Geometric (holonomic) gates

In this section the general theoretical framework of HQC is reviewed. While the exposition relies partly on Refs. \cite{hol1,hol2} some proofs have been added which clarify the physical concept of holonomic evolutions making this subject more approachable to the quantum information community. ==Quantum Evolutions== Let us suppose that we have at disposal a family \cal{F} of Hamiltonians that we can turn on and off in order to let an N-dimensional quantum system to evolve in a controllable way. Formally, we assume \cal{F}:=\{ H(\lambda)\}_{\lambda\in\cal{M}} to be a continuous family of Hermitian operators over the state-space \cal{H}\cong {\rm\kern.24em \vrule width.04em height1.46ex depth-.07ex \kern-.30em C}^N. The parameters $\backslash lambda$ on which the elements of \cal{F} depend will be referred to as control parameters and their manifold \cal{M} as the control manifold, thought to be embedded in {\rm\vrule width.04em height1.58ex depth-.0ex\kern-.04em R}^{N^2}. Indeed, one has H(\lambda)=i\,\sum_{a=1}^{N^2} \Phi_a(\lambda) \,T_a\in u(N) where the $T\_a$'s constitute a basis of the $N^2$-dimensional Lie-algebra u(N) of anti-hermitian matrices, and \Phi\colon \cal{M}\mapsto u(N) is a smooth mapping that associates to any $\backslash lambda$ in the control manifold a vector in u(N) with T-components $(\backslash Phi\_1(\backslash lambda),\backslash ldots,\backslash Phi\_\{N^2\}(\backslash lambda)).$ The evolution of the quantum system is thought of as actively driven by the parameters $\backslash lambda$, over which the experimenter is assumed to have direct access and controllability. Suppose we are able to drive by a dynamical control process the parameter configuration \lambda\in\cal{M} through a path \gamma\colon [0,\,T]\rightarrow \cal{M}. Hence, a one-parameter i.e., time-dependent family \cal{F}_\gamma:=\{H(t):=H[\Phi\circ\gamma (t)]\colon t\in [0,\,T]\}\subset \cal{F} is defined. Notice that even the converse is true: any smooth family $\backslash \{H(t)\backslash \}\_\{t\backslash in[0,\backslash ,T]\}$ defines a path in \cal{M}={{\rm\vrule width.04em height1.58ex depth-.0ex\kern-.04em R}}^{N^2}. The quantum evolution associated to the time-dependent family (\ref{1-family}) is described by the time-dependent Schr\"odinger equation $i\backslash ,\backslash partial\_t|\backslash psi(t)\backslash rangle=\; H(t)\backslash ,|\backslash psi(t)\backslash rangle$ and hence it has the operator form U_\gamma:={\bf T}\,\exp\{ -i\,\int_0^T\! dt\, H(t)\}\in U(N) where '''T''' denotes chronological ordering. The time-dependent quantum evolution (\ref{evolution}), for a given map $\backslash Phi$, depends in general on the path $\backslash gamma$ and not just on the curve $\backslash gamma([0,\backslash ,T])$ i.e., the image of $\backslash gamma$ in the control manifold. In other words the unitary transformation (\ref{evolution}) contains a dynamical as well as a geometrical contribution, the former depends even on the rate at which $\backslash gamma([0,\backslash ,T])$ is traveled along whereas the latter depends merely on the geometrical characteristics of the curve. From the physical point of view the parameters $\backslash lambda$ represent in general external fields and, for multi-partite systems, couplings among the various subsystems. To illustrate this point let us consider \cal{H}:=({\rm\kern.24em \vrule width.04em height1.46ex depth-.07ex \kern-.30em C}^2)^{\otimes\,N}\cong {\rm\kern.24em \vrule width.04em height1.46ex depth-.07ex \kern-.30em C}^{2^N} i.e., a N-qubit system. Then a basis for $u(2^N)$ is provided by the tensor products $T\_\backslash alpha:=\backslash otimes\_\{i=1\}^N\; \backslash hat\; \backslash sigma\_\{\backslash alpha\_i\}$ where $\backslash alpha\backslash