Our understanding of complex mechanical processes is by our inability to solve the equations that govern them except for simple cases. Numerical simulation of systems appears to be our best option to understand, design and improve systems. It turns out, however, that computational problems in mechanics are notoriously difficult to treat numerically. The computational time that is required often scales exponentially with the size of the problem.
One of the most radical approaches for treating problems was proposed by Feytiman in 1982 [46]: he suggested that mechanics itself showed a promising way to simulate physics. This idea, the so called computer, showed its potential convincingly in one important regime with the development of Shor's integer factorization algorithm which improves exponentially on the best known classical algorithm.
In this thesis we explore six different computational problems from mechanics, study their computational complexity and try to find ways to remedy them. In the first problem we investigate the reasons behind the improved performance of Shor's and similar algorithms. We show that the key part in Shor's algorithm, the phase estimation algorithm, achieves its good performance through the use of power queries and we give lower bounds for all phase estimation algorithms that use power queries that match the known upper bounds. Our research indicates that problems that allow the use of power queries will achieve similar exponential improvements over classical algorithms. We then apply our lower bound technique for power queries to the Sturm-Liouville eigenvalue problem and show matching lower bounds to the upper bounds of Papageorgiou and Wozniakowski [85]. It seems to be very difficult, though, to find nontrivial instances of the Sturm-Lionville problem for which power queries can be simulated efficiently.
A computer differs from a classical computer that uses randomness, because it allows negative probabilities that can cancel each other (destructive interference). Ideally we would like to transfer classical randomized algorithms to the computer and get speed improvements. One of the simplest classical randomized algorithm is the random walk and we study the behavior of the continuous-time random walk. We analyze this random walk in one dimension and give analytical formulas for its behavior that demonstrate its interference properties. Is interference or cancellation really the most important advantage that a computer has over a classical computer? To answer that question we study the class StociMA of stochastic quantum algorithms that only use classical gates, but are given a witness, i.e. an arbitrary state that can guide the algorithm in computing the correct answer, but should not be able to fool it. We show that there exists a complete problem for this class, which we call the stoquastic local Hamiltonian problem. In this problem we try to compute the lowest eigenvalue of a Hamiltonian with interactions that span only a fixed number of particles and all contribute negatively. With the help of this problem we prove that MA ⊆ StocIMA ⊆ SBP ∪ QMA. This shows that interference is one of the most important parts of computation.
The simulation of the evolution of a general system in time requires a computational time that is exponential in the dimension of the system. But maybe the problem that we ask for is too general and we can simulate special systems in polynomial time. In particular it would be interesting to study systems of limited energy, i.e. for which the state at starting time consists mainly out of components with small energy. We model this in the theory of weighted reproducing kernel Hilbert spaces with two different sets of weights: product weights and finite-order weights. We will show that the information cost of computing the evolution for starting states from these spaces is tractable, i.e. the cost does not grow exponentially with the dimension of the problem.
Finally we study a computational problem from lattice chromodynamics (QCD). In most popular algorithms that treat problems in QCD the (gauged) Dirac matrix has to be inverted numerous times. Since this matrix is large, sparse, and ill-conditioned, iterative approaches have to be used. Unfortunately a direct application of methods like conjugate gradient (CC) or minimal residual algorithms seem to give poor performance in practice. We study a newly proposed multigrid method, adaptive smoothed aggregation [29], that has promise to overcome these difficulties We show that while classical CG's convergence becomes worse as the matrix becomes almost singular, adaptive smoothed aggregation will still perform well.
Our understanding of complex mechanical processes is by our inability to solve the equations that govern them except for simple cases. Numerical simulation of systems appears to be our best option to understand, design and improve systems. It turns out, however, that computational problems in mechanics are notoriously difficult to treat numerically. The computational time that is required often scales exponentially with the size of the problem.
One of the most radical approaches for treating problems was proposed by Feytiman in 1982 [46]: he suggested that mechanics itself showed a promising way to simulate physics. This idea, the so called computer, showed its potential convincingly in one important regime with the development of Shor's integer factorization algorithm which improves exponentially on the best known classical algorithm.
In this thesis we explore six different computational problems from mechanics, study their computational complexity and try to find ways to remedy them. In the first problem we investigate the reasons behind the improved performance of Shor's and similar algorithms. We show that the key part in Shor's algorithm, the phase estimation algorithm, achieves its good performance through the use of power queries and we give lower bounds for all phase estimation algorithms that use power queries that match the known upper bounds. Our research indicates that problems that allow the use of power queries will achieve similar exponential improvements over classical algorithms. We then apply our lower bound technique for power queries to the Sturm-Liouville eigenvalue problem and show matching lower bounds to the upper bounds of Papageorgiou and Wozniakowski [85]. It seems to be very difficult, though, to find nontrivial instances of the Sturm-Lionville problem for which power queries can be simulated efficiently.
A computer differs from a classical computer that uses randomness, because it allows negative probabilities that can cancel each other (destructive interference). Ideally we would like to transfer classical randomized algorithms to the computer and get speed improvements. One of the simplest classical randomized algorithm is the random walk and we study the behavior of the continuous-time random walk. We analyze this random walk in one dimension and give analytical formulas for its behavior that demonstrate its interference properties. Is interference or cancellation really the most important advantage that a computer has over a classical computer? To answer that question we study the class StociMA of stochastic quantum algorithms that only use classical gates, but are given a witness, i.e. an arbitrary state that can guide the algorithm in computing the correct answer, but should not be able to fool it. We show that there exists a complete problem for this class, which we call the stoquastic local Hamiltonian problem. In this problem we try to compute the lowest eigenvalue of a Hamiltonian with interactions that span only a fixed number of particles and all contribute negatively. With the help of this problem we prove that MA ⊆ StocIMA ⊆ SBP ∪ QMA. This shows that interference is one of the most important parts of computation.
