# The quantum annealing gap and quench dynamics in the exact cover problem

Bernhard Irsigler  Tobias Grass  ICFO-Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain

###### Abstract

Quenching and annealing are extreme opposites in the time evolution of a
quantum system: Annealing explores equilibrium phases of a Hamiltonian with
slowly changing parameters and can be exploited as a tool for solving complex
optimization problems. In contrast, quenches are sudden changes of the
Hamiltonian, producing a non-equilibrium situation. Here, we investigate the
relation between the two cases. Specifically, we show that the minimum of the
annealing gap, which is an important bottleneck of quantum annealing
algorithms, can be revealed from a dynamical quench parameter which describes
the dynamical quantum state after the quench. Combined with statistical tools
including the training of a neural network, the relation between quench and
annealing dynamics can be exploited to reproduce the full functional behavior
of the annealing gap from the quench data. We show that the partial or full
knowledge about the annealing gap which can be gained in this way can be used
to design optimized quantum annealing protocols with a practical time-to-
solution benefit. Our results are obtained from simulating random Ising
Hamiltonians, representing hard-to-solve instances of the exact cover problem.

##  1 Introduction

Quantum annealers [1, 2, 3] are an emerging technology which combine features
from analog quantum simulators [4, 5, 6] and digital, gate-based quantum
computers [7]. Like in quantum simulators, the engineering of a Hamiltonian is
the working scheme in quantum annealers, but these devices feature also
microscopic design opportunities which allows one to choose a Hamiltonian such
that its ground state solves a specific instance of a general computational
problem. For example, any NP-hard optimization problem can be mapped onto an
Ising spin model with tunable couplings [8]. Together with a transverse field
term producing quantum fluctuations, this Ising Hamiltonian is the central
element of existing quantum annealers, specifically the commercial D-wave
device [9], as well as in atomic annealing architectures [10, 11, 12, 13, 14]
which have been proposed. In the absence of decoherence, the quantum annealing
algorithm tries to find the ground state of the model through an adiabatic
dynamical process, i.e. adiabatic state preparation or adiabatic quantum
computing [15]. In the presence of decoherence, the dynamical process to find
the ground state resembles more a conventional thermal cooling process, yet
with a quantum feature which provides control over quantum fluctuations [16,
17]. In both cases, the dynamical evolution encounters configurations in which
the energy gap above the instantaneous ground state becomes exponentially
small [18, 19, 20, 21, 22], and this becomes the bottleneck of the quantum
annealing method. Therefore, different strategies to avoid small energy gaps
have been reported, including inhomogeneous transverse fields [23, 24, 25,
26], non-stoquastic driver Hamiltonians [27, 28, 29, 30, 31], reverse or
biased annealing [32, 33, 34, 35], as well as combinations thereof [36]. There
are also different hybrid algorithms which may avoid the annealing bottleneck
[37, 38, 39, 40].

A simple strategy which can improve the fidelity of the quantum annealing
method is the deceleration of the annealing process around the minimal gap. In
this way, the efficiency of the adiabatic Grover algorithm could be improved
from $T\sim\mathcal{N}$ to $T\sim\sqrt{\mathcal{N}}$ [41], where $T$ is the
annealing time and $\mathcal{N}$ is the Hilbert space dimension of the system.
This remarkable result, however, relies on the analytical accessibility of the
problem. Strikingly, also in the case of quantum annealing far away from the
adiabatic regime, experiments on the D-wave device have shown that pausing the
evolution near the minimal gap is beneficial for the annealing fidelity [42].
This experimental observation has theoretically been explained in Ref. [43],
attributing the improvement of the quantum annealing to thermalization. No
clear time-to-solution improvement from pausing, however, has been found yet
[43]. Nevertheless, the pausing feature suggests that, in both adiabatic and
non-adiabatic scenarios, knowledge of the instantaneous energy gap above the
ground state could help to speed-up the annealing process. In the present
work, we investigate how such knowledge can be obtained through simple quench
experiments, and be refined through machine-learning techniques.

Although adiabatic evolution and quenching are two opposite extremes with a
clear discrepancy in the time-dependence of the field, relations between the
two approaches have been studied [44, 45]. A physical phenomenon which can
occur after a quantum quench is the so-called dynamical quantum phase
transition, which may be manifested from the behavior of an order parameter
after the quench [46, 47]. Specifically, it has been shown for the long-times
dynamics in long-range transverse-field Ising models that the non-zero
magnetization of a system which is initially in the ferromagnetic phase
averages to zero during the evolution after quenching with a sufficiently
strong transverse field [48, 49, 50, 51, 52, 53]. This phenomenon has also
been experimentally observed in a trapped-ion experiment [54].