colon\backslash \{1,\backslash ldots,N\backslash \}\backslash mapsto\; \backslash \{0,1,2,3\backslash \}$ and \hat \sigma_0:=\leavevmode\hbox{\small1\kern-3.8pt\normalsize1},\, \hat \sigma_1:=\sigma_x,\, \hat \sigma_2:=\sigma_y,\, \hat \sigma_3:=\sigma_z are the Pauli matrices. It is then clear that any $\backslash alpha$ which takes a non-zero value more than once e.g., $\backslash alpha\_i\backslash ,\backslash alpha\_j\backslash neq\; 0$ describes a non-trivial interaction which generates entanglement between the qubits i and j. Therefore the ability to manipulate the weight of the contribution of $T\_\backslash alpha$'s in the decomposition of $H(\backslash lambda)$, amounts to the capacity of dynamically controlling many-body couplings. This goal is, of course, even conceptually more difficult to achieve than the control of the real external fields, namely the interaction associated to single subsystem generators $T\_\backslash alpha$. Finally, we stress that there is still another possibility; the control parameters $\backslash lambda$ could represent on their own quantum-degrees of freedom e.g., nuclear coordinates in the adiabatic approximation for molecular systems, treated in some quasi-classical fashion. This situation arises when one performs an adiabatic decoupling between ``fast'' and ``slow'' degrees of freedom, getting for the former a Hamiltonian that depends parametrically on the latter \cite{SHWI}. In this case the control manifold \cal{M} is nothing but the classical configuration manifold associated with a quantum system. Within this framework the requirements for implementing universal QC \cite{UG} can be expressed in terms of the availability of paths. Universality is the experimental capability of driving the control parameters along a minimal set $\backslash \{\backslash gamma\_i\backslash \}\_\{i=1\}^g$ of paths which generate the basic unitary transformations $U\_\{\backslash gamma\_i\}$'s, i.e. the gates. By sufficiency of this set we mean the ability to approximate any $U\backslash in\; U(N)$ with arbitrarily high accuracy by means of path sequences. ==Holonomies== Now we recall some basic facts about quantum holonomies. A more mathematical approach can be found in Appendix A, where some by-now standard material has been collected aiming to make the paper as much as possible self-contained. The non-Abelian holonomies are a natural generalization of the Abelian Berry phases. We first assume that \cal{F} is an 'iso-degenerate' Hamiltonian family i.e., all the elements of \cal{F} have the same degeneracy structure. This means that a generic Hamiltonian of \cal{F} can be written as $H(\backslash lambda)=\backslash sum\_\{l=1\}^R\; \backslash varepsilon\_l(\backslash lambda)\; \backslash ,\; \backslash Pi\_l(\backslash lambda)$ where $\backslash Pi\_l(\backslash lambda)$ denotes the projector over the eigen-space \cal{H}_l(\lambda) :=\mbox{span}\{|\psi^\alpha_{l}(\lambda)\rangle\}_{\alpha=1}^{n_l}, with eigenvalues $\backslash varepsilon\_l(\backslash lambda)$, whose dimension $n\_l$ is independent on the control parameter $\backslash lambda.$ In order to preserve the $R$ degeneracies $n\_l$ we also assume that over \cal{M} there is no level-crossing i.e., l\neq l^\prime \Rightarrow\varepsilon_l(\lambda)\neq \varepsilon_{l^\prime} (\lambda), \forall \lambda\in\cal{M}. In addition, we shall restrict to 'loops' $\backslash gamma$ in the control manifold i.e., maps \gamma\colon [0,\,T]\mapsto \cal{M} such that $\backslash gamma(0)=\backslash gamma(T).