# Quantum contextual bandits and
recommender systems for quantum data

Shrigyan Brahmachari1, Josep Lumbreras1, Marco Tomamichel1,2 1Centre for
Quantum Technologies, National University of Singapore, Singapore  2Department
of Electrical and Computer Engineering, Faculty of Engineering, National
University of Singapore, Singapore

###### Abstract

We study a recommender system for quantum data using the linear contextual
bandit framework. In each round, a learner receives an observable (the
context) and has to recommend from a finite set of unknown quantum states (the
actions) which one to measure. The learner has the goal of maximizing the
reward in each round, that is the outcome of the measurement on the unknown
state. Using this model we formulate the low energy quantum state
recommendation problem where the context is a Hamiltonian and the goal is to
recommend the state with the lowest energy. For this task, we study two
families of contexts: the Ising model and a generalized cluster model. We
observe that if we interpret the actions as different phases of the models
then the recommendation is done by classifying the correct phase of the given
Hamiltonian and the strategy can be interpreted as an online quantum phase
classifier.

##  I Introduction

Recommender systems are a class of online reinforcement learning algorithms
that interact sequentially with an environment suggesting relevant items to a
user. During the last decade, there has been an increasing interest in online
recommendation techniques due to the importance of advertisement
recommendation for e-commerce websites or the rise of movies and music
streaming platforms [1, 2]. Among different settings for recommender systems,
in this work, we focus on the contextual bandit framework applied to the
recommendation of quantum data. The contextual bandit problem is a variant of
the multi-armed bandit problem where a learner at each round receives a
context and given a set of actions (also called actions) has to decide the
best action using the context information. After selecting an action the
learner will receive a reward and for the next rounds, they will use the
previous information of contexts and rewards in order to make their future
choices. As in the classical multi-armed bandit problem, the learner has to
find a balance between exploration and exploitation; exploration refers to
trying different actions in order to eventually learn the ones with the
highest reward and exploitation refers to selecting the actions that
apparently will give the highest reward immediately. For a comprehensive
review of bandit algorithms, we refer to the book by Lattimore and Szepesvári
[3]. Some real-life applications [4] of bandit include clinical trials [5],
dynamic pricing [6], advertisement recommendation [7] or online recommender
systems [8, 9]. As an example, in [8] a news article recommender system was
considered where the context is the user features, the actions are the
articles to recommend and the reward is modeled as a binary outcome indicating
that the user clicks or not on the recommended article.

Quantum algorithms for the classical multi-armed bandit problem have been
studied for the settings of best-arm identification [10, 11], exploration-
exploitation with stochastic environments [12] (uncorrelated and linear
correlated actions) and adversarial environments [13]. Also, a quantum neural
network approach was considered in [14] for a simple best-arm identification
problem. A quantum algorithm for a classical recommender system was considered
in [15] claiming an exponential speedup over known classical algorithms but
later in [16] it was proven that the price of the speedup comes from the
assumptions of the quantum state preparation part and argued that under
related classical assumptions a classical algorithm can also achieve the
speedup. There are other more general reinforcement learning frameworks beyond
bandits where actions affect the rewards in the long term such as Markov
decision process. The quantum generalization of this framework has been
considered in [17, 18], and although our model of study falls into their class
we can derive more concrete results since we study a specific setting.

We are interested in studying a recommender system for quantum data that is
modeled by a set of unknown quantum processes- which is called the
environment, and a set of tasks to perform using these quantum processes-
which is called a context set. A learner interacts sequentially with the
environment receiving at each round a task from the context set and then
choosing the best quantum process to perform this task. For example, we could
model the environments as a set of noisy quantum computers, the context set as
a set of different quantum algorithms, and then at each round, the learner is
given a quantum algorithm to run and their goal is to recommend the best
quantum computer to do this task. We note that this model exemplifies the
bandit exploration-exploitation trade-off since the learner has to try
(explore) the different quantum computers in order to decide the best one but
at the same time has to choose the best one (exploitation) to perform the
task. This trade-off is interesting in a practical scenario because it
captures settings where online decisions are important and or they have some
associated cost that makes the learner always try to perform optimally. In our
example, one could think that using a quantum computer costs money for the
learner, so at each stage, they always want to select the ones that will
output the best solutions.

Figure 1: Sketch of a recommender system for quantum data. The learner
receives sequentially quantum contexts and feed them to the classical
processing system. The context is also fed to the measurement system. The
classical processing system uses the information about the context to pick one
of the quantum processes (no information regarding these processes are known
besides from measurements). The chosen quantum process is applied to the
measurement system, and the measurement outcome is fed to the classical
processing and is added to the cumulative reward.

In our work, we extend the setting considered in [19] where they studied the
exploration-exploration trade-off of learning properties of quantum states. In
our model, the environment is a set of unknown quantum states, the context set
is a (finite or infinite) set of observables and at each round the learner
receives an observable and has to perform a measurement on one of the unknown
quantum states (the recommendation) aiming to maximize its outcome. We define
this problem as the quantum contextual bandit (QCB) and we note that it falls
into the class of linear contextual bandits [20, 21, 22]. The QCB is the basic
framework where we formulate our recommender system for quantum data. We use
as a figure of merit the regret, which is the cumulative sum of the difference
between the expected outcome of the best and selected action at each round.
Finding a strategy that minimizes regret implies finding the mentioned balance
between exploration-exploitation of the different actions. As a concrete
recommendation task captured by the QCB model, we consider the low energy
quantum state recommendation problem. In this problem, at each round, the
learner receives a quantum Hamiltonian and has to recommend from the
environment the state with the lowest energy. The ground state preparation
problem is an important ingredient of NISQ algorithms [23] and our model could
be useful in order to implement an online recommendation algorithm that helps
the learner choose the best ansatz for their energy minimization task when
they have multiple problems to solve. One of the advantages of using bandit
algorithms for this task is that they do not need to reconstruct the whole
$d$-dimensional state, just the relevant part for the recommendation which
depends on the structure of the context set. In order to do that we combine a
Grahm-Schmidt procedure with classical linear bandit strategies. This allows
our algorithm to store low-dimensional approximations of the unknown quantum
states without prior knowledge of the context set. We also perform some
numerical studies of the scaling of the regret for the cases where the context
set is an Ising model and a generalized cluster model studied in [24]. For
these models, we propose unknown actions for the algorithm corresponding to
ground states located at different phases, and then for each context received
by the algorithm we associate each action to a different phase and we
reproduce a ground state phase diagram. We observe that the recommendation of
the algorithm is done approximately by classifying the different phases of the
studied models and we are able to clearly distinguish them in the phase
diagram.

The rest of the paper is organized as follows: in Section II, we establish the
mathematical model for the quantum contextual bandit, and then define the
notation used throughout the paper; in the next section, Section III we prove
the lower bound on a performance metric (expected regret, which we define in
Section II) over all possible algorithms. In Section IV we review the linear
Upper Confidence Bound algorithm. In Section V we describe the low-energy
recommendation system and adapt the LinUCB algorithm to this setting. We
illustrate the efficiency of the algorithm through simulations of different
context sets.

##  II The model

First, we introduce some notation in order to define our model and present our
results. We define $[T]=\\{1,...,T\\}$ for $T\in\mathbb{N}$. Let
$\mathcal{S}_{d}=\\{\rho\in\mathbb{\mathbb{C}}^{d\times d}:\rho\geq
0\wedge\operatorname{Tr}(\rho)=1\\}$ denote the set of positive semi-definite
operators with unit trace, i.e _quantum states_ that act on a $d$-dimensional
Hilbert space $\mathbb{C}^{d}$. Moreover, _observables_ are Hermitian
operators acting on $\mathbb{C}^{d}$, collected in the set
$\mathcal{O}_{d}=\\{O\in\mathbb{C}^{d\times d}:O^{\dagger}=O\\}$. We denote
real $d$-dimensional column vectors as $\mathbf{v}$ and the inner product of
two of them $\mathbf{u},\mathbf{v}\in\mathbb{R}^{d}$ as
$\mathbf{u}^{\top}\mathbf{v}$ where $\mathbf{u}^{\top}$ denotes the transpose
of $\mathbf{u}$ that is a row vector. We use $\|\cdot\|_{2}$ in order to
denote the 2-norm of a real vector. For a $n$-qubit system with Hilbert space
dimension $d=2^{n}$, we denote $X_{i},Y_{i}$, and $Z_{i}$ the $x,y,z$ Pauli
operators acting on the $i$-th qubit ($1\leq i\leq n$). A Pauli observable can
be expressed as the $n$-fold tensor product of the $2\times 2$ Pauli matrices,
i.e it is an element of the set $\left\\{I,X,Y,Z\right\\}^{\otimes
n}/I_{4^{n}\times 4^{n}}$. Note that there are $4^{n}-1$ such observables and
each of them are orthogonal, and therefore form a basis, which alludes to as
the Pauli basis henceforth.

The definition of our model takes some of the conventions used for the multi-
armed quantum bandit (MAQB) problem [19].

######  Definition 1 (Quantum contextual bandit).

Let $d\in\mathbb{N}$. A $d$-dimensional quantum contextual bandit is given by
a set of observables
$\mathcal{C}=\\{O_{c}\\}_{c\in\Omega_{\mathcal{C}}}\subseteq\mathcal{O}_{d}$
that we call the context set, $(\Omega_{\mathcal{C}},\Sigma_{\mathcal{C}})$ is
a measurable space and $\Sigma_{\mathcal{C}}$ is a $\sigma$-algebra of subsets
of $\Omega_{\mathcal{C}}$. The bandit is in an environment, a finite set of
quantum states
$\gamma=\\{\rho_{1},\rho_{2},\cdots,\rho_{k}\\}\subset\mathcal{S}_{d}$, that
it is unknown. The quantum contextual bandit problem is characterized by the
tuple $(\mathcal{C},\gamma)$.

Given the environment $\gamma$ such that $|\gamma|=k$ we define the action set
$\mathcal{A}=\\{1,...,k\\}$ as the set of indices that label the quantum
states $\rho_{i}\in\gamma$ in the environment. For every observable
$O_{c}\in\mathcal{C}$ the spectral decomposition is given by

$\displaystyle O_{c}=\sum_{i=1}^{d_{c}}\lambda_{c,i}\Pi_{c,i},$ (1)

where $\lambda_{c,i}\in\mathbb{R}$ denote the $d_{c}\leq d$ distinct
eigenvalues of $O_{c}$ and $\Pi_{c,i}$ are the orthogonal projectors on the
respective eigenspaces. For each action $a\in\mathcal{A}$ we define the reward
distribution with outcome $R\in\mathbb{R}$ as the conditional probability
distribution associated of performing a measurement using $O_{c}$ on
$\rho_{a}$ given by Born’s rule

$\displaystyle
Pr\left[R=r|A=a,O=O_{c}\right]=P_{\rho_{a}}(r|a,c)=\begin{cases}\operatorname{Tr}(\rho_{a}\Pi_{c,i})\text{
if }r=\lambda_{c,i},\\\ 0\text{ else. }\end{cases}$ (2)

With the above definitions, we can explain the learning process. The learner
interacts sequentially with the QCB over $T$ rounds such that for every round
$t\in[T]$:

  1. 1.

The learner receives a context $O_{c_{t}}\in\mathcal{C}$ from some (possibly
unknown) probability measure
$P_{\mathcal{C}}:\Sigma\rightarrow[0,1]$ over the set
$\Omega_{\mathcal{\mathcal{C}}}$.

  2. 2.

Using the previous information of received contexts, actions played, and
observed rewards the learner chooses an action $A_{t}\in\mathcal{A}$.

  3. 3.

The learner uses the context $O_{c_{t}}$ and performs a measurement on the
unknown quantum state $\rho_{A_{t}}$ and receives a reward $R_{t}$ sampled
according to the probability distribution (2).