The present manuscript considers a somewhat similar scenario, but with some
“randomness” in the Ising couplings. In the absence of a transverse field,
this “randomness” makes the system behave rather like a spin glass than a
ferromagnet. In fact, we will define the “random” couplings through randomly
generated (but well characterized) instances of an NP-hard optimization
problem, specifically the exact cover problem which is frequently studied in
the context quantum annealing, see Refs. [15, 21, 35]. Analyzing the dynamical
behavior of different order parameters, related to either the local
magnetization of the system in the glassy ground state or to the paramagnetic
order in a strong transverse field, we first determine a “critical” field
strength in the quench dynamics. Assuming that the dynamical behavior in the
quench dynamics is related to the equilibrium paramagnetic and glassy states,
we establish through numerical simulations that the annealing bottleneck
coincides with the critical field value of the quench. We then show that this
connection between quench and annealing dynamics can be used to design quantum
annealing protocols with a significant fidelity enhancement, as compared to
the standard annealing protocol with a constant ramp speed. Importantly, the
proposed method to improve the annealing result does not rely on any previous
knowledge of the problem instance, but only on information which can be
collected through a few quench experiments.

In a second step, we explore how the connection between quench and annealing
dynamics can be extended beyond the matching at only one critical field
strength. To this end, we have trained a neural network which maps the order
parameter as a function of the quench field to the energy gap during the
annealing process. We note that, in comparison to reinforcement learning
strategies which in the past have been applied to problems from optimal
control [55] and quantum annealing [56], our approach has only a small amount
of training requirements, as our method is based on supervised training of
only a single-layer network. Nevertheless, after this training process, it
becomes possible to fully and almost exactly reproduce from the quench data
the annealing gap as a function of the transverse field strength. With this
information, it then becomes possible to adjust the instantaneous ramp speed
of the annealing protocol. Not surprisingly, this yields a further fidelity
improvement, as compared to our first method which had not required any
network training.

Our paper is organized in three Sections: In Sec. II, we describe the studied
computational problem, and define the annealing gap, as well as the order
parameters in the quench dynamics. We distinguish two types of quenches:
switching the transverse field on or off. In Sec. III, we present the results
which in subsection (A) consists of a relation between quenched order
parameters and annealing gap, and in subsection (B) of an application of this
connection to the quantum annealing protocol. In both subsections, we further
distinguish between (1.) the connection which can immediately be made at the
critical field strength, and (2.) the connection at arbitrary field strengths,
obtained after training a neural network. Finally, in Sec. IV, we summarize
our results, and briefly discuss the technological advances associated with
our results.

##  2 Definitions

###  2.1 Model

We define the annealing Hamiltonian as

$\hat{H}[h(t)]=(1-h(t))\hat{H}_{p}+h(t)\hat{H}_{q},$ (1)

herein, we introduced the problem Hamiltonian $\hat{H}_{p}$, the fluctuation
Hamiltonian $\hat{H}_{q}$, and the time-dependent field $h(t)$ which varies
monotonously increasing from 0 to 1. We focus on the exact cover (EC) problem
to be embedded in the problem Hamiltonian. In the language of spins, the EC
problem is described by a set of clauses $C$ containing $M$ tuples of randomly
selected spins, each point either up or down. Here, we restrict ourselves to
the EC3 problem, i.e., those tuples will be triples. In EC3, a clause is
fulfilled if exactly two of the three spins point up while one of them points
down. Since, different clauses can share the same spins, the problem can
become hard to solve. It is assumed that for $M\approx N$ the problem is hard,
where $N$ is the number of spins in the system. In this regime, many instances
have exactly one solution, dubbed as unique satisfying assignment. We focus
solely on those instances. The classical EC3 problem can be expressed in a
Hamiltonian in the following way:

$\hat{H}_{p}=J\sum_{(i,j,k)\in
C}(\hat{\sigma}^{z}_{i}+\hat{\sigma}^{z}_{j}+\hat{\sigma}^{z}_{k}-1)^{2},$ (2)

where $J$ is the energy scale. The set $C$ contains $M$ random triples, dubbed
clauses, such that

$C=\\{(i,j,k):i\neq j\neq k\in[1,2,..N]\\},$ (3)

where $N$ is the number of spins in the system. If a spin configuration
satisfies all clauses in $C$, its energy associated with Eq. (2) is zero. On
the other hand, every clause which is not fulfilled by a spin configuration
leads to an energy cost of at least $4J$. Therefore, for the instances of $C$
which have unique satisfying assignment, there is one zero-energy state, being
the non-degenerate ground state of $\hat{H}_{p}$.

For the fluctuation Hamiltonian we employ a transverse spin term as

$\hat{H}_{q}=J\sum_{i=1}^{N}\hat{\sigma}^{x}_{i}.$ (4)

Here, we define the fluctuation Hamiltonian using the same energy scale $J$ as
in the problem Hamiltonian, but in the annealing Hamiltonian, as described by
Eq. (1), the dimensionless function $h(t)$ controls the energetic weight of
$\hat{H}_{q}$ and $\hat{H}_{p}$.

Throughout this manuscript, we make use of the quspin package for the exact
diagonalization, quenches and time evolutions [57, 58]. All codes are
available at Ref. [59].

###  2.2 Annealing gap

We define the annealing gap $\Delta(h)$ as the energy gap between the two
lowest energy levels $E_{n}(h)$ of the Hamiltonian in Eq. (1) rescaled by the
bandwidth $W(h)$ for time-independent $h(t)=h$:

$\begin{split}\Delta(h)&=\frac{E_{1}(h)-E_{0}(h)}{W(h)},\\\
W(h)&=E_{2^{N}}(h)-E_{0}(h).\end{split}$ (5)

Here, $E_{0}$ and $E_{1}$ denote the energy of ground state and the first
exited state, respectively.

We define the critical field $h_{c}^{\Delta}=\mathrm{argmin}(\Delta(h))$ at
which the annealing gap is minimal. We compute $h_{c}^{\Delta}$ by determining
numerically the global minimum of the gap of the annealing Hamiltonian in Eq.
(1) for a time-independent field $h$ with $0<h<1$. The result is shown in Fig.
1 as a probability distribution. We observe that the probability distributions
become broader with increasing $N$ as more diverse configurations of clauses,
see Eq. (3), can emerge.

Figure 1: Probability distribution of the critical field $h_{c}^{\Delta}$ for
which the annealing gap is minimal for different system sizes $N$.

###  2.3 Quench dynamics

In the thermodynamic limit, the dynamical quantum phase transition is achieved
through a sudden quench of the field $h(t)$ in the Hamiltonian in Eq. (1), and
the dynamical phase transition is characterized through appropriately chosen
order parameters [46, 47]. We borrow these concepts for our study of finite
systems to define dynamical quench parameters (DQPs), in full analogy to the
order parameters in the thermodynamic limit. Assuming values between 0 and 1,
these DQPs characterize the dynamical properties of different quantum states
within the quench dynamics.

In the following, we differentiate between the instantaneous ground states
$|\psi_{h}\rangle$, which are eigenstates of the time-independent Hamiltonian
in Eq. (1) with $h(t)=h$, and the time-evolved states
$|\psi_{h(t)}(t)\rangle=\hat{U}_{h(t)}(t,0)|\psi_{h_{i}}\rangle$ where $h_{i}$
is the initial field. Here,

$\hat{U}_{h(t)}(t,0)=\mathcal{T}\exp\left[-i\int_{0}^{t}\hat{H}[h(t^{\prime})]dt^{\prime}\right]$
(6)

is the time-evolution operator. Herein, $\mathcal{T}$ is the time-ordering
operator and $\hbar=1$. In the following, we treat two different kinds of
quenches associated with different initial states and different DQPs.

####  2.3.1 “switch on”

This procedure is similar to the one used in Ref. [51] for treating the
homogeneous Ising model, i.e., spin-spin interactions between two spins only
depends on their spatial distance. The initial state is the ground state of
$\hat{H}[h=0]$ which is then quenched to the final field $h_{f}$ with finite
value $0<h_{f}\leq 1$. The associated observable is

$G(h_{f},t)=\frac{1}{N}\sum_{i=1}^{N}\langle\psi_{0}|\sigma^{z}_{i}|\psi_{0}\rangle\langle\psi_{h_{f}}(t)|\hat{\sigma}^{z}_{i}|\psi_{h_{f}}(t)\rangle.$
(7)

We note that for $t=0$, $|\psi_{h_{f}}(t)\rangle=|\psi_{0}\rangle$, and
therefore $G(h_{f},0)=1$. The corresponding DQP is defined as

$G(h_{f})=\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_{0}^{\tau}G(h_{f},t)dt.$
(8)

In practice, of course, the limit of $\tau\rightarrow\infty$ is replaced by a
finite time $\tau$ for which $G(h)$ has converged sufficiently well to its
actual long-time limit. To this end, we have determined $G(h)$ for different
values of $\tau J=10,20,50,100$. As we found the deviations to be much smaller
than the actual value of $G(h)$, we have chosen the value $\tau J=20$ for our
numerical investigations.