$ These conditions in the dynamics of the system and in the control manipulations will facilitate the generation of holonomic unitaries. Let us state the main result \cite{WIZE} on which the HQC relies. Consider a system with the above characteristics. When its control parameters are driven adiabatically i.e., slow with respect to any time-scale associated to the system dynamics, along a loop $\backslash gamma$ in \cal M any initially prepared state |\psi_{in}\rangle\in\cal{H} will be mapped after the period T onto the state |\psi_{out}\rangle= U_\gamma\,|\psi_{in}\rangle,\, U_\gamma=\oplus_{l=1}^R e^{i\,\phi_l}\,\Gamma_{A_l}(\gamma), where, $\backslash phi\_l:=\backslash int\_0^T\; d\backslash tau\backslash ,\; \backslash varepsilon\_l(\backslash lambda\_\backslash tau),$ is the dynamical phase whereas the matrices $\backslash Gamma\_\{A\_l\}(\backslash gamma)$'s represent the geometrical contributions. They are unitary mappings of \cal{H}_l onto itself and they can be expressed by the following path ordered integrals \Gamma_{A_l}(\gamma) :={\bf{P}}\exp \oint_\gamma A_{l} \in U(n_l) \,\, ,\,\,\, l=1,\ldots,R \,\, . These are the 'holonomies' associated with the loop $\backslash gamma,$ and the 'adiabatic connection forms' $A\_l.$ The latter have an explicit matrix form given by $A\_\{l\}\; =\; \backslash Pi\_l(\backslash lambda)\backslash ,d\backslash ,\backslash Pi\_l(\backslash lambda)=\backslash sum\_\backslash mu\; A\_\{l,\backslash mu\}\backslash ,d\backslash lambda\_\backslash mu,$ where \cite{SHWI} analytically (A_{l,\mu})^{\alpha\beta}:= \langle\psi_{l}^\alpha(\lambda)| \,{\partial}/{\partial\lambda^\mu}\, |\psi_{l}^\beta(\lambda)\rangle \label{conn} with $(\backslash lambda\_\backslash mu)\_\{\backslash mu=1\}^d$ the local coordinates on \cal{M}. The connection forms $A\_l$'s are nothing but the non-Abelian gauge potentials enabling the parallel transport \cite{NAK} over \cal M of vectors of the fiber \cal{H}_l. Result (\ref{conn}) is the non-Abelian generalization of the Berry phase connection presented first by Wilczek and Zee (1984) (see Appendix). Due to the decomposition of the evolution operator in (\ref{out}) into distinct evolutions for each eigen-space \cal{H}_l, we are able to restrict our study to a given degenerate eigen-space with fixed $l.$ We shall present first an intuitive proof for deriving (\ref{Hol}) and (\ref{conn}), aiming in clarifying the gauge structure interpretation of this adiabatic evolution and in providing a more physical insight. Without loss of generality we shall assume the family \cal F to be {\em iso-spectral}. This implies that for any \lambda\in\cal{M} it exists a unitary transformation \cal{U}(\lambda) such that H(\lambda)= \cal{U}(\lambda)\, H_0\,\cal{U}(\lambda)^\dagger, where $H\_0:=H(\backslash lambda\_0).$ Upon dividing the time interval $[0,\backslash ,T]$ into N equal segments $\backslash Delta\; t,$ for \cal{U}_i=\cal{U}(\gamma(\lambda(t_i))) one obtains the evolution operator in the form U_\gamma ={\bf T} e^ {-i \int_0^T \cal{U}(\lambda)\,H_0 \, \cal{U}^\dagger(\lambda) dt}= {\bf T} \!\! \lim_{N\rightarrow \infty} e^ {-i \sum_{i=1}^N \cal{U}_i\,H_0 \,\cal{U}^\dagger_i \Delta t} \\ = {\bf T} \lim_{N\rightarrow \infty} \prod_{i=1}^N \cal{U}_i e^{-iH_0 \Delta t} \cal{U}_i^\dagger The third equality holds due to the smallness of the interval $\backslash Delta\; t$ in the limit of large N. The product \cal{U}_i^\dagger \cal{U}_{i+1} of two successive unitaries, gives rise to an infinitesimal rotation of the form \cal{U}_i^\dagger \cal{U}_{i+1}\approx {\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}} +\vec{A}_i \cdot \Delta \vec{\lambda}_i, where \,\,\,\,\,\,\,\,\,\,\,\,\, ({A}_i)_\mu \equiv \cal{U}_i^\dagger {\Delta \cal{U}_i \over \Delta (\lambda_i)_\mu}. The connection A has at time $t\_i$ the components $(A\_i)\_\backslash mu$ with $\backslash mu=1,\backslash ldots,d$. Hence the evolution operator (\ref{evol}) becomes {\bf T} \lim_{N \rightarrow \infty} \cal{U}_N \left( \leavevmode\hbox{\small1\kern-3.8pt\normalsize1} -i