Application of Geometric Phase in Quantum Computations
aa r X i v : . [ qu a n t - ph ] J u l Application of Geometric Phase in Quantum Computations Application of Geometric Phase in QuantumComputations
A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. TregubovichNational Centre of High Energy Physics,220040 Bogdanovicha str. 153, Minsk, BelarusE-mail: [email protected], [email protected]
Computer Science and Quantum Computing, p.p.125-149, Nova Science Publishers 2007
This article is dedicated to memory of our dear friend, colleagueand co-author Dr. Artur Tregubovich, (1961–2007)
Abstract
Geometric phase that manifests itself in number of optic and nuclear experi-ments is shown to be a useful tool for realization of quantum computations in socalled holonomic quantum computer model (HQCM). This model is consideredas an externally driven quantum system with adiabatic evolution law and finitenumber of the energy levels. The corresponding evolution operators representquantum gates of HQCM. The explicit expression for the gates is derived bothfor one-qubit and for multi-qubit quantum gates as Abelian and non-Abeliangeometric phases provided the energy levels to be time-independent or in otherwords for rotational adiabatic evolution of the system. Application of non-adiabatic geometric-like phases in quantum computations is also discussed fora Caldeira-Legett-type model (one-qubit gates) and for the spin 3/2 quadrupoleNMR model (two-qubit gates). Generic quantum gates for these two modelsare derived. The possibility of construction of the universal quantum gates inboth cases is shown.
Keywords:
Quantum computer, Berry phase, Non-adiabatic geometric phase,Two-qubit gates
The conceptions of quantum computer (QC) and quantum computation developedin 80-th [1], [2] were found to be fruitful both for computer science and mathemat-ics as well as for physics [3]. Although a device being able to perform quantumcomputations is now far away from practical realization, there is a great number
A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich of theoretical proposals of such a construct (see e.g. [6]–[18]). Intensive investi-gations on quantum information theory (see e.g. [4], [5] for a reference source onthis subject) refreshed some interest on Berry phase [56]. The idea of using unitarytransformations produced by Berry phase as quantum computations is proposed in[19], [20] and first realized in [21], [47] in a concrete model of holonomic quantumcomputer where the degenerate states of laser beams in non-linear Kerr cell areinterpreted as qubits. For other references where Abelian Berry phase is consideredin the context of quantum computer see e.g. [22] - [25]. If the corresponding energylevel is degenerate non-Abelian phase takes place [57] that is actually a matrix mix-ing the states with the same energy. For further references on quantum computationbased on non-Abelian geometric phase see e.g. [47]–[53].On the other hand non-adiabatic analogue of Berry phase can exist and be mea-sured if transitions in a given statistical ensemble do not lead to loose of coherence[71]. For loose of coherence in quantum computations related to geometric phasesee [41]–[46]. Thus it is also possible to use the corresponding unitary operators torealize quantum gates. This fact has been noticed in [26], [27]. After that a lot ofpapers was published where the non-adiabatic phase is applied to realize the basicgates in different models of QC such as different NMR schemes [28]–[35], ion traps[36], [37], quantum dots [38], [39], and superconducting nanocirquits [40].To analyze a concrete scheme for quantum computation based on geometricphase it is desirable to be aware of analytical expression for the evolution operator ofthe system at least at the moment when the measurement is performed. This articleis concentrated on the computational aspect of geometric phase for the models whichare relevant to QC. It should be emphasized that the form of the expression for thephase and the possibility of the derivation of such a formula itself thoroughly dependon the group-theoretic structure of the corresponding Hamiltonian. Therefore amethod of the geometric phase calculation which would be more or less universal atleast in the adiabatic case can appear to be useful. The material is divided in twoparts. In section 2 the adiabatic geometric phase is considered. In subsection 2.1 weanalyze the difficulties appearing in calculation of the Abelian adiabatic geometricphase (Berry’s phase) and propose a method of its explicit derivation for the caseof the symmetric time-dependent Hamiltonian with constant non-degenerate energylevels. The symmetry of the Hamiltonian is supposed to reduce the Hamiltonianto that of a system with finite number of energy levels. In subsection 2.2 thismethod is generalized for the case when degeneration is present. In section 3 weconsider non-adiabatic phase which can only conditionally be called ”geometric” forits dependence on concrete details of the dynamics. For this reason it is not possibleto work out more or less general approach to the calculation of the non-adiabaticphase. Therefore two concrete cases are considered. In subsection 3.1 application ofthe Abelian non-adiabatic phase to one-qubit computation in a Caldeira-Legett-typemodel is cosidered. In subsection 3.2 we present an example of both non-Abelian andnon-adiabatic phase computation in spin-3/2 quadrupole NMR resonance model. pplication of Geometric Phase in Quantum Computations Here we consider a possible method of the adiabatic phase computation that seemsto be effective in a broad range of practically relevant cases. Berry phase is a conse-quence of the adiabatic (or Born–Fock) theorem [55] which states that a parametricquantum system depending on a set of slowly (adiabatically) evolving parameters R i ( t ) , i = 1 , . . . N behaves in a quasi-stationary mannerˆ H ( R ) | n ( R ) > = E n ( R ) | n ( R ) >, R = ( R , . . . , R N ) (1)where ˆ H ( R ) is the corresponding Hamiltonian and no energy level degeneration isassumed. The adiabaticity condition means that the frequencies ω n ( R ) = E n ( R ) / ~ are much greater than the characteristic Fourier frequencies of R i ( t ). Thus theeigenvectors | n ( R ) > evolve