailored to efficiently prepare MPS,
leading to a potential improvement in scalability over current VQE strategies.

We numerically simulated the ion trap implementation of VQE based on the QDB-
MPS ansatz circuit, and showed strong evidence of scalability of our approach,
even when it is realized in modular ion traps. While the ions-COM mode
sideband interactions operate in an extended Hilbert space, we demonstrated
that these resource operations can incorporate symmetries in the variational
circuit, similar to the closed-system VQE approaches. Although here we
demonstrated robustness of the method to the initial temperature of the COM
mode, further investigations are required to study other realistic
imperfections, including (i) qubits decoherence, (ii) fluctuations of the
control parameters, and (iii) shot noise resulting from a finite budget of
measurements to reconstruct the cost function.

The developed QDB-based VQE can be readily used in other currently available
experimental platforms, such as quantum dots [71] or atoms coupled by (chiral)
waveguides [72]. Moreover, we expect that the scaling and parametric
efficiency of our results, found for the MPS circuit ansatz, extend to other
efficiently-contractible tensor network states [58, 61], such as
$\text{MPS}{}^{2}$ [73, 74], thus allowing us to accomodate the entanglement
content required to explore higher dimensions. Also, by including simultaneous
operations on multiple qubits, it is possible to build many-body wavefunction
ansätze beyond tensor networks, ultimately enabling the efficient preparation
of quantum states that can not be numerically simulated. Finally, we envision
the possibility of including controllable dissipative elements to the QDB
resources [75] in order to construct optimized cooling algorithms beyond
stabilizer codes [76, 77].

We warmly thank M. Hettrich, R. Kaubruegger, C. Kokail, M. Meth, T. Monz, P.
Schindler R. van Bijnen, and P. Zoller for private discussions. Authors kindly
acknowledge support by the EU Quantum Technology Flagship project PASQUANS and
the QuantEra project QTFLAG, the Austrian Research Promotion Agency (FFG) via
QFTE project AutomatiQ, the US Air Force Office of Scientific Research (AFOSR)
via IOE Grant No. FA9550-19-1-7044 LASCEM, and by the Army Research Laboratory
and was accomplished under Cooperative Agreement Number W911NF-15-2-0060. The
views and conclusions contained in this document are those of the authors and
should not be interpreted as representing the official policies, either
expressed or implied, of the Army Research Laboratory or the U.S. Government.
The U.S. Government is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation herein. The
computational results presented have been achieved (in part) using the HPC
infrastructure LEO of the University of Innsbruck. Source codes used for our
results are available at https://github.com/Viacheslav-Kuzmin/quantum_bus_vqe,
released under a MIT license.

## Appendix

##  Appendix A Maximum bond dimension

Figure 7: (a) QDB-MPS circuit with the measurement of the bosonic QDB, which
results in probabilistic generation of MPS $\left|\Phi\right\rangle$ in
qubits. (b) Tensor network diagram of $\left|\Phi\right\rangle$.

In the section, we study the maximum bond dimension, $D_{M}$, of MPS generated
with the QDB-MPS circuit using a bosonic QDB and built with resource
operations (1) described by the Anti-Jaynes-Cummings Hamiltonian. Here we
distinguish two sets of MPS accessible by the circuit. The first set contains
all pure states which can be generated by the QDB-MPS circuit. The states of
this set can be generated probabilistically [59] after measuring (and
therefore disentangling) the QDB at the end. It defines the search space for
VQE and, therefore, is desired to be limited. The second set — a subset of the
first set — contains deterministically generated translationally invariant
MPS, which are disentangled from the QDB by the end of the circuit without
measuring the EM. This set, in opposite to the first one, is required to be
big enough to include a good approximation of the target state.

