Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.
Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.
Query complexity is a model of computation in which we have to compute a function f(x1,…,xN) of variables xi which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including: - the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring Ω(N) queries classically); - the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with M queries quantumly but requiring Ω~(M2.5) queries probabilistically); - the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with M queries probabilistically but requiring Ω~(M2) queries deterministically); - the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions); - the connections between query algorithms and approximations by low-degree polynomials.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.
The last decade witnessed a great surge of fruitful studies in the new paradigm of quantum computing. Remarkable progress has been made in all the areas including quantum algorithms, quantum complexity theory, quantum error-correcting code, quantum information theory, quantum cryptography and physical implementations of quantum computers. However, quantum algorithm design, probably the most important task in quantum computing, did not make as much progress as people expected. This thesis aims at both designing new quantum algorithms and studying why efficient quantum algorithms are so hard to design by proving many lower bounds on quantum query complexity for various problems.
We first review (and generalize) known techniques for designing quantum algorithms and for proving quantum lower bounds. Using these techniques, we then study the quantum query complexity for many problems, including some specific problems such as Graph Matching and other natural graph properties, the general tuple search, and local search. We also study the lowest possible quantum query complexity for a class of functions as a whole, such as graph properties, circular functions, and functions invariant of a transitive permutation group, where tight results are given in all these cases.
Finally for the quantum lower bounds techniques themselves, we study the main two techniques and show that they are incomparable in power. For the most widely used one, which has a lot of parameters to choose, we show limitations for it in terms of certificate complexity. This implies that we cannot use the method to prove better lower bounds for almost all known open problems in this area.
The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.
Quantum computing, based on the fundamental principles of quantum mechanics, has become a breakthrough that has the potential to revolutionize computing. At the forefront of this quantum revolution are quantum algorithms that use the origin of qubits to solve computational problems much faster than classical algorithms. This article provides a review of quantum algorithms to introduce their principles, examine famous examples such as the Grover and Shor algorithms, and delve into their applications, including quantum machine learning. Exploring the fundamentals of quantum computing begins with an overview that highlights the unique features of quantum computing compared to classical computing. We then examine several different algorithmic paradigms to highlight the unique advantages that quantum parallelism and interference bring to computing. The best knowledge of quantum algorithms focuses on Grover's blind search, Shor's efficient factorization algorithm, and the quantum phase approach for solving quantum chemistry and eigenvalue problems. As well as celebrating the achievements of quantum algorithms, this research also looks at the challenges and limitations that hinder their widespread use. Topics such as decoherence, error correction, and finding errors in quantum computing are discussed in the context of overcoming obstacles to the use of practical quantum algorithms. In addition, the latest developments are clarified by showing the successes of the experiments and their implications for the future. As we stand at the forefront of a new era in computing, this article not only provides a snapshot of the current state of quantum algorithms but also speculates on future directions. The potential applications of quantum algorithms in many fields and the ongoing quest to overcome current limitations offer exciting opportunities for further research. This research aims to contribute to the ongoing debate about quantum algorithms, gaining a deeper understanding of their impact on the future of computing.
In 1994, Peter Shor discovered a polynomial time quantum algorithm for factoring and discrete logarithm. Two years later, in 1996, Lov Grover discovered a search algorithm which is quadratically better than conventional search. By now, each of the two algorithms has developed into a line of research which goes well beyond the original algorithm. Shor's algorithm has inspired the study of quantum Fourier sampling which has resulted in more quantum algorithms for number-theoretic and group-theoretic problems. Grover's algorithm has developed into the area of quantum query algorithms.I will survey the developments in quantum query algorithms. The topics will include: applications of Grover's algorithm to element distinctness and other problems, lower bounds on quantum algorithms and the use of quantum random walks to design better search algorithm. I will also describe how some of techniques in this area can be used as "quantum black boxes" in an otherwise classical algorithm.