like | n ( R ) > = ˆ S ( R ) | n >, | n > = | n ( R (0)) >, ˆ S ˆ S † = 1 (2)with unitary rotation ˆ S describing the natural variation of | n ( R ) > due to that of R ( t ). It corresponds to the following evolution law of the Hamiltonianˆ H ( t ) = ˆ S ( R ) ˆ H ( t ) ˆ S † ( R ) (3)where H ( t ) is diagonal in the basis {| n > } . What is the solution of the non-stationary Schr¨odinger equation i ~ ∂∂t | ψ ( t ) > = ˆ H ( t ) | ψ ( t ) > (4)for this case? A natural hypothesis would be that the evolution operator for | ψ > | ψ ( t ) > = ˆ U ( t ) | ψ (0) > has the form ˆ U ( t ) = ˆ S ( R ) ˆΦ( t )where ˆ S is defined by (2) and ˆΦ simply produces the dynamic phaseˆΦ( t ) | n ( R ( t ) > = exp − i/ ~ t Z E n ( τ ) dτ | n ( R ( t ) > . (5)It is based on the analogue with the stationary case where evolution is simply rep-resented by the dynamical phase factor exp( − i/ ~ E n t ). Berry first observed [56]that the hypothesis is wrong. To see this it is sufficient to represent ˆ U in the formˆ U ( t ) = ˆ S ( R ) ˆ V ( t ) and substitute it into the Schr¨odinger equation i ~ ∂∂t ˆ U ( t ) = ˆ H ( t ) ˆ U ( t ) . (6) A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich
It gives i ~ ( ˆ˙ V ˆ V † + ˆ S † ∇ R ˆ S ˙ R ) = ˆ H ( t ) . (7)Now one can see that ˆ V cannot be simply ˆΦ because it has to cancel the secondterm in the right hand side of (7) besides of H . It follows from (7)thatˆ V ( t ) = ˆΓ( t ) ˆΦ( t )where ˆΦ( t ) is determined by (5) and the following equation is valid for ˆΓ( t ):ˆ˙ΓˆΓ † | n > = − ( ˆ S † ∇ R ˆ S ) ˙ R | n > . (8)It results in the evolution law for the state vector corresponding to the n -th energylevel | ψ n ( t ) > = e − i/ ~ Φ n ( t ) e iγ n ( t ) | n ( R ( t )) > (9)where Φ n ( t ) is the phase factor in the right-hand side of (5) and γ n ( t ) is given by γ n ( t ) = t Z A n ( R ( τ )) ˙ R ( τ ) dτ, A ( R ) = i < n ( R ) | ∇ R n ( R ) > . (10)Phase γ n ( t ) becomes purely geometric while R ( t ) evolves cyclically: R ( T ) = R (0) γ n ( T ) ≡ γ n ( C ) = I C A n ( R ) d R . (11)Here the integration contour C is a closed curve in the parameter space describedby R ( t ) as a radius-vector. It is easily seen from (11) that γ n ( C ) does not dependon the concrete details of the system’s dynamic if the adiabatic condition is held.In this article we are interested in computing of γ n ( C ) in the most general case.The problem of derivation of γ n ( C ) was solved in various particular cases in largenumber of articles some years ago. First we would like to note that the straightfor-ward formula B n = ∇ R × A n = X m = n ( ∇ R ˆ H ( R )) mn × ( ∇ R ˆ H ( R )) nm ( E n ( R ) − ( E m ( R )) (12)derived by Berry [56] by making use of the identity < m |∇ n > = < m |∇ ˆ H | n > ( E n − E m ) , m = n has not (despite of it’s beauty) much practical use because to apply it one shouldestablish the analytical dependence of all E n on R that is not a realistic task exclud-ing some special cases. To see this one should attempt to apply formula (12) to the pplication of Geometric Phase in Quantum Computations E n ( R ) of the correspondingcubic equation therein.It was first noticed in [61] that symmetries of the Hamiltonian ˆ H ( R ) play animportant role in computing of γ n . Indeed if one represents the result of the periodicmotion ˆ H (0) = ˆ H ( T ) as | ψ ( T ) > = ˆ U ( T ) | ψ (0) > (13)where as it follows from (1) ˆ U ( T ) must commute with ˆ H (0) so it can be representedas an exponent containing a linear combination of operators ˆ X k which must com-mute with ˆ H (0) as well. Thus the operators ˆ X k are integrals of motion and describecertain symmetries of the given system. Therefore in what follows we restrict our-selves with such systems whose Hamiltonian is an element of a finite Lie algebra.This assumption immediately gives the group-theoretic structure of ˆ U ( T ):ˆ U ( T ) = exp i X i a i H i ! (14)where H i are all linearly independent elements of the Cartan subalgebra and a i aresome coefficients. Thus the problem reduces to computing of the coefficients a i . Inthe simplest case of Lie algebras consisting of three elements this problem can beeasily solved [62], [64], [65] for physically relevant cases of Heisenberg-Weyl algebra, su (2) and su (1 . U ( t ) = exp (cid:16) ζ ( t ) ˆ X + − ζ ∗ ( t ) ˆ X − (cid:17) exp (cid:16) i φ ( t ) ˆ X (cid:17) (15)provided the initial Hamiltonian is proportional to ˆ X where ˆ X ± and ˆ X are thecorresponding generators of the algebras above. Their expressions for each concretecase are given in table 1. The Hamiltonian for su (2) case describes an arbitraryspin in the magnetic field so all J ’s are the angular momentum operators: ˆ J ± =1 /
2( ˆ J ± ˆ J ). su (1 .
1) case corresponds to the evolution of squeezed states [66] oflight in non-linear optics. Here ˆ K + = ˆ a + /
2, ˆ K − = ˆ a / K = ˆ a + ˆ a + 1 / a , ˆ a + are usual bosonic annihilation and creation operators. The last case representsa harmonic oscillator interacting with the time-dependent electric field. The simplecommutation relations in these three algebras admit direct computation of Berry’sphase [62], [64], [65]. γ m = m I C ω ( ξ ) = Z S d ∧ ω ( ξ ) (16)where m is an eigenvalue of the corresponding ˆ X , S is the surface in the parameterspace bounded by the closed curve C and the expressions for ω ( ξ ) and its exter-nal derivative d ∧ ω ( ξ ) are given for each case in table 2. The geometric sense ofthe derived phase factor is the integral curvature over the surface bounded by thecontour C on the manifold the evolution operator belongs to. This manifold can A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich
Table 1: Expressions for the operators ˆ X ± , ˆ X .Algebra ˆ X + ˆ X − ˆ X Commutators su (2) ˆ J + ˆ J − ˆ J [ ˆ J , ˆ J ± ] = ± ˆ J ± [ ˆ J + , ˆ J − ] = 2 ˆ J su (1 .
1) ˆ K + ˆ K − ˆ K [ ˆ K , ˆ K ± ] = ± ˆ K ± [ ˆ K + , ˆ K − ] = − K H − W ˆ a + ˆ a ˆ1 [ˆ1 , ˆ a + ] = [ˆ1 , ˆ a ] = 0[ˆ a, ˆ a + ] = ˆ1be generally expressed in the form G/H where G is the group manifold and H isthat of the stationary subgroup, i.e. the group whose Lie algebra consists of alloperators commuting with H (0) (in these three cases it is always U (1)). It is spherein the case of su (2), two-sheet hyperboloid in the case of su (1 .
