On the stability of quantum holonomic gates

We provide a unified geometrical description for analyzing the stability of holonomic quantum gates in the presence of imprecise driving controls (parametric noise). We consider the situation in which these fluctuations do not affect the adiabatic evolution but can reduce the logical gate performance. Using the intrinsic geometric properties of the holonomic gates, we show under which conditions on noise's correlation time and strength, the fluctuations in the driving field cancel out. In this way, we provide theoretical support to previous numerical simulations. We also briefly comment on the error due to the mismatch between real and nominal time of the period of the driving fields and show that it can be reduced by suitably increasing the adiabatic time.


I. INTRODUCTION
Recently, there has been a renewed interest for geometric phases and their application to quantum information [1][2][3][4][5][6][7] including several solid state experiments [8][9][10][11]. These new results could open the way for the realization of "holonomic quantum computation" in which the quantum information is manipulated only by means of geometric operators. This interesting line of research has been opened by the seminal paper of Wilczek and Zee [12] and later on by Zanardi and Rasetti [13,14].
Holonomic computation is based on an Hamiltonian H depending adiabatically and periodically on time via a set of time dependent parameters x(t) ≡ (x 1 (t), . . . , x n (t)). Interesting implementation proposals were initially based on atoms and ions [15,16] and later in many other quantum systems [17][18][19][20][21][22]. All these models have an Hamiltonian of the form where f is a real valued function of r, the norm of x ∈ R n , andx is the unit vector x/r. This structure in atomic physics is usually referred as tripod Hamiltonian. The logical operation associated with (1) depends only on the shadow that the curve x = x(t) projects onto the surface of the unit sphere S n−1 (more specifically, on the solid angle spanned by it); moreover, the dependence of the Hamiltonian on f (r) can be transformed away by suitable (time-dependent) projective transformation in the Hilbert space of the system. Thus the norm r = r(t) does not need not be periodic-only periodicity ofx =x(t) is required-and this, in its turn, ensures that any mismatch between r(0) and r(T ) does not affect at all the performance of the gate. * Electronic address: paolo.solinas@aalto.fi For sake of concreteness, one may think of the specific implementation proposal for quantum dot driven by ultrafast lasers [20], modulated in amplitude and phase with the quantum information stored in the excitonic degree of freedom. However, the same formalism and results hold for all the other proposals [15][16][17][18][19][20][21][22].
For this purpose, the relevant Hamiltonian is with x(t) ∈ R 3 and suitable matrix-valued vector b.
More precisely, we have the following structure: 1. The quantum dot has a level structure of three excited degenerate states |i (i = 1, 2, 3) at energy ǫ, and a ground state |0 , set for convenience at energy 0.
2. The system is driven by time-dependent laser fields, with frequency in resonance with ǫ, inducing transitions between ground and excited states. In the interaction representation, the Hamiltonian governing the dynamics of the system is (2), where b is the matrix-valued vector with components 3. The components of the vector x = x(t) represent the amplitudes of the three laser driving fields which are slowly varying functions of time. Moreover, the motion of the unit vectorx(t) is periodic of period T , i.e., Note that we are not assuming periodicity of r(t).
We refer to the final geometric transformation as the gate operator G, which is defined as the adiabatic limit of the evolution generated by (2), for suitable logical space L ⊂ C 4 . (We recall that "logical space" stands here for "qubit" or, equivalently, for "two dimensional complex space C 2 "), where U (t) is the evolution operator in C 4 generated by (2), i.e., the solution of ( = 1 from now on) where I is the identity operator. The main advantage in using the gate operator G to manipulate quantum information is that it has an intrinsic robustness under different kinds of errors. In fact, its performance can be optimized in presence of environmental effect and decoherence [23][24][25][26][27][28][29][30][31]. Similar properties and geometric interpretations have been discussed for the Abelian geometric phases for which a full analytical treatment exists [32]. These includes environmental induced geometric [33] and non-adiabatic non-Markovian contributions [34]. Moreover, numerical investigations show that geometric operators are robust against fluctuations of the driving fields x(t) [35][36][37], so-called parametric noise. However, no analytic treatment has been provided so far.
