# Accelerated variational algorithms for digital quantum simulation of the
many-body ground states

Chufan Lyu<EMAIL_ADDRESS>Institute of Fundamental and Frontier
Sciences, University of Electronic Science and Technology of China, Chengdu
610051, China  Victor Montenegro<EMAIL_ADDRESS>Institute of
Fundamental and Frontier Sciences, University of Electronic Science and
Technology of China, Chengdu 610051, China  Abolfazl Bayat
<EMAIL_ADDRESS>Institute of Fundamental and Frontier Sciences,
University of Electronic Science and Technology of China, Chengdu 610051,
China

###### Abstract

One of the key applications for the emerging quantum simulators is to emulate
the ground state of many-body systems, as it is of great interest in various
fields from condensed matter physics to material science. Traditionally, in an
analog sense, adiabatic evolution has been proposed to slowly evolve a simple
Hamiltonian, initialized in its ground state, to the Hamiltonian of interest
such that the final state becomes the desired ground state. Recently,
variational methods have also been proposed and realized in quantum simulators
for emulating the ground state of many-body systems. Here, we first provide a
quantitative comparison between the adiabatic and variational methods with
respect to required quantum resources on digital quantum simulators, namely
the depth of the circuit and the number of two-qubit quantum gates. Our
results show that the variational methods are less demanding with respect to
these resources. However, they need to be hybridized with a classical
optimization which can converge slowly. Therefore, as the second result of the
paper, we provide two different approaches for speeding the convergence of the
classical optimizer by taking a good initial guess for the parameters of the
variational circuit. We show that these approaches are applicable to a wide
range of Hamiltonian and provide significant improvement in the optimization
procedure.

##  1 Introduction

Simulating strongly correlated many-body systems at and out of equilibrium is
one of the key tasks in condensed matter physics. On classical computers,
simulating a general quantum many-body system is inherently intractable, due
to the exponential growth of the Hilbert space with respect to the system
size. The only genuine approach for understanding a quantum system is to
emulate its behavior on a quantum simulator, which is a quantum device with
more controllability and versatility [48, 13, 18, 28]. Quantum simulators are
now rapidly emerging in various physical systems, including optical lattices
[30, 72, 9, 26, 27], Rydberg atoms [7], ion traps [45, 89, 33, 44, 73],
photonic circuits [2, 79], quantum dot arrays [34], dopants in silicon [69,
70] and superconducting devices [33, 19, 1, 87, 84, 85, 35, 68, 3, 66, 60,
41]. There are two different approaches for implementing quantum simulation,
namely analog and digital. In the analog approach, the particles interact via
some Hamiltonian, possibly time-dependent, and the quantum simulation is the
continuous evolution of this system [9, 26, 27, 66, 84, 85]. Although,
reasonably strong noise can be tolerated [62, 86, 24], the analog quantum
simulation is not universal, in the sense that only limited types of unitary
operations can be implemented. In contrast, the digital quantum simulation is
implemented through quantum gates acting on either one or two qubits in the
system [35, 68, 3]. The main advantage of digital quantum simulation is the
capability of performing universal computation, i.e. implementing arbitrary
unitary operations [58]. The major drawback is the high fragility of such
simulators to noise and their demand for complex error correction schemes
[76].

Many important phenomena in condensed matter physics, material science, and
chemistry are explained from the ground state of a certain Hamiltonian. This
includes electronic structures of matter [32], molecular formations [38],
magnetization [20] and quantum phase transitions [67]. There has been a
significant effort to simulate the ground state of many-body systems on
classical computers. Semi-classical approaches, such as density functional
theory [39], are extensively used for characterizing the electronic structures
but fail to explain all quantum effects. Quantum Monte Carlo [25] has been
developed to go beyond such mean field approaches but they also suffer from
sign problem. Density matrix renormalization group based methods [71], are
perhaps the most successful approach for characterizing the ground state
features of many-body systems, although they are mostly limited to
1-dimensional systems. Machine learning [15, 14, 29, 17, 80, 59] has also been
exploited to capture the ground state features, but it is hard to scale them
up for large systems. The limitations of these approaches truly reveals the
limited power of classical simulations for unraveling quantum features.

The emergence of quantum simulators has opened new vistas for simulating many-
body systems. To date, there are three different methods for simulating the
ground state of many-body systems on quantum simulators: (i) imaginary time
evolution [56, 51]; (ii) adiabatic evolution [10, 24]; and (iii) Variational
Quantum Eigensolver (VQE) [61]. The first two approaches are entirely
performed on quantum simulators, and thus their success heavily relies on the
quality of the quantum hardware. In particular, for large systems, these
approaches either demand deep quantum circuits or long coherence times.
However, the current Noisy Intermediate Scale Quantum (NISQ) simulators are
prone to various types of error in their initialization, readout, and gate
operations, while their finite coherence time limits both the system size and
the depth of the circuit. To overcome these obstacles, the VQE method, a
hybrid combination of a quantum circuit and a classical optimizer, has been
theoretically investigated [88, 52, 83, 53, 4, 65, 55, 82] and experimentally
implemented on various NISQ simulators, including photonic chips [61], ion
traps [73, 33, 44], nuclear magnetic resonance systems [47], and
superconducting quantum devices [60, 19, 41]. In fact, the use of the
classical optimizer allows one to simplify the quantum hardware into a shallow
circuit. In other words, the resources needed for simulating the ground state
is divided between quantum hardware and classical optimizer, enabling the
imperfect NISQ simulators to emulate the ground state of many-body systems.
Recently, VQE has been generalized to simulate excited states [37, 40], non-
equilibrium dynamics [46], Gibbs ensemble [81], and approximate solutions for
combinatorial optimization problems [23] in many-body systems. Based on these,
it is highly desirable to have a quantitative analysis for the required
resources, either quantum or classical, in different approaches for simulating
the ground state of many-body systems (see e.g., Ref. [12] for resources in
variational methods).

In this paper, we use a digital quantum simulator to prepare the ground state
of many-body Hamiltonians. We first compare the adiabatic evolution and the
VQE with respect to their demand of quantum resources, namely the depth of the
circuit or equivalently the number of two-qubit entangling gates. Our results
quantitatively show that the VQE is far more efficient in terms of quantum
resources than the adiabatic scheme. We then focus on the classical
optimization part of the VQE to speed up its convergence and thus minimize the
total time needed for the performance of the VQE. We specifically develop two
different strategies for improving the initial guess for the circuit
parameters to start the optimization at a closer point to the global minimum.
Our analysis shows that these strategies can significantly speed up the
convergence of the optimization, in particular, for larger systems. The
results are very general and can be applied for a wide range of Hamiltonians
even without symmetries.

