We study the universality and membership problems for gate sets consisting of a finite number of quantum gates. Our approach relies on the techniques from compact Lie group theory. We also introduce an auxiliary problem called the subgroup universality problem, which helps in solving some instances of the membership problem and can be of interest on its own. The resulting theorems are mainly formulated in terms of centralizers and the adjoint representations of a given set of quantum gates.
We study the universality and membership problems for gate sets consisting of a finite number of quantum gates. Our approach relies on the techniques from compact Lie group theory. We also introduce an auxiliary problem called the subgroup universality problem, which helps in solving some instances of the membership problem and can be of interest on its own. The resulting theorems are mainly formulated in terms of centralizers and the adjoint representations of a given set of quantum gates.
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.
Consider a compact connected Lie group G and the corresponding Lie algebra L. Let {X1,...,Xm} be a set of generators for the Lie algebra L. We prove that G is uniformly finitely generated by {X1,...,Xm}. This means that every element K∈G can be expressed as K=eXt1eXt2⋅⋅⋅eXtl, where the indeterminates X are in the set {X1,...,Xm}, $t_i \in \RR$, i=1,...,l, and the number l is uniformly bounded. This extends a previous result by F. Lowenthal in that we do not require the connected one dimensional Lie subgroups corresponding to the Xi, i=1,...,m, to be compact. We discuss the consequence of this result to the question of universality of quantum gates in quantum computing.
Composing p-adic qubits: from representations of SO(3)p to entanglement and universal quantum logic gates
In the context of p-adic quantum mechanics, we investigate composite systems of p-adic qubits and p-adically controlled quantum logic gates. We build on the notion of a single p-adic qubit as a two-dimensional irreducible representation of the compact p-adic special orthogonal group SO(3)p. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)p mod pk. Then, we tackle the problem of p-adic qubit composition and entanglement, fundamental for a p-adic formulation of quantum information processing. We classify the representations of SO(3)p mod p, and analyse tensor products of two p-adic qubit representations lifted from SO(3)p mod p. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. For p=3, we construct a set of gates from 4-dimensional irreducible representations of SO(3)p mod p that we prove to be universal for quantum computation.
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.
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].