Here we provide a first-principle explanation of such a robustness. We analyze the stability of the gate operator G under different sources of error which can decrease the logical gate performance. We identify two kind of errors induced by the fluctuations of the driving fields: the error in switching off the driving fields and the one accumulated during the evolution. These errors are supposed to be weak enough to avoid any loss of adiabaticity, i.e., they do not produce any transition between nondegenerate states, but they can decrease the logical gate performance. Using the intrinsic geometric properties of the holonomic gates, we show that both errors can be reduced in the adiabatic limit, thus recovering the robustness of the holonomic transformation.
The paper is organized as follows. In Section II we set the stage for the study of the stability of G. In particular, we study the adiabatic regime using the language of scaling limits and introduce the dimensionless adiabatic scaling parameter ε. In Section III we highlight the geometrical aspects of the problem, in particular we stress the utility of adopting a geometrical extrinsic point of view in describing the geometrical features of G. In Section IV we discuss the robustness of the holonomic gate operator in the presence of parametric noise. In Section V we conclude.

II. ADIABATIC REGIME
In this section we solve (6) in three steps. First, we observe that H(t) in (2) is a unitary transformation of H(0) multiplied by a scale factor, i.e., To see how this comes about, consider the time dependence of the unit vectorx =x(t) expressed in terms of spherical coordinates θ = θ(t) and φ = φ(t) with respect to an orthonormal basis i ≡ (i 1 , i 2 , i 3 ) in the space R 3 of the driving parameters, i.e., x(t) = sin θ cos φ i 1 + sin θ sin φ i 2 + cos θ i 3 , and regard R 3 as embedded in C 4 according to the identifications i 1 = |1 , i 2 = |2 , i 3 = |3 , to which one may add, for uniformity of notations, the stipulation i 0 ≡ |0 . Let in the (i 0 , i 1 , i 2 , i 3 ) basis. Then one may easily check that (7) is satisfied for and scale factor The second step consists in writing the equation of motion in the moving frame associated with R(t). This is realized by the change of variables whence by (6) and (7), with The third step exploits the physical assumption that the external fields are slowly varying functions of time. Since the energy scale of (2) is determined by r to characterize the evolution we introduce the adiabatic parameter wherer is an estimate of the size of r(t) (say, the minimum value of r(t) during the drive). The parameter ε must, of course, be small in order to avoid transitions between non-degenerate states. For such an adiabatic regime the time dependence of the curve in R 3 should be better regarded as given by x = x(εt), with the function x(t) having a scale of variation of order one. The time for the application of the geometric transformation is of order 1/ε. Note that the derivative of R(t) leads to the rescaling A(t) → εA(εt), so that the dynamical problem in the adiabatic regime becomes that of solving the rescaled equations of motions for V (t) for ε ≪ 1. At any given time the Hamiltonian (2) can be diagonalized (the explicit time dependence plays no role in the diagonalization and we shall omit it in the notations). Consider a vector u in C 4 written as (2), the eigenvalues equation reads from which eigenvalues and the eigenvectors of H follow: take u =x, then from (18) H Thus, u 0 = 1 gives the eigenvalue λ + = r and u 0 = −1 gives the eigenvalue λ − = −r. The corresponding normalized eigenvectors are, respectively, Finally, for u 0 = 0 and u orthogonal tox (as vectors in R 3 ), we read from (18) that λ 0 = 0 is a doubly degenerate eigenvalue. Thus, we have recovered well known properties of the tripod Hamiltonian (2): there are two degenerate states, at zero energy, called "dark states" and other two, called "bright states", with one excited state (e + in (19)) and one ground state (e − in (19)) with energy r and −r, respectively. In the following, we shall denote by P β , β = +1, −1, 