We use the index $c_{t}\in[m]$ to denote the observable $O_{c_{t}}$ received
at round $t\in[T]$. The strategy of the learner is given by a set of
(conditional) probability distributions $\pi=\\{\pi_{t}\\}_{t\in\mathbb{N}}$
(policy) on the action index set $[k]$ of the form

$\displaystyle\pi_{t}(a_{t}|a_{1},r_{1},c_{1},...,a_{t-1},r_{t-1},c_{t-1},c_{t}),$
(3)

defined for all valid combinations of actions, rewards, and contexts
$(a_{1},r_{1},c_{1},...,a_{t-1},r_{t-1},c_{t-1})$ up to time $t-1$. Then, if
we run the policy $\pi$ on the environment $\gamma$ over $T\in\mathbb{N}$
rounds, we can define a joint probability distribution over the set of
actions, rewards, and contexts as

$\displaystyle
P_{\gamma,\mathcal{C},\pi}(a_{1},X_{1},C_{1},...,a_{T},X_{T},C_{T})=\int_{C_{T}}\int_{X_{T}}\cdots\int_{C_{1}}\int_{X_{1}}\prod_{t=1}^{T}\pi_{t}(a_{t}|a_{1},r_{1},c_{1},...,a_{t-1},r_{t-1},c_{t-1})\times$
$\displaystyle\times
P_{\mathcal{C}}(dc_{1})P_{\rho_{a_{1}}}(dr_{1}|a_{1},c_{1})\cdots
P_{\mathcal{C}}(dc_{T})P_{\rho_{a_{T}}}(dr_{T}|a_{T},c_{T}).$ (4)

Thus, the conditioned expected value of reward $R_{t}$ is given by

$\displaystyle\operatorname{\mathbb{E}}_{\gamma,\mathcal{C},\pi}[R_{t}|A_{t}=a,O_{c_{t}}=O_{c}]=\operatorname{Tr}(\rho_{a}O_{c}),$
(5)

where $\operatorname{\mathbb{E}}_{\gamma,\mathcal{C},\pi}$ denotes the
expectation value over the probability distribution (II). The goal of the
learner is to maximize its expected cumulative reward
$\sum_{t=1}^{T}\operatorname{\mathbb{E}}_{\gamma,\mathcal{C},\pi}\left[R_{t}\right]$
or equivalently minimizing the cumulative expected regret

$\displaystyle\text{Regret}_{T}^{\mathcal{\gamma,C},\pi}=\sum_{t=1}^{T}\operatorname{\mathbb{E}}_{\gamma,\mathcal{C},\pi}\left[\max_{\rho_{i}\in\gamma}\operatorname{Tr}(\rho_{i}O_{c_{t}})-R_{t}\right].$
(6)

For a given action $a\in\mathcal{A}$ and context $O_{c}\in\mathcal{C}$ the
sub-optimality gap is defined as

$\displaystyle\Delta_{a,O_{c}}=\max_{i\in\mathcal{A}}\operatorname{Tr}(\rho_{i}O_{c})-\operatorname{Tr}(\rho_{a}O_{c}).$
(7)

Note that the learner could try to learn the distribution of contexts
$P_{\mathcal{C}}$, however, this will not make a difference in minimizing the
regret. The strategy of the learner has to be able to learn the relevant part
of the unknown states $\\{\rho_{a}\\}_{a=1}^{k}$ that depend on the context
set and at the same time balance the tradeoff between exploration and
exploitation. We note that it is straightforward to generalize the above
setting to continuous sets of contexts $\mathcal{C}$. In order to do that we
need a well-defined probability distribution $P_{\mathcal{C}}(O)dO$ over the
context set $\mathcal{C}$.

##  III Lower bound

In this section, we derive a lower bound for the cumulative expected regret by
finding a QCB that is hard to learn for any strategy. Our regret lower bound
proof for the QCB model relies on a reduction to a classical multi-armed
stochastic bandit given in Theorem 5.1 in [25]. Now we briefly review the
multi-armed stochastic bandit problem.

The multi-armed stochastic bandit problem is defined by a discrete set of
probability distributions $\nu=(P_{a}:a\in[k])$ that is called the environment
and $\mu_{i}$ is the mean of the probability distribution $P_{i}$ for
$i\in[k]$. The learner interacts sequentially with the bandit selecting at
each round $t\in[T]$ an action $a\in[k]$ and sampling a reward $R_{t}$
distributed accordingly to $P_{a}$. The expected cumulative regret is defined
as

$\displaystyle\text{Regret}_{T}^{\nu,\pi}=\sum_{t=1}^{T}\max_{a\in[k]}\mu_{a}-\operatorname{\mathbb{E}}_{\nu,\pi}[R_{t}],$
(8)

where $\pi$ and $\operatorname{\mathbb{E}}_{\nu,\pi}$ are both defined
analogously from the definitions of the previous section accordingly to this
model. It is important to remark that in this setting the actions are
independent, meaning that when the learner samples from one action then it
cannot use this information to learn about other actions.

Using the above model we describe the multi-armed stochastic bandit studied in
Theorem 5.1 in [25] The bandit is constructed defining an environment
$\nu=(P_{a}:a\in[k])$ for $k\geq 2$ such that $P_{a}$ are Bernoulli
distributions for all $a\in[k]$ with outcomes $\\{l_{1},l_{2}\\}$. Then we set
the distributions as follows: we choose an index $i\in[k]$ uniformly at random
and assign $P_{i}(R=l_{1})=\frac{1+\Delta}{2}$ for some $\Delta>0$ and
$P_{a}(R=l_{1})=\frac{1}{2}$ for $a\neq i$. Thus, there is a unique best
action corresponding to $a=i$. Then choosing
$\Delta=\small\epsilon\smash{\sqrt{\frac{k}{n}}}\normalsize$ for some small
positive constant $\epsilon$, for $n\geq k$ the expected regret for any
strategy will scale as

$\displaystyle\text{Regret}_{T}^{\nu,\pi}=\Omega(\sqrt{kT}).$ (9)

######  Theorem 2.

Consider a quantum contextual bandit with underlying dimension $d=2^{n}$ and
$n\in\mathbb{N}$, context size $c\geq 1$ and $k\geq 2$ actions. Then, for any
strategy $\pi$, there exists a context set $\mathcal{C}$, $|\mathcal{C}|=c$, a
probability distribution over the context set $\mathcal{C}$ $P_{\mathcal{C}}$
and an environment $\gamma\in\mathcal{S}_{d}$ such that for the QCB defined by
$(\mathcal{C},\gamma)$ the expected cumulative regret will scale as

$\displaystyle\textup{Regret}_{T}^{\gamma,\mathcal{C},\pi}=\Omega\left(\sqrt{kT}\cdot\min\left\\{d,\sqrt{c}\right\\}\right),$
(10)

for $T\geq k\min\\{c,d^{2}\\}$.

###### Proof.

We use a similar technique to [20, 22] in order to analyze the regret by
dividing the problem into subsets of independent rounds. We start dividing the
$T$ rounds in $c^{\prime}=\min\\{c,d^{2}-1\\}$ groups of
$T^{\prime}=\lfloor\frac{T}{c^{\prime}}\rfloor$ elements. We say that time
step $t$ belongs to group $s$ if $\lfloor\frac{t}{T^{\prime}}\rfloor=s$. We
construct a context set $\mathcal{C}$ by picking a set of $c^{\prime}$
distinct Pauli observables (which is possible since the maximum number of
independent Pauli observables is $d^{2}-1\geq c^{\prime}$), so
$\mathcal{C}=\\{\sigma_{i}\\}_{i=1}^{c^{\prime}}$. Recall that a Pauli
observable is a $n$-fold tensor product of the 2 × 2 Pauli matrices, thus the
reward will be a binary outcome $r_{t}\in\\{-1,1\\}$. Then, the context
distribution works as follows: at each group $s$ of rounds the learner will
receive a different context $\sigma_{s}\in\mathcal{C}$, so at group $s$ the
learner only receives $\sigma_{s}$.

We want to build an environment such that for each group of rounds $s\in[m]$
all probability distributions are uniform except one that is slightly
perturbed. We associate each Pauli observable $\sigma_{i}$ to one unique
action $a\in[k]$, and we do this association uniformly at random (each action
can be associated with more than 1 Pauli observable). Then each action
$a\in[k]$ will have $\\{{\sigma_{a,1},...,\sigma_{a,n_{a}}\\}}$ associated
Paulis observables and we can construct the following environment
$\gamma=\\{\rho_{a}\\}_{a=1}^{k}$ where

$\rho_{a}=\frac{I}{d}+\sum_{j=1}^{n_{a}}\frac{\Delta}{d}\sigma_{a,j},$ (11)

$n_{a}\in\left\\{0,1,...,d^{2}-1\right\\}$, $\sum_{a=1}^{k}n_{a}=c^{\prime}$
and $\Delta$ is some positive constant. For every group $s\in[m]$ the learner
will receive a fixed context $\sigma_{s}\in\mathcal{C}$ and there will be a
unique action $a^{\prime}$ with
$P_{\rho_{a}^{\prime}}(1|A_{t}=a^{\prime},s)=\frac{1}{2}+\frac{\Delta}{2}$
(probability of obtaining $+1$) and the rest $a\neq a^{\prime}$ will have
$P_{\rho_{a}}(1|a,s)=\frac{1}{2}$ (uniform distributions). Thus, using that
the contexts are independent ($\operatorname{Tr}(\sigma_{i}\sigma_{j})$ for
$i\neq j$) we can apply (9) independently to every group $s$ and we obtain a
regret lower bound
$\Omega(\sqrt{T^{\prime}k})=\Omega(\smash{\sqrt{\frac{Tk}{c^{\prime}}}})$.
Note that in order to apply (9) we need $T^{\prime}\geq k$ or equivalently
$T\geq c^{\prime}k$. Thus, summing all the $m$ groups we obtain the total
regret scales as,

$\displaystyle\text{Regret}_{T}^{\gamma,\mathcal{C},\pi}=\Omega\left(c^{\prime}\sqrt{\frac{Tk}{c^{\prime}}}\right)=\Omega\left(\sqrt{kT}\cdot\min\left\\{d,\sqrt{c}\right\\}\right).$
(12)

∎

##  IV Algorithm

In this section, we review the linear model of multi-armed stochastic bandits
and one of the main classical strategies that can be used to minimize regret
in this model and also in the QCB model.

###  IV.1 Linear disjoint single context bandits and QCB

The classical setting that matches our problem is commonly referred to as
linear contextual bandits [22] although it has received other names depending
on the specific setting such as linear disjoint model [8] or associative
bandits [21]. The setting that we are interested uses discrete action sets and
optimal algorithms are based on upper confidence bounds (UCB). While these
algorithms use the ”principle of optimism in the face of uncertainty” there
are other approaches like a Thompson sampling [26] algorithm but they are not
optimal for discrete action sets. We use the contextual linear disjoint bandit
model from [8] where each action $a\in[k]$ has an associated unknown parameter
$\theta_{a}\in\mathbb{R}^{d}$ and at each round $t$ the learner receives a
context vector $\mathbf{c}_{t,a}\in\mathbb{R}^{d}$ for each actions. Then
after selecting an action $a\in[k]$ the sampled reward is

$\displaystyle R_{t}=\boldsymbol{\theta}^{\top}_{a}\mathbf{c}_{t,a}+\eta_{t},$
(13)

where $\eta_{t}$ is some bounded subgaussian 111A random variable $X$ is
called $\sigma$-subgaussian if for al $\mu\in\mathbb{R}$ we have
$\operatorname{\mathbb{E}}\left[\mu
X\right]\leq\exp\left(\sigma^{2}\mu^{2}/2\right)$ noise such that
$\operatorname{\mathbb{E}}[R_{t}|A_{t}=a]=\boldsymbol{\theta}_{a}\cdot\mathbb{c}_{t,a}$.

In order to map the above setting to the $d$-dimensional QCB model
$(\gamma,\mathcal{C})$ it suffices to consider a vector parametrization
(similarly done for the MAQB [19]). We choose a set
$\\{\sigma_{i}\\}_{i=1}^{d^{2}}$ of independent Hermitian matrices and
parametrize any $\rho_{a}\in\gamma$ and $O_{l}\in\mathcal{C}$ as

$\displaystyle\rho_{a}=\sum_{i=1}^{d^{2}}\theta_{a,i}\sigma_{i},\quad
O_{l}=\sum_{i=1}^{d^{2}}c_{l,i}\sigma_{i},$ (14)

where $\theta_{a,i}=\operatorname{Tr}(\rho_{a}\sigma_{i})$ and
$c_{l,i}=\operatorname{Tr}(O_{c}\sigma_{i})$ and we define the vectors
$\boldsymbol{\theta}_{a}=(\theta_{a,i})_{i=1}^{d^{2}}\in\mathbb{R}^{d^{2}}$
and $\mathbf{c}_{l}=(c_{a,i})_{i=1}^{d^{2}}\in\mathbb{R}^{d^{2}}$. Then we
note that for the QCB model the rewards will be given by (13) with the
restriction that since we only receive one observable at each round then the
context vector is constant among all actions. Thus, in our model, the rewards
have the following expression

$\displaystyle R_{t}=\boldsymbol{\theta}_{a}^{\top}\mathbb{c}_{t}+\eta_{t}.$
(15)

We denote this classical model as linear disjoint single context bandits. In
order to make clear when the classical real vectors parametrize an action
$\rho_{a}\in\gamma$ or context $O_{l}\in\mathcal{C}$ (14) we will use the
notation $\boldsymbol{\theta}_{\rho_{a}}$ and $\mathbf{c}_{O_{l}}$ respect to
the standard Pauli basis.

###  IV.2 Linear Upper Confidence Bound algorithm

Now we discuss the main strategy for the linear disjoint single context model
(15) that is the LinUCB (linear upper confidence bound) algorithm [21, 22, 28,
29, 30]. We describe the procedure of LinUCB for selecting an action and we
leave for the next section a complete description of the algorithm for the QCB
setting.

At each time step $t$, given the previous rewards
$R_{1},...,R_{t-1}\in\mathbb{R}$, selected actions $a_{1},...,a_{t-1}\in[k]$
and observed contexts $\mathbf{c}_{1},...,\mathbf{c}_{t}\in\mathbb{R}^{d}$ the
LinUCB algorithm builds the regularized least squares estimator for each
unknown parameter $\boldsymbol{\theta}_{a}$ that have the following expression

$\displaystyle\tilde{\boldsymbol{\theta}}_{t,a}=V_{t,a}^{-1}\sum_{s=1}^{t-1}R_{s}\textbf{c}_{s}\mathbb{I}\\{a_{t}=a\\},$
(16)

where
$V_{t,a}=I+\sum_{s=1}^{t-1}\textbf{c}_{s}\textbf{c}^{\top}_{s}\mathbb{I}\\{a_{t}=a\\}$.
Then LinUCB selects the following action according to

$\displaystyle
a_{t+1}=\operatorname*{\arg\\!\max}_{a\in[k]}\tilde{\boldsymbol{\theta}}^{\top}_{t,a}\mathbf{c}_{t}+\alpha\sqrt{\mathbf{c}^{\top}_{t}{V^{-1}_{t}}\mathbf{c}_{t}},$
(17)

where $\alpha>0$ is a constant that controls the width of the confidence
region on the direction of $\textbf{c}_{t,a}$. The idea behind this selection
is to use an overestimate of the unknown expected value using an upper
confidence bound. This is the principle behind UCB [31] which is the main
algorithm that gives rise to this class of optimistic strategies. The value of
the constant $\alpha$ is chosen depending on the structure of the action set.
In the next section, we will discuss the appropriate choice of $\alpha$ for
our setting. We note that this strategy can also be applied in an adversarial
approach where the context is chosen by an adversary instead of sampled from
some probability distribution.

The above procedure is shown to be sufficient for practical applications [8]
but the algorithms achieving the optimal regret bound are SupLinRel [21] and
BaseLinUCB [22]. They use a phase elimination technique that consists of each
round playing only with actions that are highly rewarding but still the main
subroutine for selecting the actions is LinUCB. This technique is not the most
practical for applications but it was introduced in order to derive rigorous
regret upper bounds. For these strategies if we apply it to a $d$-dimensional
QCB bandit $(\gamma,\mathcal{C})$ they achieve the almost optimal regret bound
of

$\displaystyle\text{Regret}_{T}^{\gamma,\mathcal{C},\pi}=O\left(d\sqrt{kT\ln^{3}(T^{2}\log(T))}\right).$
(18)

The above bound comes from [22] and it is adapted to our setting 222Our model
uses a different unknown parameter $\boldsymbol{\theta}_{a}\in\mathbb{R}^{d}$
for each action $a\in[k]$. This model can be easily adapted to settings where
they assume only one unknown parameter shared by all actions if we enlarge the
vector space and define
$\boldsymbol{\theta}=(\theta_{1},...,\theta_{k})\in\mathbb{R}^{dk}$ as the
unknown parameter. using the vector parametrization (14). Their regret
analysis works under the normalization assumptions
$\|\boldsymbol{\theta}\|_{2}\leq 1$, $\|\mathbf{c}_{t}\|_{2}\leq 1$ and the
choice of $\alpha=\sqrt{\frac{1}{2}\ln(2T^{2}k)}$. We note that it matches our
lower bound (10) except for the logarithmic terms.

##  V Low energy quantum state recommender system

In this section, we describe how the QCB framework can be adapted for a
recommender system for low-energy quantum states. We consider a setting where
the learner is given optimization problems in an online fashion and is able to
encode these problems into Hamiltonians and also has access to a set of
unknown preparations of (mixed) quantum states that they want to use in order
to solve these optimization problems. The task is broken into several rounds;
at every round, they receive an optimization problem and are required to
choose the state that they will use for that problem. As a recommendation
rule, we use the state with the lowest energy with respect to the Hamiltonian
where the optimization problem is encoded. We denote this problem as the low
energy quantum state recommendation problem. We note that our model focuses on
the recommendation following the mentioned rule. After selecting the state the
learner will use it for the optimization problem (for example the initial
ansatz state of a variational quantum eigensolver), but that is a separate
task. When a learner chooses an action, they must perform an energy
measurement using the given Hamiltonian on the state corresponding to the
chosen action. Then the measurement outcomes are used to model rewards, and
their objective is to maximize the expected cumulative reward, i.e, the
expectation on the sum of the measurement outcomes over all the rounds played.
These measurements can be done fairly simply. Any Hamiltonian can be written
as a linear combination of Pauli observables. Now by measuring each of these
Pauli observables (since these measurements are conceivable, [33]), and taking
the appropriately weighted sum of the measurement outcomes, we can simulate
such a measurement. The QCB framework naturally lends itself to this model,
where the Hamiltonians are the contexts, and the set of states that can be
prepared reliably serve as the actions.

In this paper we study some important families of Hamiltonians — specifically,
the Ising and a generalized cluster model from [24], which are linear
combinations of Pauli observables with nearest-neighbor interactions and for
$n$ qubits can be written as

$\displaystyle H_{\text{ising}}(h)=\sum_{i=1}^{n}(Z_{i}Z_{i+1}+hX_{i}),$ (19)
$\displaystyle
H_{\text{cluster}}(j_{1},j_{2})=\sum_{i=1}^{n}(Z_{i}-j_{i}X_{i}X_{i+1}-j_{2}X_{i-1}Z_{i}X_{i+1}),$
(20)

where $h,j_{1},j_{2}\in\mathbb{R}$. In the Ising model, $h$ corresponds to the
external magnetic field. Specifically, we consider QCB with the following
context sets

$\displaystyle\mathcal{C}_{\text{Ising}}=\left\\{H_{\text{ising}}(h):h\in\mathbb{R}\right\\},\quad\mathcal{C}_{\text{cluster}}=\left\\{H_{\text{cluster}}(j_{1},j_{2}):j_{1},j_{2}\in\mathbb{R}\right\\}.$
(21)

Important families of Hamiltonians like the models discussed above show
translation-invariance and are spanned by Pauli observables showing nearest-
neighbor interactions, and as a result, span a low dimensional subspace. We
illustrate the scheme described above through the example of the Ising Model
contexts. The Pauli observables that need to be measured are
$\\{X_{i}\\}_{i\in[n]}$ and $\\{Z_{i}Z_{i+1}\\}_{i\in[n]}$ . These observables
have 2 possible measurement outcomes, -1 and 1, and by the reward distribution
of a Pauli observable $M$ given by Born’s rule (2) on a quantum state $\rho$,
the reward can be modeled as

$\displaystyle
R_{M,\rho}=2\text{Bern}\left(\frac{\operatorname{Tr}(M\rho)+1}{2}\right)-1,$
(22)

where $\text{Bern}(x)\in\\{0,1\\}$ is a random variable with Bernoulli
distribution with mean $x\in[0,1]$. By performing such a measurement for all
the Pauli observables and adding the rewards, the reward for
$\mathcal{C}_{\text{Ising}}$ is

$\displaystyle
R_{\text{Ising}}=-h\sum_{M\in{X_{i},i\in[n]}}R_{M,\rho}-\sum_{M^{\prime}\in{Z_{i}Z_{i+1},i\in[n]}}R_{M^{\prime},\rho},$
(23)

where we took the negative of the sum of the measurements because we are
interested in a recommender system for the lowest energy state. A similar
formulation applies to the QCB with generalized cluster Hamiltonian contexts.
In the rest of this section, we illustrate a modified LinUCB algorithm for the
QCB setting. Then we implement this recommender system where the contexts are
Hamiltonians belonging to the Ising and a generalized cluster models (21), and
demonstrate our numerical analysis of the performance of the algorithm by
studying the expected regret. We also demonstrate that depending on the action
set, the algorithm is able to approximately identify the phases of the context
Hamiltonians.

###  V.1 Gram-Schmidt method

Similarly to the task of shadow tomography [34] and classical shadows [35], we
do not need to reconstruct the full quantum states since the algorithm has
only to predict the trace between the contexts and the unknown quantum states.
Thus, the LinUCB algorithm has only to store the relevant part of the
estimators for this computation. As the measurement statistics depend only on
the coefficient corresponding to the Pauli observables spanning the
observables in the context set, only those Pauli observables in the expansion
of the estimators are relevant. This means that our algorithm can operate in a
space with a smaller dimension than the entire spaces spanned by $n$ qubits,
which has a dimension that is exponential on the number of qubits.

In order to exploit this property to improve the space complexity of the
LinUCB algorithm, we use the Gram-Schmidt procedure in the following way. At
any round, a basis for the vector parameterizations (as shown in (14)) of all
the previously received contexts is stored. If the incoming vector
parameterization of the context is not spanned by this basis, the component of
the vector orthogonal to the space spanned by this set is found by a Gram-
Schmidt orthonormalization-like process, and this component is added to the
set, after normalization. Therefore, at any round, there will be a list of
orthonormal vectors that span the subspace of all the vector parameterizations
of the contexts received so far, and the size of the list will be equal to the
dimension of the subspace, which we call effective dimension, i.e,

$\displaystyle
d_{\text{eff},t}=\text{dim}(\\{O_{c_{t}}\in\mathcal{C}\\}:t\in[T]).$ (24)

From now on we will omit the subscript for the time step $t$ and simply denote
the effective dimension as $d_{\text{eff}}$. Instead of feeding the context
vectors directly, for any incoming context vector, we construct
$d_{\text{eff}}$-dimensional vectors, whose $i^{th}$ term is the inner product
of the context vector and the $i^{th}$ basis vector. In case the incoming
vector is not spanned by the basis, we first update the list by a Gram-Schmidt
procedure (which will result in an addition of another orthonormal vector to
the list, and an increase in $d_{\text{eff}}$ by 1), and then construct a
$d_{\text{eff}}$-dimensional vector as described before. This vector is fed to
the LinUCB algorithm. The Gram-Schmidt procedure is stated in Algorithm 1 and
the modified LinUCB algorithm is stated explicitly in Algorithm 2. The
efficiency of this method is well illustrated in the case where all the
contexts are local Hamiltonians. As an example, we discuss the case of
generalised cluster Hamiltonians. Note that the space complexity of the
standard QCB framework is $O(kd^{2})$, where k is the number of actions, and d
is the dimension of the vector parameterizations of the contexts. In the
standard LinUCB technique, the context vectors $\mathbf{c_{t}},t\in[T]$ would
be $4^{n}$-dimensional, where n is the number of qubits the Hamiltonian acts
on, in which case the space complexity of the algorithm is $O(k4^{2n})$ . In
our studies, the contexts are Ising Hamiltonians and a generalised cluster
Hamiltonian (21) with $d_{\text{eff}}\leq 2$ and $d_{\text{eff}}\leq 3$
respectively. Since the vectors fed into the modified LinUCB is
$d_{\text{eff}}$-dimensional, the space complexity is
$O(kd_{\text{eff}}^{2})$, i.e, $O(4k)$ and $O(9k)$ respectively.