H_0 \, N\,\cdot \Delta t + \sum_{i=1}^{N-1} \vec{A}_i \cdot \Delta \vec{\lambda}_i \right) \cal{U}_1^\dagger. For the case of a closed path the initial and the final transformations \cal{U}_1 and \cal{U}_N are identical as they correspond to the same point of the control parameter manifold. With a reparametrization they may be taken to be equal to the identity transformation. Now we consider an initial state $|\backslash psi\_\{in\}\backslash rangle$ belonging to an eigen-space \cal{H}_0 with associated eigenvalue e.g., $\backslash varepsilon\_0=0$. Due to the time ordering symbol the actions on the state $|\backslash psi(t)\backslash rangle$ of the Hamiltonian and of the connection A are alternated, hence in general we cannot separate them into two exponentials. On the other hand, if we demand adiabaticity, namely very slow exchange of energy during the process, this will keep the state within \cal{H}_0, then at each time $t\_i$ the state $|\backslash psi(t\_i)\backslash rangle$ will remain in the $\backslash varepsilon\_0=0$ energy level. This allows to factor out in (\ref{midd}) the action of H, thus obtaining U_{\gamma}={\bf T} \lim_{N \rightarrow \infty} \left( {\bf 1} + \sum_{i=1}^{N-1} \vec{A}_i \cdot \Delta \vec{\lambda}_i \right) = {\bf P} \exp \oint_\gamma A \,\, , \nonumber where A is projected into the subspace \cal{H}_0. Notice that we replaced the time ordering with the path ordering '''P''' as the parameter of the integration at the last expression is the position on the loop $\backslash gamma$. In this proof of the non-Abelian geometrical evolution it is clear how the holonomy appears and which physical conditions enable its formation. In the same way we could have considered in addition to the equivalent transformations of the Hamiltonian a multiplicative function $\backslash varepsilon\_0(t)$ varying the energy eigenvalue. The results would be unaltered apart from the insertion of a dynamical phase. Let us now view some of the properties the holonomies have in terms of gauge reparametrization of the connection and loop composition rules. In our context a local gauge transformation is the unitary transformation \cal{ U}(\lambda)\mapsto \cal{ U}(\lambda) g(\lambda), which does not change the Hamiltonian operator $H\_0$. Its action merely reparametrizes the variables of the control manifold. Taking into account the properties $gH\_0=H\_0g$ and $g\; \backslash Pi=\backslash Pi\; g$ we are able to obtain the transformation of the connection as $A\backslash mapsto\; g^\{\backslash dagger\}\backslash ,A\; \backslash ,g\; +g^\{\backslash dagger\}\backslash ,dg,\backslash ,(g\backslash in\; U\; (n))$. It immediately follows that the holonomy transforms as $\backslash Gamma\_A\backslash mapsto\; g^\backslash dagger\backslash ,\; \backslash Gamma\_A\backslash ,\; g$. Notice that in the new coordinates the state vectors $|\backslash psi\backslash rangle$ i.e., the sections, become $g^\{\backslash dagger\}\backslash ,|\backslash psi\backslash rangle$. This property makes it clear that the holonomy transformation has an intrinsic i.e., coordinate-free, meaning. Furthermore, the holonomy has the following property in terms of the loops. We define (setting $T=1$) the loop space at a given point \lambda_0\in\cal{M} as L_{\lambda_0}:=\{\gamma\colon [0,\,1]\mapsto \cal{M}\,/\, \gamma(0)=\gamma(1)=\lambda_0\} \nonumber over a point \lambda_0\in\cal{M}. Let us stress that, as far as the manifold \cal M is connected, the distinguished point $\backslash lambda\_0$ does not play any role. In this space we introduce a composition law for loops (\gamma_2\cdot \gamma_1)(t)=\theta( \frac{1}{2} -t)\,\gamma_1(2\,t)+ \theta(t-\frac{1}{2})\,\gamma_2(2t-1) \label{compo} and a unity element $\backslash gamma\_0(t)\; \backslash equiv\; \backslash lambda\_0,\backslash ,t\backslash in[0,\backslash ,1]$ moreover with $\backslash gamma\; ^\{-1\}$ we shall denote the loop $t\backslash mapsto\; \backslash gamma(1-t).