# Mimicking states with limited resources:
passing quantum quiz via global control

P. V. Pyshkin<EMAIL_ADDRESS>Department of Physical Chemistry,
University of the Basque Country UPV/EHU, 48080 Bilbao, Spain  E. Ya. Sherman
<EMAIL_ADDRESS>Department of Physical Chemistry, University of the
Basque Country UPV/EHU, 48080 Bilbao, Spain  EHU Quantum Center, University of
the Basque Country UPV/EHU  Ikerbasque, Basque Foundation for Science, 48011
Bilbao, Spain  A. Gábris<EMAIL_ADDRESS>Czech Technical
University in Prague, Faculty of Nuclear Sciences and Physical Engineering
Břehová 7, 115 19 Praha 1, Staré Město, Czech Republic.  Institute for Solid
State Physics and Optics, Wigner Research Centre, P.O. Box 49, H-1525
Budapest, Hungary  Lian-Ao Wu  Department of Physics, University of the Basque
Country UPV/EHU, 48080 Bilbao, Spain  EHU Quantum Center, University of the
Basque Country UPV/EHU  Ikerbasque, Basque Foundation for Science, 48011
Bilbao, Spain<EMAIL_ADDRESS>
###### Abstract

Precise control of quantum systems with moderate number of degrees of freedom,
being of interest for application of quantum technologies, becomes
experimentally feasible. Various types of quantum scenarios and protocols are
being widely discussed in scientific literatures. We propose, analyze, and
optimize a protocol which allows the fast simulation of properties of unknown
quantum states relying on minimum relevant information. Our protocol, having
common features with quantum identification and shortcuts to adiabaticity,
permits avoiding orthogonality catastrophe, where transitions between
physically very similar systems are characterized by zero or a very low
fidelity.

The problem of quantum identification [1, 2, 3, 4] can be described as
follows. Imagine a device that can produce a quantum system in a pure state
$\ket{\psi_{l}}$ belonging to a set of possible states
$\\{\ket{\psi_{1}},\ket{\psi_{2}},\dots,\ket{\psi_{L}}\\}$, where $L$ is the
total number of those states, i.e. $1\leq l\leq L$. The problem is to identify
$l$ by using the minimal possible resources. This task can be modified as
follows: instead of identifying $l$ one should prepare a state with a feature
similar to one of the features characterizing the $\ket{\psi_{l}}$ state. Such
a preparation has to use limited resources: time and information about
$\ket{\psi_{l}}$ and/or a restricted set of permitted quantum operations.
Thus, we consider a quantum control problem [5, 6]. Because time is a limited
resource we notice that our protocol may be related to shortcuts to
adiabaticity [7, 8].

In general in order to determine success metric of quantum protocol one uses
the fidelity $|\braket{\psi}{\psi^{\prime}}|^{2}$, where $\psi^{\prime}$ is a
desired state and $\psi$ is a state which is result of the protocol. However,
such a strict condition of success can be excessive for some applications and
may require an exponential overhead. Therefore we consider the following
scenario in this paper. We have two complex systems, for example, two spin
chains with the same number of spins $N$. The state of the first (second)
chain is $\ket{\psi_{1}}$ ($\ket{\psi_{2}}$), and their fidelity
$|\braket{\psi_{1}}{\psi_{2}}|^{2}=1$. Let us extend both chains by adding to
them one spin in a state $\ket{\phi_{1}}$ ($\ket{\phi_{2}}$) for the first
(second) chain, thus, we have two chains with $N+1$ spins being in the states
$\ket{\Psi_{1}}=\ket{\psi_{1}}\otimes\ket{\phi_{1}}$
($\ket{\Psi_{2}}=\ket{\psi_{2}}\otimes\ket{\phi_{2}}$) for first (second)
chain. In this scenario the common fidelity
$|\braket{\Psi_{1}}{\Psi_{2}}|^{2}=|\braket{\phi_{1}}{\phi_{2}}|^{2}$ is
defined by the fidelity of these additional spins only. If
$\braket{\phi_{1}}{\phi_{2}}=0$ the state of the first $N$ spins in each chain
becomes irrelevant. This is similar to the well-known orthogonality
catastrophe [9, 10]. However, we notice that in quantum control procedures,
especially in the tasks of quantum state preparation it is important to
operate with such a measure of closeness of quantum states which takes into
account separated similarities of different parts of the system (alongside
with the entanglement measure similar to that proposed in [11]). In this
report we introduce and propose to apply a less rigorous than the conventional
fidelity metric to a complex system which consists of distinguishable parts.
Such a metric can be considered as a “quantum quiz” consisting of several
“questions” with answers to each of them being related to the similarity of
the corresponding parts of the system. Optimization of this metric is used as
an intermediate feedback as well as the final goal.