Algorithm 1 Gram-Schmidt Algorithm
($\text{Gram}(\mathbf{c},{V_{a}},\mathbf{b}_{a},\text{CBasis})$)

Input
$[\mathbf{c},\\{V_{a}\\}_{a\in\mathcal{A}},\\{\mathbf{b}_{a}\\}_{a\in\mathcal{A}},\text{CBasis}]$

for v in CBasis do

$\textbf{c}\leftarrow\textbf{c}-(\mathbf{v}^{\top}\textbf{c})\textbf{v}$

$\mathbf{v}_{\text{ct}}=\mathbf{v}_{\text{ct}}\oplus(\mathbf{v}^{\top}\textbf{c})$

end for

if $\textbf{c}!=\textbf{0}$ then

$\mathbf{v}_{\text{ct}}=\mathbf{v}_{\text{ct}}\oplus\|\mathbf{c}\|_{2}$

Add $\textbf{c}/\|\textbf{c}\|_{2}$ to CBasis

for $a=1,2,\ldots,K$ do

Set $V_{a}=V_{a}\oplus I_{1}$,
$\mathbf{b}_{a}=\mathbf{b}_{a}\oplus\mathbf{0}_{1}$

end for

end if

Return
$\left[\mathbf{c^{\prime}},\\{V_{a}\\}_{a\in\mathcal{A}},\\{\mathbf{b}_{a}\\}_{a\in\mathcal{A}},\text{CBasis}\right]$

Algorithm 2 LinUCB with Gram-Schmidt

1:Input $\alpha\in\mathbb{R}$

2:Set CBasis = $\left[\,\,\right]$

3:Set $V_{a}=\mathbf{1},\mathbf{b}_{a}=\mathbf{0},\forall a\in\mathcal{A}$

4:for $t=1,2,\ldots$ do

5:
$\left[\mathbf{c^{\prime}}_{O_{t}},\\{V_{a}\\}_{a\in\mathcal{A}},\\{\mathbf{b}_{a}\\}_{a\in\mathcal{A}},,\text{CBasis}\right]\leftarrow\text{Gram}(\mathbf{c}_{O_{t}},\\{V_{a}\\}_{a\in\mathcal{A}},\\{\mathbf{b}_{a}\\}_{a\in\mathcal{A}},,\text{CBasis})$

6: for $a\in\mathcal{A}$ do

7: $\tilde{\boldsymbol{\theta}}_{\rho_{a}}\leftarrow V^{-1}_{a}\mathbf{b}_{a}$

8:
$p_{t,a}\leftarrow\tilde{\boldsymbol{\theta}}_{\rho_{a}}\mathbf{c^{\prime}}_{O_{t}}+\alpha\sqrt{\mathbf{c}^{{}^{\prime}\top}_{O_{t}}V^{-1}_{a}\mathbf{c^{\prime}}_{O_{t}}}$

9: end for

10: Choose action
$\mathbf{a_{t}}=\operatorname*{\arg\\!\max}_{a\in\mathcal{A}}p_{t,a}$;

11: Measure state $\rho_{a_{t}}$ with $O_{c_{t}}$ and observe reward
$R_{O_{t}}$

12: Set $V_{a_{t}}\leftarrow
V_{a_{t}}+\mathbf{c^{\prime}}_{O_{t}}\mathbf{c^{\prime}}^{{}^{\prime}\top}_{O_{t}}$

13: Set
$\mathbf{b}_{a_{t}}\leftarrow\mathbf{b}_{a_{t}}+R_{O_{t}}\mathbf{c^{\prime}}_{O_{t}}$

14:end for

###  V.2 Phase classifier

In order to implement the numerical simulations we need to choose the
environments for the QCB with context sets $\mathcal{C}_{\text{ising}}$ and
$\mathcal{C}_{\text{cluster}}$. Elements of both context sets are
parameterized by tunable parameters. We study the performance of the
recommender system by choosing a context probability distribution that is
uniform on these parameters. Then we chose the actions as ground states of
Hamiltonians that corresponded to the limiting cases (in terms of the
parameters) of these models. In order to study the performance of our strategy
apart from the expected regret (6) we want to observe how the actions are
chosen. For every action, we maintained a set, which contained all the
Hamiltonians for which that action was chosen. We observed that almost all the
elements in each of these sets belonged to the same phase of the Hamiltonian
models.

In order to study the performance of the algorithm in this respect, we define
the classifier regret as

$\displaystyle\text{ClassifierRegret}_{T}^{\mathcal{\gamma,C},\pi}=\sum_{t=0}^{T-1}\mathbb{I}\left[a_{t}\neq
a_{\text{optimal,t}}\right],$ (25)

where
$a_{\text{optimal,t}}=\operatorname*{\arg\\!\max}_{a\in[k]}\mathrm{Tr}\left(O_{t}\rho_{a}\right)$,
and $O_{t}\in\mathcal{C}$ is the context observable received in
$t^{\text{th}}$ round. Note that the above classifier regret is not guaranteed
to be sublinear, like expected regret is (18) for the LinUCB strategy. This
can be understood intuitively: consider a scenario where the bandit picks an
actions with a small sub-optimality gap (7); then the linear regret will
increase by a very small amount, the classifier regret will increase by one
unit, as all misclassifications have equal contribution to regret. These,
however, are theoretic worst-case scenarios, and this classifier regret is
useful to study the performance of the algorithm in practice in our settings.

###  V.3 Numerical simulations

Before we move into the specific cases, we note the importance of the choice
of $\alpha$ in Algorithm 2. While the theoretical analysis of the LinUCB
algorithm depends on the choice of $\alpha$ , in practice one can tune this
value to observe a better performance. We primarily use the $\alpha$ described
in [3] (Chapter 19) given by

$\displaystyle\alpha_{t}=m+\sqrt{2\log\left(\frac{1}{\delta}\right)+d\log\left(1+\frac{tL^{2}}{d}\right)}.$
(26)

Here, $L$ and $m$ are upper bounds on the 2-norm of the action vectors and
unknown parameter respectively, $d$ is the dimension and $\delta$ is once more
a probability of failure.

Finally, while we study the performance of our algorithm in our simulations
with estimates of expected regret and expected classifier regret, it is
important to note that in an experimental setup, the learner will only be able
to measure the cumulative reward at every round. However, since these are
simulations, we are able to study the regret as well, as they are standard
metrics to gauge the performance of the algorithms. In the next subsection we
discuss our simulations of the QCB bandit
$(\gamma,\mathcal{C}_{\text{cluster}})$ model and later, the QCB bandit
$(\gamma,\mathcal{C}_{\text{Ising}})$ is discussed in the Appendix Appendix.

####  V.3.1 Generalised Cluster Model

We study the performance of the recommender system for the QCB bandit
$(\gamma,\mathcal{C}_{\text{cluster}})$, where the generalised cluster
Hamiltonians [24], act on 10 qubits and 100 qubits respectively. We observe
that the performance of the algorithm is not affected by the number of qubits,
as the effective dimension of the context set remains unchanged,i.e,
$d_{\text{eff}}=3$. We study the expected regret and classifier regret for
these two cases, and illustrate the system’s performance in finding the phases
of the generalised cluster Hamiltonians. This model was also studied in [36],
where they designed a quantum convolutional neural network to classify quantum
states across phase transitions. We chose 5 actions corresponding to
approximate ground states of Hamiltonians that are the limiting cases of the
generalised cluster model; i.e, generalised cluster Hamiltonians with
parameters $j_{1},j_{2}$ in (20),
$j_{1},j_{2}\rightarrow\\{0,0\\},\\{0,\infty\\},\\{\infty,0\\},\\{0,-\infty\\}$and
$\\{0,-\infty\\}$. Note that these methods of approximating ground states is
only for simulation purposes.

Figure 2: Plots for Regret and Classifier regret for QCB bandit
$(\gamma,\mathcal{C})$, where the Hamiltonians in $\mathcal{C}$ are a specific
form of generalised cluster models acting on 10 and 100 qubits respectively.
The performance is not very different since $d_{\text{eff}}=3$ (24) for both
cases. The action set is chosen to be approx. ground states of some
generalised cluster Hamiltonians

Initially a steep growth in regret is observed, followed by sudden slower
pace. On looking closely, in the plot below, we find that the regret indeed
continues to grow, albeit at a slower pace. This was be explained by observing
that the sub-optimality gap of the second-best action is quite small in
comparison to the sub-optimality gaps of the rest of the actions. Initially
the LinUCB algorithm does not have enough information about the unknown
parameters and has to play all actions resulting in an exploration phase.
However, at some point the bandit recognizes the ”bad” actions, and plays
either the best action or the action with a small sub-optimality gap most of
the time - this is when the bandit has begun to balance exploration and
exploitation. This is illustrated by observing the growth of the regret before
and after the first 50 rounds in the insets of Fig. 2.

In the beginning of this subsection, we had mentioned that the recommendation
system picks the same action for context Hamiltonians belonging to the same
phase. We illustrate this in Figure 3. In the scatter plot, when a context
generalised cluster Hamiltonian is received, a dot is plotted with the x-axis
and y- axis coordinates corresponding to its parameters $j_{1},j_{2}$
respectively. Depending on the action picked by the algorithm, we associate a
color to the dot. The resultant plot is similar (but not exact) to the phase
diagram of the generalised cluster Hamiltonian.

Figure 3: These plots illustrate how the recommender system identifies the
phases of the generalised cluster Hamiltonian. The x and y-axis represent the
coupling coefficients of the generalised cluster Hamiltonian received as
context. Like the Ising Model simulations, we associate a color to each
action. For any context $H_{\text{cluster}}(j_{1},j_{2})$ corresponding to any
of the T rounds, one of these actions is picked by the algorithm. We plot the
corresponding colored dot (blue for ground state of
$H_{\text{cluster}}(-\infty,0)$, orange for $H_{\text{cluster}}(0,\infty)$,
red for $H_{\text{cluster}}(\infty,0)$, green for
$H_{\text{cluster}}(0,-\infty)$ and purple for $H_{\text{cluster}}(0,0)$) at
the appropriate coordinates, for rounds that follow after the bandit has
”learned” the actions, i.e, the growth in regret has slowed down.

##  VI Outlook

This work describes the first steps for recommending quantum data by
implementing the bandit framework in a rigorous fashion for practical
scenarios. We provide a recommender system based on the theory of linear
contextual bandits and show that the upper and lower-bounds on the expected
regret are tight except for logarithmic factors. We also demonstrate its
efficiency in practice through simulations. Later, we show how such a system
could also be used to recognise phases of Hamiltonians.

We restricted our attention to a model where the expected rewards follow a
linear function in terms of the context and the unknown states. While the low
energy quantum state recommendation problem uses the outcome of the
measurement as a reward, one could think of other recommendation tasks with
more complicated reward functions. Non-linear rewards have been studied in the
bandit literature and receive the name of structured bandits [37, 38, 39].
This model could be a natural extension of the QCB for other recommendation
tasks where the rewards are not in one-to-one correspondence with measurement
outcomes. Going back to the general model, the environment is modeled by a set
of unknown quantum processes which in the QCB model we assumed to be a set of
stationary unknown quantum states. In a more general scenario, we can consider
environments that change with time due to some Hamiltonian evolution or noise
interaction with an external environment. In the bandit literature, non-
stationary environments were first considered in [40, 41] where each action
was associated with a Markov chain or the restless bandit model [42] where the
Markov chain associated to each action evolves with time. More recently in
[43] they studied a contextual bandit model with non-stationary environments.
We expect that recommender systems for quantum data can also be extended to
similar settings.

Acknowledgements: This research is supported by the National Research
Foundation, Singapore and A*STAR under its CQT Bridging Grant and the Quantum
Engineering Programme grant NRF2021-QEP2-02-P05.