##  2 Model

Our goal is to prepare the ground state of a many-body system on a Digital
Quantum Simulator (DQS). For simplicity, and without loss of generality, we
consider a $1$-dimensional chain of $N$ spin-$1/2$ particles interacting via
Heisenberg Hamiltonian

$H=J\sum_{i=1}^{N-1}\boldsymbol{\sigma}^{i}\cdot\boldsymbol{\sigma}^{i+1},$
(1)

where $J>0$ is the exchange coupling and
$\boldsymbol{\sigma}^{i}=(\sigma_{x}^{i},\sigma_{y}^{i},\sigma_{z}^{i})$ is
the vector of Pauli operators at site $i$. The Heisenberg model is one of the
key models in both condensed matter and quantum information technologies, and
thus, has been extensively explored in both ground state [54] and non-
equilibrium dynamics [22, 11, 16, 5]. The Heisenberg Hamiltonian has SU(2)
symmetry and commutes with the total spin in the $\alpha=x,y,z$ direction
(i.e. $[H,S_{\alpha}]=0$ where $S_{\alpha}=\sum_{i}\sigma_{\alpha}^{i}$). In
addition, the Hamiltonian also commutes with the total spin operator
$S_{tot}^{2}=S_{x}^{2}+S_{y}^{2}+S_{z}^{2}$. This implies that every
eigenstate of the system has a specific total spin $S_{tot}$ with a fixed
$S_{z}$. In particular, for even $N$, assumed in this paper, the ground state
$|GS\rangle$ is a global singlet with both $S_{tot}=0$ and $S_{z}=0$.
Furthermore, the Heisenberg Hamiltonian supports mirror-inversion symmetry as
$[H,\mathcal{M}]=0$, where $\cal{M}$ is the mirror-inverting operator with
$\mathcal{M}|i_{1},i_{2},\cdots,i_{N}\rangle=|i_{N},\cdots,i_{2},i_{1}\rangle$.
This implies that the eigenstates of the system also follow the same symmetry.
It is worth emphasizing that our methodology is easily generalized to other
Hamiltonians, such as XXZ, which satisfy the conservation of $S_{z}$ and
mirror-inversion symmetries. The ground state of the Heisenberg Hamiltonian
has a matrix product state representation so that the results from quantum
simulators can be certified by classical simulation. This allows us to enhance
our confidence on the performance of quantum simulators, and paves the way for
solving more complex problems which are not classically simulable, such as the
ground state of $2$-dimensional systems.

##  3 Digital Quantum Simulator

A digital quantum simulator is a quantum machine which manipulates the quantum
state of its qubits through performing certain one- and two-qubit gate
operations. In order to be a universal quantum computer, a digital quantum
simulator needs to be capable of applying arbitrary single qubit unitary
operations (which can be decomposed into rotations around $x$, $y$ and $z$
axes) on all qubits together with at least one two-qubit entangling gate
between any pair of particles [58]. In this paper, we consider all the single
qubit unitary operations to be a combination of the following gates

$P(\theta)=\begin{pmatrix}1&0\\\ 0&e^{i\theta}\end{pmatrix},\text{ and
}R_{\alpha}(\theta)=e^{i\frac{\theta}{2}\sigma_{\alpha}},\text{ with
}\alpha=x,y,z\textbf{,}$ (2)

where $P(\theta)$ is the phase gate and $R_{\alpha}(\theta)$ is the rotation
around the $\alpha$-axis. For a two-qubit entangling gate we use
controlled-$x$ (also known as CNOT) gate which is defined as

$U_{CX}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
\end{pmatrix}.$ (3)

By properly combining the above one- and two-qubit gates one can make any
unitary operation. Nonetheless, there is not a straightforward recipe to
design an efficient circuit to generate a specific target state. In
particular, designing protocols for preparing an eigenstate of a Hamiltonian,
on digital quantum simulators, has attracted significant attention in recent
years [35, 68, 3, 60, 19, 41, 33]. We pursue two approaches for generating the
ground state of many-body systems, namely adiabatic approach and VQE. In both
methods, which will de discussed in the following sections, the entangling
part of the circuit is a two-qubit unitary operator in the form of

$\mathcal{N}(\theta_{x},\theta_{y},\theta_{z})=e^{i\left(\theta_{x}\sigma_{x}\otimes\sigma_{x}+\theta_{y}\sigma_{y}\otimes\sigma_{y}+\theta_{z}\sigma_{z}\otimes\sigma_{z}\right)}$
(4)

in which $\theta_{x,y,z}$ are three angles. One can actually implement this
unitary operator using a simple circuit shown in Fig. 1(a), in which three
CNOT gates are combined by five local rotations around $y$ and $z$ axes [78].
Thanks to the symmetries of the Heisenberg Hamiltonian, throughout this paper,
we consider $\theta_{x}{=}\theta_{y}{=}\theta_{z}$ and thus we omit the
dependence on three independent parameters and simply use
$\mathcal{N}(\theta)$. Note that by imposing the condition
$\theta_{x}{=}\theta_{y}{=}\theta_{z}{=}\theta$, the operator
$\mathcal{N}(\theta)$ not only conserves $S_{z}$, but also can create a
maximally entangled state when acts on $|01\rangle$ for the choice of
$\theta{=}\pi/8$. Therefore, $\mathcal{N}(\theta)$ is truly a two-qubit
entangling gate.

Figure 1: (a) Circuit for realizing
$\mathcal{N}(\theta_{x},\theta_{y},\theta_{z})$ as the entangling gate between
two qubits. (b) A single layer circuit for realizing the first order Suzuki-
Trotter approximation in a system of length $N=4$. (c) A single layer circuit
for realizing the second order Suzuki-Trotter approximation in a system of
length $N=4$.