##  1 The setting

In this section we introduce the problem in general terms. Let us assume we
have a complex quantum system which consists of $N$ distinguishable enumerated
parts, and this system is in its ground state $\ket{\psi_{0}}$ of Hamiltonian
$H=H(b_{1},b_{2},\dots,b_{K})$, where $b_{i}$ are unknown parameters, and $K$
is the number of these parameters. This system we call as target system. Let
us now introduce the second system with the same nature, i.e. a system
consisting of $N$ ($N\geq 2$) non-divisional parts with the same dimension of
the Hilbert space for all parts as the target system. For example, if the
target system is $N$ spins-$1/2$, the second system is $N$ spins-$1/2$ (or $N$
qubits) too. Here we call the second system as a candidate system. The
candidate system initially is supposed to be in the ground state
$\ket{\psi^{\prime}_{0}}$ of the Hamiltonian
$H=H(b^{\prime}_{1},b^{\prime}_{2},\dots,b^{\prime}_{K})$. In order to compare
these two states we introduce the similarity measure

$F(\ket{\psi^{[t]}},\ket{\psi^{[c]}})=\sum_{\\{\bm{k}\\}}f_{\\{\bm{k}\\}}({\bm{\rho}}_{\\{\bm{k}\\}}^{[t]},{\bm{\rho}}_{\\{\bm{k}\\}}^{[c]}),$
(1)

where $\\{\bm{k}\\}$ labels a set of subsystems (each set can contain from $1$
to $N-1$ subsystems, see Fig. 1), the maximal number of terms in sum is
$2S(N,2)$, where $S(N,2)=2^{N-1}-1$ is the number of all the possible
bipartitions of $N$ subsystems defined via Stirling number of the second kind,
${\bm{\rho}}_{\\{\bm{k}\\}}^{[t]}$ (${\bm{\rho}}_{\\{\bm{k}\\}}^{[c]}$) is the
reduced density matrix of subsystem $k$ of the target (candidate) system, and
$f_{\\{\bm{k}\\}}$ are functions which can be different for different
subsystems.

Figure 1: Example of choosing subsystems. Here have $N=6$ spins in both target
and candidate systems, and different values of $\\{\bm{k}\\}$ in (1)
correspond to sets of spins including $\\{\bm{k}\\}=\\{2,3,5\\}$ (with
$\overline{\\{\bm{k}\\}}=\\{1,4\\}$) (panel a), and set of spins
$\\{\bm{k}\\}=\\{4,5\\}$ (with $\overline{\\{\bm{k}\\}}=\\{1,2,3\\}$) (panel
b).

The reduced density matrices of subsystems $\\{\bm{k}\\}$ are defined as
partial traces over all other subsystems except the $\\{\bm{k}\\}$th set :

${\bm{\rho}}_{\\{\bm{k}\\}}^{[t,c]}=\mathop{\mathrm{tr}}_{\overline{\\{\bm{k}\\}}}\ket{\psi^{[t,c]}}\bra{\psi^{[t,c]}},$
(2)

see Fig. 1. We assume that the similarity $F$ reaches the maximum at
${\bm{\rho}}_{\\{\bm{k}\\}}^{[t]}={\bm{\rho}}_{\\{\bm{k}\\}}^{[c]},\forall\\{\bm{k}\\}$,
although this is not a necessary condition to achieve the maximum.

