Quantum algorithms and complexity for numerical problems

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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 different computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion α, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase 2πα. By estimating the Berry phase, we can approximate α, and solve the problem. For an error bound e, the algorithm can solve the problem with cost of order e−3/2 , which 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. For a concept class with d-VC dimension, the lower bound is Ω(e−1( d1−η + log(1/δ))), where e is the required error bound, δ is the maximal failure possibility and η can be an arbitrarily small positive number. The lower bound is close to the best lower bound on query complexity known for the classical PAC learning model, which is Ω(e−1(d + log(1/δ))). We also study the algorithms and the cost of simulating a system evolving with Hamiltonian H = j=1 m Hj, where the evolution of H j can be implemented efficiently. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required to approximate e −iHt with error e. Moreover, we derive the optimal order of convergence, given e and the cost of the resulting algorithm. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for simulating the evolution of Hamiltonian H. The evolution is simulated by a product of exponentials of Hj in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. First we provide a scheme to bound the error of the final quantum state in a randomized algorithm. Then we obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. Finally we provide a lower bound on the number of exponentials for both deterministic and randomized algorithms, when the evolution time is required to be positive. We also apply the improved upper bound of Hamiltonian simulation in estimating the ground state energy of a multiparticle system withrelative error e, 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 that achieves relative error e using O(d log e−1) qubits with total cost (number of quantum queries and other quantum operations) O(d e −(3+δ)), where δ > 0 is arbitrarily small. Thus, the number of qubits and the total cost are linear in the number of particles. The main result of Chapter 2 is based on the paper [127], published in Quantum Information Proceeding. The result of Chapter 3 is the same as that of the paper [126], published in Information Processing Letters. The results of Chapter 4 and Chapter 6 have been submitted, and can be found in [88] and [84] separately. Chapter 5 from a talk in the 9th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, has also been submitted and can be found in [125].