## References

  * [1] C. A. Gomez-Uribe and N. Hunt.  The netflix recommender system: Algorithms, business value, and innovation.  ACM Trans. Manage. Inf. Syst., 6(4), 2016.
  * [2] P. Dragone, R. Mehrotra, and M. Lalmas.  Deriving user-and content-specific rewards for contextual bandits.  In The World Wide Web Conference, pages 2680–2686, 2019.
  * [3] T. Lattimore and C. Szepesvári.  Bandit algorithms.  Cambridge University Press, 2020.
  * [4] D. Bouneffouf, I. Rish, and C. Aggarwal.  Survey on applications of multi-armed and contextual bandits.  In 2020 IEEE Congress on Evolutionary Computation (CEC), pages 1–8. IEEE, 2020.
  * [5] A. Durand, C. Achilleos, D. Iacovides, K. Strati, G. D. Mitsis, and J. Pineau.  Contextual bandits for adapting treatment in a mouse model of de novo carcinogenesis.  In Machine learning for healthcare conference, pages 67–82. PMLR, 2018.
  * [6] M. Cohen, I. Lobel, and R. Paes Leme.  Feature-based dynamic pricing.  Management Science, 66(11):4921–4943, 2020.
  * [7] L. Tang, R. Rosales, A. Singh, and D. Agarwal.  Automatic ad format selection via contextual bandits.  In Proceedings of the 22nd ACM International Conference on Information and Knowledge Management, page 1587–1594. Association for Computing Machinery, 2013.
  * [8] L. Li, W. Chu, J. Langford, and R. E. Schapire.  A contextual-bandit approach to personalized news article recommendation.  In Proceedings of the 19th International Conference on World Wide Web, page 661–670. ACM, 2010.
  * [9] J. McInerney, B. Lacker, S. Hansen, K. Higley, H. Bouchard, A. Gruson, and R. Mehrotra.  Explore, exploit, and explain: personalizing explainable recommendations with bandits.  In Proceedings of the 12th ACM conference on recommender systems, pages 31–39, 2018.
  * [10] B. Casalé, G. Di Molfetta, H. Kadri, and L. Ralaivola.  Quantum bandits.  Quantum Mach. Intell., 2, 2020.
  * [11] D. Wang, X. You, T. Li, and A. Childs.  Quantum exploration algorithms for multi-armed bandits.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 10102–10110, 2021.
  * [12] Z. Wan, Z. Zhang, T. Li, J. Zhang, and X. Sun.  Quantum multi-armed bandits and stochastic linear bandits enjoy logarithmic regrets.  arXiv preprint arXiv:2205.14988, 2022.
  * [13] B. Cho, Y. Xiao, P. Hui, and D. Dong.  Quantum bandit with amplitude amplification exploration in an adversarial environment.  arXiv preprint arXiv:2208.07144, 2022.
  * [14] W. Hu, Wei, and J. Hu.  Training a quantum neural network to solve the contextual multi-armed bandit problem.  Natural Science, 11(01):17, 2019.
  * [15] I. Kerenidis and A. Prakash.  Quantum recommendation systems.  arXiv preprint arXiv:1603.08675, 2016.
  * [16] E. Tang.  A quantum-inspired classical algorithm for recommendation systems.  In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 217–228, 2019.
  * [17] J. Barry, D.T. Barry, and S. Aaronson.  Quantum partially observable markov decision processes.  Phys. Rev. A, 90:032311, 2014.
  * [18] M. Ying, Y. Feng, and S. Ying.  Optimal policies for quantum markov decision processes.  International Journal of Automation and Computing, 18(3):410–421, 2021.
  * [19] J. Lumbreras, E. Haapasalo, and M. Tomamichel.  Multi-armed quantum bandits: Exploration versus exploitation when learning properties of quantum states.  Quantum, 6:749, 2022.
  * [20] N. Abe, A. W. Biermann, and P. M. Long.  Reinforcement learning with immediate rewards, linear hypotheses.  Algorithmica, 37(4):263–293, 2003.
  * [21] P. Auer.  Using confidence bounds for exploitation-exploration trade-offs.  Journal of Machine Learning Research, 3:397–422, 2003.
  * [22] W. Chu, L. Li, L. Reyzin, and R. E. Schapire.  Contextual bandits with linear payoff functions.  In AISTATS, pages 208–214, 2011.
  * [23] K. Bharti, A. Cervera-Lierta, T.H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W. Mok, S. Sim, L. Kwek, and A. Aspuru-Guzik.  Noisy intermediate-scale quantum algorithms.  Rev. Mod. Phys., 94:015004, 2022.
  * [24] R. Verresen, R. Moessner, and F. Pollmann.  One-dimensional symmetry protected topological phases and their transitions.  Physical Review B, 96(16):165124, 2017.
  * [25] P. Auer, N. Cesa-Bianchi, Y. Freund, and R. E. Schapire.  The nonstochastic multiarmed bandit problem.  SIAM J. Comput., 32(1):48–77, 2003.
  * [26] S. Agrawal and N. Goyal.  Thompson sampling for contextual bandits with linear payoffs.  In International conference on machine learning, pages 127–135. PMLR, 2013.
  * [27] A random variable $X$ is called $\sigma$-subgaussian if for al $\mu\in\mathbb{R}$ we have $\operatorname{\mathbb{E}}\left[\mu X\right]\leq\mathop{exp}\nolimits\left(\sigma^{2}\mu^{2}/2\right)$.
  * [28] D. Varsha, T. Hayes, and S.Kakade.  Stochastic linear optimization under bandit feedback.  In Proceedings of the 21st Conference on Learning Theory, pages 355–366, 2008.
  * [29] P. Rusmevichientong and J. N. Tsitsiklis.  Linearly parameterized bandits.  Mathematics of Operations Research, 35(2):395–411, 2010.
  * [30] Y. Abbasi-Yadkori, D. Pál, and Cs. Szepesvári.  Improved algorithms for linear stochastic bandits.  In Advances in Neural Information Processing Systems, volume 24. Curran Associates, Inc., 2011.
  * [31] T. L. Lai.  Adaptive Treatment Allocation and the Multi-Armed Bandit Problem.  The Annals of Statistics, 15(3):1091 – 1114, 1987.
  * [32] Our model uses a different unknown parameter $\boldsymbol{\theta}_{a}\in\mathbb{R}^{d}$ for each action $a\in[k]$. This model can be easily adapted to settings where they assume only one unknown parameter shared by all actions if we enlarge the vector space and define $\boldsymbol{\theta}=(\theta_{1},...,\theta_{k})\in\mathbb{R}^{dk}$ as the unknown parameter.
  * [33] A. Peruzzo, J. McClean, Jarrod, P. Shadbolt, M. Yung, X. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien.  A variational eigenvalue solver on a photonic quantum processor.  Nature communications, 5(1):1–7, 2014.
  * [34] S. Aaronson.  Shadow tomography of quantum states.  In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 325–338, 2018.
  * [35] H. Huang, R. Kueng, and J. Preskill.  Predicting many properties of a quantum system from very few measurements.  Nature Physics, 16(10):1050–1057, 2020.
  * [36] M. C. Caro, H. Huang, M. Cerezo, K. Sharma, A. Sornborger, L. Cincio, and P. J. Coles.  Generalization in quantum machine learning from few training data.  Nature communications, 13(1):4919, 2022.
  * [37] T. Lattimore and R. Munos.  Bounded regret for finite-armed structured bandits.  Advances in Neural Information Processing Systems, 27, 2014.
  * [38] R. Combes, S. Magureanu, and A. Proutiere.  Minimal exploration in structured stochastic bandits.  In Proceedings of the 31st International Conference on Neural Information Processing Systems, page 1761–1769, 2017.
  * [39] D. Russo and B. Van Roy.  Eluder dimension and the sample complexity of optimistic exploration.  Advances in Neural Information Processing Systems, 26, 2013.
  * [40] J. C. Gittins.  Bandit processes and dynamic allocation indices.  Journal of the Royal Statistical Society: Series B (Methodological), 41(2):148–164, 1979.
  * [41] J. Gittins, K. Glazebrook, and R. Weber.  Multi-armed bandit allocation indices.  John Wiley & Sons, 2011.
  * [42] P. Whittle.  Restless bandits: Activity allocation in a changing world.  Journal of applied probability, 25(A):287–298, 1988.
  * [43] H. Luo, C. Wei, Chen-Yu, A. Agarwal, and J. Langford.  Efficient contextual bandits in non-stationary worlds.  In Conference On Learning Theory, pages 1739–1776. PMLR, 2018.

## Appendix

In this appendix, we discuss the simulations for the QCB bandit
$(\gamma,\mathcal{C}_{\text{ising}})$ setting, in a similar fashion as
described in subsection V.3. We study the performance of the recommender
system for the QCB bandit $(\gamma,\mathcal{C}_{\text{ising}})$, where the
Ising Hamiltonians act on 10 qubits and 100 qubits respectively. We observe
that the performance of the algorithm is not affected by the number of qubits,
as the effective dimension of the context set remains unchanged. We study the
expected regret and classifier regret for these two cases, and illustrate the
system’s performance in finding the phases of the Ising Model. The action set
corresponds to the ground state of the 3 limiting cases of the Ising Model
i.e, Ising Hamiltonians for parameter $h$ in (19), $h=0,h\rightarrow-\infty$
and $h\rightarrow\infty$ .

Figure 4: Plots for Regret and Classifier regret for QCB bandit
$(\gamma,\mathcal{C})$, where the Hamiltonians in $\mathcal{C}$ are Ising
Hamiltonians acting on 10 and 100 qubits respectively. The performance is not
very different since $d_{\text{eff}}=2$ (24) for both cases. The action set is
chosen to be approx. ground states of some Ising Hamiltonians

Once more, like the generalised cluster Model simulations, we observe a
distinct ”exploration” stage, followed by a ”exploration-exploitation trade-
off” stage which we again plot separately. This is illustrated by observing
the growth of the regret before and after the first 50 rounds shown in the
insets of Fig. 4.

In the beginning of this subsection, we had mentioned that the recommendation
system picks the same action for context Hamiltonians belonging to the same
phase. We illustrate this in Figure 5. In the scatter plot, when a context
Ising Hamiltonian is received, a dot is plotted with the x-axis coordinate
corresponding to its parameter. Depending on the action picked by the
algorithm, we associate a color to the dot. The resultant plot is very similar
to that of the phase diagram of an Ising model. The Ising model is known to
have phase transitions at $h=-1,1$, resulting in 3 phases,
$(-\infty,-1],[-1,1]$ and $[1,\infty)$, and we observe that different actions
were played in each of these ranges.

Figure 5: These plots illustrate how the recommender system identifies the
phases of the Ising Hamiltonian. The x-axis represents the external field
coefficient of the Ising Hamiltonian received as context. The blue, green, or
yellow mark indicates that the algorithm plays the
$1^{\text{st}}$,$2^{\text{nd}}$ or $3^{\text{rd}}$ action. We plot the
corresponding colored dot at the appropriate coordinates, for rounds that
follow after the bandit has ”learned” the actions, i.e, the growth in regret
has slowed down.


Current address: Laboratorio Nacional de Fusión, CIEMAT, Madrid, 28040, Spain

[1]J.F. Marco

[1]Instituto de Química Física “Rocasolano”, CSIC, Madrid, 28006, Spain

2]Instituto de Cerámica y Vidrio, CSIC, 28049, Spain

3]Alba Synchrotron Light Facility, CELLS, Barcelona, 08290, Spain

# Effect of annealing in the formation of well crystallized and textured
SrFe12O19 films grown by RF magnetron sputtering

G.D. Soria  A. Serrano  J.E. Prieto  A. Quesada  G. Gorni  J. de la Figuera
<EMAIL_ADDRESS>*  [  [

###### Abstract

We have studied the influence of annealing treatment on the crystalline growth
of SrFe12O19 previously deposited on Si (100) substrates using radio frequency
(RF) magnetron sputtering. For this goal, two grown films, with and without
ex-situ heating step, have been analysed and compared to determine the
differences in their structural, compositional and magnetic properties. The
results obtained by the different analysis techniques, in particular Mössbauer
spectroscopy together with EXAFS and XANES data, suggest that the as-grown
film is composed of nanocrystalline maghemite nanoparticles and amorphous
strontium oxide. A strontium hexaferrite canonical structure with c-axis
orientation in the sample plane was found for the annealed film.

###### keywords:

annealing; crystallization; extended x-ray absorption fine structure (EXAFS);
Mösbauer effect; thin film

##  1 Introduction

Currently, M-type ferrites are among the most popular permanent magnetic
materials. This fact is mainly due to their high magnetocrystalline anisotropy
which leads to elevated coercive fields. In addition, these hard ferrites
exhibit excellent thermal and chemical stability, high Curie temperatures and
good wear resistance, as well as being environmentally friendly and
competitively priced products Brabers ; PullarProgMatSci2012 . These materials
are widely applied for different uses, such as motors, magneto-mechanical
devices and actuators, among others basak_permanent-magnet_1996 ;
gutfleisch_magnetic_2011 ; sugimoto_current_2011 . The M-hexaferrite group is
composed by strontium hexaferrite SrFe12O19 (SFO), barium hexaferrite
BaFe12O19 and lead hexaferrite PbFe12O19 smit1959ferrites ; philips .
Specifically, the hard magnetic material selected in this study has been SFO.
The SFO crystal structure (figure 1) consists of alternately arranged blocks
of spinel and rock salt units. The structure shows a hexagonal closed-packing
of oxygen ions, where 2 of the 40 sites are occupied by divalent strontium
ions while the trivalent iron ions are located in interstitial sites
smit1959ferrites ; kojima_chapter_1982 ; PullarProgMatSci2012 ;
kreisel_investigation_2001 . Such configuration confers to this material a
markedly ferrimagnetic character where iron can be found in five distinct
cationic environments: three octahedral sites (12k, 4f2, 2a), one tetrahedral
site (4f1) and one bipyramidal site (2b).

Figure 1: Polyhedral model of the SrFe12O19 structure. The drawing depicts the
chemical environments of iron and strontium in different colours. Wyckoff´s
notations are adopted for every iron site. The representation was made using
Vesta software momma_vesta_2011 .

