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Quantum Computing for Location Determination

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Quantum Computing for Location Determination Quantum Computing for Location Determination Ahmed Shokry Moustafa Youssef Alexandria University AUC and Alexandria University Alexandria, Egypt Alexandria, Egypt [email protected] [email protected] p ABSTRACT to unstructured data searches from >¹=º to >¹ =º [19]. Since Quantum computing provides a new way for approaching then, interest in QC has sparked with a number of big compa- problem solving, enabling efficient solutions for problems nies and startups investing in realizing quantum computers that are hard on classical computers. It is based on leveraging and providing tools for programmers to develop quantum how quantum particles behave. With researchers around algorithms. This materialized in currently having real cloud- the world showing quantum supremacy and the availability based quantum computers with free accounts to researchers of cloud-based quantum computers with free accounts for [16, 27] as well as a large number of quantum programming researchers, quantum computing is becoming a reality. languages and simulators [2, 17, 27]. In October 2019, Google In this paper, we explore both the opportunities and chal- announced that it has reached quantum supremacy with an lenges that quantum computing has for location determina- array of 54 qubits by performing a series of operations in tion research. Specifically, we introduce an example for the 200 seconds that would take a supercomputer about 10,000 expected gain of using quantum algorithms by providing years to complete [2]. In December 2020, a Chinese research an efficient quantum implementation of the well-known RF group reached quantum supremacy by implementing Boson fingerprinting algorithm and run it on an instance ofthe sampling on 76 photons with the Jiuzhang quantum com- IBM Quantum Experience computer. The proposed quan- puter [57]. The quantum computer generates the samples in tum algorithm has a complexity that is exponentially better 20 seconds that would take a classical supercomputer 600 than its classical algorithm version, both in space and run- million years of computation. ning time. We further discuss both software and hardware In this paper, we explore the opportunities and challenges research challenges and opportunities that researchers can of applying quantum computing to the field of location de- build on to explore this exciting new domain. termination. Specifically, we discuss both algorithmic and hardware advantages that QC provides for use with location KEYWORDS determination as well as highlight the research challenges that need to be addressed to fully leverage their potential. As quantum computing, location determination systems, quan- an example, we present a quantum RF fingerprint matching tum sensors, quantum location determination, next genera- algorithm that requires space and runs in >¹< log¹# ºº as tion location tracking systems compared to the classical algorithm that requires space and runs in >¹<# º, where # is the number of access points and 1 INTRODUCTION < is the number of fingerprinting locations. This exponential Quantum Computing (QC) is a new field at the intersection speedup in complexity and saving in space can be further of physics, mathematics, and computer science. It leverages enhanced to >¹;>6¹<# ºº using more advanced algorithms. the phenomena of quantum mechanics to improve the effi- We validate our algorithm on an instance of the IBM Quan- ciency of computation. Specifically, to store and manipulate tum Experience platform and discuss its performance. We information, quantum computers use quantum bits (qubits). end the paper by a discussion of the different opportunities arXiv:2106.11751v2 [quant-ph] 24 Jun 2021 Qubits are represented by subatomic particles properties like offered and challenges posed by quantum computing tothe the spin of electrons or polarization of photons. Quantum field of location tracking systems. computers leverage the quantum mechanical phenomena The rest of the paper is organized as follow: we start of superposition, entanglement, and interference to create with a brief background on quantum computing in Section 2. states that scale exponentially with the number of qubits, Section 3 provides the details of our quantum fingerprint potentially allowing solving problems that are traditionally matching algorithm and its evaluation. We present different hard to solve on classical computers [37]. challenges and opportunities of QC for location tracking in In 1994, Peter Shor presented a theoretical quantum al- Section 4. Finally, sections 5 and 6 discuss related work and gorithm that could efficiently break the widely used RSA conclude the paper, respectively. encryption algorithm [45]. In 1996, Lov Grover developed a quantum algorithm that dramatically sped up the solution 0 with prob. 0.5 jB>DA24i