Although the basic ideas of quantum computation are fairly straightforward, the detailed manipulations required to extract a useful result from an entangled superposition of answers can be rather complicated. There are, however, a small number of algorithms which are simple enough to explain using only elementary methods. The best example is Deutsch's algorithm: as this only requires two qubits its properties are amenable to brute-force matrix calculations, and these calculations can be simplified using a variety of short cuts, which also give some insight into how and why the algorithm works. We will also explore the simplest case of Grover's quantum search algorithm, which is once again simple enough to be tacked by brute force. Finally we will look briefly at two methods used to stabilize quantum computers against errors. Deutsch's algorithm The invention of Deutsch's algorithm can be taken as defining the start of modern quantum computation, and it remains a key example, exhibiting many of the key properties of quantum algorithms in a particularly simple form. (Note, however, that the version of Deutsch's algorithm described here is not in fact the original but a later modification which is both more powerful and easier to understand.) Consider a binary function f from one bit to one bit, that is a function which takes in either 0 or 1 as its input and returns either 0 or 1 as its output. There are four such functions which may be conveniently labeled by their outputs as shown in Table 8.1.
Although the basic ideas of quantum computation are fairly straightforward, the detailed manipulations required to extract a useful result from an entangled superposition of answers can be rather complicated. There are, however, a small number of algorithms which are simple enough to explain using only elementary methods. The best example is Deutsch's algorithm: as this only requires two qubits its properties are amenable to brute-force matrix calculations, and these calculations can be simplified using a variety of short cuts, which also give some insight into how and why the algorithm works. We will also explore the simplest case of Grover's quantum search algorithm, which is once again simple enough to be tacked by brute force. Finally we will look briefly at two methods used to stabilize quantum computers against errors. Deutsch's algorithm The invention of Deutsch's algorithm can be taken as defining the start of modern quantum computation, and it remains a key example, exhibiting many of the key properties of quantum algorithms in a particularly simple form. (Note, however, that the version of Deutsch's algorithm described here is not in fact the original but a later modification which is both more powerful and easier to understand.) Consider a binary function f from one bit to one bit, that is a function which takes in either 0 or 1 as its input and returns either 0 or 1 as its output. There are four such functions which may be conveniently labeled by their outputs as shown in Table 8.1.
Deutsch's algorithm and Grover's quantum search are theoretically interesting, but it is not obvious that these two algorithms are actually useful for anything important. We next consider a selection of more advanced algorithms, several of which may have real-life applications. Some of these will be too complicated to explain fully, and their properties will only be sketched briefly. The Deutsch–Jozsa algorithm Deutsch's algorithm is simple, but important, as it shows that a quantum algorithm can find a property of an unknown function (its parity) with a smaller number of queries than any possible classical algorithm (one rather than two). For this reason we can say that quantum computing is more efficient than classical computing within the oracle model of function evaluation. (It is widely believed that quantum computing is more efficient than classical computing in general, but this is a surprisingly hard thing to prove.) The simplicity of the algorithm is also an advantage, as it can be implemented on very primitive quantum computers. Beyond this, however, Deutsch's algorithm is also important as the simplest member of a large family of quantum algorithms, including most notably Shor's quantum factoring algorithm. The second simplest algorithm in the family is the Deutsch–Jozsa algorithm, which solves a very closely related problem. Consider an unknown binary function with n input bits, giving N = 2 n possible inputs, and a single output bit.
Quantum computing is evolving so rapidly that it forces us to revisit, rewrite, and update the foundations of the theory. \emph{Basic Quantum Algorithms} revisits the earliest quantum algorithms. The journey began in 1985 with Deutsch attempting to evaluate a function at two domain points simultaneously. Then, in 1992, Deutsch and Jozsa created a quantum algorithm that determines whether a Boolean function is constant or balanced. The following year, Bernstein and Vazirani realized that essentially the same algorithm could be used to identify a specific Boolean function within a set of linear Boolean functions. In 1994, Simon introduced a novel quantum algorithm that determines whether a function is one-to-one or two-to-one exponentially faster than any classical algorithm for the same problem. That same year, Shor developed two groundbreaking quantum algorithms for integer factoring and calculating discrete logarithms, posing a threat to widely used cryptographic methods. In 1995, Kitaev proposed an alternative formulation based on phase estimation that proved valuable in numerous applications. The following year, Grover devised a quantum search algorithm that is quadratically faster than its classical counterpart. More than a decade later, Harrow, Hassidim, and Lloyd proposed a quantum algorithm for solving systems of linear equations, now known as the HHL algorithm. With an emphasis on the circuit model, this work provides a detailed description of all these remarkable algorithms.
Abstract Alice and Bob introduce and explain in detail some of the algorithms that convinced physicists and computer scientists that there might be something to this crazy idea of quantum computing. Each algorithm provides a case study of how the properties of quantum states can be used to provide an advantage for using a quantum computer compared to a classical computer. Alice and Bob describe the Deutsch algorithm, designed to answer a simple question about the properties of functions of binary digits (0 and 1), and the Simon algorithm, which finds a “secret code.” The Simon algorithm, like many other quantum algorithms, is a mix of quantum state algorithms and classical computational methods. Alice and Bob also introduce Hadamard gates that act on multi-qubit systems. Those gates play an important role in many quantum algorithms.
In quantum computers a unitary transformation is applied to a given initial state of a set of qubits through several quantum gates. Tests were conducted using Grover's algorithm and implemented in a variety of different ways on the publically available IBM quantum computers. Essentially, quantum computers manipulate quantum states instead of classical bits. Quantum information can be represented by the elementary units called quantum bits abbreviated as qubits or qbits. This chapter discusses the fundamental properties of quantum systems that are relevant to information processing. Quantum computation needs something much more than that quantum parallelism to be useful. This is achieved in Deutsch's algorithm which combines quantum parallelism with interference. One can simulate a classical circuit using a quantum circuit. That is, it is possible to perform classical computations on quantum computers. But the advantage of quantum computing is that powerful functions may be computed by making use of quantum parallelism, the fundamental feature of many quantum algorithms.
Quantum computing takes fully advantage of the superposition principle to increase greatly (even exponentially) the speed of calculations, relative to the classical approach. The Deutsch-Jozsa algorithm is the simplest quantum algorithm illustrating this power. Unfortunately, the standard derivation involves several ingenious steps which usually leave students feeling that they could never have figured out the algorithm by themselves. We present here a different formulation of the problem which allows students to derive the algorithm using only basic knowledge of quantum mechanics and linear algebra.
Computer graphics relies on computing hardware for everything from animation to rendering. An emerging new technology exploits the properties of quantum objects to offer radically new ways to think about, create, and run algorithms. For example, quantum computers can evaluate many different input values simultaneously, where "many" can be larger than the number of atoms in the visible universe. The catch is that only one output can be obtained from each run. Quantum computing may change how we think about computer graphics. For example, future quantum computers may be able to intersect astronomical numbers of rays with a similarly massive database of objects in a single execution, or evaluate enormous numbers of simulation parameters in parallel to return the one set that produces a desired result. This course describes, without advanced math, the core ideas of quantum computing. We start with the quantum version of the classical bit (called a qubit) and the basic operators that modify qubits. We introduce the four essential properties that distinguish quantum computers from familiar classical computers: superposition, interference, entanglement, and measurement. We show how these properties are orchestrated to build quantum algorithms. We conclude with a brief overview of some of the most well-known quantum algorithms, and some of their possible applications in computer graphics. Quantum computers are already here, and are increasing in size and reliability at a rapid pace. Open-source simulators abound, and small quantum computers are even available for free use on the web. Classical hardware has served computer graphics well. Quantum computing offers a fundamentally new way to design and execute algorithms, which could change our field in fundamental ways. Now is the perfect time to starting thinking about quantum computing for graphics.