### Set of probabilistically generated MPS

We start with the set of MPS generated probabilistically. We characterize this
set of MPS by considering a possible range of their bond dimensions. Bond
dimension $D_{M}$ of MPS of size $N$ is the maximum of bond dimensions $D_{n}$
between two blocks of qubits $[1,n]$ and $[n+1,N]$. $D_{n}$ corresponds to
entanglement entropy between the blocks [57], such that the Schmidt
decomposition of MPS $\left|\Phi\right\rangle$ in the basis of two blocks
$\\{|\phi_{[1,n]}^{(i)}\rangle\\}$ and $\\{|\phi_{[n+1,N]}^{(i)}\rangle\\}$ is

$\left|\Phi\right\rangle=\sum_{i=1}^{D_{n}}a_{i}|\phi_{[1,n]}^{(i)}\rangle|\phi_{[n+1,N]}^{(i)}\rangle.$

The entanglement entropy of the blocks is
$S_{[1,n]}=-\text{Tr}\left[\rho_{[1,n]}\text{log}(\rho_{[1,n]})\right]\leq\text{log}D_{n}$,
with $\rho_{[1,n]}$ the state of qubits $[1,n]$.

Using singular value decomposition, one can show that, for an arbitrary state
of size $N$, it is valid that $D_{n}\leq 2^{\text{min}(n,N+1-n)}$, and,
therefore, $D_{M}\leq 2^{\left\lfloor N/2\right\rfloor}$. By using a not
universal set of resource operations (9) we limit $D_{M}$ and, therefore,
restrict the set of achievable states. As presented in Appendix B,
interactions (1) preserve an extended symmetry
$Z=1/2\sum_{j=1}^{N}\sigma_{j}^{z}-a^{\dagger}a$. In the following, we explain
that this symmetry imposes $D_{n}\lessapprox\left\lfloor n/2\right\rfloor\cdot
2^{l-1}$, where $l$ is the size of the bulk boxes in the QDB-MPS circuit, and
limits the maximum bond dimension as $D_{M}\lessapprox\left\lfloor 2/3\cdot
N\right\rfloor 2^{l-1}$. Therefore, in comparison with the general case, the
bond dimension of the MPS generated by the QDB-MPS circuit is exponentially
suppressed.

Figure 7, demonstrates entanglement distribution in the QDB-MPS circuit
between the qubits by a subsystem, highlighted by orange color, which includes
the QDB and $l-1$ qubits shared by neighbor virtual boxes. Such a subsystem
can be considered as a virtual ancilla that sequentially interacts with the
qubits distributing entanglement among them [59]. The effective dimension of
this ancilla (dimension of the populated Hilbert space) bounds bond dimension
$D_{n}$.

Let us consider the QDB initially prepared in state
$\left|p_{0}\right\rangle_{a}=\left|0\right\rangle_{a}$, and the qubits
prepared in state
$\left|\psi_{\text{in}}\right\rangle=\left|\downarrow\right\rangle_{1}\left|\uparrow\right\rangle_{2}\left|\downarrow\right\rangle_{3}...\left|\uparrow\right\rangle_{N}$.
Then, because of the preserved symmetry, the maximum possible population of
the QDB after interaction with $n$ ions can not be higher than
$\left|\left\lfloor n/2\right\rfloor\right\rangle_{a}$. Multiplying the
growing dimension of the QDB by the dimension of $l-1$ qubits in the ancilla,
we obtain that $D_{n}\lessapprox\left\lfloor n/2\right\rfloor\cdot 2^{l-1}$.
From the other side, as explained in [59], $D_{n}$ has to decrease as $n$
approaches to $N$ to guarantee that the ancilla (with QDB measured in
$\left|p_{N}\right\rangle_{a}$) is decoupled by the end of the state
preparation. Therefore, there exists some $n_{M}$, for which $D_{n_{M}}$
achieves its maximum. To obtain $n_{M}$, we study how $D_{n}$ can decrease
while the ancilla interacts with the qubits after qubit $n_{M}$. Intuitively,
this can be done other way around by considering the growth of bond dimension
$\tilde{D}_{N-n}$ if MPS would be generated from the other side of the qubits
chain, starting from $n=N$. Since the measured output state
$\left|p_{N}\right\rangle_{a}$ can contain number of excitations $p_{N}\geq
0$, the number of excitations in the QDB can not only increase but also
decrease as the QDB interacts with the qubits, populating Hilbert space with
the twice bigger dimension than if the QDB would be initially prepared in
$\left|0\right\rangle_{a}$. Therefore, $\tilde{D}_{N-n}\approx
2D_{\tilde{n}}\big{|}_{\tilde{n}=N-n}\lessapprox(N-n)2^{l-1}$. Solving
$D_{n_{M}}=\tilde{D}_{N-n_{M}}$ we obtain that, after
$n_{M}\lessapprox\left\lfloor 2/3\cdot N\right\rfloor$, $D_{n}$ has to
decrease with $n$. Thus, we find that $D_{M}=D_{n_{M}}\lessapprox\left\lfloor
2/3\cdot N\right\rfloor 2^{l-1}$.

### Set of deterministically generated translationally invariant MPS

In order to generate MPS with translationally invariant $D_{n}$ in the bulk,
the virtual ancilla has to own dimension which repeats itself with a target
period along the state preparation. This includes the ancilla dimension after
realization of the last box in the bulk, after which the edge box is aimed to
disentangle the QDB from $l-1$ ancilla qubits as required by deterministic
preparation of a pure state. This can be done only if the effective dimension
of the ancilla does not exceed dimension $2^{l-1}$ of the qubits. Hence, the
limit of the bond dimension of the translationally invariant MPS
deterministically generated in the considered QDB-MPS circuit is $\sim 2^{l}$.
Here, we defined the limit not exactly, since the actual bond dimension can
vary within one period of the bulk.

##  Appendix B Symmetries

In this section, we discuss the symmetries protected by the operations

$H_{j}=i\tilde{\Omega}\left(a\sigma_{j}^{-}-a^{\dagger}\sigma_{j}^{+}\right),$
(9)

with $\tilde{\Omega}$ the Rabi frequency, $a$ is the destruction operator for
the bosonic QDB, and
$\sigma_{j}^{-}=(\sigma_{j}^{+})^{\dagger}=\left|\downarrow\right\rangle\left\langle\uparrow\right|_{j}$
the lowering operator for qubit $j$, with
$\\{\left|\downarrow\right\rangle_{j},\left|\uparrow\right\rangle_{j}\\}$ the
internal logical states of the qubit.

Magnetization $-$ A number symmetry, forming an Abelian $U(1)$ group of
symmetric transformations $W(\varphi)=\text{exp}(-i\varphi Z)$, is generated
by the integer operator

$Z=1/2\sum_{j=1}^{N}\sigma_{j}^{z}-a^{\dagger}a=M-a^{\dagger}a,$ (10)