####  2.3.2 “switch off”

This procedure is more relevant for quantum annealing since here the
fluctuation field is initially maximal, $h=1$, and is then quenched to $h_{f}$
with $0\leq h_{f}<1$. Also, it does not require any prior knowledge about the
initial ground state $|\psi_{0}\rangle$ of the problem Hamiltonian. For the
“switch off” procedure the associated observable is defined as

$X(h_{f},t)=\frac{1}{N}\sum_{i=1}^{N}\langle\psi_{h_{f}}(t)|\hat{\sigma}^{x}_{i}|\psi_{h_{f}}(t)\rangle$
(9)

As in in the example above, for $t=0$,
$|\psi_{h_{f}}(t)\rangle=|\psi_{1}\rangle$ and therefore $X(h_{f},0)=1$. The
corresponding DQP is defined as

$X(h_{f})=\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_{0}^{\tau}X(h_{f},t)dt.$
(10)

As for $G(h_{f})$, also here, in practice, the limit $\tau\rightarrow\infty$
has to be replaced by a sufficiently large time scale.

In Fig. 2, we present the DQPs as they appear from the numerical quenches.
Therein, $G(h_{f})$ and $X(h_{f})$, defined in Eqs. (8) and (10),
respectively, are plotted for 100 different instances of the problem
Hamiltonian with $N=10$.

Figure 2: DQPs as a function of the quench field $h_{f}$ for 100 different
instances of the problem Hamiltonian in Eq. (2) with $N=10$. Thick lines
correspond to the respective averages.

##  3 Results

We will now report our results of numerical experiments in which, as described
in the previous Section, the EC3 problem Hamiltonian and a transverse magnetic
field term are quenched with respect to each other. Our goal is to use the
data obtained in this way in order to optimize quantum annealing protocols.
Therefore, we proceed in the following two-fold way: First, in Sec. 3.1, we
show that there exists a sizable correlation between the minimal annealing gap
and the DQP. This result establishes that the two very different, dynamical
approaches are indeed connected. It provides the possibility to predict the
annealing gap from the result of quench experiments, either via linear
regression (Sec. 3.1.1) or using a properly trained neural network (Sec.
3.1.2). While the linear regression approach is sufficient to obtain an
estimate of the position $h_{c}^{\Delta}$ of the minimal gap, the neural
network approach is able to reproduce well the full annealing gap $\Delta(h)$
as a function of the field $h$. Second, in Sec. 3.2, we exploit this
predictability of the gap in order to gain a time-to-solution improvement over
the standard annealing procedure. We distinguish between protocols which are
restricted to knowledge about the position of the minimum gap (Sec. 3.2.1),
and protcols which make use of the full gap function (Sec. 3.2.2).

###  3.1 Relation between annealing gap and quench dynamics

Annealing and quenching are at the opposite sides in terms of time dependence,
but they may actually reveal the same information about a quantum state. In
the following, we investigate how the annealing gap can be estimated from the
information drawn from the DQP. To this end, we have simulated the quench
dynamics using exact diagonalization and calculated the DPQs, as described in
the previous Section, and we have calculated the low part of the energy
spectrum of the system as a function of $h$.

####  3.1.1 Critical field from multivariate linear regression

In this Section, we employ a multivariate linear regression in order to
determine the critical field $h^{\Delta}_{c}$ at which the annealing gap
$\Delta(h)$ is minimal from the DQPs. To this end, we have discretized the
quench field $h_{f}$ into 20 points $h_{f}^{n}$, for $n=1,2,...20$, with
$h_{f}^{0}=0$ and $h_{f}^{20}=1$. The values which the DQP assumes at these
points, e.g., $G(h_{f}^{n})$, are then used in the multivariate regression.
This means that we assume the following functional relation:

$\mathbf{h}_{c}^{\Delta}=\beta_{0}\mathbf{1}+\sum_{n=0}^{20}\beta_{n}\mathbf{G}(h_{f}^{n}).$
(11)

Here, the bold symbols denote vectors containing the 100 instances of Fig. 2
as elements. The 21 parameters $\beta_{n}$ determine the linear contribution
of the different $\mathbf{G}(h_{f}^{n})$ to the critical field
$\mathbf{h}_{c}^{\Delta}$. The $100\times 21$ matrix

$A=[\mathbf{1},\mathbf{G}(h_{f}^{1}),\mathbf{G}(h_{f}^{2}),\dots,\mathbf{G}(h_{f}^{20})]$
(12)

is called design matrix. Together with the parameter vector
$\vec{\beta}=(\beta_{0},\beta_{1},\dots,\beta_{20})^{T}$ a least-squares
estimator $||A\vec{\beta}-\mathbf{h}_{c}^{\Delta}||^{2}$ can be employed which
can be minimized with respect to $\vec{\beta}$ by standard routines.

The predictions from the $G(h_{f}^{n})$ are presented in Fig. 3(a) and those
of the $X(h_{f}^{n})$ are presented in Fig. 3(b). The predictions are both
plotted versus the true values of the minimal gap.

