An Introduction to Quantum Algorithms Contents, 1 What are quantum algorithms 3. 1 1 Background 3, 1 2 Caveats 4, 2 Mathematical representation 5. 2 1 Fundamental differences 5, 2 2 Hilbert spaces and Dirac notation 6. 2 3 The qubit 9, 2 4 Quantum registers 11, 2 5 Quantum logic gates 12. 2 6 Computational complexity 19, 3 Grover s Algorithm 20. 3 1 Quantum search 20, 3 2 Grover s algorithm How it works 22. 3 3 Grover s algorithm Worked example 24, 4 Simon s Algorithm 28. 4 1 Black box period finding 28, 4 2 Simon s algorithm How it works 29. 4 3 Simon s Algorithm Worked example 32, 5 Conclusion 33. References 34, Page 2 of 35, An Introduction to Quantum Algorithms 1 What are quantum algorithms. 1 What are quantum algorithms, 1 1 Background, The idea of a quantum computer was first proposed in 1981 by Nobel laureate Richard. Feynman who pointed out that accurately and efficiently simulating quantum mechanical. systems would be impossible on a classical computer but that a new kind of machine a. computer itself built of quantum mechanical elements which obey quantum mechanical. laws 1 might one day perform efficient simulations of quantum systems Classical. computers are inherently unable to simulate such a system using sub exponential time. and space complexity due to the exponential growth of the amount of data required to. completely represent a quantum system Quantum computers on the other hand exploit. the unique non classical properties of the quantum systems from which they are built. allowing them to process exponentially large quantities of information in only polynomial. time Of course this kind of computational power could have applications to a multitude. of problems outside quantum mechanics and in the same way that classical computation. quickly branched away from its narrow beginnings facilitating simulations of Newtoninan. mechanics the study of quantum algorithms has diverged greatly from simply simulating. quantum physicical systems to impact a wide variety of fields including information. theory cryptography language theory and mathematics. Probably the most widely known development in quantum computation was Peter. Shor s 1997 publication of a quantum algorithm for performing prime factorization of. integers in essentially polynomial time 2 Shor s algorithm was a monumental discovery. not only because it provides exponential speedup over the fastest classical algorithms but. because a number of algorithms for public key cryptography including the commonly used. RSA algorithm depend on the fact that there is no known efficient classical algorithm. to factor integers into prime numbers 3 Shor demonstrated that the realization of. a full scale quantum computer would have the potential to provide a truly significant. increase in computing speed at the same time pointing out the possible implications of. such an increase in computational power to the field of cybersecurity For both reasons. Shor s discovery sparked a great deal of interest in the design of quantum algorithms and. computers that endures today, In addition to Shor s algorithm there is a wealth of other interesting and important. algorithms that have been developed for quantum computers Two of those algorithms. will be described in detail in this tutorial in order to better elucidate the study of quantum. computing theory and quantum algorithm design These two algorithms are good models. for our current understanding of quantum computation as many other quantum algorithms. use similar techniques to achieve their results whether they be algorithms to solve linear. systems of equations 4 or quickly compute discrete logarithms Shor for example. credits one of these algorithms as the inspiration for his own 2. Page 3 of 35, An Introduction to Quantum Algorithms 1 2 Caveats. The first algorithm that will be explored in this tutorial is Lov Grover s quantum. database search 5 Grover s algorithm searches for a specified entry in an unordered. database employing an important technique in quantum algorithm design known as. amplitude amplification to achieve a polynomial speedup over the best classical algo. rithms In fact Grover s algorithm is optimal for any quantum algorithm for performing. such a search 6 Of course searching for an unique element in an unordered set can. be generalized to apply to a wide variety of problems in computer science such as the. problem of boolean satisfiability SAT Determining whether there exists a set of truth. values that satisfy a given set of clauses is equivalent to searching for an element the set. of truth values that satisfies a certain condition the set of clauses in an unordered search. space the set of all 2n possible truth assignments of n boolean variables Although this. algorithm does not provide an extreme increase in efficiency over its classical counterparts. the speedup is still certainly significant with large input. The second algorithm that this tutorial will present is Daniel Simon s algorithm. for determining the exclusive or XOR mask over which a given black box function is. invariant 7 Simon s was the first quantum algorithm found to have exponential speedup. over any equivalent classical algorithm and the runtime of his algorithm is optimal 8. Although the problem may at first seem very abstract Simon s problem corresponds. to an important problem with a number of applications in computer science known as. the hidden Abelian subgroup problem which includes factoring integers into primes and. calculating discrete logarithms 9, 1 2 Caveats, Now if quantum computers are so much faster than classical machines how is it that. their use is not yet widespread Unfortunately the engineering required to construct a. quantum computer is very difficult although it has come a long way For example the. largest number that has been factored by a quantum computer using Shor s algorithm. is 15 and the circuit was hard wired to factor only the number 15 not any other input. as the algorithm is designed to do To put this in perspective in order for Shor s algo. rithm to break 1024 bit RSA keys the length currently recommended for corporate use. 10 the algorithm would have to be applied to a 1024 bit integer which would require. a quantum computer orders of magnitude larger than that needed to factor 15 into 5 and 3. Engineers continue to experiment with many different physical implementations of. quantum computers the details of which are beyond the scope of this tutorial Currently. the most popular implementation known as an ion trap quantum computer works by. using lasers to change the spin states of supercooled ions trapped in an electromagnetic. field This implementation has lead to some of the largest quantum machines boasting. registers of up to 8 quantum bits qubits 11 The difficulty faced by all of the architec. Page 4 of 35, An Introduction to Quantum Algorithms 2 Mathematical representation. tures considered to date is foremost