# Bi-Quadratic Improvement in Conditional Quantum Search

Akankshya Dash  School of Applied Sciences, Kalinga Institute of Industrial
Technology (KIIT) University,
Bhubaneswar, 751024, Odisha, India<EMAIL_ADDRESS>Biswaranjan Panda
<EMAIL_ADDRESS>Indian Institute of Science Education and
Research (IISER),
Berhampur, 760005 Odisha, India  Arun K Pati<EMAIL_ADDRESS>Center for
Quantum Science and Technology (CQST), International Institute of Information
Technology (IIIT),
Hyderabad, 500032, Telangana

###### Abstract

The Grover search algorithm performs an unstructured search of a marked item
in a database quadratically faster than classical algorithms and is shown to
be optimal. Here, we show that if the search space is divided into two blocks
with the local query operators and the global operators satisfy certain
condition, then it is possible to achieve an improvement of bi-quadratic
speed-up. Furthermore, we investigate the bi-quadratic speed-up in the
presence of noise and show that it can tolerate noisy scenario. This may have
potential applications for diverse fields, including database searching, and
optimization, where efficient search algorithms play a pivotal role in solving
complex computational problems.

††preprint: APS/123-QI

##  I Introduction

Quantum computing has emerged as a transformative paradigm, promising to
revolutionize various domains by solving problems that were once considered
computationally intractable for classical computers. Quantum computers utilize
the unique properties of quantum mechanics to perform computations at an
unprecedented speed and efficiency compared to classical computers [1, 2, 3].
However, the technology is still in its infancy, and there remain significant
challenges to be addressed, including error correction [4], scalability [5],
and reliability [6]. Despite of these challenges, the rapid pace of innovation
and development in quantum computing suggests that it will undoubtedly
revolutionize the way we process and communicate information in the future.

Quantum algorithms have showcased their performance in quantum speed up, big
data analysis, optimization, simulation and cryptographic security. Quantum
search algorithm [7, 25, 9, 10, 11] can be potentially applied for efficiently
counting the number of solutions for a given search problem, speedup the
solution of NP-complete problems, speed up the search for keys to
cryptosystems [12], finding the shortest path between two cities [13], and
searching problems or extracting statistics in unstructured databases [14]
faster than the classical search.

The classical brute-force approach to searching an unsorted database requires
examining each item one by one, leading to a linear time complexity. Grover’s
algorithm [7], in stark contrast, offers a quadratic speedup, fundamentally
altering the landscape of search algorithms. By utilizing quantum
superposition and interference, it efficiently identifies the target item from
an unstructured database with remarkable speed-up, making it a promising
solution to some of the most complex search problems.

Over the year, there have been several development in the realm of Grover’s
algorithm. One noteworthy development is the incorporation of structured
search [15], where the overall success rate is contingent upon the
multiplication of individual search success rates. This is particularly
pivotal in scenarios characterized by relatively modest dimensions, as the
conventional Grover’s algorithm may exhibit suboptimal performance under such
circumstances. Consequently, to address this challenge, a series of adapted
search algorithms have been introduced [18, 19, 20, 21, 22, 23, 24]. Also, the
role of quantum entanglement in the Grover algorithm has been explored and it
was shown that the quantum feature such as entanglement is neccessary for the
desired speed-up [25, 26].

In the pursuit of enhancement of algorithmic efficiency, our study delves into
the intricate realm of the Grover search algorithm. Our quest involves
strategically partitioning of the search space into two discrete blocks,
coupled with the imposition of specific condition upon local query operators.
The perceptive outcome of our study unveils a noteworthy prospect with an
attainment of bi-quadratic speed-up. Moreover, our analysis extends to
scrutinize the algorithm’s resilience in the achievement of bi-quadratic
speed-up in the presence of noise, demonstrating its capacity to tolerate
noisy scenarios. Our inquiry is directed towards the examination of the
behavior of the ultimate state when subjected to the perturbing effects of a
noisy channel, exemplified by the amplitude damping channel [27, 28, 29, 30].
Considering the intrinsic multilevel nature of our system, we utilize the
multi-level amplitude damping channel [31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47], specifically designed to illuminate the decay of
energy levels within higher dimensional of quantum system. These advancements
resonate across diverse applications in fields such as database searching and
optimization, underscoring the algorithm’s pivotal role in addressing complex
computational challenges.