Figure 3: Predicted values for the field of the minimal gap $h_{c}^{\Delta}$
through multivariate regression of the (a) $G(h_{f}^{n})$ and (b)
$X(h_{f}^{n})$ estimators, respectively, versus true values of
$h_{c}^{\Delta}$.

In order to estimate the quality of the prediction, we introduce the Pearson’s
$r$ as a measure for the linear correlation between to random variables $x$
and $y$:

$r=\frac{\left\langle(x-\langle x\rangle)(y-\langle
y\rangle)\right\rangle}{\sqrt{\langle(x-\langle
x\rangle)^{2}\rangle}\sqrt{\langle(y-\langle y\rangle)^{2}\rangle}}$ (13)

In particular, we want to quantify how well our predictive methods work by
interpreting $x$ as the true values and $y$ as the predicted values of
$h_{c}^{\Delta}$ in Fig. 3. In Fig. 4, we show the Pearson’s $r$ quantifying
the predictive power of the $G(h_{f}^{n})$ and $X(h_{f}^{n})$ as a function of
the number of spins in the system. The fact that the system size dependence is
found to be non-monotonic makes it difficult to estimate the scaling with $N$.
Although a tendency of decreasing with the system size can be seen, the
correlation between the predicted and the true minimal gap remain strong
($r\gtrsim 0.75$) thoughout the considered range of $N$. We conclude that the
DQP contains information about the quantum state change during the annealing
process.

Figure 4: The Pearson’s $r$, computed from the data presented in Fig. 3 using
Eq. (13), as a function of the system size $N$ for both quench approaches
introduced in Sec. 2.3.1 and 2.3.2, respectively.

####  3.1.2 Full gap function from neural network

In the previous section, we have shown that the value for the critical field
at the minimal gap $h_{c}^{\Delta}$ can be obtained from the DQP as a function
of the quench field $h_{f}$. In the present section, we employ modern machine-
learning techniques [60] to acquire not only the value of $h_{c}^{\Delta}$ but
the full function $\Delta(h)$, see Eq. (5), in the entire regime $0<h<1$.

Compared to conventional algorithms, neural networks work best for a large
amount of data. Therefore, we restrict ourselves to the system size of $N=7$
as well as the data from $X(h_{f})$ only and create 5000 random instances of
the Hamiltonian in Eq. (2). We then compute $\Delta(h)$ as well as $X(h_{f})$,
see Eqs. (5) and (10), respectively. Both functions are evaluated on an
equidistant grid of 128 points. Note that it is not required by the algorithm
that both functions are discretized by the same number of data points.

We make use of the established high-level package Keras [61] in order to
construct a simple neural network with one single dense layer exhibiting a
linear activation function. We use the mean square error as the loss function,
a batch size of 64 and the ADAM optimizer [60]. Out of the full data set of
5000 instances, we use 4000 as the train/validation data set and 1000 as the
test data set. The validation data set is another split of 20% of the 4000
instances.

We show the training of the neural network in Fig. 5. After 600 epochs, we
achieve a loss of $\sim 10^{-6}$. In order to make sure that there is no
overfitting occurring during the training, we plot the loss for the validation
set, i.e., the loss on the data the neural network has not seen before. Since
validation loss behaves exactly like the loss function itself, we can be
certain that there is no overfitting happening. The only free parameter in our
neural network is the learning rate which enters in the optimizer. We have
found through a grid search that $5.5\cdot 10^{-4}$ leads to the lowest loss.

Figure 5: Training of the neural network: evolution of the loss and the
validation loss over the episodes. We plot the validation loss in order to
rule out possible overfitting.

Finally, we plot some instances for the gap function as predicted by the
trained neural network from the test data set in Fig. 6. We compare these
predictions with the true gap functions out of the test set shown as dashed
curves in Fig. 6.

Figure 6: Predictions for the gap of the test data as solid lines and the
respective true gap functions as dashed lines. The different instances are
labeled by A1 to A10. The diagonal of the respective Hamiltonian
$\hat{H}_{p}$, see Eq. (2), is provided in the supplemental material [59].

###  3.2 Time-to-solution improvement for annealing

In this section, we want to draw a technical benefit out of the knowledge of
the annealing gap. Specifically, we design annealing protocols with non-linear
time dependencies which, at a fixed annealing time, lead to a better final
fidelity than the protocol with homogeneous ramp speed. For the construction
of these inhomogeneous protocols, we first restrict ourselves to exploiting
the correlation between minimal gap and the DQP. As shown in the previous
Section, this information is readily obtained from quench experiments. In a
second step, we will further refine the annealing strategy by exploiting the
full knowledge of the annealing gap, which we have shown to be accessible
through a combination of quench data and machine learning.