$ The holonomy can be considered as a map $\backslash Gamma\_\{A\}\backslash colon\; L\_\{\backslash lambda\_0\}\backslash mapsto\; U(n\_l)$, whose basic properties can be easily derived from eq. (\ref{Hol}): #$\backslash Gamma\_A(\backslash gamma\_2\backslash cdot\backslash gamma\_1)=\backslash Gamma\_A(\backslash gamma\_2)\backslash ,\backslash Gamma\_A(\backslash gamma\_1)$ by composing loops in \cal M one obtains a unitary evolution that is the product of the evolutions associated with the individual loops, #\Gamma_A(\gamma_0)=\leavevmode\hbox{\small1\kern-3.8pt\normalsize1} staying at rest in the parameter space corresponds to no evolution at all, #$\backslash Gamma\_A(\backslash gamma^\{-1\})=\backslash Gamma\_A^\{-1\}(\backslash gamma)$ in order to get the inverse holonomy one has to traverse the path $\backslash gamma$ with reversed orientation, #$\backslash Gamma\_A(\backslash gamma\backslash circ\; \backslash varphi)=\backslash Gamma\_A(\backslash gamma),$ where $\backslash varphi$ is any diffeomorphism of $[0,\backslash ,1]$; as long as adiabaticity holds the holonomy does not depend on the speed at which the path is traveled but just on the path geometry. From the properties listed above it is easy to show that the set $\backslash mbox\{Hol\}(A):=\backslash Gamma\_A(L\_\{\backslash lambda\_0\})$ is a 'subgroup' of U(n). Such a subgroup is known as the 'holonomy group' of the connection A. When the holonomy group coincides with the whole U(n) then the connection A is called 'irreducible'. The notion of irreducibility plays a crucial role in HQC in that it corresponds to the computational notion of 'universality' \cite{UG}. In order to evaluate if this condition is fulfilled by a given connection it is useful to consider the 'curvature' 2-form $F=\backslash sum\_\{\backslash mu\backslash nu\}F\_\{\backslash mu\backslash nu\}\backslash ,dx^\backslash mu\backslash wedge\; dx^\backslash nu$ associated with the 1-form connection A whose components F_{\mu\nu} =\partial_\mu A_\nu-\partial_\nu A_\mu + [A_\mu,\,A_\nu]. \label{curvature} The relation of the curvature with irreducibility is given by the following statement \cite{NAK}: the linear span of the $F\_\{\backslash mu\backslash nu\}$'s is the Lie algebra of the holonomy group. It follows in particular that when the $F\_\{\backslash mu\backslash nu\}$'s span the whole u(n) the connection is irreducible. ==Holonomic Quantum Computation== The unitary holonomies (\ref{Hol}) are the main ingredient of our approach to QC. From now on we shall consider a given subspace \cal{H}_l (accordingly the label l will be dropped). Such a subspace, denoted by \cal C, will represent our quantum 'code', whose elements will be the quantum information encoding codewords. The crucial remark \cite{hol1} is that when the connection is irreducible, for any chosen unitary transformation U over the code there exists a path $\backslash gamma$ in \cal M such that $\backslash |\backslash Gamma\_A(\backslash gamma)-U\backslash |\; \backslash le\; \backslash epsilon\; ,$ with $\backslash epsilon$ arbitrarily small. This means that any computation on the code \cal C can be realized by adiabatically driving the control parameter configuration $\backslash lambda$ along a suitable closed path $\backslash gamma.