##  4 Adiabatic Evolution

Adiabatic theorem [10] is a well-known procedure for the preparation of the
ground state of a complex Hamiltonian starting from a simpler one. The system
is initially prepared in the ground state of a simple Hamiltonian $H_{ad}(0)$
which is then slowly varied in time to a more complex desired Hamiltonian
$H_{ad}(T_{max})$ after time $t=T_{max}$. If the variation of the Hamiltonian
is slow enough, the quantum state of the system always remains in the ground
state of the instant Hamiltonian $H_{ad}(t)$ [57]. In this paper, we are
focused in the preparation of the ground state of the Hamiltonian $H$ in Eq.
(1) using a digital quantum simulator. In Ref. [24], in an analog approach, a
dimerized Heisenberg Hamiltonian is adiabatically evolved to the uniform
Hamiltonian $H$ according to

$H_{ad}(t)=H_{odd}+\frac{t}{T_{max}}H_{even}$ (5)

where $0\leq t\leq T_{max}$ and

$H_{odd}=J\sum_{\text{odd
}i}\boldsymbol{\sigma}^{i}\cdot\boldsymbol{\sigma}^{i+1},\text{ and
}H_{even}=J\sum_{\text{even
}i}\boldsymbol{\sigma}^{i}\cdot\boldsymbol{\sigma}^{i+1}.$ (6)

At $t=0$ the Hamiltonian is fully dimerized with only odd couplings switched
on and the ground state of the system is simply
$|\Psi(0)\rangle=|\psi^{-}\rangle\otimes\cdots\otimes|\psi^{-}\rangle$, where
$|\psi^{-}\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$. By choosing $T_{max}\sim
N^{2}$ the condition of adiabatic theorem [57] is satisfied and at
$t=T_{max}$, where $H_{ad}(T_{max})=H$, the quantum state of the system
becomes the ground state of the desired Hamiltonian $H$ [24].

To obtain the time evolution of the system, governed by $H_{ad}(t)$, on a
digital quantum simulator one has to adopt two essential steps: (i) discretize
the time evolution into $M_{ad}$ steps over which the Hamiltonian remains
fixed for a time interval of $\Delta t=T_{max}/M_{ad}$; and (ii) exploit the
Suzuki-Trotter expansion for evolving the system in each time step. In each
discretized time step $k$ (with $k$ going from $1$ to $M_{ad}$) the
Hamiltonian is considered to be fixed, namely $H_{ad}(k\Delta t)$. The time
evolution operator at time step $k$ is thus written as

$U_{k}(\Delta t)=e^{-iH_{ad}(k\Delta t)\Delta t}=e^{-i(H_{odd}+\frac{k\Delta
t}{T_{max}}H_{even})\Delta t}.$ (7)

Notice that, in order to emulate the true time evolution of the $H_{ad}(t)$
the discrete time steps have to be small or equivalently the number of steps
$M_{ad}$ has to be large. This will be discussed in more details in the
following. As mentioned above, the second essential step for realizing the
time evolution of a many-body system on a digital quantum simulator is to use
Suzuki-Trotter expansion [31]. This will allow us to simulate the dynamics
only through one- and two-qubit gate operations. We consider both the first
and the second order realization of the Suzuki-Trotter expansion for the
unitary operator $U_{k}(\Delta t)$, given in Eq. (7), at the time step $k$ ,
namely

$\displaystyle U_{k}^{\text{ST1}}(\Delta t)$ $\displaystyle=$ $\displaystyle
e^{-iH_{even}\frac{k\Delta t^{2}}{T_{max}}}e^{-iH_{odd}\Delta t},$ (8)
$\displaystyle U_{k}^{\text{ST2}}(\Delta t)$ $\displaystyle=$ $\displaystyle
e^{-iH_{odd}\frac{\Delta t}{2}}e^{-iH_{even}\frac{k\Delta
t^{2}}{T_{max}}}e^{-iH_{odd}\frac{\Delta t}{2}}.$ (9)

While the first order Suzuki-Trotter operator $U_{k}^{\text{ST1}}(\Delta t)$
approximates the discrete time evolution $U_{k}(\Delta t)$ by a quadratic
error as $U_{k}(\Delta t)\approx U_{k}^{\text{ST1}}(\Delta t)+O(\Delta
t^{2})$, the second order Suzuki-Trotter operator $U_{k}^{\text{ST2}}(\Delta
t)$ improves the approximation as $U_{k}(\Delta t)\approx
U_{k}^{\text{ST2}}(\Delta t)+O(\Delta t^{3})$. In Figs. 1(b)-(c) we present
the quantum circuit for each time step of the first and the second order
Suzuki-Trotter evolution, respectively.

In order to speed up the adiabatic protocol, we first determine the minimum
time $T_{max}$ which is required to prepare the system in its ground state for
a given threshold fidelity $F=|\langle\Psi(T_{max})|\text{GS}\rangle|^{2}$
between the quantum state of the system $|\Psi(T_{max})\rangle$ and the target
state $|\text{GS}\rangle$. By fixing the threshold fidelity $F$ and system
size $N$, one can solve the Schrödinger equation with time dependent
Hamiltonian $H_{ad}(t)$ and find the minimum $T_{max}$ which results in a
fidelity above the threshold $F$. As shown in Ref. [24], the minimum time
needed scales as $T_{max}\sim N^{2}$ where the proportionality coefficient
gets larger as the threshold fidelity increases. By specifying the minimum
value of $T_{max}$ one can then apply the above adiabatic protocol on a
digital quantum simulator to find the minimum required Suzuki-Trotter steps
$M_{ad}^{*}$ to achieve the threshold fidelity $F$. In Figs. 2(a)-(b), we plot
$M_{ad}^{*}$ as a function of $N$ for various threshold fidelities when the
first and the second order Suzuki-Trotter circuits are used respectively.
Notably, for a given threshold fidelity, the number of layers in the the
second order Suzuki-Trotter approximation is almost one order of magnitude
smaller than the the first order one. However, the number of gates in each
layer of the second order Suzuki-Trotter circuits are more than the number of
gates in their corresponding first order. Hence, it is more meaningful to
compare the number of gates needed for achieving the threshold fidelity rather
than the number of layers. In practice, the two-qubit gates are far more
challenging than the one-qubit gates. Therefore, in Figs. 2(c)-(d) we plot the
number of CNOTS versus the system size $N$ for various threshold fidelities in
both first and second order Suzuki-Trotter circuits, respectively. Remarkably,
the second order Suzuki-Trotter circuit shows a clear superiority over the
first order by demanding significantly less number of CNOT gates for
delivering the same fidelity. This clearly shows that the second order Suzuki-
Trotter expansion is significantly more resource efficient than the first
order for realizing the ground state of a many-body system on a digital
quantum simulator.

Figure 2: The minimum circuit depth $M_{ad}^{*}$ as a function of $N$ for
various threshold fidelities using: (a) first order Suzuki-Trotter
approximation; and (b) Second order Suzuki-Trotter procedure. The number of
CNOT gates required for the adiabatic evolution as a function of length $N$
for required for various threshold fidelities using: (c) first order Suzuki-
Trotter approximation; and (d) Second order Suzuki-Trotter procedure.

##  5 Variational Method

As shown in the previous section, the adiabatic approach for creating the
ground state of many-body systems demands deep circuits, i.e. large
$M_{ad}^{*}$, with a considerable number of gates. The near-term quantum
simulators cannot provide such deep circuits due to their imperfect noisy
quantum gates, in particular the two-qubit gates. Therefore, it is worthy of
exploring the possibilities for the use of shallow circuits to simulate the
ground state of many-body systems. Variational methods, such as variational
quantum eigensolver (VQE) algorithm, provide an alternative to adiabatic
evolution which can be realized in shallow circuits. In these methods, the
post processed measurement outcomes of a fairly shallow quantum circuit has to
be fed into a classical optimizer iteratively to find the optimal circuit
parameters for realizing the ground state of a many-body system. In the VQE
algorithm, the quantum device prepares a quantum state
$|\psi(\vec{\theta})\rangle$, which is often called the ansatz, with
$\vec{\theta}=(\theta_{1},\theta_{2},\cdots,\theta_{L})$ being some
controllable parameters in the circuit. Then one can measure the average
energy $\langle
H\rangle=\langle\psi(\vec{\theta})|H|\psi(\vec{\theta})\rangle$ for this
quantum state through appropriate measurements on the qubits of the simulator.
For the Heisenberg model in Eq. (1), this reduces to measure correlation
functions between adjacent qubits and then add them up, namely $\langle
H\rangle=\sum_{i=1}^{N-1}\langle\psi(\vec{\theta})|\boldsymbol{\sigma}^{i}\cdot\boldsymbol{\sigma}^{i+1}|\psi(\vec{\theta})\rangle$.
The measured average energy can be used as an input for a classical optimizer
to find its minimum by adaptively adjusting the parameters $\vec{\theta}$ on
the simulator. The minimization algorithm eventually finds the optimal
parameters $\vec{\theta}_{min}$ for which $|\psi(\vec{\theta}_{min})\rangle$
is the ground state with minimum average energy. Therefore, the resources
required for VQE is divided between the quantum simulator and the classical
optimizer. In fact, in each Classical Optimization Iteration (COI) a new
experiment on the quantum simulator is needed, making the VQE very time
consuming. Thus, for the success of the VQE, two steps are essential: (i) a
quantum circuit, as shallow as possible with minimal number of parameters; and
(ii) an efficient classical optimizer which can find the minimum of average
energy using least number of iterations.

Designing a quantum circuit, with minimal resources, which is capable of
realizing the ground state of the Hamiltonian, here Heisenberg interaction, is
a key step in a VQE algorithm [36]. The output of the circuit will be
$|\psi(\vec{\theta})\rangle$ and by varying the parameters $\vec{\theta}$ one
can explore the Hilbert space for finding the quantum state with minimum
average energy. Ideally, the quantum circuit should be able to explore the
whole Hilbert space as $\vec{\theta}$ vary. However, using symmetries of the
system one can simplify the circuit, i.e. decreasing the number of parameters
in $\vec{\theta}$, and instead of exploring the whole Hilbert space only focus
on the relevant part of the Hilbert space. For instance, in the case of the
Heisenberg Hamiltonian, considering the preservation of $S_{z}$ and the
mirror-inversion symmetries can significantly reduce the number of parameters
in $\vec{\theta}$ which not only simplifies the quantum circuit but also makes
the convergence of the classical optimization procedure faster. Considering
these two symmetries, we suggest to use $\mathcal{N}(\theta)$, introduced in
Eq. (4), as the entangling gates of the circuit because they do not create any
extra excitation in the system. These gates can be combined in a geometry
similar to the first order Suzuki-Trotter steps. In order to be able to
locally manipulate the phases of each qubit we can also add phase gates to
every qubit of the circuit as shown in Fig. 3(a). The mirror-inversion
symmetry implies that for even (odd) $N$ the angles of the phase gates acting
on site $k$ and $N-k+1$ are equal with the opposite (the same) signs.
Therefore, one can easily show that the total number of parameters $L$ in a
VQE circuit is

$L=\left(\frac{3N}{2}-1\right)M_{VQE}$ (10)

where, $M_{VQE}$ represents the number of layers. The proposed circuit is
capable to explore a part of Hilbert space which includes the ground state. In
principle, one can chose the parameters such that this circuit replicates the
adiabatic circuit in Fig. 1(b)-(c), e.g. by putting the angles of the phase
gates to zero. However, in the adiabatic circuit every layer is close to
identity operator as it evolves the system only for small time $\Delta t$.
This results in a deep circuit demanding a considerable number of gates as
shown in Fig. 2. The idea behind the VQE is to exploit an optimization
algorithm to find the optimal set of parameters for which a shallow circuit
can implement the ground state of a many-body system.

Figure 3: (a) The schematic picture of a single layer VQE circuit for a chain of length $N=4$. (b) The qubit recursive algorithm for creating a large circuit using the optimized parameters of two subsystem with smaller sizes, here half size. Apart from the optimized gates, one entangling gate, shown in green, initialized by a random number is added between the two subsystems for connecting them. If the larger system needs more layers than the smaller ones then the extra layers are initialized with the parameters copied from the previous layer. (c) The layer recursive algorithm for making the circuit deeper step by step after optimizing each layer. The optimized parameters of the previous layers are used as the initial guess for the new circuit when a new layer is added. The parameters in the new layer are copied from the previous one. | $ST\leavevmode\nobreak\ 1st\leavevmode\nobreak\ order$ | $ST\leavevmode\nobreak\ 2nd\leavevmode\nobreak\ order$ | $VQE$
---|---|---|---
N | $M^{*}_{ad}$ | $\\#CNOT$ | $M^{*}_{ad}$ | $\\#CNOT$ | $M^{*}_{VQE}$ | $\\#CNOT$
4 | 15 | 135 | 3 | 45 | 2 | 18
8 | 113 | 2373 | 18 | 594 | 3 | 63
10 | 247 | 6669 | 32 | 1344 | 3 | 81
16 | 770 | 34650 | 79 | 5451 | 5 | 225
20 | 1330 | 75810 | 132 | 11484 | 6 | 342
Table 1: A comparison for the circuit depth $M^{*}$ between the adiabatic
approach (with both first and second order Suzuki-Trotter approximation) and
the VQE algorithm for various system sizes when the threshold fidelity is
chosen to be $F=0.99$.