####  3.2.1 Protocols with slow-down at critical field strength

We perform calculations for the full time evolution in order to simulate the
annealing procedure at finite times. In this way, we compare linear annealing
protocols, i.e., with a constant ramp speed, and annealing protocols with a
varying ramp speed. The linear protocol has the following form

$h_{\text{lin}}(t)=1-\frac{t}{T},$ (14)

where $T$ is the duration of the annealing procedure. For the protocol with
variable ramp speed, we choose a combination of two quadratic functions

$h_{\text{quad}}(t,t_{c})=\begin{cases}1+b_{1}t+c_{1}t^{2},\text{ if
}t<t_{c}\\\ a_{2}+b_{2}t+c_{2}t^{2},\text{ if }t\geq t_{c}.\end{cases}$ (15)

Here, we have introduced a time $t_{c}$ at which the annealing protocol should
slow down. Therefore, we impose the condition of a vanishing velocity at
$t_{c}$, i.e., $\partial_{t}h_{\text{quad}}(t,t_{c})|_{t=t_{c}}=0$. Moreover,
we demand that $h_{\text{quad}}(t,t_{c})$ is a smooth function, and that it
assumes the values $h_{\text{quad}}(t,t_{c})=1$ for $t=0$ and
$h_{\text{quad}}(t,t_{c})=0$ at $t=T$. These conditions define the
coefficients $b_{1},c_{1},a_{2},b_{2},$ and $c_{2}$ in Eq. (15). Finally, with
the knowledge of a field strength $h_{c}$ at which the protocol is supposed to
slow down (ideally the field strength at which the gap becomes smallest), we
are able to associate the time $t_{c}$ with $h_{c}$ via $t_{c}=(1-h_{c})T$. In
this way, the construction of the quadratic protocol incorporates a slow-down
of the annealing speed in the vicinity of $h_{c}$, at fixed duration $T$ of
the annealing procedure. With $t_{c}$ being determined by $h_{c}$, we denote
the quadratic protocol by $h_{\text{quad}}(t,h_{c})$.

To evaluate the quality of the different protocols, we define the fidelity as
the overlap

$F(t)=|\langle\psi_{h}|\psi_{h(t)}(t)\rangle|.$ (16)

Herein, $|\psi_{h}\rangle$ is the instantaneous ground state of the
Hamiltonian $H[h(t)=h]$ defined in Eq. (1). The time-evolved state
$|\psi_{h(t)}(t)\rangle$ is computed through the time evolution in Eq. (6)
including the time variation of the protocol $h(t)$. Of course, the interest
in this quantity is largest at the end of the annealing procedure, $t=T$,
where the squared fidelity corresponds to the success rate of the annealer. As
a functional of the protocol $h(t)$, we define the final fidelity
$F_{T}[h(t)]=F(T)$, and analyze it for different protocols in Figs. 7-12.

In particular, we are interested how the choice of the slow-down field value
$h_{c}$ affects the fidelity. The simplest choice of $h_{c}$ is to set it to a
constant, e.g., $h_{c}=0.5$. This choice will serve as a reference. Of course,
a better choice would be to use the instance-specific field strength
corresponding to the minimal annealing gap as the critical field. We denote
this choice by $h_{c}=h_{c}^{\Delta}$. However, this information is typically
unknown from an experimental point of view. As established in the previous
Section, data from quench experiments allows to make instance-specific guesses
of the region with small gap, and therefore $h_{c}$ can be coupled to the
previously defined order parameters $G$ and $X$ (see Eqs. (8) and 10)). We
define

$\displaystyle h_{c}^{G},\text{ where }G(h_{f}=h_{c}^{G})=0.5$ (17)
$\displaystyle h_{c}^{X},\text{ where }X(h_{f}=h_{c}^{X})=0.5.$ (18)

Figure 7: Fidelity for five different annealing protocols for $N=10$ and
$T=1$, see Eqs. (14) and (15).

In Fig. 7, we show exemplarily the results for the fidelity of four different
quadratic annealing protocols compared to the fidelity of the linear protocol
$h_{\text{lin}}(t)$ defined in Eq. (14). While one quadratic protocol is
chosen independent of the respective instance of the problem Hamiltonian, the
other quadratic protocols depend on an instance-specific field value $h_{c}$
as an input parameter. We observe that these last three protocols yield a
significant improvement with respect to the instance-independent protocol.
Even though all these three protocols perform similarly well, it is curious to
note that choosing the parameter $h_{c}$ to be exactly at the minimal gap
$h_{c}^{\Delta}$ is not the best choice. Instead, the choice of $h_{c}$ such
that $X(h_{c})=0.5$ is seen to outperform all other protocols for almost all
instances. We note that with this choice, $h_{c}$ tends to be shifted towards
slightly smaller values, i.e. the slowest annealing occurs slightly after the
minimal gap has been passed.

