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 λ 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 N²-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 λ in the control manifold a vector in u(N) with T-components (Φ₁(λ), …, Φ_N²(λ)).

The evolution of the quantum system is thought of as actively driven by the parameters λ, 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 {H(t)}_{t ∈ [0,  T]} defines a path in \cal{M}=^{N^2}. The quantum evolution associated to the time-dependent family (\ref{1-family}) is described by the time-dependent Schr\"odinger equation i ∂_t∣ψ(t)⟩ = H(t) ∣ψ(t)⟩ 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 Φ, depends in general on the path γ and not just on the curve γ([0,  T]) i.e., the image of γ 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 γ([0,  T]) is traveled along whereas the latter depends merely on the geometrical characteristics of the curve.

From the physical point of view the parameters λ 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 $T_\alpha:=\otimes_{i=1}^N \hat \sigma_{\alpha_i}$ where α: {1, …, N} ↦ {0, 1, 2, 3} 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 α which takes a non-zero value more than once e.g., α_i α_j ≠ 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_α's in the decomposition of H(λ), 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_α. Finally, we stress that there is still another possibility; the control parameters λ 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 {γ_i}_i = 1^g of paths which generate the basic unitary transformations U_γi's, i.e. the gates. By sufficiency of this set we mean the ability to approximate any U ∈ 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(λ) = ∑_l = 1^Rɛ_l(λ) Π_l(λ) where Π_l(λ) denotes the projector over the eigen-space \cal{H}_l(\lambda) :=\mbox{span}\{|\psi^\alpha_{l}(\lambda)\rangle\}_{\alpha=1}^{n_l}, with eigenvalues ɛ_l(λ), whose dimension n_l is independent on the control parameter λ. 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' γ in the control manifold i.e., maps \gamma\colon [0,\,T]\mapsto \cal{M} such that γ(0) = γ(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 γ 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, ϕ_l :  = ∫₀^Tdτ ɛ_l(λ_τ),  is the dynamical phase whereas the matrices Γ_Al(γ)'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 γ,  and the 'adiabatic connection forms' A_l. The latter have an explicit matrix form given by A_l = Π_l(λ) d Π_l(λ) = ∑_μA_l, μ dλ_μ,  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 (λ_μ)_μ = 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 H₀ :  = H(λ₀). Upon dividing the time interval [0,  T] into N equal segments Δt,  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 Δt 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)_μ with μ = 1, …, 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 ∣ψ_in⟩ belonging to an eigen-space \cal{H}_0 with associated eigenvalue e.g., ɛ₀ = 0. Due to the time ordering symbol the actions on the state ∣ψ(t)⟩ 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 ∣ψ(t_i)⟩ will remain in the ɛ₀ = 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 γ. 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 ɛ₀(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 H₀. Its action merely reparametrizes the variables of the control manifold. Taking into account the properties gH₀ = H₀g and gΠ = Πg we are able to obtain the transformation of the connection as A ↦ g^† A g + g^† dg,  (g ∈ U(n)). It immediately follows that the holonomy transforms as Γ_A ↦ g^† Γ_A g. Notice that in the new coordinates the state vectors ∣ψ⟩ i.e., the sections, become g^† ∣ψ⟩. 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 λ₀ 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 γ₀(t) ≡ λ₀,  t ∈ [0,  1] moreover with γ^− 1 we shall denote the loop t ↦ γ(1 − t).

The holonomy can be considered as a map Γ_A: L_λ0 ↦ U(n_l), whose basic properties can be easily derived from eq. (\ref{Hol}):

1. Γ_A(γ₂ ⋅ γ₁) = Γ_A(γ₂) Γ_A(γ₁) by composing loops in \cal M one obtains a unitary evolution that is the product of the evolutions associated with the individual loops,
2. \Gamma_A(\gamma_0)=\leavevmode\hbox{\small1\kern-3.8pt\normalsize1} staying at rest in the parameter space corresponds to no evolution at all,
3. Γ_A(γ^− 1) = Γ_A^− 1(γ) in order to get the inverse holonomy one has to traverse the path γ with reversed orientation,
4. Γ_A(γ ∘ φ) = Γ_A(γ),  where φ is any diffeomorphism of [0,  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 Hol(A) :  = Γ_A(L_λ0) 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 = ∑_μνF_μν dx^μ ∧ dx^ν 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_μν's is the Lie algebra of the holonomy group. It follows in particular that when the F_μν'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 γ in \cal M such that ∥Γ_A(γ) − U∥ ≤ ε,  with ε arbitrarily small. This means that any computation on the code \cal C can be realized by adiabatically driving the control parameter configuration λ along a suitable closed path γ.

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⟩ state of m qubits. The statement of irreducibility of the connection A relates a particular unitary U with the loop γ<sub>U</sub> over which the connection is integrated to give Γ<sub>A</sub>(γ<sub>U</sub>) = 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 (d − 1)/2 ≥ n² 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 ^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 (^2)^{\otimes\,N}.

category:Evolutions and Operations