In SFO, the easy magnetization axis is the c crystallographic axis. Such
characteristic of SFO is very promising for both longitudinal and
perpendicular recording media applications where magnetic thin films are often
required. Strontium hexaferrite films have been grown by pulsed laser
deposition khaleeq-ur-rahman_deposition_2013 , chemical solution deposition
bursik_oriented_2011 , metal-organic chemical vapor deposition kato_high-
speed_2020 , spin coating sol-gel process masoudpanah_effect_2011  and RF
magnetron sputtering acharya_effect_1996 . For the latter deposition method,
it has been reported that the growth of the SFO crystalline phase needs two
steps: deposition and post-annealing treatment at a high temperature (800∘C -
900∘C) eva ; acharya_sputter_1994 ; acharya_effect_1996 . Multiple previous
works have established the requirement for the annealing step either ex-situ
or in-situ the deposition chamber morisako_effect_1997 ;
ramamurthy_acharya_preparation_1993 ; hui_effect_2014 . Such studies pointed
out by X-ray diffraction (XRD) analysis that the as-grown sample is of
amorphous nature. However, only a few studies discuss the character of the as-
grown film and how this is affected by the subsequent annealing step. Snyder
et al. snyder_local_1995 ; snyder_local_1996 ; snyder_determination_1996
published on the local structural anisotropy in the as-grown film before
annealing for barium ferrite thin films by extended X-ray absorption fine
structure (EXAFS). The data suggested networks of iron atoms surrounded by
their oxygen nearest neighbors, with barium atoms fitting into spaces in-
between. Apparently, this determines the final crystalline texture (and the
resulting magnetic anisotropy), which forms upon annealing. Therefore, this
study aims to unravel the role of temperature on the aggregation/diffusion of
atoms and the consequent formation of the crystalline phase in SFO films
deposited by radio frequency (RF) magnetron sputtering. To this end, we have
focused on the characterization of a typical as-grown sample and on a sample
subject to an annealing treatment after being deposited in identical
conditions in the RF magnetron chamber. The characterization includes an ample
range of structural, spectroscopic and magnetic techniques such as Rutherford
backscattering spectrometry (RBS), XRD, Raman spectroscopy, Mössbauer
spectroscopy, vibrating-sample magnetometry (VSM), X-ray absorption near edge
structure (XANES) and EXAFS.

##  2 Experimental methods

Two films were deposited by radio frequency (RF) magnetron sputtering of a
sintered SrFe12O19 target made from a commercial SFO powder commercial .
Silicon (100) wafers 1 mm thick were used as substrates. They were kept at
room temperature (RT) during the whole sputtering process. The target-
substrate distance was approximately 60 mm and the base pressure in the
sputtering chamber was 1 $\times$ 10-6 mbar. In order to start from clean
targets, a 15 min pre-sputtering process was performed prior to each
deposition. A working pressure of 7 $\times$ 10-3 of an (Ar/O2) mixture
containing 2$\%$ oxygen was maintained inside the sputtering chamber during
the deposition process. Samples were grown using a sputtering power of 260 W
for five hours. After deposition one of the films was subjected to an
annealing treatment at 850 ∘C in air for three hours in a separate furnace.

Figure 2: RBS experimental spectra (red circles) of a) an as-grown film and b)
an annealed film. The spectrum simulated by the SIMRA code for each sample is
given by the blue solid line. The signals of the different elements are
identified in each spectrum.

The composition and thickness of the different films were determined by RBS
using 4He+ at 3.045 MeV to resonantly enhance the O signal. Experiments were
performed at the standard beamline of the 5 MV tandem accelerator Centre for
Micro-Analysis of Materials (CMAM) in Madrid redondo-cubero_current_2021 . The
ion fluence for the Rutherford backscattering and resonant backscattering
experiments was set at 10 $\mu$C. A silicon surface barrier detector, at a
scattering angle of 170.0∘, collected the backscattering ions while a three-
axis goniometer was employed to control the crystal position.

The crystal structure of the films was examined by X-ray diffraction using a
Bruker D8 Advance diffractometer and Cu-K$\alpha$ (1.54 $\r{A}$) radiation in
a $\theta$/2$\theta$ configuration. The measuring step was 0.02∘/s with 0.5 s
measuring time per step.

Raman spectra were acquired with a micro-Raman Via Renishaw spectrophotometer,
equipped with an electrically refrigerated CCD camera, and coupled to a Leica
microscope. Radiation of 532 nm from air-cooled Nd: YAG laser was employed for
excitation using a diffraction grating of 1800 l/mm. The laser power at the
sample was up to 2.0 mW and the spectral resolution was 2 cm-1. The
acquisition time was 20 sec, the Raman signal was collected over the range
100-1000 cm-1 and the objective employed for the measurements was a long-
distance 50x Leica.

Mössbauer spectra were collected in the conventional transmission mode using a
constant acceleration spectrometer, a 57Co(Rh) source, and a helium closed-
cycle cryorefrigerator GancedoHI1994 . Spectra were recorded at 298 K and 35 K
and computer-fitted. The velocity scale was calibrated using a metallic
$\alpha$-Fe foil 6 $\mu$m thick. The isomer shifts were referred to the
centroid of the spectrum of $\alpha$ Fe at room temperature.

Sr and Fe K-edge XANES and EXAFS data were recorded in the fluorescence mode
at the CLAESS beamline of the Alba synchrotron CLAESS . Data were collected at
298K and different incident angles of the incoming X-rays (5∘ and 85∘). The
energy scale was calibrated using a 6 $\mu$m iron foil and strontium(II)
oxide. The position of the first inflection point on the iron foil edge was
taken at 7112.3 eV and that of strontium(II) oxide at 16110.2 eV. All the
iron-XANES and strontium-XANES data are reported here relative to these
values. The edge profiles were separated from the EXAFS data and, after
subtraction of a linear pre-edge background, normalized to the edge step. The
position of the absorption edge was obtained from the first most intense
maximum of the first derivative of the edge profile. The EXAFS oscillations
were isolated after background subtraction of the raw data using the software
package Athena and converted into k space. The data were weighted by k2, where
k is the photoelectron wave vector, to compensate for the diminishing
amplitude of EXAFS at high k.

Magnetic properties were studied by recording hysteresis loops at RT with a
homemade VSM applying a magnetic field of 1.25 T.

##  3 Results and Discussion

RBS analysis allowed for determining the chemical composition of both samples.
The data (figure 2) were fitted using the SIMRA code mayer_simnra_nodate
obtaining good agreement with the experimental spectra. The as-grown sample
and the annealed film presented the same elements in their spectra: O, Si, Fe,
Sr and Ba (Fig. 2). Their signals come from the SFO film as well as from the
substrate. The barium signal stems from a small Ba impurity in the commercial
powder soria_influence_2020 . The concentrations of the different elements
were not significantly different between both films. The atomic Fe/O ratio
expected for pure SFO is 0.63. In the case of the non-annealed film, the Fe/O
ratio was 0.66, while in the annealed film was 0.63. Thus, it follows that the
small percentage of missing oxygen atoms during deposition are recovered after
the annealing treatment. In addition, the film thicknesses calculated were
1.86 $\mu$m and 1.62 $\mu$m for the as-grown and annealed films, respectively.
The observed difference in thickness (13$\%$) is probably due to a deviation
in the deposition gradient arising from the samples being placed at a slightly
different position in the sample stage of the magnetron device.

The crystal structure, texture and chemical composition of the films were also
studied by XRD and Raman spectroscopy. Fig. 3a shows the XRD patterns recorded
from each film together with a reference XRD pattern recorded from a
commercial SFO powder. The annealed film shows all the diffraction peaks that
correspond to a crystalline strontium hexaferrite structure, besides the Si
peaks arising from the silicon substrate. The highest intensity peaks appear
at 30.4∘, 35.2∘, 37.1∘, 54.0∘ and 63.2∘ which match with the (110), (201),
(203), (300), (220) Miller indices, respectively of SFO kiani_synthesis_2013 ;
azim_indexing_2014 . The most prominent peak is the one at 30.4∘, i.e, that
corresponding to the (110) diffraction plane. This indicates a preferential
growth of the SFO structure with the c-axis parallel to the film plane. The
crystallite size was determined by the Scherrer equation
scherrer_bestimmung_1918 , and the lattice parameters by the distance formula
between adjacent planes in the (hkl) set of the hexagonal crystal structure
cullity_elements_1978 . The calculated average particle size is 55 nm whilst
the a and c lattice parameters are 0.589 nm and 2.349 nm, respectively. The
unit cell values are close to those reported in the literature
PullarProgMatSci2012 ; azim_indexing_2014 . The pattern of the as-grown sample
shows only a few broad diffraction peaks, indicating a low crystallinity of
the film, which, besides, are not coincidental with those expected for SFO. In
fact, the diffraction peaks at 35.1∘ and 66.5∘, are compatible with the
presence of the spinel-related phases magnetite (Fe3O4) or maghemite
($\gamma$-Fe2O3) aliahmad_synthesis_2013 ; ruiz-baltazar_effect_2015 ;
zulfiqar_static_2014 ; kim_new_2012 ; chernyshova_size-dependent_2007 . Since
XRD does not allow unambiguous identification of these two phases, we resorted
to Raman and Mössbauer spectroscopy in order to achieve a more accurate
assignment (see below).

Figure 3: a) XRD diffraction patterns and b) Raman spectra recorded from both
films together with the data recorded from a SFO reference powder.

Figure 4: Transmission Mössbauer spectra recorded from both films at RT and 35 K. T | Site | $\delta$ | 2$\varepsilon$ | H | $\Gamma$ | Area
---|---|---|---|---|---|---
(K) | ($\pm$ 0.03 mms-1) | ($\pm$ 0.05 mms-1) | ($\pm$ 0.05 T) | ($\pm$ 0.05 mms-1) | ($\%$) |
298 | Fe3+ | 0.34 | 0.80 | - | 0.40 | 100
35 | Fe${}^{3+}_{Td}$ | 0.39 | -0.03 | 44.2 | 0.40 | 28
| Fe${}^{3+}_{Oh}$ | 0.46 | 0.01 | 49.3 | 0.40 | 72
Table 1: 57Fe Mössbauer parameters obtained from the fit of the spectra shown
in figure 4a. The symbols $\delta$, $\Delta$, 2$\varepsilon$, H, $\Gamma$
correspond to isomer shift, quadrupole splitting (applies to doublets),
quadrupole shift (applies to sextets), hyperfine magnetic field and linewidth,
respectively.

The corresponding Raman spectra are collected in figure 3b. As in the XRD
analysis, a reference SFO powder pattern was used for identification purposes.
The spectrum of the annealed sample presents the bands characteristic of
strontium hexaferrite elansary_new_2020 ; morel_sublattice_2002 ;
kreisel_raman_1999 ; nethala_investigations_2018 ; kreisel_raman_1998 . Such
Raman modes arise from the iron cations in different crystallographic sites of
the SFO structure. The as-grown sample shows a vibration mode with a broad
peak at 690-700 cm-1. According to the published literature, the broad and
most intense peak for maghemite appears around 703 cm-1, and can be assigned
to the A1g vibration mode (O-Fe-O bridge) dar_single_2013 ;
santillan_optical_2017 ; lopez_sensitivity_2009 . This fact correlates well
with the XRD data. A presumed amorphous SrO phase was not detected by Raman
spectroscopy due to its lower scattering cross-section SERRANO20221014 .

In figure 4a the transmission Mössbauer spectra recorded from the as-grown SFO
film at 298 K (RT) and 35 K are presented. The RT spectrum does not exhibit
the characteristic magnetic hyperfine pattern of SFO BerryJMSL2001  but a
paramagnetic doublet instead. This doublet presents an isomer shift ($\delta$
= 0.34 mm/s) and a quadrupole splitting ($\Delta$ = 0.80 mm/s) which are
characteristic of high spin Fe3+ menil_systematic_1985 . The lack of magnetic
ordering at RT together with the broad signals shown by both the XRD and Raman
data suggests that the as-grown film could be of superparamanetic/low
crystallinity nature associated to the reduced size of the film grains. The 35
K spectrum shows two relatively well-separated magnetic contributions whose
Mössbauer parameters (Table 1) are close to those shown by Fe3+ cations
occupying the octahedral and tetrahedral sites of a spinel-related structure.
The low-temperature spectrum registered from the as-grown sample does not
correspond, then, with that expected for SFO and resembles clearly that shown
by many oxides having a spinel-related structure. In this particular case, the
35 K spectrum, and the corresponding hyperfine parameters, are very similar to
those reported for 5 nm maghemite nanoparticles measured at comparable
temperatures (40 K) roca_effect_2007 . Both the RT and 35 K Mössbauer spectra
endorse the XRD and Raman results and confirm that the iron phase contained in
the as-grown film is, most likely, superparamagnetic maghemite. In contrast,
the RT and 35 K spectra recorded from the annealed film (figure 4b) are fully
typical of SFO BerryJMSL2001 , except by the small Fe3+ paramagnetic doublet
(3$\%$) seen at RT, which possibly arises from a minor fraction of
SFO/maghemite particles of nanometer dimensions.

Figure 5: a) and b) XANES spectra at the Sr and Fe K-edge, respectively,
recorded from the as-grown and annealed films at different setup orientations.
The data recorded from different reference compounds are also shown.

The intensity ratio of the absorption lines of the sextets (3:x:1:1:x:3) can
be used to determine the average orientation of the magnetization in each
sample. The spectrum of the as-grown sample (4a) at 35 K is best fitted using
a value of x = 2, indicating a random orientation of the magnetization.
However, for the annealed sample a x = 3.5 value is obtained which corresponds
to an average magnetization mostly within the plane (15∘) and indicates that
the film is highly magnetically textured.

Since the data so far have indicated that the as-grown film contains maghemite
of nanometric dimensions, a pertinent question which is still unanswered
refers to the chemical state of strontium in that film. In order to understand
this point, we recorded both Sr- and Fe-K edge XANES and EXAFS data. Figure 5a
shows the Sr-K edge XANES of the reference compounds SrO and SFO together with
those of the as-grown and annealed films at different incident angles of the
incoming X-rays (5∘ and 85∘). The two different orientations were used in
order to look into the existence of a local structural orientation preference
in the films. Inspection of this figure indicates that the position of the
edge, defined as the first inflexion point of the edge, i.e., the point where
the first derivative has its first maximum, is the same in all the cases
(16112.7 eV) and, therefore, indicative of the presence of Sr2+ in both films.
The Sr K-edge XANES data of the as-grown and the annealed film are clearly
different, while those corresponding to the as-grown film resemble that of
SrO, and the spectra of the annealed film match the spectrum of SFO. As
compared with the SrO spectrum, is clear that the oscillations beyond the
white line in the spectra of the as-grown film are considerably attenuated
what suggests a more disordered character of this sample. This does not occur
in the case of the spectra of the annealed film when compared with the
reference spectrum of SFO. Finally, it appears that there are no significant
differences between the spectra taken at different incident angles of the
X-ray beam. This result is contrary to prior studies which pointed out that
the anisotropic local structure in the as-grown sample determines the
crystalline orientation of the annealed film snyder_local_1995 ;
snyder_local_1996 ; snyder_determination_1996 .

Figure 6: a) and b) EXAFS data at the Sr and Fe K-edge, respectively, recorded
from the as-grown and annealed films at different setup orientations. The data
recorded from different reference compounds are also shown. c) and d) Fourier-
transformed data at the Sr and Fe K-edge, respectively, recorded from the as-
grown and annealed films at different setup orientations together with the
Fourier-transform data recorded from different reference compounds.

Figure 5b collects the Fe K-edge XANES recorded from the reference compounds
magnetite, maghemite, hematite and SFO together with the spectra recorded
(again at two different incidence angles of the X-ray beam) from the as-grown
and annealed films. The position of the edge of the spectra recorded from the
two films is located at 7123.2 eV, a value which coincides with that of the
edge of the spectra of maghemite and hematite, therefore confirming, as
Mössbauer spectroscopy had anticipated, the presence of Fe3+ in the samples.
The position of the edge of magnetite, in which 50$\%$ of the octahedral sites
are occupied by Fe2+, appears at the lower energy of 7121.0 eV, what is
consistent with the mentioned presence of Fe2+ in this material. The spectra
of the as-grown film show a very intense pre-edge peak at 7114.5 eV indicating
the existence of tetrahedral environments in the sample. This endorses the
tetrahedral component observed in the 35K Mössbauer spectrum. Similarly to
that observed in the Sr K-edge XANES of this sample, the oscillations beyond
the white line are very much attenuated which is consistent with its low
crystallinity as evidenced by the rest of the diffraction and spectroscopic
techniques. Because of the strong attenuation of the mentioned oscillations,
it is hard to associate the Fe K-edge XANES spectrum of the as-grown-film with
any of the reference compounds. However, the Fe K-edge XANES data recorded
from the annealed film are very similar to that recorded from the reference
SFO.

Figure 6a shows the Sr K-edge EXAFS recorded from SrO, SFO, the as-grown film
and the annealed film. The data recorded from the as-grown film resemble very
much those recorded from SrO, both in terms of the number of maxima and
minima, their positions and the corresponding frequencies. However, the
amplitudes of the oscillations are significantly smaller in the spectra of the
as-grown film, the data becoming structureless beyond 6 $\AA^{-1}$. This
strongly suggests the existence in the as-grown films of SrO-like domains
having no long-range order what would explain why they have not been observed
in the XRD data. The spectra recorded from the annealed film are totally
consistent with the presence of a well crystalline SFO phase.

The Fe K-edge recorded from the different iron compounds is shown in Figure
6b. It is interesting to compare the data recorded from the as-grown film with
those recorded from maghemite and hematite. Inspection of Figure 6b shows that
the oscillations in the EXAFS of the as-grown film follow those of maghemite
although with lower amplitude and being, in general, much structureless.
However, they differ considerably from the data recorded from hematite. Again,
this result is consistent with the Mössbauer data since it clearly points out
the presence of poorly crystallized domains of maghemite in this film. As
expected, the data recorded from the annealed film are fully consistent with
well crystallized SFO.

Figures 6c and 6d collect the Sr- and Fe-K edge EXAFS transforms recorded from
all these samples. We will not go into detail in the description of these
since the conclusions which can be extracted are the same as those obtained
from the EXAFS. The data recorded from the as-grown film shows the decreased
intensity of the first coordination shell (both in the case of the strontium
and iron data) and the absence of shells at longer distances, all the facts
being indicative of the occurrence of a poorly crystalline material. This is
not the case with the annealed film whose Fourier transforms are
characteristic of SFO.

Figure 7: RT hysteresis loops registered from the SFO annealed film. The blue
curve was recorded with a magnetic field applied normal to the sample while
the red one was acquired with a magnetic field applied parallel to it.

Finally, the magnetization curves of both films were recorded at RT under an
applied external magnetic field of 1.25 T. For the as-grown sample, no
magnetic easy axis or hysteresis loops were observed (not shown), in good
agreement with the lack of magnetic interactions observed by Mössbauer
spectroscopy at RT. Figure 7 shows the RT hysteresis loops recorded from the
annealed film under an applied magnetic field parallel and normal to the
sample. The results show clearly the existence of a magnetization easy axis
within the plane of the sample. This is consistent with the preference of SFO
to grow with the c-axis oriented parallel to the film plane, as indicated by
XRD, and the fact that magnetization occurs in SFO along that axis. These data
also confirm the orientation of the in-plane magnetization determined by
Mössbauer spectroscopy at RT and 35 K (figure 4b). The saturation
magnetization and coercive field determined with the in-plane configuration
were 63 emu/g and 0.5 T, respectively. These values are within the range of
magnetization values reported for this compound PullarProgMatSci2012 ;
fernandez_topical_2020 . When the magnetization curve was recorded
perpendicular to the plane of the sample, a different saturation magnetization
value was found. This shows the need for a higher applied field to reach the
threshold magnetization value observed previously in the in-plane measurement.
The noisy signal is related both to the relatively small thickness of the
films (a reduced amount of material results in a poorer signal) and the
particular acquisition conditions of our homemade VSM device.

From the inspection of the results above, the precursor phase in the as-grown
sample is quite isotropic. On one hand, the iron phase detected is locally
cubic with a low magnetic anisotropy. On the other, the magnetization
direction determined from Mössbauer at low temperature is compatible with a
random orientation of the magnetization. However, once the film is annealed,
it presents a clear uniaxial anisotropy with the easy axis within the film
plane (see the VSM and Mössbauer results). We suggest that the origin of the
in-plane magnetization is due to the film morphology that in the annealing
stage promotes the growth of the SFO crystals with the c-axis in the in-plane
direction.

##  4 Conclusions

In this work, we have confirmed that the annealing treatment is crucial for
obtaining a genuine SFO film having the right composition and crystal
structure. To this aim, two samples grown by RF magnetron sputtering with and
without heating step were characterized by distinct diffraction and
spectroscopy techniques. The XRD and Raman data showed the poor crystallinity
of the as-grown sample as well as the possible presence of maghemite.
Mössbauer spectroscopy corroborated the presence of nanometric $\gamma$-Fe2O3
domains in the non-annealed film while the XANES and EXAFS results revealed
the existence of SrO-like domains having no long-range order. The EXAFS and
XANES data taken at different incident angles of the incident X-ray beam in
the as-grown films do not show significant differences and therefore do not
support previous claims snyder_local_1995 ; snyder_local_1996 ;
snyder_determination_1996  that the as-grown film acts as a “c-axis oriented”
precursor of SFO. All the results gathered from the annealed film confirm that
it corresponds to a canonical SFO phase which besides is highly textured from
both the structural and magnetic points of view, having the c-axis and
consequently, the easy magnetization axis parallel to the film plane.

##  5 Acknowledgments

This work is supported by the Spanish Ministry of Science and Innovation
through Projects RTI2018-095303-B-C51, RTI2018-095303-B-C53,
MAT2017-86450-C4-1-R, RTI2018-095303-A-C52, through the Ramon y Cajal Contract
RYC-2017-23320 and by the Regional Government of Madrid through project
S2018-NMT-4321 and by the European Commission through the H2020 Project
no.720853 (AMPHIBIAN). We gratefully acknowledge the support from CMAM for the
beamtime proposal with code STD009/20. Part of these experiments were
performed at the CLAESS-BL22 beamline at ALBA Synchrotron and in collaboration
with ALBA staff. A.S. acknowledges financial support from the Comunidad de
Madrid for an “Atracción de Talento Investigador” contract No. 2017-t2/IND539.
We thank Dr S. Cortés for providing access to the Raman spectrometer and for
his help in recording these data.

##  6 CRediT authorship contribution statement

G.D. Soria: conceptualization, investigation, formal analysis, software,
visualization, writing-original draft; A. Serrano: investigation, resources,
writing-review $\&$ editing; J.E. Prieto: methodology, investigation,
software, writing-review $\&$ editing; A. Quesada: methodology, supervision,
writing-review $\&$ editing, project administration, funding acquisition; G.
Gorni: investigation, resources; J. de la Figuera: conceptualization,
methodology, supervision, writing-review $\&$ editing, project administration,
funding acquisition; J.F. Marco: conceptualization, investigation, formal
analysis, visualization, writing-original draft.

##  7 Additional Information

Competing Interests: The authors declare that they have no known competing
financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data Availability: Data will be made available on reasonable request.