that of creating a machine that is minimally effected. by its surroundings while at the same time responsive to control by the engineer or. user The quality of a quantum computer implementation is measured in terms of its. decoherence error caused by the inherent coupling of artificial quantum systems to the. unpredictable natural world that we use to construct them For example in the ion trap. implementation possible sources of undesirable interference are any electric and magnetic. fields in the vicinity including those necessary to operate the machine itself which can. cause the ions to change spin states at random and ultimately collapse into a classical. system 12 Current research focuses on minimizing such sources of decoherence but. eliminating decoherence completely is impossible So while it is currently possible to. decrease decoherence enough to allow for relatively accurate computation on the scale of. 8 qubits as the number of qubits increases so does the decoherence to the extent that. properly functioning quantum computers with more than 8 qubits have yet to be realized. Still many universities and corporate research centers around the world are working. to build the first scalable quantum computer as quantum computation with its promises. of speed proven unattainable by classical machines remains a good candidate for the. next computational paradigm Even if general purpose quantum computers might seem. a long way off it is likely that small scale quantum computers might soon be used by. classical computers to perform certain calculations more quickly such as the calculations. required to simulate complex biological systems or to identify the content of a raster. image for improved web search 13 For these reasons it is important to become familiar. with the basics of quantum computation now in order to be prepared for the quantum. computational tools that will likely be available in the not so distant future. 2 Mathematical representation, 2 1 Fundamental differences. Quantum computers employ the laws of quantum mechanics to provide a vastly different. mechanism for computation than that available from classical machines Fortunately. for computer scientists interested in the field of quantum computing a deep knowledge. of quantum physics is not a prerequisite for understanding quantum algorithms in the. same way that one need not know how to build a processor in order to design classical. algorithms However it is still important to be familiar with the basic concepts that. differentiate quantum mechanical systems from classical ones in order to gain a better. intuitive understanding of the mathematics of quantum computation as well as of the. algorithms themselves, The first distinguishing trait of a quantum system is known as superposition or more. formally the superposition principle of quantum mechanics Rather than existing in one. distinct state at a time a quantum system is actually in all of its possible states at the. Page 5 of 35, An Introduction to Quantum Algorithms 2 2 Hilbert spaces and Dirac notation. same time With respect to a quantum computer this means that a quantum register. exists in a superposition of all its possible configurations of 0 s and 1 s at the same time. unlike a classical system whose register contains only one value at any given time It is. not until the system is observed that it collapses into an observable definite classical state. It is still possible to compute using such a seemingly unruly system because probabili. ties can be assigned to each of the possible states of the system Thus a quantum system. is probabilistic there is a computable probability corresponding to the liklihood that that. any given state will be observed if the system is measured Quantum computation is. performed by increasing the probability of observing the correct state to a sufficiently. high value so that the correct answer may be found with a reasonable amount of certainty. Quantum systems may also exhibit entanglement A state is considered entangled if. it cannot be decomposed into its more fundamental parts In other words two distinct. elements of a system are entangled if one part cannot be described without taking the. other part into consideration In a quantum computer it is possible for the probability of. observing a given configuration of two qubits to depend on the probability of observing. another possible configuration of those qubits and it is impossible to describe the. probability of observing one configuration without considering the other An especially. interesting quality of quantum entanglement is that elements of a quantum system may. be entangled even when they are separated by considerable space The exact physics of. quantum entanglement remain elusive even to professionals in the field but that has not. stopped them from applying entanglement to quantum information theory Quantum. teleportation an important concept in the field of quantum cryptography relies on. entangled quantum states to send quantum information adequately accurately and over. relatively long distances1, 2 2 Hilbert spaces and Dirac notation. Before delving into the formal mathematics used to describe quantum algorithms it is. important to first become familiar with the notation that is commonly used to describe. more general quantum mechanical systems as well as the states of quantum registers. This notation is known as Bra ket or Dirac notation named after the Nobel laureate Paul. Dirac Dirac notation is just another way of describing vectors In dirac notation. As of May 2010 researchers were able to teleport quantum information over a distance of about 16. kilometers with 89 accuracy 14, Page 6 of 35, An Introduction to Quantum Algorithms 2 2 Hilbert spaces and Dirac notation. The column vector vi is referred to as ket v The dual vector of vi has the following. Dirac notation, hv vT v0 v1 vn, Where v is the complex conjugate2 of v This dual vector is called bra v. Dirac notation is a convenient way to describe vectors in the Hilbert space Cn which. is the vector space that is most useful for reasoning about quantum systems For the. purposes of this tutorial3 a Hilbert space is a vector space with an inner product and a. norm4 defined by that inner product The inner product of a vector space is an operation. that assigns a scalar value to each pair of vectors u and v in the vector space and the. inner product of two vectors in a Hilbert space in Cn is denoted using the Dirac notation. Shor s 1997 publication of a quantum algorithm for performing prime factorization of integers in essentially polynomial time 2 Shor s algorithm was a monumental discovery not only because it provides exponential speedup over the fastest classical algorithms but because a number of algorithms for public key cryptography including the commonly used RSA algorithm depend on the fact that

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