Next we assume that there is a limited resource- “black-box” device which can
compare the candidate with target systems, and output values of the similarity
(1). The task is to implement some unitary operation $U$ to candidate system
in order to have
$F(\ket{\psi_{0}^{[t]}},U\ket{\psi_{0}^{[c]}})>F(\ket{\psi_{0}^{[t]}},\ket{\psi_{0}^{[c]}})$
with choosing $U$ in the most optimal way to have

$F(\ket{\psi_{0}}^{[t]},U\ket{\psi_{0}^{[c]}})=\max_{U^{\prime}}F(\ket{\psi_{0}^{[t]}},U^{\prime}\ket{\psi_{0}^{[c]}}),$
(3)

where $U^{\prime}$ belongs to a family of allowed for implementation unitary
transformations. We assume that $U=U_{M}U_{M-1}\dots U_{1}$. This means, that
it is possible to apply $M$ different unitary operations, and after applying
each of them we use the black-box in order to estimate the current value of
similarity (1) which can be used to choose the next unitary operation. We
assume all these unitary operators $U_{j}$ ($j=1,2,\dots M$) are generated by
special-types of Hamiltonians with fixed or limited evolution time or energy.
In analogy with “quantum quiz”, we allow to reexaminate $M$ times, and our
goal is to increase similarity (1) as much as possible with $M$ being as less
as possible.

The definition (1) can be considered as a final grade of “quantum quiz”
consisting of $2S(N,2)$ questions, where each “question” is related to a
certain part of complex quantum system. Similar kind of measure which involves
density matrices of parts of quantum system has been presented earlier in Ref.
[11] to analyze entanglement properties. In the next section we present and
consider in detail an example of the problem described above.

##  2 Mimicking a spin chain

###  2.1 Analytical procedure

As an example we consider periodic spin-1/2 chain of $N$ spins with the
following Hamiltonian

$H=\sum_{k=1}^{N}(X_{k}+b_{k}Y_{k}+JZ_{k}Z_{k+1}),$ (4)

where $X,Y,Z$ are Pauli matrices and the first and last sites to be adjacent
$N+1\equiv 1$. Further we assume antiferromagnetic $J=1$.

Figure 2: $N=4$ periodic spin chains as a target (upper panel) and candidate
(lower panel) systems.

In this example we will only compare each spin in the target system with the
corresponding one in the candidate system, thus we can mark $\\{\bm{k}\\}=k$,
where $1\leq k\leq N$ in (1),

$F(\ket{\psi^{[t]}},\ket{\psi^{[c]}})=\sum_{k=1}^{N}\cos\theta_{k},$ (5)

where $\theta_{k}$ is the angle between the Bloch vectors of the $k$th target
spin and $k$th candidate spin (see Fig. 2). Below we show that
$\cos\theta_{k}$ is a function of ${\bm{\rho}}_{k}^{[t]}$ and
${\bm{\rho}}_{k}^{[c]}$ as in definition (1).

The density matrices (2) of spins are expressed in terms of their Bloch
vectors

$\theta_{k}=\widehat{\bm{r}_{k}\bm{c}_{k}},\quad{\bm{\rho}}^{[t]}_{k}=\frac{\mathbb{I}+\bm{r}_{k}\bm{\sigma}_{k}}{2},\quad{\bm{\rho}}^{[c]}_{k}=\frac{\mathbb{I}+\bm{c}_{k}\bm{\sigma}_{k}}{2},$
(6)

where $\bm{r}_{k}$ ($\bm{c}_{k}$) is the Bloch vector of the $k$th spin of the
target (candidate) system, and $\bm{\sigma}_{k}=(X_{k},Y_{k},Z_{k})$ is the
Pauli vector for the $k$th spin.

We now consider a family of the allowed unitary operations in (3) determined
by $U=\exp(-iH_{1}t)$ generated by the Hamiltonian

$H_{1}=\bm{B}\cdot\sum_{k=1}^{N}\bm{\sigma}_{k},$ (7)

where $\bm{B}$ is a vector parameter the same for all spins [12]. Since the
Hamiltonian (7) is not assumed to be able to provide complete control [13],
and thus we do not expect the maximal possible similarity as a result of our
algorithm. We assume $\bm{B}=B\bm{b}$, with $B\geq 0$ and $|\bm{b}|=1$.

Notice that for our setting since the Bloch vectors do not have $z$-component,
this allows us to use $\bm{b}=(0,0,\pm 1)$ in order to have spin rotations
generated by (7) only in the $XY$ plane. Here we present main algebraic
relations which will be used to understand the optimization and the numerical
simulation in the next subsection.