## References

  *   * (1) Brabers, V.A.M.: Progress in Spinel Ferrite Research vol. 8, (1995)
  * (2) Pullar, R.: Hexagonal Ferrites: A Review of the Synthesis, Properties and Applications of Hexaferrite Ceramics vol. 57, (2012)
  * (3) Basak, A.: Permanent-magnet DC linear motors (1996)
  * (4) Gutfleisch, O., Willard, M.A., Brück, E., Chen, C.H., Sankar, S.G., Liu, J.P.: Magnetic materials and devices for the 21st century: stronger, lighter, and more energy efficient. Adv. Mater. 23(7), 821–842 (2011). https://doi.org/10.1002/adma.201002180
  * (5) Sugimoto, S.: Current status and recent topics of rare-earth permanent magnets. Journal of Physics D: Applied Physics 44(6), 064001 (2011). https://doi.org/10.1088/0022-3727/44/6/064001
  * (6) Smit, J., Wijn, H.: Ferrites, Philips technical library. Eindhoven, The Netherlands 278 (1959)
  * (7) Went, J.J., Rathenau, G.W., E.W, G., Van Oosterhout, G.W.: Ferroxdure, a class of new permanent magnet materials. Philips Technical Review 13(7), 194–208 (1952). https://doi.org/621.318.12:538.246.2
  * (8) Kojima, H.: Chapter 5 Fundamental properties of hexagonal ferrites with magnetoplumbite structure 3, 305–391 (1982). https://doi.org/10.1016/S1574-9304(05)80091-4
  * (9) Kreisel, J., Vincent, H., Tasset, F., Paté, M., Ganne, J.P.: An investigation of the magnetic anisotropy change in BaFe12-2xTixCoxO19 single crystals. J. Magn. Magn. Mater. 224(1), 17–29 (2001). https://doi.org/10.1016/S0304-8853(00)01355-X
  * (10) Momma, K., Izumi, F.: VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. Journal of Applied Crystallography 44(6), 1272–1276 (2011). https://doi.org/10.1107/S0021889811038970
  * (11) Khaleeq-ur-Rahman, M., Bhatti, K.A., Rafique, M.S., Latif, A., fou-uz-Zia, S.: Deposition and characterization of strontium hexaferrite (SrFe12O19) by PLD technique. Opt Laser Technol. 47, 361–365 (2013). https://doi.org/10.1016/j.optlastec.2012.07.025
  * (12) Buršík, J., Drbohlav, I., Frait, Z., Knížek, K., Kužel, R., Kouřil, K.: Oriented SrFe12O19 thin films prepared by chemical solution deposition. J Solid State Chem. 184(11), 3085–3094 (2011). https://doi.org/10.1016/j.jssc.2011.09.024
  * (13) Kato, K., Watanuki, R., Ito, A.: High-speed epitaxial growth of M-type Strontium hexaferrite films on sapphire using metal–organic chemical vapor deposition and their magnetic property. Mater. Lett. 274, 128046 (2020). https://doi.org/10.1016/j.matlet.2020.128046
  * (14) Masoudpanah, S.M., Ebrahimi, S.A.S.: Effect of citric acid content on the structural and magnetic properties of SrFe12O19 thin films. Thin Solid Films 520(1), 199–203 (2011). https://doi.org/10.1016/j.tsf.2011.07.021
  * (15) Acharya, B.R., Prasad, S., Venkataramani, N., Shringi, S.N., Krishnan, R.: The effect of deposition and annealing conditions on textured growth of sputter-deposited strontium ferrite films on different substrates. Journal of Applied Physics 79(1), 478–484 (1996). https://doi.org/10.1063/1.360854
  * (16) Cespedes, E., Rodriguez-Rodriguez, G., Navio, C., et al.: Inter-grain effects on the magnetism of M-type strontium ferrite. J. Alloys Compd. 692(692), 280–287 (2017). https://doi.org/10.1016/j.jallcom.2016.08.318
  * (17) Acharya, B.R., Krishnan, R., Prasad, S., Venkataramani, N., Ajan, A., Shringi, S.N.: Sputter deposited strontium ferrite films with c-axis oriented normal to the film plane. Appl. Phys. Lett. 64(12), 1579–1581 (1994). https://doi.org/%****␣SFOthinfilm.bbl␣Line␣300␣****10.1063/1.111845
  * (18) Morisako, A., Matsumoto, M., Takei, S., Yamazaki, T.: Effect of substrate temperature on magnetic properties of strontium ferrite thin films. IEEE Trans. Magn. 33(5), 3100–3102 (1997). https://doi.org/10.1109/20.617857
  * (19) Ramamurthy Acharya, B., Venkatramani, N., Prasad, S., Shringi, S.N., Krishnan, R., Tessier, M., Dumond, Y.: Preparation and magnetic properties of strontium ferrite thin films. IEEE Trans. Magn. 29(6), 3370–3372 (1993). https://doi.org/10.1109/20.280818
  * (20) Hui, Y., Cheng, W., Lin, G., Miao, X.: Effect of heat treatments on the structural and magnetic properties of La-Co substituted strontium ferrite films. J. Electron. Mater. 43(9), 3640–3645 (2014). https://doi.org/10.1007/s11664-014-3266-3
  * (21) Snyder, J.E., Harris, V.G., Koon, N.C., Sui, X., Kryder, M.H.: Local anisotropic structure in amorphous Ba-Fe-O films and its role in determining magnetic anisotropy in crystallized Ba-hexaferrite films. IEEE Trans. Magn. 31(6), 3844–3846 (1995). https://doi.org/10.1109/20.489791
  * (22) Snyder, J.E., Harris, V.G., Koon, N.C., Sui, X., Kryder, M.H.: Local structure of the amorphous precursor to Ba-Hexaferrite thin films: an anisotropic octahedral Fe-O glass network. Phys. Rev. Lett. 77(16), 3383–3386 (1996). https://doi.org/10.1103/PhysRevLett.77.3383
  * (23) Snyder, J.E., Harris, V.G., Das, B.N., Koon, N.C., Sui, X., Kryder, M.H.: Determination of local order in the amorphous precursor to Ba-hexaferrite thin-film recording media. J. Appl. Phys. 79(8), 4890–4892 (1996). https://doi.org/10.1063/1.361641
  * (24) Baermann GmbH, M.: Commercial powder of Strontium Hexaferrite (2018)
  * (25) Redondo-Cubero, A., Borge, M.J.G., Gordillo, N., Gutiérrez, P.C., Olivares, J., Pérez Casero, R., Ynsa, M.D.: Current status and future developments of the ion beam facility at the centre of micro-analysis of materials in Madrid. Eur. Phys. J. Plus 136(2), 175 (2021). https://doi.org/10.1140/epjp/s13360-021-01085-9
  * (26) Gancedo, J.R., Gracia, M., Marco, J.F.: Practical aspects of Mössbauer spectroscopy instrumentation. Hyperfine Interact. 83(1), 71–78 (1994). https://doi.org/10.1007/BF02074260
  * (27) Simonelli, L., Marini, C., Olszewski, W., Ávila Pérez, M., Ramanan, N., Guilera, G., Cuartero, V., Klementiev, K.: ClÆss: The hard x-ray absorption beamline of the alba cells synchrotron. Cogent Physics 3(1), 1231987 (2016). https://doi.org/10.1080/23311940.2016.1231987
  * (28) Mayer, M.: SIMNRA User’s Guide, 67
  * (29) Soria, G.D., Marco, J.F., Mandziak, A., Sánchez-Cortés, S., Sánchez-Arenillas, M., Prieto, J.E., Dávalos, J., Foerster, M., Aballe, L., López-Sánchez, J., Guzmán-Mínguez, J.C., Granados-Miralles, C., Figuera, J.d.l., Quesada, A.: Influence of the growth conditions on the magnetism of SrFe12O19 thin films and the behavior of Co/SrFe12O19 bilayers. J Phys D Appl Phys 53(34), 344002 (2020). https://doi.org/10.1088/1361-6463/ab8d70
  * (30) Kiani, E., Rozatian, A.S.H., Yousefi, M.H.: Synthesis and characterization of SrFe12O19 nanoparticles produced by a low-temperature solid-state reaction method. J. Mater. Sci. Mater. Electron. 24(7), 2485–2492 (2013). https://doi.org/10.1007/s10854-013-1122-5
  * (31) Azim, M., Atiq, S., Riaz, S., Naseem, S.: Indexing the structural parameters and investigating the magnetic properties of lanthanum doped strontium hexaferrites. IOP Conference Series: Mater. Science and Engineering 60, 012045 (2014). https://doi.org/10.1088/1757-899X/60/1/012045
  * (32) Scherrer, P.: Bestimmung der Größe und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1918, 98–100 (1918)
  * (33) Cullity, B.D.: Elements of x-ray diffraction (1978)
  * (34) Aliahmad, M., Nasiri Moghaddam, N.: Synthesis of maghemite ($\gamma$-Fe2O3) nanoparticles by thermal-decomposition of magnetite (Fe3O4) nanoparticles. Mater. Sci-Poland 31(2), 264–268 (2013). https://doi.org/10.2478/s13536-012-0100-6
  * (35) Ruíz-Baltazar, A., Esparza, R., Rosas, G., Pérez, R.: Effect of the surfactant on the growth and oxidation of iron nanoparticles. J. Nanomater. (2015). https://doi.org/10.1155/2015/240948
  * (36) Zulfiqar, Rahman, M.U., Usman, M., Hasanain, S.K., Zia-ur-Rahman, Ullah, A., Kim, I.W.: Static magnetic properties of Maghemite nanoparticles. J Korean Phys Soc. 65(11), 1925–1929 (2014). https://doi.org/10.3938/jkps.65.1925
  * (37) Kim, W., Suh, C.-Y., Cho, S.-W., Roh, K.-M., Kwon, H., Song, K., Shon, I.-J.: A new method for the identification and quantification of magnetite–maghemite mixture using conventional X-ray diffraction technique. Talanta 94, 348–352 (2012). https://doi.org/10.1016/j.talanta.2012.03.001
  * (38) Chernyshova, I.V., Jr, M.F.H., Madden, A.S.: Size-dependent structural transformations of hematite nanoparticles. 1\. Phase transition. Phys. Chem. Chem. Phys. 9(14), 1736–1750 (2007). https://doi.org/10.1039/B618790K
  * (39) Elansary, M., Belaiche, M., Ferdi, C.A., Iffer, E., Bsoul, I.: New nanosized Gd–Ho–Sm doped M-type strontium hexaferrite for water treatment application: experimental and theoretical investigations. RSC Adv. 10(42), 25239–25259 (2020). https://doi.org/10.1039/D0RA04722H
  * (40) Morel, A., Le Breton, J.M., Kreisel, J., Wiesinger, G., Kools, F., Tenaud, P.: Sublattice occupation in Sr1-xLaxFe12-xCoxO19 hexagonal ferrite analyzed by Mössbauer spectrometry and raman spectroscopy. J. Magn. Magn. Mater. 242-245, 1405–1407 (2002). https://doi.org/10.1016/S0304-8853(01)00962-3
  * (41) Kreisel, J., Lucazeau, G., Vincent, H.: Raman study of substituted barium ferrite single crystals, BaFe12-2xMexCoxO19 (Me=Ir, Ti). J. Raman Spectrosc. 30(2), 115–120 (1999). https://doi.org/10.1002/(SICI)1097-4555(199902)30:2<115::AID-JRS354>3.0.CO;2-D
  * (42) Nethala, G.P., Tadi, R., Gajula, G.R., Chidambara Kumar, K.N., Veeraiah, V.: Investigations on the structural, magnetic and mossbauer properties of cerium doped strontium ferrite. Physica B Condens. Matter 550, 136–144 (2018). https://doi.org/10.1016/j.physb.2018.08.035
  * (43) Kreisel, J., Lucazeau, G., Vincent, H.: Raman spectra and vibrational analysis of BaFe12O19 hexagonal ferrite. J. Solid State Chem. 137(1), 127–137 (1998). https://doi.org/10.1006/jssc.1997.7737
  * (44) Dar, M.I., Shivashankar, S.A.: Single crystalline magnetite, maghemite, and hematite nanoparticles with rich coercivity. RSC Adv. 4(8), 4105–4113 (2013). https://doi.org/10.1039/C3RA45457F
  * (45) Santillan, J.M.J.e.a.: Optical and magnetic properties of Fe nanoparticles fabricated by femtosecond laser ablation in organic and inorganic solvents. Chem. Phys. Chem 18(9), 1192–1209 (2017). https://doi.org/10.1002/cphc.201601279
  * (46) Lopez, L.B.M., Pasteris, J.D., Biswas, P.: Sensitivity of micro-Raman spectrum to crystallite size of electrospray-deposited and post-annealed films of iron-oxide nanoparticle suspensions. Appl. Spectrosc. 63(6), 627–635 (2009)
  * (47) Serrano, A., García-Martín, E., Granados-Miralles, C., López-Sánchez, J., Gorni, G., Quesada, A., Fernández, J.F.: Effect of organic solvent on the cold sintering processing of SrFe12O19 platelet-based permanent magnets. J. Eur. Ceram. Soc. 42(3), 1014–1022 (2022). https://doi.org/10.1016/j.jeurceramsoc.2021.10.062
  * (48) Berry, F.J., Marco, J.F., Ponton, C.B., Whittle, K.R.: Preparation and characterization of rare earth-doped strontium hexaferrites Sr1-xMxFe12O19 (M=La, Eu). J. Mater. Sci. Lett. 20(5), 431–434 (2001). https://doi.org/10.1023/A:1010950430100
  * (49) Menil, F.: Systematic trends of the 57Fe Mössbauer isomer shifts in (FeOn) and (FeFn) polyhedra. Evidence of a new correlation between the isomer shift and the inductive effect of the competing bond T-X (→ Fe) (where X is O or F and T any element with a formal positive charge). J. Phys. Chem. Solids 46(7), 763–789 (1985). https://doi.org/10.1016/0022-3697(85)90001-0
  * (50) Roca, A.G., Marco, J.F., Morales, M.d.P., Serna, C.J.: Effect of nature and particle size on properties of uniform magnetite and maghemite nanoparticles. J. Phys. Chem. C. 111(50), 18577–18584 (2007). https://doi.org/10.1021/jp075133m
  * (51) Fernandez, C.d.J., Sangregorio, C., Figuera, J.d.l., Belec, B., Makovec, D., Quesada, A.: Topical review: Progress and prospects of hard hexaferrites for permanent magnet applications. J Phys D Appl Phys (2020). https://doi.org/%****␣SFOthinfilm.bbl␣Line␣875␣****10.1088/1361-6463/abd272