1) and plane in thecase of Heisenberg-Weyl group. To complete the computation one has to establishcorrespondence between the complex parameter ξ and physical parameters of theHamiltonian. Let us do that for SU (2). It is worth to notice that the result iscompletely determined by the geometric properties of the group and does not de-pend on the concrete representation. For this reason one can chose the fundamentalrepresentation of SU (2) to simplify the derivation. Thus we take H (0) = ω B σ which corresponds to the initial eigenvectors |± > = (1(0) , T ( T denotes trans-position). The evolution operator generally parametrized by the spherical as cos ϑ − sin ϑ e − iϕ sin ϑ e iϕ cos ϑ (17)rotates H (0) into H ( t ) = ω B nσ where n = (cos ϕ sin ϑ/ , sin ϕ sin ϑ/ , cos ϑ/ ζ ˆ J + − ζ ∗ ˆ J − ) gives for this representation | ζ | = ϑ/ , arg ζ = ϕ + π .It leads to the well known expressions for the fictitious ”strength field” B ± = ∓ R R (18)where R denotes the true magnetic field vector in order not to confuse it with the pplication of Geometric Phase in Quantum Computations ω and d ∧ ω .Algebra ω ( ξ ) d ∧ ω ( ξ ) Relation to ζsu (2) ξdξ ∗ − ξ ∗ dξ | ξ | dξ ∧ dξ ∗ (1 + | ξ | ) | ξ | = tan( | ζ | ) , arg ξ = arg ζsu (1 . ξdξ ∗ − ξ ∗ dξ − | ξ | dξ ∧ dξ ∗ (1 − | ξ | ) | ξ | = tanh( | ζ | ) , arg ξ = arg ζ H-W ξdξ ∗ − ξ ∗ dξ dξ ∧ dξ ∗ ξ = ζ fictitious one which determines the resulting Berry’s phase. It should be noted thatthe correspondence R → ξ realizes the stereographic projection of the sphere withthe coordinates ϑ, ϕ on the plane that points are labeled by ξ . The other two casesof SU (1 .
1) and Heisenberg-Weyl groups can be considered in a similar manner.Re-derivation of these simplest results has the intention to extract a universalidea of computing the geometric phase in more or less general case. For the sake ofcertainty let us assume the symmetry algebra of the Hamiltonian to be semisimple.It means that the generic evolution operator can be represented in the formˆ U ( t ) = Y α ∈ ∆ + ˆ U α ( t ) (19)where ∆ + denotes the set of the positive roots α and each U α is analogous to (15)(see also table 1 for su (2) and su (1 .
1) cases):ˆ U α ( t ) = exp (cid:16) ζ α ( t ) ˆ E α − ζ ∗ α ( t ) ˆ E − α (cid:17) (20)where the standard notations for the Cartan basis [67][ H β , ˆ E ± α ] = ± α ( H β ) ˆ E ± α [ ˆ E α , ˆ E − α ] = H α , [ ˆ E α , ˆ E β ] = N αβ ˆ E α + β (21)are used. The pairs of generators ˆ E ± α are analogous for ˆ J ± in su (2) and H α ’s arethat of ˆ J . Here α ( H β ) and N αβ are constants that can be chosen rational andinteger correspondingly. Taking account of the consideration above leads to somemore detailed form for one-cycle evolution operator ˆ U ( T ) (14)ˆ U ( C ) = exp i X α ∈ ∆ + a α ( C ) H α . (22) A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich
Each pair ( ˆ E α , ˆ E − α ) makes besides of the trivial group-theoretic contribution H α which produces the corresponding quantum number also a non-trivial one reflectingadiabatic dynamics of the system a α ( C ) = I C θ α ( ζ ) (23)where θ α generally depends on all ζ α ( t ). Thus to solve the problem one has to findthis dependence making use of commutation relations (21) and then establish theconnection between the parameters ζ α ( t ) and the natural set of parameters R ofthe Hamiltonian. Unfortunately the hope to obtain a solution of even one of thesetwo tasks that would be a non-trivial generalization of the above examples is notrealistic. Neither the first part of the problem nor the second one could be solved ina way resulting in physically relevant explicit formulas having practical use. Firstthe 1-forms θ α fulfill Maurer-Cartan equations that express the quantity ˆ U † d ˆ U interms of the 1-forms ω α and θ i ˆ U † ( ζ ) d ˆ U ( ζ ) = i ( ω α ( ζ ) ˆ E α + θ i ( ζ ) H i ) (24)where the index i labels all linearly independent generators of the Cartan subalgebra(not all H α are so). For commutation relations (21) these equations take the form d ∧ ω α = C αβk ω β ∧ θ k + 1 / C αβλ ω β ∧ ω λ (25) d ∧ θ i = 1 / C iβλ ω β ∧ ω λ (26)Equations (25), (26) describe the parallel transport on the coset manifold G/H .The possibility to solve them depends on the manifold’s symmetry and of courseis entirely determined by the structure constants C ·· · that are built from the rootvectors α ( H β ) and the constants N αβ . The general solution of this system canbe constructed for very high symmetry of symmetric spaces [68] where the wholealgebra can be split in two subsets X and Y such that[ Y, Y ] ⊂ Y, [ Y, X ] ⊂ X, [ X, X ] ⊂ Y. It is seen from (21) that the last condition is generally speaking not valid for ourcase because not all N αβ are zeroes. Its geometric sense is that the considered cosetspaces G/H are of more general symmetry type than symmetric spaces. Thus for G = SU ( n ) the space SU ( n ) / U (1) × U (1) . . . × U (1) | {z } n − (27)belongs to the more general class of K¨ahlerian spaces. The general solution of (25),(26) for the types of spaces we are interested in is not obtained so far. Thereforethe practical use of these equations is not high. Moreover the solution of the secondpart of the problem discussed is not possible for the same reason. pplication of Geometric Phase in Quantum Computations
9A simple and effective method of practical computation of geometric phase whereit is not necessary to find the forms ω α , θ i is proposed in [75]. For this purpose wehave to make some assumptions. First we regard the Hamiltonian to belong toa finite irreducible representation of a semisimple Lie algebra therefore ˆ H in (3)can always be represented as a finite matrix ˆ H = R i ( t ) H i where the set { H i } is a basis of the Cartan subalgebra and R i ( t ) are parameters. Then we supposethe energy levels E m to be constants. It corresponds to a rotation-type evolution(3) where ˆ H does not depend on t . Such a situation takes place practically in allexperiments on the geometric phase measurement. This makes it possible to regard E m as additional secondary parameters to be found just once (may be numerically).The third assumption is that the spectrum remains always non-degenerate i.e., nocrossing of energy levels occurs. As the spectrum of the Hamiltonian is finite, thestate vector | ϕ m i is a unit vector m in C n , so A m is A m = i m ∗ d m − m d m ∗ ) . (28)As the evolution is adiabatic, the spectrum of H ( t ) remains always non-degenerateif it was so at the initial time. Then there is always a nonzero main minor of H − E m which we assume to consist always of the first n − H − E m .Denoting the matrix consisting of the first n − H by H ⊥ wecome to the condition det( H ⊥ − E m ) = 0 (29)Making use of this condition one can represent n in the uniform coordinates m = ( ξ m , p | ξ m | and express ξ m in terms of H ij for 1 ≤ i, j ≤ n − E m : ξ m = ( H ⊥ − E m ) − h , h i = − H in , (30)where h is a vector in C n − but not in C n . Thus we have for A m A m = i ξ m ∗ d ξ m − ξ m d ξ m ∗ ) . | ξ m | , (31)where ξ m is completely determined by (30). Note that the result obtained is purelygeometrical because it can be expressed of the K¨ahlerian potential F = log(1 + | z | )where z is a vector in C N consisting of n ( n − / ξ m . The function F ( z , z ∗ ) determines all the geometrical properties of the statespace (27). Particularly its metric tensor is g ij = ∂ F ( z , z ∗ ) ∂z i ∂z ∗ j . A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich
It should be emphasized that the simplification of the problem reached here is basedon the fact that ˙ E m = 0 so one can include it in new parameters and use them ratherthan R i . Therefore the dependence of E m on R is not required. One can calculate E m numerically and substitute it into the formulas regarding this quantity as onemore external parameter. Moreover to find γ m one needs only the energy E m butnot the whole spectrum as in (12). It can become an important issue if one considerspartially solvable models. The requirement ˙ E m = 0 is sufficient because otherwiseone has to solve the secular equation at each moment t that is equivalent to thenumerical solution of the non-stationary Schr¨odinger equation itself and therefore itmakes the discussed method useless.Let us now consider some simple applications of the proposed method. Firstlet us see how it works for the trivial case n = 2. (1) reduces then to two linearlydependent equations ( B ∓ B ) ξ + ( B − iB ) = 0( B + iB ) ξ + ( − B ∓ B ) = 0Here we returned to the usual notations of the magnetic field components B i and ± B = ±| B is the energy of the state |± > . Choosing one of the equations andtaking the spherical coordinates we come to one of the relations ξ = − tan ϑ/ e − iϕ ξ = cot ϑ/ e − iϕ . for the upper and lower sign correspondingly. Thus these are the coordinates ofstereographic projection made from the north (south) pole of the sphere. Substitut-ing it into the formula for ω ± ( ξ ) (see Table 2 ) and integrating over a contour C weget the well known result [56] γ ± = ± I C ξ ∗ dξ − ξdξ ∗ | ξ | = ∓
12 Ω( C ) , where Ω( C ) is the solid angle corresponding to the closed contour C on the sphere.One more example which is less trivial is a generic three-level system. The k -th eigenvector ξ k is then two-dimensional and some trivial algebra gives for itscomponents ξ = ∆ / ∆ , ξ = ∆ / ∆ , ∆ = ( H − E k )( H − E k ) − | H | ∆ = H H − H ( H − E k )∆ = H H ∗ − H ( H − E k ) (32)Here we have omitted where possible the index k . Substitution of these expressionsinto (31) gives the final formula for this case. Note that the use of formula (12) pplication of Geometric Phase in Quantum Computations H ij , i ≤ j but H donot depend on t . Then substitution of (32) into (31) gives ω k ( ξ ) = i C k [ A − D k sin( φ + φ − φ ) ] dφ , (33) C k = | H || H || H | ∆ ( E k ) + | ∆ ( E k ) | + | ∆ ( E k ) | , (34) A = 1 / | H | − / | H | , (35) D k = H + H − E k (36)where φ ij are arguments of the complex numbers H ij . As it was discussed abovethe condition ∆ ( t ) = 0 is supposed to be held everywhere on C . For other cases ofthe geometric phase in the 3-level system see [63]. Now we proceed with a more general case of degenerate spectrum. Quantum com-putation for this case generated by an adiabatic loop in the control manifold is de-termined by the same quasi-stationary Schr¨odinger equation (1) where each energylevel E m corresponds to a set of eigenstates | m a >, a = 1 , . . . d m . Cyclic evolutionof the parameters results in | m a ( T ) > = U ab ( T ) | m b (0) > (37)where the matrix U is presented by a P -ordered exponent U ( C ) = P exp (cid:18)I C A m (cid:19) , ( A m ) ab = i < m b | dm a > . (38)In this section we generalize the proposed approach to the geometric phase compu-tation for the generic case of degenerate energy levels [75].The set of eigenvectors ξ ma , a = 1 , ..., d m must obey the equation( H ( d m ) ⊥ − E m ) ξ ma = h c a . (39)Here the matrix H ( d m ) ⊥ is constructed from the first n − d m lines and columns of H , c a are arbitrary d m -dimensional vectors and h is the following ( n − d m ) × d m -matrix: h = − H ,n − d m +1 . . . H ,n ... . . . ...H n − d m ,n − d m +1 . . . H n − d m ,n A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich
Of course it has sense only if the conditiondet( H ( d m ) ⊥ ( t ) − E m ) = 0 (40)is valid along the evolution process. The set of vectors ξ ma must be orthogonalizedby the standard Gram algorithm and after that we get the orthonormal set of theeigenvectors z a (here and below we has omitted the index m ) in the form z a = 1det Γ a − x Γ a − ... h ξ a | ξ i . . . h ξ a (cid:12)(cid:12) ξ a − i x a , (41)where x b = ( ξ b , c b ) and c a is chosen to be the standard orthogonal set c a = (0 ... a z}|{ ... a are determined byΓ a = h ξ | ξ i . . . h ξ | ξ a i ... . . . ... h ξ a | ξ i . . . h ξ a | ξ a i = 1 + Z † a Z a , (42)where the ( n − d m ) × a -matrix Z a consists of a first lines of the ( n − d m ) × d m -matrix Z = ( H ( d m ) ⊥ − E m ) − h . Using (42) and (41) we come to the final expression for thematrix-valued 1-form A : A = i g ijab ( ξ j ∗ d ξ i − d ξ j ∗ ξ i ) + 2 ω ab det(1 + Z † a − Z a − ) det(1 + Z † b − Z b − ‘ ) , ≤ i ≤ a, ≤ j ≤ b, (43)where g ijab = Γ ia Γ ∗ jb , ω ab = h ξ j (cid:12)(cid:12)(cid:12) d Im( g ijab ) (cid:12)(cid:12)(cid:12) ξ i i + min ( a,b ) X i =1 d Im( g iiab ) , and Γ ia is the cofactor of ξ i in Γ a . Note that the change of our basis c a by c ′ a = U ab ( λ ) c b leads to a standard gauge transformation of AA ′ = U A U † + i ( dU ) U † . The formula (43) is the desired expression of A in terms of the matrix elementsof the Hamiltonian. It is correct if condition (40) is valid. It is not neverthelessa principal restriction because d m does not depend on time due to adiabaticity ofthe evolution and there is always at least one nonzero n − d m -order minor of H .Then, if the minor we choose vanishes somewhere on the loop C one can alwaystake local coordinates such that the techniques