In order to show that our choice of similarity (5) does not contradict the
definition (1) we show that the cosine function of the angle $\theta_{k}$
between average spins number $n$ of the target and candidate systems can be
expressed in the following way. From the definition of the scalar product we
have

$\cos\theta_{k}=\frac{\bm{r}_{k}\bm{c}_{k}}{{r}_{k}{c}_{k}}.$ (8)

With using (6) and algebra of the Pauli matrices we write

$\mathop{\mathrm{tr}}{\bm{\rho}}^{[t]}_{k}{\bm{\rho}}^{[c]}_{k}=\frac{1+\bm{r}_{k}\bm{c}_{k}}{2}.$
(9)

Then we combine (8) and (9) and have

$\cos\theta_{k}=\frac{2\mathop{\mathrm{tr}}({\bm{\rho}}^{[t]}_{k}{\bm{\rho}}^{[c]}_{k})-1}{\sqrt{2\mathop{\mathrm{tr}}\left({\bm{\rho}}^{[c]}_{k}\right)^{2}-1}\sqrt{2\mathop{\mathrm{tr}}{\left({\bm{\rho}}^{[t]}_{k}\right)}^{2}-1}},$
(10)

where all rotations generated by (7) do not affect the denominator in (10).

For a global rotation governed by (7) we introduce $\chi=Bt$ and obtain
$V=\exp(-iH_{1}t)$ $=\prod_{k=1}^{N}V_{k}$
$=\prod_{k=1}^{N}\exp(-i\bm{b}\bm{\sigma}_{k}\chi)$, and thus

$\mathop{\mathrm{tr}}_{\overline{k}}(V\ket{\psi}\bra{\psi}V^{\dagger})=V_{k}{\bm{\rho}}_{k}V_{k}^{\dagger}.$

This allows us to express the changes of similarity after the rotation applied
to the candidate system in terms of the Bloch vectors before the rotation and
parameters ${\bm{b}}$, $t$ and $B$

$F(\ket{\psi^{[t]}},V\ket{\psi^{[c]}})-F(\ket{\psi^{[t]}},\ket{\psi^{[c]}})=\\\
2\sin\chi\sum_{k=1}^{N}\sqrt{\left(1-\left(\frac{\bm{c}_{k}}{c_{k}}\bm{b}\right)^{2}\right)\left(1-\left(\frac{\bm{r}_{k}}{r_{k}}\bm{b}\right)^{2}\right)}\sin(\chi+\gamma_{k}),$
(11)

with

$\gamma_{k}=\arccos\frac{(\bm{c}_{k}\bm{b})(\bm{r}_{k}\bm{b})-\bm{c}_{k}\bm{r}_{k}}{\sqrt{\left(c_{k}^{2}-\left({\bm{c}_{k}}\bm{b}\right)^{2}\right)\left(r_{k}^{2}-\left({\bm{r}_{k}}\bm{b}\right)^{2}\right)}}$
(12)

where we use (10) and
$\exp(-i\bm{b}\bm{\sigma}_{k}Bt)=\mathbb{I}\cos(Bt)-i(\bm{b}\bm{\sigma}_{k})\sin(Bt)$.
In our system we have $\bm{b}\perp\bm{c}_{k},\bm{r}_{k}$, and thus square root
in (11) is equal to $1$ and
$\gamma_{k}=\arccos(-(\bm{c}_{k}\bm{r}_{k})/(c_{k}r_{k}))=\pi-\theta_{k}$, so
that (11) can be rewritten as

$F(\ket{\psi^{[t]}},V\ket{\psi^{[c]}})-F(\ket{\psi^{[t]}},\ket{\psi^{[c]}})=\\\
2\sin\chi\sum_{k=1}^{N}\sin(\theta_{k}-\chi).$ (13)

Notice that if $\theta_{k}=\theta,\forall k$, then (13) gives the condition
for the maximal increase in the similarity: $\chi=\theta/2$, meaning that we
have to consider positive angles $\theta_{k}$ for the case when vector
$\bm{c}_{k}$ must be rotated counter clock-wise in order to decrease
$\theta_{k}$.