In order to finalize the VQE circuit, apart from designing each layer, one has
to also determine the number of layers $M_{VQE}$. To find out the number of
layers needed to successfully produce the ground state, one can use the
fidelity between the quantum state of the circuit and the real ground state at
the end of optimization procedure. The layers are increased one by one so that
the attainable fidelity exceeds a threshold value. Note that fidelity is not
an experimentally friendly quantity as one needs to know the quantum state.
Therefore, in practice, instead of fidelity one can focus on the convergence
of optimized average energy as the number of layers increases. For any system
size $N$ and a chosen value for the layer number $M_{VQE}$ we use $50L$ (for
$L$ being the number of parameters in $\vec{\theta}$) COI to see whether the
obtained fidelity (or average energy) reaches the threshold value. If the
optimization fails to reach the threshold fidelity then we increase the layer
number $M_{VQE}$ by one unit until the algorithm is successful.

In every circuit, we initially start with a random guess for all the
parameters in $\vec{\theta}$ and use Adam algorithm (with amsgrad applied)
[43, 64], as an extension to stochastic gradient descent, to minimize our cost
function, namely the average energy of the system. In Adam algorithm there are
four different hyper parameters which have to be set as the input of the
optimizer: (i) the learning rate $\alpha$, which controls how fast the
parameters are going to be updated in each iteration. (ii) $\beta_{1}$, the
exponential decay rate for the first moment estimation; (iii) $\beta_{2}$, the
exponential decay rate for the second-moment estimation; and (iv) a very small
number $\epsilon$ to prevent any division by zero. We set the hyper parameters
to be $\alpha=0.01$, $\beta_{1}=0.9$, $\beta_{2}=0.999$ and
$\epsilon=10^{-8}$.

In order to be independent of the initial random guess for the parameters
$\vec{\theta}$ we repeat the procedure for several ($\sim 100$) different
random samples. Only if the fidelity of all of them exceeds the threshold
after $50L$ iterations we call it a success. For our numerical simulations, we
use a cluster equipped with 192 cpu cores and 512 GB memory. For the largest
system size considered here $N=20$, the simulations take nearly a week to
finish 100 different random samples using such hardware. Similar to the
adiabatic approach, the minimum number of layers which can successfully
generate the ground state within the threshold fidelity is called
$M_{VQE}^{*}$. In Table. 1, we compare $M_{VQE}^{*}$ with $M_{ad}^{*}$ in both
first and second order Suzuki-Trotter approximation and their corresponding
CNOT gate numbers for different system sizes when the threshold fidelity is
chosen to be $F=0.99$. As it is clear from the Table, the number of required
layers and the CNOT gate numbers in the VQE approach is significantly smaller
than the adiabatic procedure. Even for a system of size $N=20$, a fairly
shallow circuit of $M_{VQE}^{*}=6$ layers suffices to produce the ground state
of the Heisenberg chain with the fidelity exceeding $F=0.99$. This remarkable
observation shows the superiority of the VQE over the adiabatic approach with
respect to the complexity of the circuit. The simplification in quantum
resources, i.e. the circuit, is achieved thanks to the employment of a
classical optimizer. Thus, in order to truly investigate the complexity of the
VQE approach, one has to consider the required resources for the classical
optimization procedure as well.

Figure 4: (a) The average energy as a function of COI in random strategy (i.e.
strategy 1) for various circuit layers in a chain of length $N=20$. As the
circuit layer increases the optimized average energy approaches the real
ground state energy. (b) The obtainable fidelity averaged over 100 different
random samples as a function of COI for circuits of different depth. (c) The
average obtainable fidelity, and its corresponding error bars, as a function
of COI when the whole VQE algorithm is repeated for $100$ times.  Figure 5:
(a) The average energy as a function of COI in qubit recursive strategy (i.e.
strategy 2) for various circuit layers in a chain of length $N=20$. As the
circuit layer increases the optimized average energy approaches the real
ground state energy. (b) The obtainable fidelity averaged over 100 different
random samples as a function of COI for circuits of different depth. (c) The
average obtainable fidelity, and its corresponding error bars, as a function
of COI when the whole VQE algorithm is repeated for $100$ times. Figure 6: (a)
The average energy as a function of COI in layer recursive strategy (i.e.
strategy 3) for various circuit layers in a chain of length $N=20$. As the
circuit layer increases the optimized average energy approaches the real
ground state energy. (b) The obtainable fidelity averaged over 100 different
random samples as a function of COI for circuits of different depth. (c) The
average obtainable fidelity, and its corresponding error bars, as a function
of COI when the whole VQE algorithm is repeated for $100$ times.

##  6 Strategies for accelerating the classical optimization

A conventional classical optimization procedure starts with a random guess for
$\vec{\theta}$ and iteratively minimizes the cost function, here the average
energy. However, the random initial guess can be very far from the real
minimum of the cost function. This may lead to two unwanted outcomes: (i) many
optimization iterations are needed to get to the global minimum; and (ii) it
is likely that the optimization gets trapped in a local minimum and thus fails
to reach the right answer. Therefore, it is of high interest to see whether
one can start the optimization with a smart guess which is relatively close to
the real minimum of the cost function. In the following, we first explain the
random initial guess in details, as the first strategy, and then provide two
different strategies with which we can speed up the convergence and improve
the precision.

Strategy 1: random initialization. We first provide a detailed introduction
for the basic random initialization strategy. A single circuit layer which is
used in this strategy is shown in Fig. 3(a). One has to cascade several layers
in order to achieve a high quality output. In a circuit of $N$ qubits and
$M_{VQE}$ layers there are $L$ number of parameters, which is given in Eq.
(10). In this strategy, the initial values of the parameters are sampled from
a Gaussian distribution of mean $0$ and variance $1$. In Fig. 4(a) we plot the
average energy as a function of optimization iterations for circuit of
different layers. As the figure shows by increasing the iterations the average
energy decreases steadily, however, if the number of layers are small the
final converged value is far from the ground state energy. This shows that if
the circuit is too shallow then it has no capacity to reach the ground state
merely by classical optimization. As the number of layers increases the
converged value is closer to the real ground state energy. In the Fig. 4(b) we
give the corresponding obtainable fidelity averaged over 100 different random
samples as a function of COI for circuits of different circuit layers
$M_{VQE}$. As the average energy approaches the real ground state the fidelity
also goes to $1$. In order to see the stability of the protocol, we repeat the
optimization for 100 different random samples. In Fig. 4(c) we plot the
average of obtained fidelities together with the error bars, given by the
standard deviation, as a function of iterations for a system of length $N=20$
and a circuit with $M_{VQE}=6$ layers. Even after $\sim 7500$ iterations the
average fidelity does not reach $0.95$ showing that the random initial choice
is not an efficient guess for the optimizer.