This paper is structured as follows: Firstly, in section-II, an overview of
the Grover algorithm is presented and concurrently we introduce a resilient
variant of the conditional quantum search, featuring a bi-quadratic speed-up.
This speed-up is achieved when the search space is partitioned into two
blocks, and the local query operators adhere to specific conditions for both
pure and mixed states. In section-III, we delve into an examination of the bi-
quadratic speed-up in the presence of noise. We showcase its capacity to
endure and remain effective in a noisy scenario. In section-IV, we conclude
our findings.

##  II Bi-Quadratic Improvement in Quantum Search

In search algorithm, we have a function $f(x)$ which results a value $f(x)=1$,
if $x$ is the search item. If the item $x$ is not a solution to the search
problem, then $f(x)=0$. The search problem aims to find find an item $y$ such
that $f(y)=1$. In the Grover algorithm we have a classical function
$f(x):\\{0,1,2,\cdots,N-1\\}\rightarrow\\{0,1\\}$, where $N$ is the size of
the database. Our goal is to find $y$ such that $f(y)=1$.

The Grover algorithm fundamentally entails the execution of a series of Grover
operators, ultimately finding the desired target state in the requisite number
of iterations. Each Grover operator corresponds to a subtle rotation within
the two-dimensional subspace. For $n$ qubit system, we take the initial state
as $|000...0\rangle$ and to make it equal amplitude we apply the Hadamard
transformation to maintain superposition on every qubit as expressed by

$|\Psi_{0}\rangle=H^{\otimes
n}|000...0\rangle=\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}|x\rangle,$ (1)

where $N$ = 2n and $H^{\otimes n}$ is the Hadamard transformation. After
applying the Hadamard transformation, we apply the Grover Operator $G$ as
given by

$G=-I_{0}I_{y},$ (2)

where $I_{y}=(I-2|y\rangle\langle y|)$ and
$I_{0}=(I-2|\Psi_{0}\rangle\langle\Psi_{0}|)$. Geometrically, the Grover
operator, $G$ rotates the initial state towards the target state $|y\rangle$.
As we proceed through the $k$th iteration of the Grover operator, we achieve
convergence towards our desired state an approximate order of $O$($\sqrt{N}$)
steps. This outcome is particularly remarkable when contrasted with classical
search algorithms. The ensuing state is depicted as

$|\Psi_{k}\rangle=\frac{cos\theta_{k}}{\sqrt{N-1}}\sum_{x\neq
y}^{N-1}|x\rangle+sin\theta_{k}|y\rangle,$ (3)

where $\theta_{k}$ = $(2k+1)\theta$ and $\sin\theta=\frac{1}{\sqrt{N}}$. For
$k$ = $O(\sqrt{N})$, we reach the target state $|y\rangle$. This gives
quadratic speed up for the Grover algorithm.

Since it has previously been established that surpassing the quadratic speedup
achieved by Grover’s algorithm in a general case is unattainable, one might
ponder: What happens if we were to introduce specific conditions in the search
item? To answer this question, we find that it is possible to go beyond
$O(\sqrt{N})$ by imposing a particular condition, which is succinctly
elucidated in the ensuing sections.

###  II.1 Conditional Quantum Search for Pure State

Here, we will introduce the Conditional Quantum Search (CQS) to enhance the
speed up of our search within an unsorted database. This constraint imparts a
notable improvement, significantly amplifying the speed-up achieved in our
search endeavors. This augmentation assumes a heightened significance,
especially when our quest aligns with the prescribed criteria, as it empowers
us to harness the capabilities of the conditional quantum search algorithm.

Imagine that the marked item has an inherent structure and we are able to
partition our target state into two discernible components. For example, the
telephone directory has a person’s name which has the first name and the
surname. One can directly search the full name and also search the first name
and the surname separately. In the later case, the search space is divided
into two blocks and the combination of marked items from both the database
give the marked item in the larger database. Under such scenario, we can then
effectively apply our tailored searching algorithm. We have provided one
example here, but practical instances are abound where this condition can be
met. To expound upon this, in any scenario where our target item is denoted as
$ab$, should we possess the capability to disentangle and isolate the
constituents $a$ and $b$ within our search problem, the efficacy of our
algorithm becomes readily applicable and will be notably advantageous.