For simplification, we further assume possible values of parameters in (4) are
discrete (each $b_{k}$ can take only $D$ different values) and relatively
small, for example, $|b_{k}|\leq 1$, and thus we expect $\theta_{k}<2\arctan
1=\pi/2.$

Figure 3: Optimal rotation angle as a function of similarity from the black-
box query.

We now propose the following protocol for increasing $F$ with just a single
query $M=1$ to the black-box:

1\. Let us choose the initial state $\ket{\psi_{0}^{[c]}}$ with all the Bloch
vectors having direction corresponding to $b_{k}=b_{\min}\;\forall k$, where
$b_{\min}$ is the minimal possible value of parameters $b_{k}$.

Before the query to black-box for the value of $F$ (5) we organize a special
statistics for all possible target states. This can be done without using
resources of black-box. For each possible target state $\psi_{0}^{[t]}$ we
calculate angle $\chi_{\rm opt}$ which corresponds to the optimal global
rotation

$U=\exp\left(-i\chi_{\rm opt}\sum_{k}Z_{k}\right),$ (14)

which maximizes $F(\ket{\psi_{0}}^{[t]},U\ket{\psi_{0}^{[c]}})$. The optimal
angle is given by the circular mean

$\chi_{\rm
opt}=\frac{1}{2}\arctan\left(\frac{1}{F}\sum_{k=1}^{N}\sin\theta_{k}\right),$
(15)

where $F$ is the similarity from (5). Expression (15) can be obtained by
setting the derivative of (13) to zero with respect to $\chi$. For large
possible values of $|b_{k}|>1$ the similarity $F$ can become negative, and one
should use the function ${\mathrm{atan2}}(\sum_{k=1}^{N}\sin\theta_{k},F)$
instead of (15).

2\. Ask black-box for the similarity $F$.

3\. After obtaining the result, by using previously calculated data set to
find the corresponding $\chi_{\rm opt}=\chi_{\rm opt}(F)$ and produce rotation
(14).

###  2.2 Numerical analysis

We restrict ourself by choosing $N=K=4$ and the values of $b_{k}$ from the set
$\\{-0.5,-0.25,0,0.25,0.5\\}$. Following the protocol we choose the initial
state $\ket{\psi^{[c]}_{0}}$ corresponding to $b_{k}=-0.5\;\forall k$.

We gave numerical simulation for all possible $D^{N}=625$ target states and
for each of these states we found optimal rotation angle $\chi$ as shown in
Fig. 3. For each target state we calculated the increase in the similarity
$\Delta F$ after optimal rotation (4). Increase in the similarity averaged
over all possible target states approximates to $0.45$. It is obvious that for
some target states the increase in the similarity is maximally possible, and
corresponding points are lying on the line $\Delta F+F=N$.

It is hard to see from Fig. 3 that the realization $\chi_{1}(F)=\chi_{2}(F)$
cannot occur for $\chi_{1}\neq\chi_{2}$. However, the black-box operation can
involve some noise, i.e. instead of the exact similarity $F$ we can get a
somewhat different $F^{\prime}$ with $|F-F^{\prime}|<\varepsilon$, where
$\varepsilon$ is an error. In such a case the increase of similarity will not
be a maximal $\Delta F$ as in Fig.4. In other words, we might erroneously use
an imprecise point $(F,\chi)$ from Fig. 3 or corresponding table, and make
non-optimal rotation (14). However, our estimation shows that the average
error in $\chi$ is only $\approx 0.06$ for $\varepsilon=0.05$, and $\approx
0.08$ for $\varepsilon=0.1$ which is much less than the median value of
$\chi$, $\chi_{\rm max}/2=\arctan 0.5$.

Figure 4: Increase in the similarity $\Delta\,F$ after optimal global
rotation.

