Determining whether a given algebra is commutative or not is important in the study of these algebraic objects in general and in the classification of semifields in particular. The best classical (ie, nonquantum) algorithm for this task has a running time that is of order O(n3), where n is the dimension of the algebra. To reduce this cost, in this paper, we study an approach to test the commutativity of a finite-dimensional algebra using quantum adiabatic computing. Previous quantum algorithms solving the same problem were based on Grover's quantum search. The algorithm is built from a quantum oracle for the multiplication constants of the algebra. Results of the experiments carried out on a quantum computer simulator, based on two different annealing schedules, are presented, showing that a quantum adiabatic algorithm for the problem can determine the commutativity of finite-dimensional algebras with one-side bounded error with a running time of order , achieving a quadratic speedup over the classical case.
Determining whether a given algebra is commutative or not is important in the study of these algebraic objects in general and in the classification of semifields in particular. The best classical (ie, nonquantum) algorithm for this task has a running time that is of order O(n3), where n is the dimension of the algebra. To reduce this cost, in this paper, we study an approach to test the commutativity of a finite-dimensional algebra using quantum adiabatic computing. Previous quantum algorithms solving the same problem were based on Grover's quantum search. The algorithm is built from a quantum oracle for the multiplication constants of the algebra. Results of the experiments carried out on a quantum computer simulator, based on two different annealing schedules, are presented, showing that a quantum adiabatic algorithm for the problem can determine the commutativity of finite-dimensional algebras with one-side bounded error with a running time of order , achieving a quadratic speedup over the classical case.
A quantum procedure for testing the commutativity of a finite dimensional algebra is introduced. This algorithm, based on Grover's quantum search, is shown to provide a quadratic speed-up (when the number of queries to the algebra multiplication constants is considered) over any classical algorithm (both deterministic and randomized) with equal success rate and shown to be optimal among the class of probabilistic quantum query algorithms. This algorithm can also be readily adapted to test commutativity and hermiticity of square matrices, again with quadratic speed-up. The results of the experiments carried out on a quantum computer simulator and on one of IBM's 5-qubit quantum computers are presented.
Fixed-point quantum search algorithms succeed at finding one of M target items among N total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property---the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.
The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an unstructured search problem with a quadratic speedup over a classical search, just as Grover's algorithm. In this paper, we study how the structure of the search problem may be exploited to further improve the efficiency of these quantum adiabatic algorithms. We show that by nesting a partial search over a reduced set of variables into a global search, it is possible to devise quantum adiabatic algorithms with a complexity that, although still exponential, grows with a reduced order in the problem size.
When classifying a collection of finite algebras (for instance, in the computational classification of finite semifields), an important task is the determination of substructures such as the right, middle, and left nuclei, the nucleus, and the center. Finding these structures may become computationally expensive when there is no additional information about the algebra properties. In this paper, we introduce quantum algorithms than solve this task efficiently, by formulating it as an instance of the Hidden Subgroup Problem (HSP) {over Abelian groups}. We give detailed constructions of the quantum circuits involved in the process and prove that the overall (quantum) complexity of our algorithm is polynomial in the dimension of the algebra, while a similar approach with classical computers would require an exponential number of queries to the HSP function
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.
Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each other or not with as few queries to the oracle as possible. In order to solve this problem, we use a theorem of Mario Szegedy (quant-ph/0401053) that relates a hitting time of a classical random walk to that of a quantum walk. We also take a look at another method of quantum walk by Andris Ambainis (quant-ph/0311001). We apply both walks into triangle finding problem (quant-ph/0310134) and matrix verification problem (quant-ph/0409035) to compare the powers of the two different walks. We also present Ambainis's method of lower bounding technique (quant-ph/0002066) to obtain a lower bound for this problem. It turns out Szegedy's algorithm can be generalized to solve similar problems. Therefore we use Szegedy's theorem to analyze the problem of matrix set commutativity. We give an O(k^{4/5}n^{9/5}) algorithm as well as a lower bound of Omega(k^{1/2}n). We generalize the technique used in coming up with the upper bound to solve a broader range of similar problems. This is probably the first problem to be studied on the quantum query complexity using quantum walks that involves more than one parameter, here, k and n.