where $N$ is the number of qubits, $M=1/2\sum_{j}\sigma_{j}^{z}$, and
$\sigma_{j}^{z}=\left|\uparrow\right\rangle\left\langle\uparrow\right|_{j}-\left|\downarrow\right\rangle\left\langle\downarrow\right|_{j}$.
It is straightforward to check that $Z$ is a symmetry since $[H_{j},Z]=0$ for
every site $j\in\\{1..N\\}$. $Z$ thus defines a quantum number, which is
conserved during the controlled variational dynamics. Consider the initial
state of the system of qubits plus the QDB as
$\left|\Psi_{\text{in}}\right\rangle=\left|\psi_{\text{in}}\right\rangle\otimes|s\rangle_{a}$,
with
$\left|\psi_{\text{in}}\right\rangle=\left|\downarrow\right\rangle_{1}\left|\uparrow\right\rangle_{2}\left|\downarrow\right\rangle_{3}...\left|\uparrow\right\rangle_{N}$
the state of the qubits and $|s\rangle_{a}$ the Fock state of the QDB,
$a^{\dagger}a|s\rangle_{a}=s|s\rangle_{a}$, $s\in\mathbb{N}$. This state has
well-defined quantum number
$Z\left|\Psi_{\text{in}}\right\rangle=s\left|\Psi_{\text{in}}\right\rangle$.
Due to the symmetry of Eq. (10), the evolution of this state will preserve its
quantum number, and the state will always be of the form
$\left|\Psi_{\text{out}}\right\rangle=\sum_{q,m,t}\delta_{m,q-s}c_{m,t}|m,t_{m}\rangle\otimes|q\rangle_{a}$,
where the qubit states are labeled by the magnetization $m$ (eigenvalues of
$M$) and degeneracy $t_{m}$. At the end of the variational quantum state
preparation, after optimization of the process, the output state will be
disentangled between phonon and qubits, and reads
$\left|\Psi_{\text{opt}}\right\rangle=\left(\sum_{t}c_{m,t}|m,t_{m}\rangle\right)\otimes|m+s\rangle_{a}$.
The final qubit state will thus have a well-defined magnetization
$m\in\mathbb{N}$. This requirement identifies the class of target Hamiltonians
$H$ targetable with the chosen resources: which must contain the magnetization
symmetry: $[H,M^{z}]=0$.

Conjugation $-$ A second symmetry is nested within our controls, and it
becomes evident after gauging the resource Hamiltonians as in Eq. (1). In this
format, the resource Hamiltonians are purely imaginary $H_{j}=-\bar{H}_{j}$
when expressed in the canonical basis, and thus antisymmetric
$H_{j}=-H_{j}^{t}$ and traceless $\text{Tr}[H_{j}]=0$. It follows that the
unitary operators $U_{j}(t)=e^{-iH_{j}t/\hbar}$ of time evolution under
$H_{j}$ must be real, as they are imaginary exponentials of imaginary
matrices, and have determinant $\text{det}(U_{j})=e^{-it\text{Tr}[H_{j}]}=1$.
This means that, with the chosen resources, we can only perform
transformations in the special orthogonal group SO($2^{d}$), a subgroup of the
unitaries U($2^{d}$), which, in turn, limits the set of states we can prepare
to real states (vectors with real coefficients in the canonical basis). In
conclusion, the class of models targetable with our variational scheme is
restricted to models expressed by real, or symmetric, Hamiltonians, which
always possess real eigenbases. We stress that the complex conjugation
symmetry can be relaxed by adding a detuning $\delta_{j}=\nu\pm\epsilon$ to
one or more of the control lasers, which can be helpful if we aim to quantum
simulate a non-real Hamiltonian $H^{T}$.

##  Appendix C Controllability

We now discuss the controllability problem and show that, within the
constraints set in the symmetry section, the selected controls are
theoretically able to prepare any quantum state, in the limit of infinite
available resources. First of all, we argue that starting from the state
$\left|\psi_{\text{in}}\right\rangle\otimes|q\rangle_{a}$ it is possible to
shift the system qubits state to any magnetization sector. This is shown
recursively, by highlighting that the state
$\left|\downarrow\right\rangle_{j}|q\rangle_{a}$ under the action of $H_{j}$
for a time equal to $t=\pi/(\tilde{\Omega}\sqrt{q+1})$ is completely mapped
into the state $\left|\uparrow\right\rangle_{j}|q+1\rangle_{a}$, ultimately
increasing the magnetization by +1. Secondly, we show that our controls
$H_{j}$ provide strong controllability within each magnetization sector $m$
(of dimension $d_{m}$) of real states, meaning that the Lie algebra generated
by $\\{H_{j}\\}$ contains the Lie algebra of imaginary, hermitian matrices
onto the $m$ sector, generators of SO$(d_{m})$, as we will show later in this
section. From this observation, it follows that any two real states
$|\psi_{1}\rangle|q\rangle_{a}$ and $|\psi_{2}\rangle|q\rangle_{a}$ can be
dynamically connected, i.e.,
$|\psi_{2}\rangle=e^{-iH^{\prime}t/\hbar}|\psi_{1}\rangle$ for some $t$,
simply because the effective Hamiltonian
$H^{\prime}=i|\psi_{1}\rangle\langle\psi^{\prime}|-i|\psi^{\prime}\rangle\langle\psi_{1}|$,
with
$|\psi^{\prime}\rangle\propto|\psi_{2}\rangle-|\psi_{1}\rangle\langle\psi_{1}|\psi_{2}\rangle$,
can be generated with our resources.

