The goal of finding a commutative group of gates from the set of unitary matrices is to simplify the process of quantum computing. If a set of gates is commutative, then we can apply the gates in any order without affecting the final outcome. This is important because quantum systems are very sensitive to errors and decoherence, and any disruption to the system can cause errors in the computation.By having a commutative group of gates, we can simplify the process of designing quantum algorithms and reduce the risk of errors. This is particularly important for solving complex computational problems that cannot be solved by classical computers.
The goal of finding a commutative group of gates from the set of unitary matrices is to simplify the process of quantum computing. If a set of gates is commutative, then we can apply the gates in any order without affecting the final outcome. This is important because quantum systems are very sensitive to errors and decoherence, and any disruption to the system can cause errors in the computation.By having a commutative group of gates, we can simplify the process of designing quantum algorithms and reduce the risk of errors. This is particularly important for solving complex computational problems that cannot be solved by classical computers.
A 1-qubit gate is defined as an element of the unitary group U(2), the group of complex valued matrices whose conjugate transposes equal their inverses. In quantum computing, it is important to be able to find good approximations of 1-qubit gates by using a dense group generated by a universal subset of the projective special unitary group G=PSU(2). Here, PSU(2) is the group formed by taking all elements of U(2) with determinant 1 and taking the quotient of this subgroup by the group {I,−I} where I is the identity matrix. In this http URL [1], Sarnak recently studied this problem motivated by developing a continued fraction type algorithm for this problem. The continued fraction algorithm is the fastest known algorithm to approximate real numbers by the dense subset of rational numbers.
In this report, we study the efficiency of the approximation of G by using a dense group generated by a universal subset of G. The measure of the efficiency K(g) of an arbitrary universal subset g is defined in [1] and satisfies K(g)⩾1. g is more efficient when K(g) is closer to 1. We address some open problems posed by Sarnak in [1]. This report forms the basis of the shorter preprint [LD].
This work is devoted to the construction and study of commutative gates for a two-qubit quantum system. Using four-dimensional algebra developed by the Kazakh mathematician Abenov M.M. all groups of commutative gates have been constructed, and among all states of a two-qubit quantum system, unitary states with which a specific gate is connected have been identified. An explicit type of gate is described that transfers a quantum system from one unitary state to another unitary state. The proposed approach opens up new possibilities for the design of quantum algorithms not only for two-qubit quantum systems, but also for n-qubit quantum systems.
Quantum circuit synthesis is one of the major areas of current research in the field of quantum computing.Analogous to its Boolean counterpart, the task involves constructing arbitrary quantum gates using only those available within a small set of universal gates that can be realized physically.However, unlike the latter, there are an infinite number of single qubit quantum gates, all of which constitute the special unitary group SU(2).Realizing any given single qubit gate using a given universal gate family is a complex task.Although gates can be synthesized to arbitrary degree of precision as long as the set of finite strings of the gate family is a dense subset of SU(2), it is desirable to accomplish the highest level of precision using only the minimum number of universal gates within the string approximation.Almost all algorithms that have been proposed for this purpose are based on the Solovay-Kitaev algorithm.The crux of the Solovay-Kitaev algorithm is the use of a procedure to decompose a given quantum gate into a pair of group commutators with the pair being synthesized separately.The Solovay-Kitaev algorithm involves group commutator decomposition in a recursive manner, with a direct approximation of a gate into a string of universal gates being performed only at the last level, i.e. in the leaf nodes of the search tree representing the execution of the Solovay-Kitaev algorithm.The main contribution of this research is in integrating conventional optimization procedures within the Solovay-Kitaev algorithm.Two specific directions of research have been studied.Firstly, optimization is incorporated within the group commutator decomposition, so that a more optimal pair of group commutators are obtained.As the degree of precision of the synthesized gate is explicitly minimized by means of this optimization procedure, the enhanced algorithm allows for more accurate quantum gates to be synthesized than what the original Solovay-Kitaev algorithm achieves.Simulation results with random gates indicate that the obtained accuracy is an order of magnitude better than before.Two versions of the new algorithm are examined, with the optimization in the first version being invoked only at the bottom level of Solovay-Kitaev algorithm and when carried out across all levels of the search tree in the next.Extensive simulations show that the second version yields better results despite equivalent computation times.Theoretical analysis of the proposed algorithm is able to provide a more formal, quantitative explanation underlying the experimentally observed phenomena.The other direction of investigation of this research involves formulating the group commutator decomposition in the form of bi-criteria optimization.This phase of research relaxed the equality constraint in the previous approach and with relaxation, a bi-criteria optimization is proposed.This optimization algorithm is new and has been devised primarily when the objective needs to be relaxed in different stages.This bi-criteria approach is able to provide comparably accurate synthesis as the previous approach.
Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group and stabilizing gates within an extension group . In this communication we explore the nice adequacy between group theoretical concepts such as commutators, normal subgroups, groups of automorphisms, short exact sequences, wreath products etc. and the coherent quantum computational primitives. The structure of the single-qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, we discover that M 20 , the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. We recover similar results by looking at the automorphisms of a complete set of mutually unbiased bases.
Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decompositions of Toffoli gate, the amplification step of the Grover search algorithm, the quantum Fourier transform, and the sender part of the quantum teleportation. Using this procedure, we present the circuit designs for the simulation of the unitary propagators of the Hamiltonians for the hydrogen and the water molecules. The approach is general and can be applied to generate the sequence of quantum gates for larger molecular systems.
We consider the problem of deciding if a set of quantum one-qudit gates S={g1,…,gn}⊂G is universal, i.e. if <S> is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in S we assign the orthogonal matrix Adg that is image of g under the adjoint representation Ad:G→SO(g) and g is the Lie algebra of G. The necessary condition for the universality of S is that the only matrices that commute with all Adgi ’s are proportional to the identity. If in addition there is an element in <S> whose Hilbert–Schmidt distance from the centre of G belongs to ]0,21[ , then S is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem.