We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in information-based complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.
We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in information-based complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.
Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this type of problem. In this paper we give a short introduction to the basic ideas of quantum computing and survey recent results on high dimensional integration. We discuss connections to the Monte Carlo methology and compare the optimal error rates of quantum algorithms to those of classical deterministic and randomized algorithms.
A challenging question in the overlap of computer science, mathematics, and physics, is the exploration of potential capabilities of quantum computers. Milestones were the factoring algorithm of Shor (1994) and the search algorithm of Grover (1996). So far, major research was concentrated on discrete and algebraic problems. Much less was known about computational problems of analysis, including such a prominent example as high dimensional numerical integration, which is well-studied in the classical settings. We seek to understand how efficiently this and related problems can be solved in the quantum model of computation (i.e., on a quantum computer) and how the outcome compares to the efficiency of deterministic or randomized algorithms on a classical (i.e. non-quantum) computer. In this paper we give a survey of the state of the art in this field, including also a brief introduction to the general ideas of quantum computing. Introduction A quantum computer is a computing device based on quantum mechanical laws of the (sub)atomic world. The idea of such a computer was developed by Feynman [8] in 1982. He emphasized that simulating quantum mechanics on a classical computer is extremely hard, probably infeasible. So why not try to simulate quantum mechanics using quantum devices themselves. (Thoughts in this direction were also expressed by Manin [21] in 1980, see also [22].) In 1985 Deutsch [6] presented a formal model of computation for quantum computing.
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two major motivations for studying quantum algorithms and complexity for continuous problems. 1. Are quantum computers more powerful than classical computers for important scientific problems? How much more powerful? 2. Many important scientific and engineering problems have continuous formulations. To answer the first question we must know the classical computational complexity of the problem. Knowing the classical complexity of a continuous problem we obtain the quantum computation speedup if we know the quantum complexity. If we know an upper bound on the quantum complexity through the cost of a particular quantum algorithm then we can obtain a lower bound on the quantum speedup. Regarding the second motivation, in this article we'll report on high-dimensional integration, path integration, Feynman path integration, the smallest eigenvalue of a differential equation, approximation, partial differential equations, ordinary differential equations and gradient estimation. We'll also briefly report on the simulation of quantum systems on a quantum computer.
Quantum computing has attracted a lot of attention in different research fields, such as mathematics, physics and computer science. Quantum algorithms can solve certain problems significantly faster than classical algorithms. There are many numerical problems, especially those arising from quantum systems, which are notoriously difficult to solve using classical computers, since the computational time required often scales exponentially with the size of the problem. However, quantum computers have the potential to solve these problems efficiently, which is also one of the founding ideas of the field of quantum computing. In this thesis, we explore five computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm based on the Berry phases for the counting problem. For the running time, it is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result should be robust to decoherence and resilient to certain kinds of noise. Since the counting problem is the foundation of many other numerical problems, such as high-dimensional integration and path integration, our adiabatic algorithms can be directly generalized to these kinds of problems. In addition, we study the quantum PAC learning model, offering an improved lower bound on the query complexity. The lower bound is very close to the best lower bound on query complexity known for the classical PAC learning model. We also study the algorithms and the cost of simulating a system evolving under a given Hamiltonian. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required. Moreover, we derive the optimal order of convergence given the required error bound. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for Hamiltonian simulation. The evolution is simulated by a product of exponentials in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. We provide a scheme to bound the error of the final quantum state in a randomized algorithm, and obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. We also apply Hamiltonian simulation in estimating the ground state energy of a multiparticle system, which is also known as the multivariate Sturm-Liouville eigenvalue problem. Since the cost of this problem grows exponentially with the number of particles using deterministic classical algorithms, it suffers from the curse of dimensionality. Quantum computers can vanquish the curse, and we exhibit a quantum algorithm whose total cost are linear in the number of particles.
We study the complexity of a two-point boundary value problem. We concentrate on the linear problem of order k with separated boundary conditions. Right-hand side functions are assumed to be r times differentiable with all derivatives bounded by a constant. We consider three models of computation: deterministic with standard and linear information, randomized and quantum. In each setting, we construct an algorithm for solving the problem, which allows us to establish upper complexity bounds. In the deterministic setting, we show that the use of linear information gives us a speed-up of at least one order of magnitude compared with the standard information. For randomized algorithms, we show that the speed-up over standard deterministic algorithms is by 1/2 in the exponent. For quantum algorithms, we can achieve a speed-up by one order of magnitude. We also provide lower complexity bounds. They match upper bounds in the deterministic setting with the standard information, and almost match upper bounds in the randomized and quantum settings. In the deterministic setting with the linear information, a gap still remains between the upper and lower complexity bounds.
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’.