Strategy 2: qubit recursive. The first strategy to improve the classical
optimization procedure is to use the optimized value of a smaller system as
the initial guess for the larger one. Here, we focus on the case that we use
the optimized parameters in a circuit of size $N/2$ as the initial guess for a
system of size $N$. Let’s first consider the case that the number of layers
$M_{VQE}$ is identical for both the systems of size $N/2$ and $N$. In this
case, we simply replicate the optimized circuit of size $N/2$ twice to make a
larger system of size $N$. To have a complete circuit one has to add one
entangling gate $\mathcal{N}(\theta)$ to each layer at the connection between
the two copies acting on odd couplings, as shown in Fig. 3(b). Therefore, in
total, one has to add $M_{VQE}$ different entangling gates between the copies
which are initialized with random guesses. Apart from these newly added gates,
the rest of the one- and two-qubit gates are all initialized with the
optimized value from the system of size $N/2$. The motivation behind this
choice is that, the reduced density matrix of the subsystems from the ground
state of the large system has large overlap with the ground state of the local
system. Thus, using this strategy, one expects to converge to the real global
minimum faster. In the case that the number of layers in the small chain is
less than the layers required for the large system we simply repeat the same
parameters of the last layer for the added layers. The convergence of the
average energy versus the COI is shown in Fig. 5(a) for different circuit
layers. In Fig. 5(b) the corresponding obtainable fidelity averaged over 100
different random samples are depicted as a function of COI. Indeed, the
results show faster convergence for each choice of layers. In order to see the
overall performance, we repeat the optimization for 100 different initial
random guesses in a system of size $N=20$ with $M_{VQE}=6$ layers. The average
obtained fidelities, with corresponding error bars, are plotted in Fig. 5(c)
as a function of iterations. The performance shows clear improvement in
comparison with the random strategy as the average fidelity exceeds the
threshold fidelity of $F=0.95$ only after $\sim 3500$ iterations.

Strategy 3: layer recursive. In this strategy, we start a circuit with only
one layer. Obviously this circuit is too shallow to provide the ground state
for large systems. Nonetheless, one can perform the optimization using random
initial guess for minimizing the average energy. The minimization converges
normally fairly quick, as the circuit is shallow and the number of parameters
are not that large. However, the output state may have a small fidelity with
the real ground state as the circuit is shallow. Hence, in the next step we
add one more layer, which means that the circuit becomes deeper and more
parameters exist to be optimized. The initial values for the parameters of the
newly added layer are copied from the previous layer. At the end of the
optimization, one can add a new layer and repeat this procedure. A schematic
of the procedure is shown in Fig. 3(c). In Fig. 6(a) we plot the average
energy versus the number of iterations for different layers. As the figure
shows while the energy converges by increasing the iterations the final value
reaches the ground state energy only for circuits with adequate depth. In
order to see the overall performance, we repeat the procedure $100$ times and
compute the final fidelities. In Fig. 6(b), we illustrate the obtainable
fidelity averaged over 100 different random samples as a function of COI. As
evident from the figure, this strategy shows faster convergence as contrasted
with the previous ones. In Fig. 6(c) we plot the average fidelities, with the
corresponding error bar, as a function of COI in a circuit of size $N=20$ and
depth $M=6$. As the figure shows the speed of convergence is sensibly faster
than the random initial guess, discussed in the strategy 1, and the average
fidelity exceeds the threshold fidelity $F=0.95$ only after $\sim 2500$
iterations.

| 1000 iter. | 3000 iter. | 6000 iter.
---|---|---|---
Random | 0.648 $\pm$ 0.090 | 0.786 $\pm$ 0.098 | 0.849 $\pm$ 0.087
Qubit | 0.810 $\pm$ 0.123 | 0.898 $\pm$ 0.084 | 0.923 $\pm$ 0.064
Layer | 0.916 $\pm$ 0.049 | 0.939 $\pm$ 0.024 | 0.947 $\pm$ 0.023
Table 2:  Comparing the obtainable fidelities between the three different VQE
strategies for given number of iterations in a system of size $N=20$.

In order to compare the three VQE strategies, in Table 2, we report the
average fidelities obtained in each strategy for three different iterations in
a system of size $N=16$ and a circuit of $M=6$ layers. The data clearly shows
significant improvement of both qubit and layer recursive strategies over the
random one, in particular, when the number of iterations are smaller.
Interestingly, the layer recursive strategy also shows modest superiority over
the qubit recursive strategy. This means that optimizing the system layer by
layer and use their optimized data as the initial guess for building a deeper
circuit significantly enhances the convergence of the classical optimization
in a VQE algorithm. This is the main result of our paper.

Figure 7: Simulating the ground state of a XYZ Hamiltonian using the mirror
symmetry rule. (a) The average energy as a function of COI in random
initialization strategy (i.e. strategy 1) for various circuit layers in a
chain of length $N=10$. As the circuit layer increases the optimized average
energy approaches the real ground state. (b) The average obtainable fidelity,
and its corresponding error bars, as a function of COI when the whole VQE
algorithm is repeated for $100$ times in random initialization strategy. (c)
The average energy as a function of COI in layer recursive strategy (i.e.
strategy 3) for a chain of $N=10$. As the circuit layer increases the
optimized average energy approaches the real ground state faster than in
strategy 1. (d) The average obtainable fidelity, and its corresponding error
bars, as a function of COI when the whole VQE algorithm is repeated for $100$
times in layer recursive strategy. Figure 8: Simulating the ground state for a
spin chain Kondo Hamiltonian. (a) The average energy as a function of COI in
random initialization strategy (i.e. strategy 1) for various circuit layers in
a chain of length $N=10$. As the circuit layer increases the optimized average
energy approaches the real ground state energy. (b) The average obtainable
fidelity, and its corresponding error bars, as a function of COI when the
whole VQE algorithm is repeated for $100$ times in random initialization
strategy. (c) The average energy as a function of COI in layer recursive
strategy (i.e. strategy 3) for various circuit layers in a chain of length
$N=10$. As the circuit layer increases the optimized average energy approaches
the real ground state energy faster than in the strategy 1. (d) The average
obtainable fidelity, and its corresponding error bars, as a function of COI
when the whole VQE algorithm is repeated for $100$ times in layer recursive
strategy.