We are now interested in the scaling of these protocols with respect to the
number of spins in the system. The average fidelity as a function of the
number of spins is shown in Fig. 8 for the five different protocols.

Figure 8: Scaling of the average fidelity as a function of the number os spins
in the system for $T=1$.

In order to quantify the time-to-solution improvement we compute the annealing
time evolution for different annealing times $T$. We then determine the
isobaric lines of constant average fidelity in order to find the scaling of
$T$ as a function of $N$. Those isobaric lines are shown in Fig. 9(a) for the
best and the worst protocol. We observe that both scale approximately
linearly, $T\approx aN$, at least in the regime $5\leq N\leq 10$, with the
linear coefficient $a$. By linearly fitting the isobaric lines of constant
average fidelity, we determine the respective linear coefficient $a$ shown in
Fig. 9(b). We observe that the linear scaling coefficient $a$ of the linear
annealing protocol and the quadratic protocol exhibit a similar functional
behavior. However, the parameter $a$ of the linear protocol seems to be offset
by a constant, i.e., the linear protocol is always less efficient than the
quadratic protocol.

Figure 9: Scaling of the worst and the best annealing protocol shown through
the isobaric lines of constant average fidelity in (a). Linear fitting
coefficients of the scaling in (b).

####  3.2.2 Protocols utilizing the full gap function

In the previous section, we have observed that the knowledge of a critical
field obtained from, e.g., the DQP, can improve the time-to-solution of the
annealing process. In this section, we will investigate whether we can improve
this even further, assuming we have knowledge about the full gap function. As
shown in Sec. 3.1.2, the combination of quench data and neural network
training might be an accurate source of this information.

Inspired by the adiabatic theorem, we assume that the optimal protocol should
scale with the size of the gap

$\frac{dh_{\mathrm{opt}}(t,\alpha)}{dt}\propto|E_{1}(t)-E_{0}(t)|^{\alpha}.$
(19)

Here, we have introduced a free parameter $\alpha$ which in the most standard
version of the adiabatic theorem [2] would be $\alpha=2$. We note that the
adiabatic theorem contains a matrix element of the Hamiltonian which couples
the two states of interest. As it has been done in Ref. [41], we approximate
this matrix element by the time derivative of the protocol
$h_{\mathrm{opt}}(t,\alpha)$. Furthermore, we note that Eq. (19) considers
only transitions between the energetically lowest two states. In fact, our
systems of interest have $2^{N}$ states, and generally all transitions
depleting the ground state contribute to the loss of annealing fidelity. Thus,
Eq. (19) is certainly to be understood as an approximation. This also means
that there might not be a universal $\alpha$ which is optimal for all the spin
glass instances we investigate.

We now use Eq. (19) in order to find an optimized annealing protocol. In a
discretized form, this means

$\Delta
t_{i}=\frac{|E_{1}(t_{i})-E_{0}(t_{i})|^{-\alpha}}{\sum_{j}|E_{1}(t_{j})-E_{0}(t_{j})|^{-\alpha}}T,$
(20)

where we have introduced the discretized time $t_{i+1}=t_{i}+\Delta t_{i}$.
Also note that this protocol is normalized to the duration of the annealing
process $\sum_{i}\Delta t_{i}=T$. For $\alpha=0$, the protocol becomes the
standard linear protocol as Eq. (14).

Figure 10: Full gap annealing protocols according to Eq. (20) for different
values of $\alpha$ and a single Hamiltonian instance for $N=7$ and $T=1$. The
diagonal of the respective Hamiltonian $\hat{H}_{p}$, see Eq. (2), is provided
in the supplemental material with the label B1 [59].

In Fig. 10, we show annealing protocols $h_{\mathrm{opt}}(t,\alpha)$
constructed according to Eq. (20) for different values of $\alpha$ as a
function of time. We also show the evolution of the fidelity with respect to
time. First, the protocols $h_{\mathrm{opt}}(t,\alpha)$ with finite $\alpha$
improve the fidelity at the end of the annealing, as compared to the linear
protocol $\alpha=0$. Secondly, we observe that the fidelity for larger values
of $\alpha$ is similar at $t=T$. In Fig. 11, we show the fidelity at the end
of the annealing process as a function of $\alpha$ for a single instance. We
observe that the maximum is achieved for $\alpha=3$. As we have mentioned
before, and as our discussion in the next paragraph will make more explicit,
this value is not universal and differs from instance to instance. We also
include in Fig. 11 the values for the fidelity of the quadratic protocols
investigated in the section before. We conclude that knowledge of the full gap
function can indeed lead to a further improvement of the annealing process.

Figure 11: Final fidelity as a function of $\alpha$ for a single instance. The
diagonal of the respective Hamiltonian $\hat{H}_{p}$, see Eq. (2), is provided
in the supplemental material with the label B1 [59].