$ In particular we aim to constructing specific logical gates by moving along their corresponding loops. Initially, the degenerate states are prepared in to a ``ground'' state, interpreting the $|0...0\backslash rangle$ state of m qubits. The statement of irreducibility of the connection A relates a particular unitary U with the loop $\backslash gamma\_U$ over which the connection is integrated to give $\backslash Gamma\_A(\backslash gamma\_U)=U$. Hence, there are loops in the control space such that the associated holonomies give, for example, a one qubit Hadamard gate or a two qubit ``controlled-not'' gate. Let us emphasize the fact that one can perform {\em universal} QC by only using quantum holonomies is remarkable. Indeed this kind of quantum evolutions is quite special, yet it contains in a sense the full computational power. On the other hand one has to pay the price given by the restriction of the computational space from \cal H to its subspace \cal C. Notice that, for the irreducibility property to hold, a necessary condition is clearly given by $d\backslash ,(d-1)/2\backslash ge\; n^2$ where d:=\mbox{dim}\,\cal{M}. In particular this implies that for an exponentially large code \cal C one has to be able to manipulate an exponentially large number of control parameters. Moreover like in any other scheme for QC, once the computation is completed a final state measurement is performed. To this aim it could be useful to lift the energy degeneracy in order to be able address energetically the different codewords \cite{hol2}. This can be done by switching on an external perturbation in a coherent fashion. We conclude this section by discussing the 'computational complexity' issue. The computational subspace \cal C does not have in general a tensor product structure. This means that it cannot be viewed in a natural way as the state-space of a multi-partite system for which the notion of quantum entanglement makes sense. The latter, on the other hand, is known to be one of the crucial ingredients that provides to QC its additional power with respect to classical computation. It follows that, from this point of view, the scheme for HQC described so far is potentially incomplete. Indeed -- as it will be illustrated later by explicit examples -- if N=\mbox{dim}\cal{C} =2^k i.e., we encode in \cal C k qubits, then for obtaining with a multi-partite structure a universal set of gates one needs $O(N)$ elementary holonomic loops. Thus in general one has an 'exponential' slow-down in computational complexity. In Ref. \cite{hol2} we argued how one can in principle overcome such a drawback by focusing on a class of HQC models with a multi-partite structure given from the very beginning. The basic idea is simple: one considers an holonomic family \cal F associated to a genuine multi-partite quantum system such that local (one- and two-qubit) gates can be performed by holonomies. Then from standard universality results of QC \cite{UG} stems that efficient quantum computations can be performed. An explicit example of the above strategy is formalized as follows \cite{hol2}. Let us consider N 'qu-trits'. The state space is then given by \cal{H}_j\cong {{\rm\kern.24em \vrule width.04em height1.46ex depth-.07ex \kern-.30em C}}^3=\mbox{span}\{ |\alpha\rangle_j \, /\,\alpha=0,1,2\}. The holonomic (iso-spectral) family has the built-in local structure \cal{F} =\{ H_{ij}(\lambda_{ij})\} where the local Hamiltonians $H\_\{ij\}$ have a non trivial actions only on the ith and jth factors of \cal H. Moreover, $H\_\{ij\}$ admits a four-dimensional degenerate eigen-space \cal{ C}_{ij}:= \mbox{span}\{ |\alpha\rangle_i \otimes |\beta\rangle_j\,/\, \alpha,\beta =0,1\} \subset \cal{H}_i\otimes\cal{H}_j\cong {\bf{C}}^9. If the $H\_\{ij\}$'s allow for universal HQC over \cal{C}_{ij} then universal QC can be 'efficiently' implemented over \cal{C}:= \mbox{span}\{\otimes_{i=1}^N |\alpha_i\rangle_i\,/\, \alpha_i=0,1\}\cong ({{\rm\kern.24em \vrule width.04em height1.46ex depth-.07ex \kern-.30em C}}^2)^{\otimes\,N}. [[category:Evolutions and Operations]]