The simulation of the evolution of a general system in time requires a computational time that is exponential in the dimension of the system. But maybe the problem that we ask for is too general and we can simulate special systems in polynomial time. In particular it would be interesting to study systems of limited energy, i.e. for which the state at starting time consists mainly out of components with small energy. We model this in the theory of weighted reproducing kernel Hilbert spaces with two different sets of weights: product weights and finite-order weights. We will show that the information cost of computing the evolution for starting states from these spaces is tractable, i.e. the cost does not grow exponentially with the dimension of the problem.
Finally we study a computational problem from lattice chromodynamics (QCD). In most popular algorithms that treat problems in QCD the (gauged) Dirac matrix has to be inverted numerous times. Since this matrix is large, sparse, and ill-conditioned, iterative approaches have to be used. Unfortunately a direct application of methods like conjugate gradient (CC) or minimal residual algorithms seem to give poor performance in practice. We study a newly proposed multigrid method, adaptive smoothed aggregation [29], that has promise to overcome these difficulties We show that while classical CG's convergence becomes worse as the matrix becomes almost singular, adaptive smoothed aggregation will still perform well.
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 approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schroedinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schroedinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query ("power queries"). They showed that the Sturm-Liouville equation can be solved with O(log(1/e)) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1/e. This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou's and Wozniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of log(1/e) power queries, therefore showing that log(1/e) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.
We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schroedinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schroedinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query ("power queries"). They showed that the Sturm-Liouville equation can be solved with O(log(1/e)) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1/e. This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou's and Wozniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of log(1/e) power queries, therefore showing that log(1/e) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.
Multivariate continuous problems are widely encountered in physics, chemistry, finance and in computational sciences. Unfortunately, interesting real world multivariate continuous problems can almost never be solved analytically. As a result, they are typically solved numerically and therefore approximately. In this thesis we deal with the approximate solution of multivariate problems. The complexity of such problems in the classical setting has been extensively studied in the literature. On the other hand the quantum computational model presents a promising alternative for dealing with multivariate problems. The idea of using quantum mechanics to simulate quantum physics was initially proposed by Feynman in 1982. Its potential was demonstrated by Shor's integer factorization algorithm, which exponentially improves the cost of the best classical algorithm known. In the first part of this thesis we study the tractability of multivariate problems in the worst and average case settings using the real number model with oracles. We derive necessary and sufficient conditions for weak tractability for linear multivariate tensor product problems in those settings. More specifically, we initially study necessary and sufficient conditions for weak tractability on linear multivariate tensor product problems in the worst case setting under the absolute error criterion. The complexity of such problems depends on the rate of decay of the squares of the singular values of the solution operator for the univariate problem. We show a condition on the singular values that is sufficient for weak tractability. The same condition is known to be necessary for weak tractability. Then, we study linear multivariate tensor product problems in the average case setting under the absolute error criterion. The complexity of such problems depends on the rate of decay of the eigenvalues of the covariance operator of the induced measure of the one dimensional problem. We derive a necessary and sufficient condition on the eigenvalues for such problems to be weakly tractable but not polynomially tractable. In the second part of this thesis we study quantum algorithms for certain eigenvalue problems and the implementation and design of quantum circuits for a modification of the quantum NAND evaluation algorithm on k-ary trees, where k is a constant. First, we study quantum algorithms for the estimation of the ground state energy of the multivariate time-independent Schrodinger equation corresponding to a multiparticle system in a box. The dimension d of the problem depends linearly to the number of particles of the system. We design a quantum algorithm that approximates the lowest eigenvalue with relative error ε for a non-negative potential V, where V as well as its first order partial derivatives are continuous and uniformly bounded by one. The algorithm requires a number of quantum operations that depends polynomially on the inverse of the accuracy and linearly on the number of the particles of the system. We note that the cost of any classical deterministic algorithm grows exponentially in the number of particles. Thus we have an exponential speedup with respect to the dimension of the problem d, when compared to the classical deterministic case. We extend our results to convex non-negative potentials V, where V as well as its first order partial derivatives are continuous and uniformly bounded by constants C and C' respectively. The algorithm solves the eigenvalue problem for a sequence of convex potentials in order to obtain its final result. More specifically, the quantum algorithm estimates the ground state energy with relative error ε a number of quantum operations that depends polynomially on the inverse of the accuracy, the uniform bound C on the potential and the dimension d of the problem. In addition, we present a modification of the algorithm that produces a quantum state which approximates the ground state eigenvector of the discretized Hamiltonian within Δ. This algorithm requires a number of quantum operations that depends pollynomially on the inverse of ε, the inverse of Δ, the uniform bound C on the potential and the dimension d of the problem. Finally, we consider the algorithm by Ambainis et.al. that evaluates balanced binary NAND formulas. We design a quantum circuit that implements a modification of the algorithm for k-ary trees, where k is a constant. Furthermore, we design another quantum circuit that consists exclusively of Clifford and T gates. This circuit approximates the previous one with error ε using the Solovay-Kitaev algorithm.
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’.
Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations.