Abstract.We consider the problem of deciding if a set of quantum onequdit gates S = {g1, . . . , gn} ⊂ G is universal, i.e. if 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 Adg i 's are proportional to the identity. If in addition there is an element in whose Hilbert-Schmidt distance from the centre of G belongs to ]0, 1 √ 2 [, 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.
We consider the problem of deciding if a set of quantum one-qudit gates S = {U1, . . . , Un} is universal. We provide the compact form criteria leading to a simple algorithm that allows deciding universality of any given set of gates in a finite number of steps. Moreover, for a non-universal S our criteria indicate what type of gates can be added to S to turn it into a universal set.Universal quantum gates play an important role in quantum computing and quantum optics [15,24,30]. The ability to effectively manufacture gates operating on many modes, using for example optical networks that couple modes of light [9,29], is a natural motivation to consider the universality problems not only for qubits but also for higher dimensional systems, i.e. qudits (see also [27,28] for fermionic linear optics and quantum metrology). For quantum computing with qudits, a universal set of gates consists of all one-qudit gates together with an additional two-qudit gate that does not map separable states onto separable states [10] (see [11,[35][36][37] for recent results in the context of universal Hamiltonians). The set of all one-qudit gates can be, however, generated using a finite number of gates [23]. We say that one-qudit gates S = {U 1 , . . . , U n } ⊂ SU (d) are universal if any gate from SU (d) can be built, with an arbitrary precision, using gates from S. It is known that almost all sets of qudit gates are universal, i.e. non-universal sets S of the given cardinality are of measure zero and can be characterised by vanishing of a finite number of polynomials in the gates entries and their conjugates [17,23]. Surprisingly, however, these polynomials are not known and it is hard to find operationally simple criteria that decide one-qudit gates universality. Some special cases of optical 3-mode gates have been recently studied in [5,32] and the approach providing an algorithm for deciding universality of a given set of quantum gates that can be implemented on a quantum automata has been proposed [13] (see also [1,2,14] for algorithms deciding if a finitely generated group is infinite). The main obstruction in the problems considered in [5,32] is the lack of classification of finite subgroups of SU (d) for d > 4. Nevertheless, as we show in this paper one can still provide some reasonable conditions for universality of one-qudit gates without this knowledge.The efficiency of universal sets is typically measured by the number of gates that are needed to approximate other gates with a given precision ǫ. The Solovay-Kitaev theorem states that all universal sets are roughly the same efficient. More precisely, the number of gates needed to approximate any gate U ∈ SU (d) is bounded by O(log c (1/ǫ)) [26], where c may depend only on d and c ≥ 1. Recently there has been a bit of flurry in the * E-mail: a.sawicki@cft.edu.pl, karnas@cft.edu.pl area of single qubit gates [3,21,22,34] showing that using some number theoretic results and conjectures one can construct universal sets with c = 1. The approach presented in these contribution...
We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as identification of states with 'the same correlations properties' i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question 'what can be transformed into what via available means?'. Exactly such an interpretation, i.e, in terms of mutual transformability, can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods. more element to "the mysteries of quantum mechanics" as seen from the classical point of view.Although typically a quantum system, as, e.g., a harmonic oscillator or a hydrogen atom, is described in terms of an infinite-dimensional Hilbert space, for most quantuminformation applications the restriction to finite dimensions suffices, since usually the active role in information processing play only spin degrees of freedom or only few energy levels are excited during the evolution.From the mathematical point of view such finite-dimensional quantum mechanics seems to mount a smaller challenge than in the infinite-dimensional case -the tool of choice here is linear algebra rather than functional analysis. Nevertheless, understanding of correlations in multipartite finite dimensional quantum systems is still incomplete, both for systems of distinguishable particles [4] as well as for ones consisting of nondistinguishable particles like bosons and fermions [5,6,7,8].The statistical interpretation of quantum mechanics disturbs a bit the simple linearalgebraic approach to quantum mechanics -vectors corresponding to a state (elements of a finite-dimensional Hilbert space H) should be of unit norm. Obviously, physicist are accustomed to cope with this problem in a natural way by "normalizing the vector and neglecting the global phase". Nevertheless, it is often convenient to implement this prescription by adopting a suitable mathematical structure, the projective space P(H), already from the start 1 . The projective space is obtained from the original Hilbert space H by identifying vectors 2 differing by a scalar, complex, non-zero f...
We consider a product of two finite order quantum -gates U1, U2 and ask when has an infinite order. Using the fact that is a double cover of we actually study the product of two rotations and about axes , . In particular, we focus on the case when , and are rational multiples of π and show that γ is not a rational multiple of π unless . The proof presented in this paper boils down to finding all pairs that are solutions of .
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