##  7 Generality of the protocol

In the previous sections, we have focused on the Heisenberg Hamiltonian for
which we have exploited two major symmetries of the system, namely the
conservation of $S_{z}$ and the mirror symmetry. Here, we generalize our
analysis and show that our protocol and the developed accelerated algorithms
work well for more general Hamiltonians where these symmetries are not
applicable. As the first example, we consider the general XYZ Hamiltonian

$H_{XYZ}=\sum_{i=1}^{N-1}J_{x}\sigma_{x}^{i}\sigma_{x}^{i+1}+J_{y}\sigma_{y}^{i}\sigma_{y}^{i+1}+J_{z}\sigma_{z}^{i}\sigma_{z}^{i+1},$
(11)

where $J_{\alpha}$ stands for the coupling in $\alpha$ direction. Clearly this
Hamiltonian does not commute with $S_{z}$ for general choices of
$J_{\alpha}$’s. Nonetheless, the XYZ Hamiltonian still supports the mirror
symmetry. To perform our VQE analysis, we use the circuit in Fig. 3(a) with
the difference that $\theta_{x}$, $\theta_{y}$, and $\theta_{z}$ in
$\mathcal{N}$ (see Fig. 1(a)) are replaced by
$\theta_{\alpha}=J_{\alpha}\theta$. Note that by this choice, the number of
parameters remains the same as the case for the Heisenberg Hamiltonian.

In Fig. 7(a) we plot the average energy $\langle H\rangle$ as a function of
COI for a circuit of different depths in a chain of length $N=10$ using the
strategy 1, namely random initialization of parameters. Clearly, the average
energy converges as the number of COI increases and the converged value gets
closer to the real ground state energy as the circuit gets deeper. In Fig.
7(b), we plot the fidelity for a circuit with 6 layers. By increasing the COI,
the fidelity increases above the $0.95$ threshold, showing that the protocol
still works even in this more general case. To speedup the convergence, in
Fig. 7(c) we plot the average energy $\langle H\rangle$ as a function of COI
employing the layer recursive strategy which shows faster convergence in
contrast to the random strategy case. To evidence this more clearly, in Fig.
7(d) we plot the fidelity as a function of COI when the layer recursive
strategy is exploited. As evident from the figures, the layer recursive
strategy shows significant improvement over the random strategy, which
demonstrates that our protocols for speeding up the convergence are still
applicable in more general Hamiltonians.

In the XYZ case, as the Hamiltonian supports the mirror symmetry, we can still
design the circuit using this constraint, i.e. $\theta_{i}=-\theta_{N-i+1}$.
To go beyond this symmetry, we now consider the spin chain Kondo Hamiltonian
[75, 74, 6] as one of the key models in impurity physics. The Hamiltonian can
be written as

$H_{Kondo}=J\left(J^{\prime}\boldsymbol{\sigma}^{1}\cdot\boldsymbol{\sigma}^{2}+\sum_{i=2}^{N-1}\boldsymbol{\sigma}^{i}\cdot\boldsymbol{\sigma}^{i+1}\right),$
(12)

where the dimensionless parameter $J^{\prime}<1$ determines the first qubit as
the impurity. The Kondo Hamiltonian commutes with $S_{z}$ but obviously does
not have mirror symmetry. In order to simulate the ground state of this
Hamiltonian using the VQE algorithm, we again exploit the circuit of Fig. 3(a)
with the difference that the coupling between the qubits 1 and 2 is different
and the local rotations do not follow the mirror symmetry rule, i.e.
$\theta_{i}=-\theta_{N-i+1}$. Due to this, the number of parameters to be
optimized increases.

In Fig. 8(a) we plot the average energy $\langle H\rangle$ as a function of
COI for a system length of $N=10$ and different circuit layers using the
random initialization strategy. As shown from the figure, increasing the depth
of the circuit one can fairly get closer to the ground state energy of the
system. To observe this in more detail, in Fig. 8(b), we plot the fidelity as
a function of COI for the case of 4 layers. As evident, in the absence of the
mirror symmetry, the ground state can still be simulated with high fidelity.
We now attempt to compare the above strategy with the layer recursive one. In
Fig. 8(c), we plot the average energy $\langle H\rangle$ as a function of COI
for different circuit layers. As the figure shows, the convergence of the
average energy is clearly accelerated for equals circuit depth. Finally, in
Fig. 8(d) we plot the fidelity as a function of COI for the case of 4 layers,
showing that the threshold fidelity of $0.95$ is reached faster than the
random strategy case.

The results presented above, i.e. simulating the ground state of the XYZ and
Kondo Hamiltonian, show that not only our VQE algorithm works well under more
general Hamiltonians but also that the layer recursive strategy can still be
applied to accelerate the optimization procedure.

Figure 9: Average obtainable fidelity, and its corresponding error bars, as a
function of COI for the XYZ Hamiltonian in a chain of length $N=10$. (a)
Simulating the ground state energy of the system using the random strategy in
the presence of mirror symmetry. (b) Same simulation in the absence of the
mirror symmetry rule. In panels (c) and (d), we compute the ground state
energy using the layer recursive strategy in the presence and in the absence
of the mirror symmetry rule, respectively.

##  8 The role of mirror symmetry

As mentioned before, the presence of mirror symmetry can simplify the circuit
through reducing the number of parameters for the local rotations. This has
been implemented by the specific choice of rotation angles, i.e.
$\theta_{i}=-\theta_{N-i+1}$. The use of mirror symmetry in designing the
circuit has clear advantages as reducing the number of parameters implies the
need of less number of measurements for computing the gradient in the
optimization procedure. However, one may argue that having a larger parameter
space may provide a shorter route to the global minimum. To investigate this,
we compare the VQE algorithm for the XYZ Hamiltonian in the presence and in
the absence of the mirror symmetry. In Fig. 9(a), we plot the fidelity as a
function of COI when the mirror symmetry is employed in our random strategy.
While the fidelity improves by increasing the iterations, the final fidelity
is not very high. In Fig. 9(b), we plot the same quantity, still using the
random strategy, when no mirror symmetry is employed in the circuit and thus
the parameter space is larger. Surprisingly, the fidelity shows significant
improvement showing that the mirror symmetry may make the convergence slower.
To see the performance of our layer recursive strategy, in Fig. 9(c) and (d)
we plot the same quantity as a function of iterations in the presence and in
the absence of mirror symmetry. Both figures show significant improvement in
contrast to the case of random strategy. However, the remarkable observation
is that even in the layer recursive strategy still the absence of mirror
symmetry in our circuit can reduce the number of iterations.

It is, however, not easy to conclude that the mirror symmetry is not useful.
We have to emphasize that in practical situations for obtaining the gradient
of $\langle H\rangle$ with respect to the parameters of the system demands one
measurement setup (with thousands of repeatations) for every single parameter.
Therefore, from an experimental perspective, exploiting the mirror symmetry is
beneficial. In addition, if the parameters are too many, the classical
optimization may indeed fail. Based on this, although for smaller system
sizes, not using the mirror symmetry may demand less iterations, in larger
systems it will be more useful to implement such symmetry in the circuit.