0, the spectral projectors of H(0) corresponding respectively to the eigenvalues λ + = r(0), λ − = −r(0) and λ 0 = 0. P 0 projects onto the plane L orthogonal tox(0) which soon will be identified with the logical space in (5). Consider now the interaction representation of the time evolution operator V (t), whose dynamics is governed by (17), with respect to free dynamics generated by B with where (see (12)) Then from (17) it follows that V I (t) satisfies or, equivalently Equation (24) shows that the effect of A on the evolution manifests itself only on the adiabatic time scale (25) Since h(t) is positive and never equal to zero, standard stationary phase approximation gives β ′ = β and By multiplying both sides of the above equation by P α we see that the evolution separates into autonomous spectral components. In particular, since P 0 projects onto the plane L, (26) defines a non-trivial dynamics on it. In other words, the operator evolves autonomously in the adiabatic limit (modulo corrections of order ε) according to the equation where, recalling (15), III. GEOMETRY Equation (29) can be solved analytically using a geometric approach. By defining the operator-valued vectors a = (a 1 , a 2 , a 3 ) and a * = (a * 1 , a * 2 , a * 3 ), having components a i = |0 i| and a * i = |i 0|, i = 1, 2, 3, respectively. The Hamiltonian in (2) can be written as H(t) = a · x(t) + a * · x(t). Another operator-valued vector that it is useful to introduce is J = a * × a, whose components are indeed the generators of an SO(3) algebra with commutation relations [J i , J j ] = −ǫ ijk J k and in terms of which D(t) in (10) can be expressed as The frame e(t) = (e 0 , e θ (t), e φ (t), e r (t)), where is indeed a moving frame adapted to the surface of the unit sphere S 2 on whichx(t) moves in the course of time, i.e., e θ (t) and e φ (t) are tangent to S 2 , and e r (t) is perpendicular to it. Then (31) is the operator transforming the frame i = (i 0 , i 1 , i 2 , i 3 ) into the frame e(t), Note that D(0) −1 is not the identity and, in particular, that D(0) −1 i 3 =x(0), whence from (11) Then (7) is nothing but an expression of the usual duality between action of the operators on vectors and on operators, i.e., The unit vectors e θ (t) and e φ (t) (the "dark states") are a natural basis in the moving degenerate space-the plane orthogonal to e r (t) =x(t), which is, according to Section II, the degenerate eigenspace of the eigenvalue 0 of H(t). Let P 0 (t) denote the projector onto such a plane, then P 0 (0) is the projector P 0 onto the L plane. Note that P 0 is just a D(0)-rotation of the projector P (i 1 i 2 ) = a * 1 a 1 + a * 2 a 2 onto the i 1 -i 2 plane; thus, Putting all the pieces together, from (11), (28) and (29) we obtain with (see (31)) Using (31) and the above definition of P (i 1 i 2 ), we can calculate the projection P (i 1 i 2 )J k P (i 1 i 2 ) = δ k3 J k and we are left with Observing (31) at time t = 0, and using the rotational properties of J k we have and, finally, we arrive to a geometric expression We have at our disposal all the ingredients to determine the geometric operator that is used to manipulate the quantum state. Now, since A(t) at different times commutes, equation (28) can be solved by direct exponentiation One may recognize that G(t) is a rotation in the L plane of an angle given by the integral multiplying J ·x(0) in (36). At this point starting from (27) and going back from (36) to the moving frame by means of equation (20) is immediate: W (t) is just the identity on L since the corresponding eigenvalues λ 0 is zero. To go back to the laboratory frame, we follow (13) and obtain From the assumption (4) of periodicity ofx(t), it follows that R(T ) = I. Therefore If we choose a logical basis directly in L we can write G(t) in (36) where This is the expected result: the G(T ) operator depends only on the solid angle spanned by x(t) on the Hamiltonian parameter space. Note that this result does not rely on the assumption of full periodicity (x(0) = x(T )) made by Wilczek-Zee and Zanardi-Rasetti, although it holds only for a restricted class of Hamiltonians.

IV. STABILITY
We shall now consider the effects of external perturbations on the system, in order to understand the robustness of quantum evolution in presence of noise. Several are the sources of undesired energy exchange due to coupling with impurities, and/or external environments, or due to imprecise control of the system parameters during the evolution. In the following we will focus on this parametric error. First of all, we observe that the good performance of the gate G relies on the validity of the adiabatic limit, and since the dimensionless constant ε is finite, this fact alone introduces an error of order ε, i.e., O(ε), which is the order of magnitude of the off-diagonal terms (β = β ′ ) in (25) which are neglected in the limit ε → 0. Then we shall say that the gate G is stable if the perturbations produce corrections of higher order in ε. Letting aside corrections not following a power law, we can say that stability is ensured if the corrections on G due to the perturbations are O(ε r ), with r > 1. To simplify the analysis, we will work with dimensionless quantities: energy scale in units ofr, so that (16) becomes ε = 1/T .
We consider the error induced by the inaccuracy of the control field. We start with the case in which the actual curve in the parameter space is not x(t) but instead x ′ (t) = x(t) + δx(t), fulfilling still the periodicity requirement (4) on the unit vectorx ′ (0) =x ′ (T ). The error δx(t) is assumed small with respect to x(t) in the sense of some suitable functional norm. It should be regarded as rapidly fluctuating random process whose scale of variation is very small on the adiabatic scale.
From the above periodicity it follows that the holonomic operator U ′ (T ), related to x ′ (t) is still of the form The geometric operator G[x ′ (t)] is now a functional of x ′ (t) and then the error is accumulated during the whole evolution.
In the following we will evaluate at lowest order in δx(t) the error on the holonomic operator U ′ (T ) = G[x(t) + δx(t)] induced by the fluctuations of the driving fields along the path. We have where σ is a measure of the size of the (mean) variation of δx(t). To do that, let us start to rewrite the solid angle in (40) for path x ′ (t) as whereγ is the shadow on the unit sphere S 2 of the curve γ in R 3 given by the parametric equations x ′ = x ′ (t), and satisfying the conditions (4) of partial periodicity, Σ is the surface on S 2 bounded byγ, i.e.,γ = ∂Σ. Thus, Ω is the area of Σ, that is, the solid angle spanned by curve γ (i.e., the solid angle from which γ is seen from the origin in R 3 ). Accordingly, different unitary geometric transformations and then quantum logical gates can be constructed traversing different loop in the parameters space.
Since the North Pole θ = φ = 0 is a singularity of spherical coordinates, one is naturally lead to consider the 1-form on S 2 , locally defined by ω = cos θdφ and extended it to all S 2 in a coordinate independent way. Accordingly (43) should be replaced by the Stokes theorem on S 2 expressed in an "intrinsic" geometrical way (i.e., coordinate independent), This is the approach usually adopted in holonomic computation [13,14]. To take care of the singularity problem one can rewrite (43) in terms of the auxiliary vector field A = e φ (1 − cos θ)/(r sin θ) where We can now rewrite (45) as an integral over time Noticing that the right-hand side is analogous to the Lagrangian of a particle in a magnetic field given by (46), we have (see (36) and (39)) Note, as expected, that radial fluctuations give no contribution to the variation since B ×ẋ is tangent to the sphere. We now define the statistical properties of δx(t) which describes the parametric noise perturbing the external field. As already discussed it should have a scale of variation very small on the adiabatic scale, yet, if we wish to ensure the validity of the adiabatic approximation, we should demand, at the same time, that its scale of variation be sufficiently long on the microscopic scale. The simplest possibility to ensure this is to regard the components of δx(t) as independent mean-zero stationary Gaussian processes with "white-noise" correlation function, i.e., with i, j = 1, 2, 3 and where σ 2 i is a measure of the strength of the noise components (time-independent, as the processes are stationary), τ i s are the correlation times of the noise components. Since the Gaussian processes are stationary, we can take σ 2 i ≡< δx i (0) 2 >. Again, it is convenient to consider τ i and σ i as