The thesis contains an analysis of two computational problems. The first problem is discrete quantum Boolean summation. This problem is a building block of quantum algorithms for many continuous problems, such as integration, approximation, differential equations and path integration. The second problem is continuous multivariate Feynman-Kac path integration, which is a special case of path integration. The quantum Boolean summation problem can be solved by the quantum summation (QS) algorithm of Brassard, Hí yer, Mosca and Tapp, which approximates the arithmetic mean of a Boolean function. We improve the error bound of Brassard et al. for the worst-probabilistic setting. Our error bound is sharp. We also present new sharp error bounds in the average-probabilistic and worst-average settings. Our average-probabilistic error bounds prove the optimality of the QS algorithm for a certain choice of its parameters. The study of the worst-average error shows that the QS algorithm is not optimal in this setting; we need to use a certain number of repetitions to regain its optimality. The multivariate Feynman-Kac path integration problem for smooth multivariate functions suffers from the provable curse of dimensionality in the worst-case deterministic setting, i.e., the minimal number of function evaluations needed to compute an approximation depends exponentially on the number of variables. We show that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables. The exponents of these polynomials are 2 in the randomized setting and 1 in the quantum setting. These exponents can be lowered at the expense of the dependence on the number of variables. Hence, the quantum setting yields exponential speedup over the worst-case deterministic setting, and quadratic speedup over the randomized setting.
The thesis contains an analysis of two computational problems. The first problem is discrete quantum Boolean summation. This problem is a building block of quantum algorithms for many continuous problems, such as integration, approximation, differential equations and path integration. The second problem is continuous multivariate Feynman-Kac path integration, which is a special case of path integration. The quantum Boolean summation problem can be solved by the quantum summation (QS) algorithm of Brassard, Hí yer, Mosca and Tapp, which approximates the arithmetic mean of a Boolean function. We improve the error bound of Brassard et al. for the worst-probabilistic setting. Our error bound is sharp. We also present new sharp error bounds in the average-probabilistic and worst-average settings. Our average-probabilistic error bounds prove the optimality of the QS algorithm for a certain choice of its parameters. The study of the worst-average error shows that the QS algorithm is not optimal in this setting; we need to use a certain number of repetitions to regain its optimality. The multivariate Feynman-Kac path integration problem for smooth multivariate functions suffers from the provable curse of dimensionality in the worst-case deterministic setting, i.e., the minimal number of function evaluations needed to compute an approximation depends exponentially on the number of variables. We show that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables. The exponents of these polynomials are 2 in the randomized setting and 1 in the quantum setting. These exponents can be lowered at the expense of the dependence on the number of variables. Hence, the quantum setting yields exponential speedup over the worst-case deterministic setting, and quadratic speedup over the randomized setting.
The Feynman-Kac path integration problem was studied in the worst case setting by Plaskota et al. (J. Comp. Phys. 164 (2000) 335) for the univariate case and by Kwas and Li (J. Comp. 19 (2003) 730) for the multivariate case with d space variables. In this paper we consider the multivariate Feynman-Kac path integration problem in the randomized and quantum settings. For smooth multivariate functions, it was proven in Kwas and Li (2003) that the classical worst case complexity suffers from the curse of dimensionality in d. We show that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the number of function evaluations and/or quantum queries required to compute an e-approximation has a bound independent of d and depending polynomially on 1/e. The exponents of these polynomials are at most 2 in the randomized setting and at most 1 in the quantum setting. Hence we have exponential speedup over the classical worst case setting and quadratic speedup of the quantum setting over the randomized setting. However, both the randomized and quantum algorithms presented here still require extensive precomputing, similar to the algorithms of Plaskota et al. (2000) and Kwas and Li(2003).
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.
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 study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^{-k} with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an $\e$-approximation to path integrals whose integrands are at least Lipschitz. We prove: 1. Path integration on a quantum computer is tractable. 2. Path integration on a quantum computer can be solved roughly $\e^{-1}$ times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. 3.The number of quantum queries is the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most $4.22 \e^{-1}$. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved. 4.The number of qubits is polynomial in $\e^{-1}$. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands.
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.
Classical approximation complexity asks what solution quality can be guaranteed with polynomial-time computation. The classes APX, PTAS, and FPTAS distinguish a fixed approximation ratio, approximation to any fixed accuracy, and approximation schemes whose running time is also polynomial in inverse accuracy. Their randomized counterparts are R-APX, R-PTAS, and R-FPTAS. We define bounded-error quantum counterparts BQ-APX, BQ-PTAS, and BQ-FPTAS. Membership requires a uniform quantum algorithm that, on every input, returns a feasible classical solution achieving at least the claimed approximation ratio with probability at least 2/3. Scores (objective values) must be efficiently classically computable. Running time includes parameter selection, preparation, measurement, decoding, and repetition. Many quantum optimization methods are used heuristically, and high benchmark scores alone do not establish these guarantees. We further establish a conditional hierarchy for logarithmic, polynomial, and exponential approximation factors.
Assuming NP ⊈ BQP, the quantum classes form a strict hierarchy. Problems based on prime factorization and discrete logarithms give conditional quantum-classical separations. Certified Maximum Order has an exact quantum algorithm, while any randomized polynomial-time algorithm guaranteeing at least an inverse-polynomial approximation ratio on every input would yield efficient factoring. Discrete-Logarithm Fitting has an exact quantum algorithm and a deterministic one-half approximation, but any fixed improvement over one half would give a randomized polynomial-time algorithm for the safe-prime discrete logarithm problem. Our results show, under explicit complexity assumptions, that quantum computation can improve worst-case approximation guarantees. A quantum-classical gap for common problems such as MaxCut or MaxSAT remains open.