##  9 Designing the VQE circuit

The key step for the success of the VQE algorithm is to design a proper
circuit such that explores the right part of the Hilbert space which contains
the state of interest. In addition, due to practical constraints, e.g.
imperfect gates and decoherence, the designed circuit has to be as shallow as
possible. Interestingly, the adiabatic circuits can always be considered as a
special case of the VQE such that each layer changes the quantum state
minimally (as $\Delta t$ is typically small). Motivated by this, one can
always adapt the adiabatic circuit, either first or second order Suzuki-
Trotter, and add local rotations to all the qubits of each layer, see Fig.
3(a). By properly optimizing the local rotations and the parameters of the
entangling gates, one can make the operation of each layer significantly
different from identity and thus reduce the number of layers. In other words,
the adiabatic circuit is an inefficient realization of the VQE, with no local
rotations and small action of the entangling gates. Consequently, the
operation of each layer in adiabatic circuit is close to identity. Therefore,
one can conclude that the VQE always outperforms the adiabatic evolution with
respect to the circuit complexity.

##  10 Decoherence

Figure 10: (a) The obtainable VQE fidelity as a function of disorder strength
$h$ when the CNOT gates suffer a unitary error, given in Eq. (13). (b) The
obtainable VQE fidelity as a function of dephasing rate $\gamma\Delta t$. In
both cases the system size is $N=10$.

In the NISQ era, the quantum simulators are noisy and their performance is
affected by the presence of decoherence. There are several attempts to
simulate the effect of noise in existing digital quantum simulators [46, 41,
42, 77, 21, 50, 63, 8]. Here, although in a very simplistic and rather naive
fashion, we model some errors that can serve as a first glimpse into a more
realistic scenario.

We consider two types of errors: (i) unitary ones due to imperfect gate
pulses; and (ii) non-unitary errors due to decoherence. Regarding the gate
errors, the operations of single qubit rotations are very precise. The
performance of CNOT gates, however, may not be as perfect as it expected due
to imperfect pulses which couples the two qubits. To see the quality of such
errors we consider a particular realization of the CNOT gate through Ising
type interaction of the two qubits

$\widetilde{U}_{CX}(\phi)=\sqrt{i}\hat{H}_{2}e^{i\sigma^{z}_{1}\sigma^{z}_{2}(\pi/4+\phi)}e^{-i\sigma^{z}_{1}\pi/4}e^{-i\sigma^{z}_{2}\pi/4}\hat{H}_{2}$
(13)

where $\hat{H}_{2}=(\sigma_{2}^{x}+\sigma_{2}^{z})/\sqrt{2}$ is the Hadamard
gate, and $\phi$ should ideally be zero but due to imperfect pulses in the
device is a random number uniformly distributed over the interval $[-h,+h]$
with zero average. To see the impact of this error, we consider a VQE circuit
for a chain of length $N=10$ with $M_{VQE}=3$ layers using optimized
parameters, obtained for the ideal case. In reality each CNOT gate will have a
random phase $\phi$ in the circuit. We consider $100$ different realization of
such random phases and average the final fidelity for each value of $h$, as
the noise strength. The average fidelity, using $100$ different random
realizations, versus noise strength $h$ is plotted in Fig. 10(a). As the
figures shows even for a relatively strong noise of $h=0.1$ the fidelity still
remains above $0.8$.

A more serious form of noise in NISQ simulators is dephasing which appears as
a non-unitary operation and destroys the superposition of quantum states.
Dephasing cannot be digitalized and thus it is difficult to incorporate it in
our circuit picture. However, in order to have an approximation of its effect,
after each layer of gates we evolve the system according to a Lindblad master
equation for a certain time $\Delta t$ which is equal to the time needed to
perform the gates in the previous layer. Thus the output of the system after
$M$ layers will be

$\rho^{(M)}=e^{\mathcal{L}\Delta t}\circ U^{(M)}[\rho^{(M-1)}]$ (14)

where, $\rho^{(M)}$ ($\rho^{(M-1)}$) represents the density matrix of the
system at the output of the layer $M$ ($M-1$), $U^{(M)}$ represents the
unitary operation of all the gates in layer $M$ and $\mathcal{L}$ represents
the Lindblad master equation $\partial_{t}\rho=\mathcal{L}(\rho)$ with

$\mathcal{L}(\rho)=\gamma\sum_{k=1}^{N}(\rho-\sigma_{k}^{z}\rho\sigma_{k}^{z}).$
(15)

The coefficient $\gamma$ shows the strength of the decoherence. In Fig. 10(b),
we plot the fidelity as a function of $\gamma\Delta t$ for a VQE circuit of
length $N=10$ and depth $M_{VQE}=3$ with all the parameters already optimized.
As the figure shows up to $\gamma\Delta t=0.0125$ the fidelity remains above
$0.5$ which reveals considerable contribution of the ground state in the
output of the quantum simulator.

##  11 Conclusion

In this paper, we have investigated two different strategies, namely the
adiabatic evolution and the VQE, for simulating the ground state of many-body
systems on digital quantum simulators. Our results are multifold. First, for
implementing the adiabatic algorithm on a digital quantum simulator, our
results show that the second order Suzuki-Trotter circuit demonstrates a clear
superiority over the first order by demanding significantly less number of
CNOT gates for delivering the same fidelity. Second, we demonstrate that the
VQE approach demands shallower circuits with significantly less number of two-
qubit entangling gates in comparison with the adiabatic evolution. This makes
the VQE more suitable for the NISQ simulators. However, the simplicity in the
VQE circuit comes with the price of the necessity for a classical optimizer
which may demand a significant amount of iterations to succeed. Third, as our
main results, we have developed two approaches, namely qubit and layer
recursive methods, for accelerating the convergence of the classical
optimizer. Both of these approaches, try to start the optimization procedure
with an initial guess close to the global minimum and thus reduce the total
number of iterations as well as the chance of getting trapped in local
minimums. These strategies, indeed, significantly reduce the COI required for
the convergence of the VQE algorithms.

The protocols for accelerating the optimization are very general and work
perfectly well for a wide range of Hamiltonians. This includes the
Hamiltonians which do not preserve the number of excitations, such as the XYZ,
and the ones without the mirror inversion symmetry, such as the single
impurity Kondo model. In addition, we have provided a detailed analysis for
exploiting symmetries in designing the VQE circuit as well as the destructive
role of decoherence. All the codes can be found in [49].

## Acknowledgements

AB acknowledges the National Key R&D Program of China, Grant No.
2018YFA0306703. VM thanks the Chinese Postdoctoral Science Fund for grant
2018M643435.

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