considered is applicable on eachsegment of C . It should also be noted at the end of this section that the idea ofphysical realization of the quantum gates based on the concrete system driven by pplication of Geometric Phase in Quantum Computations A n E α can be realized by means of ordinary bosonic creation and annihilation operators,namely E α = a i † a j for some 1 ≤ i, j ≤ n . Then E α represents nothing but two-modesqueezing operator. Thus the model considered can be applied to optical HQC with n laser beams (the case n=2 is considered in [21]) and the logical gates U α are justtwo-qubit transformations realized by transformation of two laser beams.The method presented here enables one to build in principal any computationfor HQC described by a Hamiltonian with a stationary spectrum in terms of ex-perimentally measured values exactly the matrix elements of the Hamiltonian. Themethod depends weakly on the dimension of the qubit space which other modelsbased on various parameterizations of the system’s evolution operator are very sen-sitive to. Application of this method to a concrete physical model will be discussedelsewhere. The adiabatic condition of quantum system’s evolution is strong enough to restrictsufficiently the scope of the search for realistic candidates for practical realizationof quantum computations despite of some attractive features of the adiabatic casesuch as fault tolerance due to independence of the evolution law on the details of theparameters’ dynamics etc. Therefore it is desirable to find physically relevant casesfor which on the one hand this condition would be not necessary but on the otherhand the coherency in such a system would be not yet violated so that the notionof the phase shift itself could have physical sense. As the adiabatic theorem is nolonger valid the property of universality of the system’s dynamics (independence ofthe concrete form of the functions R i ( t )) is no more preserved and the evolutionlaw is sufficiently more complicated. Then one cannot hope to carry out a generalapproach to derivation of the corresponding phase shift because in each case itdepends on the fine details of the parameters variation. For the same reason thephase can be called ”geometric” only conditionally because geometric intuition isno more helpful for this case e.g. the result can be represented as an integral over t rather than over a contour that expresses mathematically the thesis above. Onthe other hand non-adiabatic conditional geometric phase that was theoreticallypredicted in [60] can be measured if transitions taking place in the system do notlead to decoherence [71]. Therefore it is also possible to use the correspondingunitary operators to perform quantum calculations. This fact has been noticed in[26], [27] (see also [28]–[35] for further references).Let us consider a parametric quantum system described by the Hamiltonian H ( R ), where R ( t ) is a set of arbitrarily evolving parameters. We suppose thatevolution of the Hamiltonian is determined by unitary rotation (3) Looking for4 A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich particular solutions of the Schr¨odinger equation (4) (here we supposed ~ = 1) wetake a rotating frame by assigning ˜ ψ ( t ) = U ( t ) ψ ( t ) and get in such a way i ∂ ˜ ψ∂t = ( H ( t ) − i U † ( t ) ˙ U ( t )) ˜ ψ ( t ) . (44)Of course, this transformation generally does not help to solve equation (4) due tothe fact that the algebraic structure of the coupling term − i U † ˙ U can appear to berather complicated and the last generally does not commute with H . However ifa receipt is known how to evaluate the last term in (44), further solution of thisequation is straightforward: ψ ( t ) = e − i φ n ( t ) T exp (cid:18) − i Z t U † ( τ ) ˙ U ( τ ) dτ (cid:19) ψ (0) , (45)where φ n ( t ) = R t E n ( τ ) dτ is so called dynamic phase, T denotes time-ordering and E n are elements of H that is by definition diagonal. Of course if there is no way tofind U † ( τ ) ˙ U ( τ ), expression (45) is useless.Let us illustrate it for the simplest case of spin 1 / ϑ = const is representedby H ( t ) = e ± iω R t ˆ J e − iϑ ˆ J (Ω ˆ J ) e − iϑ ˆ J e ∓ iω R t ˆ J (46)where the sign + ( − ) corresponds to the left (right) polarization. Application of(44) to (46) gives for the Hamiltonian in the rotating frame H ( t ) = e − iϑ ˆ J (Ω ˆ J ) e − iϑ ˆ J ± ω R ˆ J . (47)To diagonalize Hamiltonian (47) one has to apply one more rotation to it H ( t ) = V H ( t ) V † , V = e iϑ ∗ ˆ J where the angle ϑ ∗ does not coincide with ϑ due to the second non-adiabatic term.It should rather fulfill the conditiontan ϑ ∗ = sin ϑ cos ϑ ± ω R / Ω . (48)The second term in the denominator of (48) is the measure of non-adiabaticity ofthe motion. It is clear that the angle ϑ ∗ replaces the usual azimuthal angle ϑ in theformula for the geometric-like phase: γ ± = ∓ m π (1 − cos ϑ ∗ ) (49)where m is the third spin projection. Formula (49) is a natural generalization ofthe usual Berry’s formula for the adiabatic case and coincides with it in the limit pplication of Geometric Phase in Quantum Computations ω R / Ω →
0. Note that the dependence of the result on ω R reflects the fact the phaseis no longer truly geometric because ω R characterizes the rotation velocity and thusthe velocity of the motion along the contour in the parameter space.As an example of the application of the non-adiabatic formula above we proposea realization of quantum gates for a concrete 4-level quantum system driven byexternal magnetic field [77]. Let us consider a system of two qubits in a bosonicenvironment described by the Hamiltonian H = H S + H B + H SB , (50)where H S is the Hamiltonian of two coupled spins H S = H (0) S + H int S = ω σ z ⊗ + ω ⊗ σ z + J σ z ⊗ σ z , (51)where J is the coupling constant, H B is the Hamiltonian of the bosonic environment H B = X k ω bk (ˆ b + k ˆ b k + 1 / , (52)and H SB is the Hamiltonian of the spin- enviroment interaction. H SB = H (1) SB + H (2) SB , (53) H ( a ) SB = S ( a ) z X k ( g ak ˆ b + k + g ∗ ak ˆ b k ) a = 1 , . (54)Here S (1) z = σ z ⊗ , S (2) z = 1 ⊗ σ z ,σ z is the third Pauli matrix, 1 is 2 × b + k , ˆ b k are bosonic creationand annihilation operators and g ak are complex constants. We assume that thetwo spins under consideration are not identical so that ω = ω . The Hamiltoniandetermined by (50) – (54) is a natural generalization of Caldeira-Legett Hamiltonian[74] for the case of two non-interacting spins. Let such a system be placed in themagnetic field affecting the spins but not the phonon modes. The only change tobe made in the spin part (51) is the substitution ω s σ z −→ Bσ , Three components of B represent a control set for the qubits under consideration.Evolution of B ( t ) generates evolution of the reduced density matrix ρ s ( t ) thatdescribes the spin dynamics i ∂ρ s ( t ) ∂t = H S ρ s ( t ) , ρ s ( t ) = U ( t ) ρ (0) U + ( t ) . (55)6 A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich
Thus given curve in the control space corresponds to a quantum calculation inwhich each qubit is to be processed independently. To obtain such a calculation asa function of control parameters we first recall some common issues of spin dynamics.We consider the external magnetic field as a superposition of a constant componentand a circular polarized wave: B = B + B e iω R t , (56)where B is perpendicular to B . It is well known that the case of the circularpolarization is exactly solvable. The evolution of an individual spin correspondingto the Hamiltonian H = − µB (57)is determined by (15) where ˆ X ± , ˆ X are replaced by S ± = S x ± iS y , S z correspond-ingly and ζ ( t ) = | ζ ( t ) | exp ( i ∆ ωt + iα ( t ) + iπ/ , (58) | ζ ( t ) | = ω ⊥ sin(Ω t/ q (∆ ω ) + ω ⊥ ,α ( t ) = arctan (cid:18) ∆ ω Ω tan(Ω t/ (cid:19) ,φ ( t ) = − ω ⊥ ( ξ n + ξ n ) , (59)where ∆ ω = ω k − ω R , Ω = (∆ ω ) + ω ⊥ , ω ⊥ and ω k are Rabi frequencies corre-sponding to B and B respectively and finally n is the unit vector along B .It is known [71] that the pure states acquire within the rotating wave approxi-mation a phase factor that after one complete cycle T = 2 π/ω R is: | m ( T ) > = exp( − iφ D + iγ ) | m (0) >, (60)where m is the azimuthal quantum number and the phase is split in two parts:dynamic φ D = 2 πm Ω ω R cos( θ − θ ∗ )and geometrical γ = − πm cos θ ∗ , (61)where cos θ = B /B ( B = B + B ) and θ ∗ is determined by formula (49). Thephase shift between the states | ± / > results then in∆ φ g = − π cos θ ∗ (62) pplication of Geometric Phase in Quantum Computations B (0) = B ( T ) on theBloch sphere. If the rotation is slow such that ω R / Ω → θ ∗ → θ and phaseshift(62) coincides with the usual Berry phase.Thus the adiabaticity condition is not really necessary for obtaining of the ge-ometrical phase in an ensemble of spins if the decoherence time is much greaterthan T . Therefore one can attempt to use this phase to get quantum gates suchas CNOT. Calculation of the corresponding phase factors is rather straightforwardbecause the free and the coupling parts of the spin Hamiltonian commute with eachother h H (0) S , H int S i = 0 . Therefore the coupling part can be diagonalized simultaneously with the free partby applying of the transformation U = U ⊗ U where U , are the diagonalizingmatrices for each single-spin Hamiltonian respectively. This simple fact togetherwith the following obvious identity U † ˙ U = U † ˙ U ⊗ + 1 ⊗ U † ˙ U the final formula for the part of the evolution operator that stands for the non-adiabatic geometric phase U g = exp( − πi cos θ ∗ S z ) ⊗ exp( − πi cos θ ∗ S z ) , (63)where tan θ ∗ = sin θ cos θ + ω R / Ω , tan θ ∗ = sin θ cos θ + ω R / Ω and cos θ = ω / Ω , Ω = ω + ω , cos θ = ω / Ω , Ω = ω + ω . Note that gate (63) is symmetric with respect to the spin transposition as it shouldbe and does not depend on J that is typical for geometrical phase in spin systemswhere the phase depends only on the position drawn by the vector B on the Blochsphere. As J does not affect this position, it is absent in the final result. We do notconsider here the dynamic phase determining by the factor U d = exp (cid:18) − i ~ ˆ H S T (cid:19) . It is so because one can eliminate it by making use of the net effect of the compoundtransformation proposed in [22]. After this transformation that is generated by twodifferent specifically chosen contours the dynamic phase acquired by the different8
A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich spin states becomes the same and the geometric phase of each state is counted twice.After that we get (up to a global phase) the following quantum gate U g = e i ( γ + γ ) e i ( γ − γ ) e i ( − γ + γ )
00 0 0 e − i ( γ + γ ) . (64)Thus we have constructed the quantum gate, which possess the advantage to befault tolerant with respect to some kinds of errors such as the error of the amplitudecontrol of B . On the other hand this approach makes it possible to get rid of theadiabaticity condition that strongly restricts the applicability of the gate. Instead ofthis condition one needs some more weak one: τ ≫ ω − R , where τ is the decoherencetime. In this section we give an example of both non-Abelian and non-adiabatic phase fora concrete 4-level quantum system driven by external magnetic field [76]. Let us con-sider a spin-3 / ± / ± / A that is the subject of computation in this section.We assume the condition of the Xe NMR experiment to be held so one doesnot need to trouble about the coherency in the system. The last is described by thefollowing Hamiltonian in the frame where the magnetic field is parallel to the z-axis( ~ = 1) H = ω ( J − / j ( j + 1)) . (65)Here and in what follows we omitted the hat symbol over all J ’s for the sake ofsimplicity. For a spin-3 / J = / − / / − / = (cid:18) / σ
00 1 / σ (cid:19) , (66) pplication of Geometric Phase in Quantum Computations J = √ / √ / √ / √ / = √ √ σ ! , (67) J = −√ /
20 0 √ / √ / − i −√ / i = − i √ σ i √ σ σ ! . (68)In the laboratory frame the Hamiltonian takes the form H = ω (( J n ) − / j ( j + 1)) = e − iϕJ e − iθJ H e iϕJ e iθJ . (69)Rotation around the z-axis means that ϕ = ω t and one should perform the unitarytransformation | ψ > = U | ˜ ψ >, U = e − iω tJ . (70)In the rotating frame we get H = e − iθJ ( ω J − ω ˜ J ) e iθJ − ω , (71)where ˜ J = e iθJ J e − iθJ Expression (71) is equivalent to H = ω − ω cos θ σ ω √ / ω √ / − ω − ω cos θ σ + ω sin θ σ (72)It is convenient to diagonalize this matrix in two steps. First we get rid of σ in thelast matrix element by performing of the block-diagonal transformation U = diag(1 , e − iα σ ) , (73)where tan α = 2 tan θ . Thereafter the Hamiltonian H reads H = ω − ω cos θ σ ω √ / ω √ / − ω − ω cos θ cos α σ . (74)At the second step we apply the transformation (cid:18) β β − β ∗ β ∗ (cid:19) , (75)0 A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich where β , β are diagonal 2 × | β | + | β | = 1 . (76)Supposing β , to be real and performing transformation (75) we come to the diag-onalization condition in the form ξ ( β − β ) + ( λ − λ ) β β = 0 , (77)where λ , λ are 2 × ξ is a parameter λ = ω − / ω cos θ σ , (78) λ = − ω − / ω cos θ cos α σ (79) ξ = ω √ / θ (80)Assuming β = µ β where µ is a diagonal 2 × µµ i = k i + q k i , (81)where k i = ∆ λ i ξ , ∆ λ i = λ i − λ i . Finally we get for the matrix elements of β , β i = 1 / k i ) − / (cid:18) k i + q k i (cid:19) − (82) β i = 1 / k i q k i (83)Now one can evaluate the connection 1-form. It is convenient to represent it asfollows: A = i U + dU = A dφ = A / A tr ˜ A tr A / dφ, (84)where all matrix elements of A denote 2 × U = U U U and U i are determined by (70), (73), (75) correspondingly. Here tilde denotes a pplication of Geometric Phase in Quantum Computations A tr = 12 β β (3 − cos α ) σ + 12 sin α β σ β , (85) A / = ( a / + b / σ + c / σ ) dφ, (86) a / = 14 (cid:0) β − β + β cos α − β cos α (cid:1) , (87) b / = 14 (cid:0) β + 3 β + β cos α + β cos α (cid:1) , (88) c / = −
12 sin α β β , (89) A / = ( a / + b / σ + c / σ ) dφ, (90) a / = 14 (cid:0) β − β + β cos α − β cos α (cid:1) , (91) b / = 14 (cid:0) β + 3 β + β cos α + β cos α (cid:1) , (92) c / = −
12 sin α β β , (93)where dφ = ω dt . Note that as A does not depend on time, the final solution doesnot require T -ordering. It should be also emphasized here that in the non-adiabaticcase we discuss the term A / contains non-diagonal terms that is not the case whenthe adiabaticity condition is held [58]. Now the solution of the problem takes aparticular form of (45): ψ ( t ) = e − i φ n ( t ) e − iω t A ψ (0) , (94)Formula (94) solves the problem of the evolution control for the system underconsideration. The resulting quantum gate is entirely determined by A and theevolution law of the magnetic field, i.e. by a contour in the parameter space. Ofcourse it is always possible to choose the parameters so that A turns out to generate a2-qubit transformation that produces a superposition of basis states. For this reasonthe gate can be thought of as a universal one [14]. Of course, a suitable speed ofthe parameters evolution can not be reached by rotation of the sample as it tookplace in the experiment by authors of [71]. Nevertheless it is clear that this mannerof control is not principle and one could imagine a situation where the parametersevolution is provided by the controlling magnetic field by adding a non-stationarytransverse component. It should be also noted here that general formulas (85) do2 A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich not provide an apparent way to realize CNOT gate or another common 2-qubit gate.They just give the evolution law of the system provided that the external parametersvary as shown above. To knowledge of the authors other examples of computationof a conditional geometric phase that would be both non-Abelian and non-adiabaticare absent. To provide the gates of common interest one has to invent some specialcase of the parameters variation which makes the generic evolution operator moresimple and transparent. This subject is out of the scope of this article.
The approach developed in [75]– [77] is to be applied in the models where it is notpossible to reduce the computation of the geometric phase to the case of 2-level sys-tem. Among those relevant to QC one can point out e.g. the model with anisotropicHeisenberg ferromagnetism where the exchange term in (51) is determined by a ma-trix of constants J ab rather than by a single constant J . In this case the Zeemanterms H (0) S no longer commute with the exchange term J ab S a ⊗ S b and to derivethe expression for the geometric phase it is necessary to consider a more generalcase of 4-level system. The problem becomes more complicated also if the superfineelectron-nucleus spin interaction must be taken into account. It is the case for theKane model of silicon QC [79]. The effective dimension of the system’s Hamiltonianis then 16. It is hopeless to attempt to obtain an exact analytic expression for thesystem’s dynamics which should be investigated numerically (see e.g. [78] ) but itis nevertheless possible to derive an exact formula at least for the adiabatic phase.One more problem to be mentioned here is interaction with the environment.It can appear to be important not only for such issue as decoherence but it alsocan in principle contribute to the geometric phase. The simplest way to see it is toconsider the model described by (50)– (54). If the external electromagnetic wavefield can affect not only the qubits but the phonons as well. The phonon degrees offreedom can produce Heisenberg–Weyl-like geometric phase that can fill the sign ofthe spin projection due to electron-phonon term (53), (54). Some more complicatedinteraction between the qubits and the environment makes it necessary to computethe geometric phase for a system with the symmetry algebra which is larger than su (2) and cannot be reduced to the last one (in the sense of the phase derivation).Other field of application could be multi-beam optical schemes for quantumcomputations where several energy levels must be included in the scheme to providetwo-qubit operations. Besides of some special cases [80] it can require more generalmethods for the geometric phase computation. References [1] Deutsch, D.
Proc. Roy. Soc. London
Ibid . 1989, A425,73-90 pplication of Geometric Phase in Quantum Computations
Int. J. Theor. Phys.
Rep. Progr. Phys.
Quantum Computing. quant-ph/9708022[4] Cabello, A. (2000)
Bibliographic guide to the foundations of quantum mechanicsand quantum information . quant-ph/0012089[5] Kilin, S.Ya.
Progres in Optics
Science
Ibid . 1994, 263, 695-697[7] Bermen, G.P.; Doolen, G.D.; Holm, D.D.; Tsifrinovich, V.I.
Phys. Lett.
Phys. Rev. Lett.
74 (1995) 4083-4086[9] Di Vincenzo, D.P.
Science
Phys. Rev.Lett.
Phys.Rev. Lett.
Proc. Nat. Acad. Sci.
Physica
Phys. Rev. Lett.
Proc. Roy. Soc. London
Fault-tolerant quantum computation by anyons .quant-ph/9707021[16] Preskill, J. (1999)
Quantum information and physics: some future directions .quant-ph/9904022[17] Corac, J.I.; Zoller, P.
Phys. Rev. Lett.
Science
Phys. Lett.
Phys. Rev.
Phys. Rev.
A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich [22] Ekert, A. Ericsson, M. and Hayden, P. (2000)
Geometric Quantum Computa-tion . quant-ph/0004015[23] Fuentes-Guridi I., Bose S. and Vedral V. (2000) Proposal for measurment ofharmonic oscillator Berry phase in ion traps. quant-ph/0006112;
Phys. Rev.Lett.
Phys. Rev. Lett.
75 (1995)3788-3791[25] Averin, D.V.
Solid State Commun.