##  3 Discussion

A possible experimental realization of our model is described below. To begin
with, let us assume that we have two copies of candidate system. In the limit
$J\ll 1$ the states of the spin chains are almost disentangled, and in the
limit $N\gg 1$ our black-box can be realized via projective measurements on
the candidate system. In this situation the state of the $k$th spin of the
candidate (as well as of the target) system is approximately pure, i.e.
$\ket{\psi_{0}^{[c,t]}}\approx\ket{\psi_{1}^{[c,t]}}\otimes\ket{\psi_{2}^{[c,t]}}\otimes\dots\ket{\psi_{N}^{[c,t]}}$,
and for the candidate system we can write

$\ket{\psi_{k}^{[c]}}=\cos(\theta_{k}/2)\ket{\psi_{k}^{[t]}}+\sin(\theta_{k}/2)e^{i\phi_{k}}\ket{\psi_{\perp
k}^{[t]}},$

with $\braket{\psi_{k}^{[t]}}{\psi_{\perp k}^{[t]}}=0$, and where $\phi_{k}$
is a unimportant phase for our purposes. Notice, in this case
$\cos\theta_{k}=2f_{k}-1$, where $f_{k}$ is a commonly defined fidelity
$f_{k}=|\braket{\psi_{k}^{[t]}}{\psi_{k}^{[c]}}|^{2}$. We apply projective
measurement $\ket{\psi_{k}^{[t]}}\bra{\psi_{k}^{[t]}}$ to each spin number $k$
of the candidate system. The result of each measurement is stored in a
variable $m_{k}$: if the result of measurement of the $k$th spin is “yes” then
$m_{k}=1$ otherwise $m_{k}=0$. The probability of getting $m_{k}=1$ is
$\cos^{2}(\theta_{k}/2)=(\cos\theta_{k}+1)/2$. Thus we have

$F\approx 2\sum_{k=1}^{N}m_{k}-N.$

Due to the probabilistic nature of the black-box we can expect the dispersion
(or the error) of $F$ being of the order of $\sqrt{N}\ll N$, where we assume
that the black-box does not require to have target system physically. This
system can be “programmed” into the black-box by the input of the $b_{k}$ set.
The proposed scenario can be assumed as some kind of the “global quantum state
tomography” which requires only one copy of the large system and which output
is the similarity value. Notice that under the limit $J\ll 1$ it is easy to
compute data $\chi(F)$ (see Fig. 3) for large $N$, considering spins being
independent.

Such a black-box for small systems can be realized experimentally by quantum
state tomography [14] with many copies of the target-candidate systems. The
reduced density matrices are therefore given, so is the similarity $F$. As a
result, the operations with this black-box become a limited resource.

The above discussed problem can be utilized, for example, in the scenario when
the corresponding spins in two chains (candidate and target) become connected
to each other while internal interaction $J$ inside each chain is switched off
(see Fig. 5), thus forming ladders of dimers (see, e.g., Ref. [15]). If the
Hamiltonian of each dimer is ferromagnetic, we see that the maximal similarity
before the transformation corresponds to the minimal energy of the system of
dimers.

Figure 5: Spin chains transformed into array of dimers.

We also would like to mention that the similarity (1) which takes into account
some entanglement properties can use, for example, these functions
$f_{\\{\bm{k}\\}}$

$f_{\\{\bm{k}\\}}=1-\left[\mathop{\mathrm{tr}}\left({\bm{\rho}}_{\\{\bm{k}\\}}^{[t]}\right)^{2}-\mathop{\mathrm{tr}}\left({\bm{\rho}}_{\\{\bm{k}\\}}^{[c]}\right)^{2}\right]^{2},$
(16)

corresponding to the maximal similarity when entropy (purity) of each subset
of the candidate system is equal to entropy (purity) of the corresponding
subset of the target.

##  4 Conclusion

We have studied the manipulation of similarity of quantum states in spin
systems by external magnetic field and demonstrated that even a single step of
global rotation can improve similarity related to an unknown state in reality.
More significantly, the only information required from the unknown state of a
target system is single time we asking the value of similarity (5) which does
not provide us information about the structure of the target state. Thus, we
demonstrated here by a model analysis the possibility of relatively “blind”
accurate mimicking of the initially unknown state by using limited resources.
Another important goal of this research is to formulate a problem of
increasing the specially defined similarity of a complex system. Such a
similarity definition using a summation over subsystems can avoid
orthogonality catastrophe which can appear with the commonly defined fidelity,
and thus it can be useful for quantum protocols. Finding the optimal solution
of the proposed problem without limitation of simplified global control can be
of interest for the future analytical and numerical analysis.

## Acknowledgments

This work is financially supported through the Grant PGC2018-101355-B-I00
funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”,
and by the Basque Government through Grants No. IT986-16 and IT1470-22.

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