The non-monotonic behavior of the annealing protocol
$h_{\mathrm{opt}}(t,\alpha)$ in Fig. 11 leads to the focus on the maximal
reachable fidelity. We define the value $\alpha_{\text{max}}$ at which the
fidelity for the protocol $h_{\mathrm{opt}}(t,\alpha)$ is maximal. To answer
the question whether the maximum fidelity, i.e. the complexity of a certain
instance, is correlated to $\alpha_{\text{max}}$, we consult Fig. 12(a). By
plotting the data of 100 instances, we see no tendency to a correlation
between $\alpha_{\text{max}}$ and the respective maximum fidelity
$F_{T}[h_{\mathrm{opt}}(t,\alpha)]$. Moreover, we deploy a histogram for the
values of $\alpha_{\text{max}}$ shown in Fig. 12(b). We fit a Poisson
distribution $P_{\lambda}(\alpha_{\text{max}})$ to the data in the range
$0\leq\alpha_{\text{max}}\leq 10$ leading to a parameter $\lambda\approx 4.6$.
In the Poisson distribution, this parameter at the same time represents the
mean value as well as the variance. Remarkably, a mean value of 4.6 is much
larger than the commonly encountered value of 2 in adiabatic theorems [2].

Figure 12: Statistics of $\alpha_{\text{max}}$ for which the fidelity is
maximal: (a) fidelity at $\alpha_{\text{max}}$ versus $\alpha_{\text{max}}$,
(b) histogram of $\alpha_{\text{max}}$. Values larger than 10 are grouped and
highlighted. The orange curve is a Poisson distribution fitted to the values
$\alpha_{\text{max}}\leq 10$.

##  4 Summary and Outlook

We have investigated the annealing and quench dynamics of a spin model
incorporating the exact cover problem. While the quench dynamics exhibits a
jump of the dynamical quench parameters (DQP), the annealing dynamics
undergoes an equilibrium transition between a paramagnetic and a glassy state.
We have shown that the DQP can be used to determine the minimal annealing gap
of the system through multivariate regression, see Figs. 3 and 4. Moreover, by
means of a one-layer neural network, it even becomes possible to fully predict
the gap function of a given instance, based on the field dependence of the
DQP, see Fig. 6.

These findings enable the optimized design of quantum adiabatic protocols with
significantly enhanced performance. In the present article, we have presented
a practical advantage by feeding annealing protocols with the information of
the DQP, see Figs. 7 and 11. At least if one disregards the time needed to
perform the quench experiments, our procedure leads to a time-to-solution
benefit in annealing algorithms which has not been achieved yet, e.g., with
pausing [42, 43].

Our study suggests that the annealing gap and the DQP are related in spin
glass systems. This enables to draw instance-specific information about the
problem Hamiltonian and its energy spectrum in a transverse field from simple
quench experiments. Specifically, the magnetization after a quench which is
readily measured can reveal the energy gap of the system in a transverse
field. Moreover, it would be interesting to find a relation between the
dynamical quantum phase transition and the annealing gap in the thermodynamic
limit. This, however, requires a proper scaling of the system size which is
not trivial to perform for random Hamiltonians. In this context, we have also
given an example how modern machine learning algorithms can be used to support
quantum annealing protocols and thus enhance their efficiency.

This new insight into the annealing gap is expected to trigger further
research on optimal annealing protocols and the adiabatic theorem. We already
have explicitly demonstrated that partial or full knowledge of the annealing
gap can be used to improve the protocol. Interestingly, as our simulation of
the protocol defined in Eq. (20) has shown, simple adiabatic theorems are not
sufficient to determine the best protocol, that is, the protocol which for a
given annealing time would maximize the final fidelity.

The authors are grateful for enlightening discussions with Gabriel Fernández
Fernández, Alexandre Dauphin, and Maciej Lewenstein. The authors acknowledge
funding from “la Caixa” Foundation (ID 100010434, fellowship code
LCF/BQ/PI19/11690013); ERC AdG NOQIA; Agencia Estatal de Investigación (R$\&$D
project CEX2019-000910-S, funded by MCIN/ AEI/10.13039/501100011033, Plan
National FIDEUA PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2,
Proyectos de I+D+I “Retos Colaboración” RTC2019-007196-7); Fundació Cellex;
Fundació Mir-Puig; Generalitat de Catalunya through the CERCA program, AGAUR
Grant No. 2017 SGR 134, QuantumCAT U16-011424, co-funded by ERDF Operational
Program of Catalonia 2014-2020; EU Horizon 2020 FET-OPEN OPTOLogic (Grant No
899794); National Science Centre, Poland (Symfonia Grant No.
2016/20/W/ST4/00314).

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