Let us consider a quantum computer with $2n$ qubits. With this quantum
resister, we can encode $2^{2n}$ = $N^{2}$ entries of a database. Let the
marked item is denoted as $|Y\rangle$. If we apply the Grover search operator
in this context, knowing that it offers quadratic speed-up, we can achieve our
target state in just $O$($N$) steps. In the sequel, we can achieve a bi-
quadratic speed-up if the target state can be decomposed into two separate
parts using conditional quantum search (CQS).

Now, we divide the $2n$ qubits into two blocks, each with $n$ qubits. Let us
imagine that the target item encoded in the state
$|Y\rangle=|1\rangle^{\otimes 2n}$ has an inherent structure, i.e., the target
item $|Y\rangle$ is combination of target items for two small blocks. For
simplification, let us denote the target item for each block as $|y\rangle$ =
$|111.....11\rangle=|1\rangle^{\otimes n}$ and $|y\rangle$ =
$|111.....11\rangle=|1\rangle^{\otimes n}$ with the condition that $|Y\rangle$
= $|y\otimes y\rangle$. We also equipped with local query operators, $I_{y}$ =
$I-2|y\rangle\langle y|$ in each block as well as the global query, $I_{Y}$ =
$I-2|Y\rangle\langle Y|$. Note that $I_{y}\otimes I_{y}$ $\neq I_{Y}$.
Nevertheless, we have

$I_{y}\otimes I_{y}|Y\rangle=-I_{Y}|Y\rangle,$

i.e., $I_{Y}\otimes I_{Y}|Y\rangle$ = $I_{Y}|Y\rangle$ up to a phase. This
symmetry matching condition helps us to obtain a bi-quadratic improvement in
the conditional quantum search.

Now, consider the initial state of $2n$ qubits as given by

$|\Psi_{0}\rangle=H^{\otimes n}|0\rangle^{\otimes n}\otimes H^{\otimes
n}|0\rangle^{\otimes n}.$ (4)

Next, we apply $G\otimes G$ operator $k$ times on the initial state
$|\Psi_{0}\rangle$ and obtain

$\displaystyle|\Psi_{k}\rangle=$
$\displaystyle\left(\cos\theta_{k}|\overline{y}\rangle+\sin\theta_{k}|y\rangle\right)$
(5)
$\displaystyle\otimes\left(\cos\theta_{k}|\overline{y}\rangle+\sin\theta_{k}|y\rangle\right),$

where $|\overline{y}\rangle=\frac{1}{\sqrt{N-1}}\sum_{x\neq y}|x\rangle$.

At last, we apply the $I_{Y}$ operator, which acts as a reflection operator
for the target state $|Y\rangle=|y\rangle\otimes|y\rangle$. Thus, the final
state is given by

$\displaystyle|\Psi_{k}\rangle$ $\displaystyle=I_{Y}|\Psi_{k}\rangle$
$\displaystyle=(\cos\theta_{k}|\overline{y}\rangle+\sin\theta_{k}|y\rangle)\otimes(\cos\theta_{k}|\overline{y}\rangle+\sin\theta_{k}|y\rangle)$
$\displaystyle\quad-2\sin^{2}\theta_{k}|Y\rangle$ (6)
$\displaystyle=cos^{2}\theta_{k}|\overline{y}\rangle|\overline{y}\rangle-
sin^{2}\theta_{k}|Y\rangle+sin\theta_{k}cos\theta_{k}(|\overline{y}y\rangle+|y\overline{y}\rangle).$
(7)

Here, the application of $I_{Y}$ entangles two blocks. If we measure the
target state $|Y\rangle$, we will find the marked item. From Eq.(II.1), we see
that to find the measured item $|Y\rangle$, we have used the oracle $I_{y}$
$O$($2\sqrt{N}$) times and $I_{Y}$ once. Therefore this shows that by
application of Grover Oracle $O$($2\sqrt{N}+1$) $\approx$ $O$($\sqrt{N}$) we
are able to find marked item. Without the matching condition it will take
$O$($N$) steps. Classically, it takes $O$($N^{2}$) steps. Hence, our approach
has the capability to identify designated items with a speed enhancement of
$O$($N^{1/4}$).