functions of ε, e.g., with suitable exponents p > 0 and q > 0. Recalling that the intrinsic error of the adiabatic approximation is O(ε), we should then inquire whether there is a range of p, q-values for which the mean error is below the O(ε) upper bound, i.e., O(ε r ) with r > 1. Of course, ∆ would then provide an estimate of the (mean) first order correction δG in (42). Since the fluctuations δx i have zero mean, < δΩ >= 0. Moreover, from (49) and (50) it follows Recalling (46), we find Since the velocityẋ i scales as 1/T , the integral in the right-hand side of (54) is O(ε), and the mean error is then From (51) it follows that with this constraint there are many solutions p/2 + q + 1/2 > 1, in the desired range of p, q-values, which ensure stability of the gate. Moreover, from (55), we also read the answer to the questions about the smallness of τ i and largeness of σ i , namely, that the fluctuations can indeed be quite large with respect to ε, provided that the τ i s are small on the macroscopic scale (but large on the microscopic scale). This is the cancellation effect already discussed in Refs. [35,36]. For example, let ε = 10 −4 . Then a noise with correlation time of order, say, τ = 10 −2 (p = 1/2) can have fluctuations of order, say, σ = 10 −2 (q = 1/2) while producing, at the same time, an error on the gate of order 10 −5 (p/2 + q + 1/2 = 5/4), well within the bound ε = 10 −4 of the adiabatic regime. We conclude this section with two observations. The first concerns the geometric interpretation of solid angle perturbation, with a simple geometrical formula for the right-hand side of (53). We focus on the case of noise components δx i (t) with the same statistical properties, i.e., σ i = σ and τ i = τ (a condition which is indeed quite reasonable from a physical point of view). Since Ω is invariant under re-parametrization of time, it is convenient to use as invariant parameter the arc length s with origin inx 0 and rewrite (47) as where L is the length of the curve andẋ = dx/ds. Then (53) becomes where ℓ is the correlation length of the noise. (Note that the integral is now O(1), since L is the length of the curve on S 2 , and therefore ℓ = O(ε p+1 )). But, from (46), B = n, the normal to the curve lying on S 2 , i.e., Let us now turn to the second observation. It concerns how to deal with the difference between nominal and real value of the period of the laser. We recall that in the tripod model two control fields are turned off at the initial and final configurations [15][16][17][18][19][20][21][22]. However, in a practical implementation, there is uncertainty in the real time T in which the fields are turned off since in general this does not correspond to the nominal time T 0 , with T = T 0 + ∆T , being ∆T a statistical fluctuation time. This can be due, for example, to latent times or delays in the control fields of the experimental set-up. In the parameter space this error corresponds to an evolution that, if referred to the nominal time, has R −1 (T 0 ) = I, since the periodicity requirement is fulfilled at time T , e.g., R −1 (T ) = I. This means a path non-periodic at the nominal time [39]. Thus, from (37) and therefore at the lowest order in ∆T For the partial derivative in the right-hand side of the previous equation, one has The ∂R −1 /∂x depends only on geometrical properties of the path and thus is of order one, while the velocity is of order 1/T (since we are analyzing the system in the adiabatic regime). Thus the error due to the mismatch between real and nominal time of the period of the laser is of order 1/T ≈ 1/T 0 and can therefore be reduced by suitably increasing the nominal time T 0 .

V. CONCLUSIONS
We have provided a unified geometrical description for analyzing the stability of holonomic quantum gates in the presence of parametric noise affecting their time evolution. We have identified two main critical parameters: the correlation time of the noise and its strength with respect to the driving field. In this way, we have recovered what was already obtained numerically for Abelian [35] and non-Abelian gates [36], namely, that even strong fluctuations in the driving field can lead to accurate logical gates if the correlation time is short enough to let the fluctuations cancel out. This is an effect of the geometric dependence of the holonomic operator. In addition, we have shown that the error due to the mismatch between real and nominal time of the period of the laser can be reduced by suitably increasing the adiabatic time.