j0i 퐻 1 with prob. 0.5 jC0A64C1i Figure 1: A simple quantum circuit. Single lines carry jC0A64C2i quantum information while double lines carry classi- cal information. Figure 2: An example of controlled gates. The two tar- 2 BACKGROUND get qubits are swapped, if and only if, the source line j1i In this section, we give a brief background on the basic con- is . cepts of quantum computing that we will build on in the rest amplitude of the measured state). The state collapses to the of the paper. observed classical bit value. A quantum bit (qubit) is the basic unit of information and It is important to note that the concept of quantum inter- is analogue to the classical bit. Contrary to classical bits, a ference is at the core of quantum computing. Using quantum qubit can exist in a superposition of the zero and one states. interference, one uses gates to cleverly and intentionally bias This superposition is what allows quantum computations to the content of the qubits towards the needed state, hence work on both states at the same time. This is often referred achieving a specific computation result. to as quantum parallelism. Qubits can have various physical The notion of qubit can be extended to higher dimen- implementations, e.g. the polarization of photons. sions using a quantum register. A quantum register jki, Formally, the Dirac notation is commonly used to describe consisting of = qubits, lives in a 2=-dimensional complex k U V U V k Í2=−1 U 8 the state of a qubit as j i = j0i ¸ j1i, where and are Hilbert space H. Register j i = 0 8 j i is specified Í 2 complex numbers called the amplitudes of classical states j0i by complex numbers U0, ..., U2=−1, where jU8 j = 1. Ba- and j1i, respectively. The state of the qubit is normalized, i.e. sis state j8i denotes the binary encoding of integer 8. We U2 ¸ V2 = 1. When the state jki is measured, only one of j0i use the tensor product ⊗ to compose two quantum sys- or j1i is observed, with probability U2 and V2, respectively. tems. For example, we can compose the two quantum states The measurement process is destructive, in the sense that the jki = U j0i ¸ V j1i and jqi = W j0i ¸X j1i as jli = jki ⊗ jqi = state collapses to the value j0i or j1i that has been observed, UW j00i ¸ UX j01i ¸ VW j10i ¸ VX j11i. losing the original amplitudes U and V [37]. Gates can also be defined on multiple qubits. For example, Operations on qubits are usually represented by gates, Figure 2 illustrates a frequently encountered gate in quan- similar to a classical circuit. An example of a common quan- tum circuits, the control gate. In a control gate, the operation tum gate is the NOT gate (also called Pauli-X gate) that is (e.g. Swap) is performed on the target wire(s), if and only if, analogous to the not gate in classical circuits. In particular, the source line is j1i. This can be used to “entangle” qubits when we apply the NOT gate to the state jk0i = U j0i ¸ V j1i, together. Entangled qubits are correlated with one another, we get the state jk1i = V j0i ¸ U j1i. Gates are usually repre- in the sense that information on one qubit will reveal in- sented by unitary matrices while states are represented by formation about the other unknown qubit, even if they are 0 1 separated by large distance [37]. column vectors1. The matrix for the NOT gate is and 1 0 A common way to describe a quantum algorithm is to use the above operation can be written as jk1i = #$) ¹jk0iº = a quantum circuit, which is a combination of the quantum U 0 1 . gates (e.g as in Figure 1). The input to the circuit is a number 1 0 V of qubits (in quantum registers) and the gates act on them Another important gate is the Walsh–Hadamard gate, 퐻, to change the combined circuit state using superposition, that maps j0i to p1 ¹j0i ¸ j1iº, i.e. a superposition state with 2 entanglement, and interference to reach a desired output equal probability for j0i and j1i; and maps j1i to p1 ¹j0i−j1iº. state that is a function of the algorithm output. The final step 2 Figure 1 shows a simple quantum circuit. Single lines carry is to measure the output state(s), which reveals the required quantum information while double lines carry classical in- information. formation (typically after measurement). The simple circuit Finally, the no-cloning theorem [51] indicates that, counter applies an 퐻 gate to state j0i, which produces the state to classical bits, quantum bits cannot be cloned. Therefore, p1 ¹j0i¸j1iº at the output of the gate. The measurement step one cannot assume that a quantum bit can be copied as 2 needed (i.e. there is no fan-out as in classical circuits). This produces either 0 or 1 with equal probability (the squared has a number of implications on designing quantum algo- rithm. For example, the no cloning theorem is a vital ingre- 1The ket notation j.i is used for column vectors while the bra notation h.j dient in quantum cryptography as it forbids eavesdroppers is used for row vectors.
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