###  II.2 Conditional Quantum Search For Mixed State

Here, we will discuss how the conditional quantum search is affected if the
initial state is not pure but a mixed state. For the sake of illustration, we
consider a pseudo-pure state which is an admixture of pure state and a random
state. This kind of state arises naturally in many noisy scenarios. Suppose
the noise parameter, denoted as $\epsilon$, is introduced into the density
matrix of our quantum system. The density matrix of the pseudo-pure state for
$2n$ qubits can be expressed as

$\rho_{0}=\epsilon|\Psi_{0}\rangle\langle\Psi_{0}|+(1-\epsilon)\frac{I}{N^{2}},$
(8)

where the initial state is given in Eq. (4). Now, we apply the Grover operator
$(G\otimes G)$ on the density operator $\rho_{0}$ $k$ times. After applying
the Grover operator to the above density operator, we obtain

$\rho_{k}=(G\otimes G)\rho_{0}(G\otimes G).$ (9)

At last we apply the global query operator $I_{Y}$ which entangles the two
block of quantum registrars. The final state is given by

$\displaystyle\rho_{k}^{\prime}=$ $\displaystyle I_{Y}(G\otimes
G)\rho_{0}(G\otimes G)I_{Y}$ (10)
$\displaystyle=\epsilon|\Psi_{k}\rangle\langle\Psi_{k}|+(1-\epsilon)\frac{I}{N^{2}}.$

Now we measure our target state $|Y\rangle$. The probability of finding the
target state $|Y\rangle$ can be calculated by applying the measurement
operator on $\rho_{k}^{\prime}$ and can be expressed as

$P(|Y\rangle)=Tr[\Pi_{Y}(\rho_{k}^{\prime})]=Tr[|Y\rangle\langle
Y|(\rho_{k}^{\prime})]$ $=(1-\epsilon)+\epsilon sin^{4}\theta_{k}.$ (11)

The above equation expresses the effect of noise at $k$th iteration. For
$k=O(\sqrt{N})$, the effect of noise cancels and we can find the marked item
with unit probability.

##  III Bi-Quadratic Speed-up in Noise Scenario

In this section, we delve into the behavior of the final state
$|\Psi_{k}\rangle$ in the presence of a noisy channel, such as the amplitude
damping channel (ADC). Given that our system is inherently multi-level, we
employ the multi-level amplitude damping (MAD) channel (referenced as [24]),
specifically tailored to elucidate the energy level decay within a higher
dimensional quantum system. The general basis of this channel is represented
as $\\{|i\rangle\\}$, where $i$ = 0, 1, 2…, $d-1$. An MAD channel
$\mathcal{D}$ is a completely positive trace-preserving mapping that operates
on the set $L({\cal H}_{A})$ of linear operators on the quantum system ${\cal
H}_{A}$. This channel is characterized by a set of Kraus operators, which
define the evolution of the system under the influence of the channel. The MAD
channel $\mathcal{D}$ and Kraus operators are given by

$\mathcal{D}(\rho)=K_{0}\rho K_{0}^{\dagger}+\sum_{0\leq i\leq j\leq
d-1}\eta_{ji}|i\rangle\langle i|\langle j|\rho|j\rangle,$ (12)

where $K_{0}=|0\rangle\langle 0|+\sum_{1\leq j\leq
d-1}\sqrt{1-\kappa_{j}}|j\rangle\langle j|$ and
$K_{ij}=\sqrt{\eta_{ji}}|i\rangle\langle j|$.

Here, $\rho$ is the density matrix and $\eta_{ji}$ describes about the decay
rate from the j-th to i-th level which follows the conditions: 0 $\leq$
$\eta_{ji}$ $\leq$ 1, $\forall$ $i,j$ s.t. 0 $\leq$ $i$ $<$ $j$ $\leq$ $d-1$.
and $\kappa_{j}$ = $\sum_{0\leq j<j}\eta_{ji}$ $\leq$ 1, $\forall j$ = 1, 2,
…, $d-1$.

If this noisy channel acts on the density matrix $\rho_{k}$ =
$|\Psi_{k}\rangle\langle\Psi_{k}|$, then the output state
$\mathcal{D}(\rho_{k})$ is given by

$\displaystyle\mathcal{D}(\rho_{k})=$ $\displaystyle\left|\langle
0|\Psi_{k}\rangle\right|^{2}|0\rangle\langle 0|+\sum_{1\leq J\leq
N^{2}-1}\left|\langle J|\Psi_{k}\rangle\right|^{2}|J\rangle\langle J|+$ (13)
$\displaystyle\sum_{0\leq\overline{Y}<J\leq
N^{2}-1}\eta_{J\overline{Y}}\left|\langle
J|\Psi_{k}\rangle\right|^{2}|\overline{Y}\rangle\langle\overline{Y}|.$

Now, measuring the output state of MAD channel $\mathcal{D}(\rho_{k})$ in the
target state $|Y\rangle$, gives us

$\displaystyle\langle Y|\mathcal{D}(\rho_{k})|Y\rangle=$
$\displaystyle\left|\langle 0|\Psi_{k}\rangle\right|^{2}\left|\langle
Y|0\rangle\right|^{2}+$ $\displaystyle\sum_{1\leq J\leq
N^{2}-1}(1-\kappa_{J})\left|\langle JH|\Psi_{k}\rangle\right|^{2}\left|\langle
Y|J\rangle\right|^{2}+$ $\displaystyle\sum_{0\leq\overline{Y}<J\leq
N^{2}-1}\eta_{J\overline{Y}}\left|\langle
J|\Psi_{k}\rangle\right|^{2}\left|\langle Y|\overline{Y}\rangle\right|^{2}.$

By performing the calculations for all the terms, we ultimately obtain the
transformed expression of $\langle Y|\mathcal{D}(\rho_{k})|Y\rangle$ as given
by

$\displaystyle\langle Y|\mathcal{D}(\rho_{k})|Y\rangle=$
$\displaystyle(1-\kappa_{Y})sin^{4}\theta_{k}+$
$\displaystyle(\sum_{\overline{Y}\neq
Y}\eta_{Y\overline{Y}}cos^{2}\theta_{k}-\eta_{YJ}sin^{2}\theta_{k}+$
$\displaystyle
2\eta_{yy^{{}^{\prime}}\overline{y}y}cos\theta_{k}sin\theta_{k})^{2}.$

Finally, invoking the definition of the Grover algorithm, for $k=O(\sqrt{N})$
it becomes evident that $\sin\theta_{k}=1$ and all other terms vanish.
Consequently, the probability of successfully finding the target item is given
by

$P(\mathcal{D}(\rho_{k}))=1-(\kappa_{Y}+\eta_{YJ}).$ (14)

Eq. (14) articulates the fact that for $(\kappa_{Y}+\eta_{YJ})=0$, we have the
noise-less scenario and the target state can be found with certainty. When the
noise is present, we have $(\kappa_{Y}+\eta_{YJ})$ is nonzero and the success
probability reduces.

##  IV Conclusion

The Grover search algorithm has undeniably been a cornerstone in the evolution
of quantum algorithms, providing a quadratic speed-up. This paper has
contributed to the discourse by demonstrating that under specific conditions,
leveraging the intrinsic structure of marked items and employing local query
operators, the algorithm’s prowess can be elevated to a bi-quadratic speed-up.
Our exploration into the algorithm’s resilience in the face of noise
underscores its ability to tolerate a noisy scenario. The implications of our
findings reverberate across various domains, particularly in conditional
database searching and optimization, reaffirming the algorithm’s crucial role
in unraveling intricate computational challenges. As quantum computing
advances, our study propels the Grover search algorithm from a catalyst of
innovation to a versatile tool poised to shape the landscape of future quantum
applications.

###### Acknowledgements.

AD and BP acknowledge IIIT, Hyderabad for the invaluable support and enriching
academic environment during our visit, which significantly enhanced the
research presented in this project.

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