Nonadiabatic conditional geometric phase shiftwith NMR . quant-ph/0101038[27] Xiang-Bin, W.; Kieji, M. (2001)
NMR C-NOT gate through Aharonov-Anandan’s phase shift . quant-ph/0105024[28] Xiang-Bin, W.; Kieji, M. (2001)
On the nonadiabatic geometric quantum gates .quant-ph/0108111[29] Shi-Liang Zhu; Wang, Z.D.
Implementation of universal quantum gates basedon nonadiabatic geometric phases . quant-ph/0207037[30] Du, J.; Shi, M.; Wu, J.; Zhou, X.; Han, R. (2002)
Implementation of nonadia-batic geometric quantum computation using NMR . quant-ph/0207022[31] Oshima, K; Azuma, K. (2003)
Proper magnetic fields for nonadiabatic quantumgates in NMR . quant-ph/0305109[32] Blais, A.; Tremblay, A.M.S. (2003)
Effect of noise on geometric logic gates forgeometric quantum computation . quant-ph/0105006[33] Marinov, M.S.; Strahov, E. (2000)
A geometrical approach to non-adiabatictransitions in quantum theory: application to NMR, over-barrier reflectionparametric excitayion of quantum oscillator . quant-ph/0011121[34] Zhang, X.D.; Zhu, S.L.; Hu, L.; Wang, Z.D. (2005)
Non-adiabatic geometricquantum computation using a single-loop scenario . quant-ph/0502090[35] Das, R.; Kumar, S.K.K.; Kumar, A. (2005)
Use of non-adiabatic ge-ometric phase for quantum computing by nuclear magnetic resonance .quant-ph/0503032[36] Solinas, P.; Zanardi, P.; Zanghi, N.; Rossi, F. (2003)
Non-adiabatic geometricalquantum gates in semiconductor quantum dots . quant-ph/0301089 pplication of Geometric Phase in Quantum Computations
Holonomic quantum gates:a semiconductor-based implementation . quant-ph/0301090[38] Li, X.; Cen, L.; Huang, G.; Ma, L.; Yan, Y.
Non-adiabatic quantum computationwith trapped ions . quant-ph/0204028[39] Scala, M.; Militello, B.; Messina, A. (2004)
Geometric phase accumulation-based effects in the quantum dynamics of an anisotropically traped ion .quant-ph/0409168[40] Zhu, S.L.; Wang, Z.D. (2002)
Geometric phase shift in quantum computationusing superconducting nanocircuits: nonadiabatic effects . quant-ph/0210175[41] Nazir, A.; Spiller, T.P.; Munro, W.J. (2001)
Decoherence of geometric phasegates . quant-ph/0110017[42] Brion, E.; Harel, G.; Kebaili, N.; Akulin, V.M.; Dumer, I. (2002)
Decoherencecorrection by the Zeno effect and non-holonomic control . quant-ph/0211003[43] Carollo, A.; Fuentes-Guridi, I.; Santos, M.F.; Vedral, V. (2003)
Spin-1/2 geo-metric phase driven by quantum decohering field . quant-ph/0306178[44] Fuentes-Guridi, I; Girelli, F.; Livine, E. (2003)
Holonomic quantum computa-tion in the presence of decoherence . quant-ph/0311164[45] Gaitan, F. (2003)
Noisy control, the adiabatic geometric phase and destructionof the efficiency of geometric quantum computation . quant-ph/0312008[46] Yi, X.X.; Wang, L.C.; Wang, W. (2005)
Geometric phase in dephasing systems .quant-ph/0501085[47] Pachos, J.; Zanardi, P. (2000)
Quantum holonomies for quantum computing .quant-ph/0007110[48] Lucarelli, D. (2002)
Control algebra for holonomic quantum computation withsquizeed coherent states . quant-ph/0202055[49] Niskanen, A.O.; Nakahara, M.; Salomaa, M.M. (2002)
Realization of arbitrarygates in holonomic quantum computation . quant-ph/0209015[50] Tanimura, S.; Hayashi, D.; Nakahara, M. (2003)
Exact solutions of holonomicquantum computation . quant-ph/0312079[51] Nordling, M.; Sj¨oqvist, E. (2004)
Mixed-state non-Abelian holonomy for sub-systems . quant-ph/0404162[52] Zhang, P.; Wang, Z.D.; Sun, J.D.; Sun, C.P. (2004)
Holonomic quan-tum computation using Rf-SQUIDs Coupled through a microwave cavity .quant-ph/04070696
A.E. Shalyt-Margolin, V.I. Strazhev and A.Ya. Tregubovich [53] Yi, X.X.; Chang, J.L. (2004)
Off-diagonal geometric phase in composite sys-tems . quant-ph/0407231[54] Whitney, M.S.; Makhlin, Yu.; Shnirman, A.; Gefen, Y. (2004).
Geomet-ric nature of the environment-induced Berry phase and geometric dephasing .quant-ph/0405267[55] Messiah, A.
Quantum Mechanics ; North Holland: Amsterdam, 1961; V.2[56] Berry, M.V.
Proc Roy Soc London
Phys. Rev. Lett.
Phys. Rev. Lett.
Geometric Phases in Physics ; World Scientific:Singapore, 1989[60] Aharonov, Y.; Anandan, J.
Phys. Rev. Lett.
Phys. Rev.
Topological phases inQuantum Theory ; Dubovik, V.M.; Markovski, B.L.; Vinitski, S.I.; Ed.; WorldScientific: Singapore, 1989; pp 119-128[63] Tolkachev, E.A.; Boukanov, I.V.; Tregubovich A.Ya.
Phys. Atom. Nucl.
J. Phys.
Phys. Lett.
Phys. Rev.
Semisimple Lie algebras ; Marcel Dekker,Inc.: New Yorkand Basel, 1978[68] Helgason, S. Differential Geometry and Symmetric Spaces; Academic Press:New York and London, 1962[69] Korenblit, S.E.; Kuznetsov, V.E.; Naumov, V.A. In
Quantum systems: newtrends and methods, Proceedings ; Barut, A.O. et al.Ed.; World Scientific: Sin-gapore, 1995; pp 209-217[70] Tycko, R.
Phys. Rev. Lett.
Phys. Rev. Lett. pplication of Geometric Phase in Quantum Computations
Phys. Lett.
A204 (1995) 210-216[73] Appelt, S.; W¨ackerle, G.; Mehring, M.
Z. Phys. D , 1995, 34-45[74] Caldeira, A.O.; Legett, A.J.
Phys. Rev. Lett.
Phys. Lett.
Phys. Lett.
Optica and Specroscopia
A non-adiabatic controlledNOT gate for the Kane solid state quantum computer . quant-ph/0108103[79] Kane, B.E.
Nature