We now prove that, by means of linear combination and commutation, we can
generate the whole algebra of imaginary Hermitian matrices for the sector $m$
by starting from the bare resources
$H_{j}=i(\sigma_{j}^{+}a^{\dagger}-\sigma_{j}^{-}a)$. We start by considering
the commutator
$i[H_{j},H_{j^{\prime}}]=i(\sigma_{j}^{-}\sigma_{j^{\prime}}^{+}-\sigma_{j}^{+}\sigma_{j^{\prime}}^{-})$,
for sites $j\neq j^{\prime}$, which is an endomorphism of magnetization sector
$m$. Moreover, it is easy to show that
$i[H_{j},i[H_{j},H_{j^{\prime}}]]=\sigma_{j}^{z}H_{j^{\prime}}$, for $j\neq
j^{\prime}$. By recursion and locality, it follows that we can generate any
string operator of the form

$i(\sigma_{j}^{-}\sigma_{j^{\prime}}^{+}-\sigma_{j}^{+}\sigma_{j^{\prime}}^{-})\prod_{k\neq
j,j^{\prime}}(\sigma^{z})^{q_{k}}$ (11)

for any binary string $q_{k}$. We run one additional commutator, namely
$i[(\sigma_{j}^{-}\sigma_{j^{\prime}}^{+}-\sigma_{j}^{+}\sigma_{j^{\prime}}^{-})\sigma_{k}^{z},(\sigma_{k}^{-}\sigma_{k^{\prime}}^{+}-\sigma_{k}^{+}\sigma_{k^{\prime}}^{-})]=2i(\sigma_{j}^{-}\sigma_{j^{\prime}}^{+}-\sigma_{j}^{+}\sigma_{j^{\prime}}^{-})(\sigma_{k}^{-}\sigma_{k^{\prime}}^{+}+\sigma_{k}^{+}\sigma_{k^{\prime}}^{-})$
where all sites are different. By recursion, we conclude that we can generate
operators $\bar{H}$ which are tensor product of one operator
$i(\sigma_{j}^{-}\sigma_{j^{\prime}}^{+}-\sigma_{j}^{+}\sigma_{j^{\prime}}^{-})$
at a pair of sites $j\neq j^{\prime}$, any amount of $\sigma_{\ell}^{z}$, and
any amount of
$(\sigma_{k}^{-}\sigma_{k^{\prime}}^{+}+\sigma_{k}^{+}\sigma_{k^{\prime}}^{-})$
(each one acting on separate sites). We remark that these operators form a
basis for the imaginary Hermitian endomorphisms of the symmetry sector $m$. To
see this, consider the canonical basis $i(|q_{1}q_{2}q_{3}\ldots\rangle\langle
r_{1}r_{2}r_{3}\ldots|-H.c.)$, where $\vec{q}$ and $\vec{r}$ are any two-bit
strings with total magnetization $m$. This operator can be easily decomposed
in operators of the form $\bar{H}$: for bits $j$ that do not flip
($q_{j}=r_{j}=\pm 1$) just insert a $(1\pm\sigma_{j}^{z})$ in the tensor
product string, while for pairs of bits that flip-flop, insert a
$(\sigma_{k}^{-}\sigma_{k^{\prime}}^{+}+\sigma_{k}^{+}\sigma_{k^{\prime}}^{-})$,
except one, which is substituted by
$i(\sigma_{j}^{-}\sigma_{j^{\prime}}^{+}-\sigma_{j}^{+}\sigma_{j^{\prime}}^{-})$.
This concludes the proof.

##  Appendix D Parameters optimization

In the section, we describe the method used for the optimization of
variational parameters $\boldsymbol{\theta}$ in the numerical simulations of
VQE with the circuits given in the main text. Since the purpose of the paper
is to study the achievable accuracy of the proposed approaches, we employ the
exact value of cost function $\langle
H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}$ in the optimization. The
expectation value $\langle H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}$ is
obtained by simulating the circuits in terms of density matrixes without
considering shot noise caused by the finite number of measurements of the
trial ions states. To optimize parameters $\boldsymbol{\theta}$ we use a
gradient-based optimization algorithm, which relies on numerically accessible
approximate values of $\nabla_{\theta_{i}}\langle
H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}=(\langle
H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}\mid_{\theta_{i}\to\theta_{i}+\Delta}-\langle
H_{\text{SSH}}\rangle_{\boldsymbol{\theta}})/\Delta$$\mid_{\Delta\to 0}$. To
be more precise, we use the basin-hopping method [78, 79], which is a two-
phase method that combines a global stepping algorithm with local minimization
at each step. As a local minimization algorithm, we use the quasi-newton
method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [80, 81, 82, 83].

To decrease noise in the data presented in the result section, we ensured that
$\langle H_{\text{SSH}}\rangle_{\boldsymbol{\theta}}$ is monotonically
decreases with decrease of temperature $T$ of the COM mode (or equivalently
number of excitations $n_{0}$), with decrease of state size $N_{\text{ions}}$,
and with growth of the number of variational parameters $N_{p}$. We begin by
independently optimizing parameters for all data points (defined by one or two
values of $n_{0}$, $N_{\text{ions}}$, and $N_{p}$), each of which starts from
several sets of initial parameters $\boldsymbol{\theta}$ with values $\sim
0.001-0.1$. Then, the optimized parameters of the best result in one data
point is used as the initial parameters set for the neighbor data point, and
the obtained result is compared with the existing ones. In the case when the
neighbor data point requires less initial parameters, the extra parameters are
removed from the inserted parameters set; otherwise, parameters with $0$ can
be inserted. To obtain the presented data, we iteratively re-optimize the
parameters from one side of data points range to another and, afterward, in
the reverse direction until convergence.

##  Appendix E Gates sequence in the circuit

Figure 8: Strategy for increasing of the number of the variational parameters
in circuits: (a) QDB-MPS, (b) CSD-MPS, and (c) CSA. The numbers indicate the
order in which the new variational operations are inserted. The operations
without the numbers represent the minimum set. In (a), blue links in the left
box represent interactions between QDB (red line) and qubits (black lines)
and, for simplicity, are replaced by the blue squares in the following boxes.

In the section, we consider in detail the sequences of variational operations,
Fig. (8), used to construct circuits considered in our work. The number of
variational operations in the edge boxes of the QDB-MPS and CSD-MPS circuits
is fixed to $2$ and 3 correspondingly, as shown in Fig. (8). In these
circuits, the number of variational parameters is increased by progressively
inserting operations in the first bulk box (orange) or second (green),
alternately. In the QDB-MPS circuit, we increase this number by 1, arranging
the operations such that the last operation in one box and the first in the
next box do not act to the same qubit. In the CSD-MPS circuit, we add a single
layer, consisting of a local MS operation and $\sigma_{i}^{z}$ rotations at
each ion. To increase the number of variational operations in the CSA circuit,
we add a single layer consisting of $H_{XY}$ operations and $\sigma_{i}^{z}$
per ion.

The trial states in the CSA circuit are obtained as
$\left|\psi_{\text{out}}(\boldsymbol{\theta})\right\rangle=\prod_{i}\text{exp}\\{-i\theta_{i}H_{i}^{(r)}\\}\left|\psi_{\text{in}}\right\rangle$,
with $\\{H_{i}^{(r)}\\}$ the resource operations. Afterward, the required
correlation functions can be obtained from
$\left|\psi_{\text{out}}(\boldsymbol{\theta})\right\rangle$. Since this method
requires numerical operation with the full quantum state, the number of the
qubits in the CSA circuit in our study is limited by $12$. On the contrary,
one can see from Fig. 8 that, in order to obtain correlation functions of the
states generated by the QDB-MPS and CSD-MPS, the full state is not required to
be kept, but only a reduced density matrix of at maximum $l$ qubits plus the
QDB, with $l$ the size of the bulk boxes. The qubits which do not more
participate in the evolution until the end of the circuit can be measured in
the required bases, and the next qubit can be added to the remained qubits
state by using the Kronecker product operation.

##  Appendix F Energy bounds on fidelity and purity

In this section, we show that the final energy $\langle
H\rangle=\text{Tr}[H\rho(\boldsymbol{\theta})]$, in respect to a target
Hamiltonian $H$, measured on the optimized variational quantum state
$\rho(\boldsymbol{\theta})$ imposes a bound on its fidelity
$F=\langle\psi_{\text{targ}}|\rho(\boldsymbol{\theta})|\psi_{\text{targ}}\rangle$,
with $|\psi_{\text{targ}}\rangle$ the exact ground state. Establishing this
bound requires a good estimate of the two lowest energy levels $E_{0}$ and
$E_{1}$, which can be estimated on the quantum device via a subspace expansion
technique for mixed states, illustrated in detail in the Supplementary
Information of Ref. [84]. This analysis provides self-verification of the
variational quantum state preparation algorithm [23]. Alternatively, for 1D
lattice Hamiltonians, $E_{0}$ and $E_{1}$ can be obtained using numerical
method DMRG.

Let us consider the following inequality

$\langle H\rangle-E_{0}=\text{Tr}[H\rho(\boldsymbol{\theta})]-E_{0}\\\
=\sum_{j\geq 0}\langle
e_{j}|(H-E_{0}I)\rho(\boldsymbol{\theta})|e_{j}\rangle=\sum_{j\geq
1}(E_{j}-E_{0})\langle e_{j}|\rho(\boldsymbol{\theta})|e_{j}\rangle\geq\\\
\geq(E_{1}-E_{0})\sum_{j\geq 1}\langle
e_{j}|\rho(\boldsymbol{\theta})|e_{j}\rangle\\\ =(E_{1}-E_{0})\left(1-\langle
e_{0}|\rho(\boldsymbol{\theta})|e_{0}\rangle\right)=(E_{1}-E_{0})\,(1-\mathcal{F})$
(12)

stating that the state fidelity $F$ is bound from below by the ratio

${F}\geq\frac{E_{1}-\langle H\rangle}{E_{1}-E_{0}},$ (13)

delivering an actual (positive) bound only when the variational state energy
is below the first excited energy, i.e. $\langle H\rangle<E_{1}$. A lower
bound on the fidelity, in turn, infers a bound on the purity
$\text{Tr}[\rho^{2}]$. In fact

$\text{Tr}[\rho^{2}]=\sum_{j\geq 0}\langle
e_{j}|\rho^{2}|e_{j}\rangle\geq\langle e_{0}|\rho^{2}|e_{0}\rangle\\\
=\sum_{j\geq 0}\langle e_{0}|\rho|e_{j}\rangle\langle
e_{j}|\rho|e_{0}\rangle\geq\langle e_{0}|\rho|e_{0}\rangle^{2}={F}^{2}.$ (14)

In conclusion, as long as $\langle H\rangle\leq E_{1}$, we have that

$\text{Tr}[\rho^{2}]\geq\left(\frac{E_{1}-\langle
H\rangle}{E_{1}-E_{0}}\right)^{2},$ (15)

approaching $\text{Tr}[\rho^{2}]=1$ (certified pure state) in the limit
$\langle H\rangle\to E_{0}$.

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