Contemporary Mathematics

Quantum Computation by Geometrical Means

Jiannis Pachos

Abstract. A geometrical approach to quantum computation is presented, where a non-abelian connection is introduced in order to rewrite the evolution operator of an energy degenerate system as a holonomic unitary. For a simple geometrical model we present an explicit construction of a universal set of gates, represented by holonomies acting on degenerate states.

1. Prologue

Abelian [Sha] and non-abelian [Wil] geometrical phases in quantum theory have been considered as a deep and fascinating subject. They provide a natural connection between the evolution of a physical system with degenerate structure and differential geometry. Here we shall present a model where these concepts can be explicitly applied for quantum computation [Zan].

The physical setup consists of an energy degenerate quantum system on which we perform an adiabatic isospectral evolution described by closed paths in the parametric space of external variables. The corresponding evolution operators acting on the code-state in the degenerate eigenspace are given in terms of holonomies and we can use them as quantum logical gates. This is a generalization of the Berry phase or geometrical phase, to the non-abelian case, where a non-abelian adiabatic connection, A, is produced from the geometrical structure of the degenerate spaces. In particular, on each point of the manifold of the external parameters there is a code-state attached and a transformation between these bundles of codes is dictated by the connection A.

In order to apply this theoretical construction to a concrete example we employ a model with CP^{2} geometry, that is a complex projective manifold with two complex coordinates. This is interpreted as a qubit [Pac]. A further generalization with the tensor product of m CP^{2} models and additional interaction terms parametrized by the Grassmannian manifold, G(4, 2), is interpreted as a model of quantum computer.

The initial code-state is written on the degenerate eigenspace of the system. The geometrical evolution operator is a unitary acting on it and it is interpreted as

\begin{matrix}c \\ ◯ \end{matrix}2002 American Mathematical Society

2000 Mathematics Subject Classification. Primary 81P68.

The author was supported in part by TMR Network under the contract no. ERBFMRXCT96 - 0087.

a logical gate. Due to adiabaticity the geometrical part of the evolution operator has a dimensionality equal to the degree of degeneracy of the eigenspace. Specific logical gates given by holonomies are constructed for a system with a tensor product structure resulting in universality while at the end a quantum optical application is sketched.

2. Coset Space Geometry

A transformation U(n) between the states |α⟩, α = 1, ..., n can be realized by all possible sub-U(2) transformations between any two of those states, |i⟩ and |j⟩. A coset space can be produced as the factor with respect to some particular U(2) symmetries of these transformations.

Examples of such constructions are given in the following:

• the CP^{2} projective space:

CP^{2} ≅ \frac{U(3)}{U(2) × U(1)}

|1⟩ U_{1}(z_{1}) |˜2⟩  |2⟩ U_{2}(z_{2})

The lines denote U(2) transformations between the states represented here by “holes”. The U(3) group could be interpreted by three lines connecting all the holes together. The distinction between “filled” and “unfilled” holes is due to the coset structure, which factors out the symmetry transformations, between |1⟩ and |2⟩, and denotes explicitly the non-symmetric ones between |1⟩ or |2⟩ and |˜2⟩.

• the (CP^{2})^{×m} × (G(4, 2)_{int})^{×(m−1)} product space:

⋯ \frac{U(3)}{U(2) × U(1)} × \frac{U(3)}{U(2) × U(1)} , \frac{U(4)}{U(2) × U(2)}|_{int} ⋯

|1⟩  |˜2⟩  |2⟩  |3⟩  |˜4⟩  |4⟩

Transformations can be performed between the states {|1⟩, |2⟩} and {|3⟩, |4⟩} due to their connections with the states {|˜2⟩, |˜4⟩}, while an interaction in the tensor product space between |24⟩ and |˜2˜4⟩ gives transformations between those two sets. The transformations between only two states may be performed by linear operations with respect to U(2) generators, while the combined transformations between the two qubits can be produced by bilinear generators which act simultaneously on the states of both of the CP^{2} models. The latter is denoted in the previous figure by the dashed lines, where the connection between the black dots indicates the simultanious action.

3. Degeneracy, Adiabaticity and Holonomies

Let us introduce the degenerate Hamiltonians H_{0}^{1} and H_{0}^{m} as follows

H_{0}^{1} = [ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ] , H_{0}^{m} = ∑_{m} H_{0}^{1} .

The orbit, that is the parametric manifold of the unitary transformations which preserves the degenerate spectrum of H_{0}^{1} is given by CP^{2}. A sub-manifold of the orbit of H_{0}^{m} in which we are interested in is given by the (CP^{2})^{×m} × (G(4, 2)_{int})^{×(m−1)} product manifold.

A general transformation parametrized by the CP^{2} space is given by U(z) := U_{1}(z_{1})U_{2}(z_{2}), with U_{α}(z_{α}) = exp G_{α}(z_{α}) = exp(z_{α}|α⟩⟨˜2| − z̄_{α}|˜2⟩⟨α|). The complex parameter z_{α} may be decomposed as z_{α} = θ_{α} exp iϕ_{α}. Due to the 2 × 2 sub-form of G_{α}(z_{α}) we can rewrite U_{α}(z_{α}) as

U_{α}(z_{α}) = 1_{α}^{⊥} + cos θ_{α}1_{α} + \frac{sin θ_{α}}{θ_{α}} G_{α}(z_{α}) ,

where 1_{α}^{⊥} = 1 − 1_{α} and 1_{α} = |α⟩⟨α| + |˜2⟩⟨˜2|. For the Grassmannian manifold G(4, 2) we have, for example, the U(2) rotation in the tensor product basis of two qubits, between the states |24⟩ and |˜2˜4⟩, given by U(z) = exp(z|24⟩⟨˜2˜4|−z̄|˜2˜4⟩⟨24|), with z = θ exp iϕ. The coordinates {λ^{a}} = {θ, ϕ} provide the parametric space which the experimenters control.

In the four dimensional manifold CP^{2} with coordinates {λ^{a}} a closed path, C, is drawn on a two-submanifold. Consider this evolution to be adiabatic as well as isospectral which is provided by the formula H(λ(t)) = U(λ(t))H_{0}U^{†}(λ(t)). As a result the state of the system, |ψ(t)⟩, stays on the same eigenvalue of the Hamiltonian, taken in our example to be E_{0} = 0, without level-crossing.

At the end of the loop C, spanned in time T = NΔt, when divided in N equal time intervals, we obtain

|ψ(T)⟩  = Te^{−i ∫_{0}^{T} UH_{0}U^{†}dt} |ψ(0)⟩

= T lim_{N→∞} ∏_{i=1}^{N} U_{i}e^{−iH_{0}Δt}U_{i}^{†} |ψ(0)⟩

= P lim_{N→∞} (1 + ∑_{i=1}^{N} A_{i}Δλ_{i}) |ψ(0)⟩

with  A_{i} = U_{i}^{†} \frac{ΔU_{i}}{Δλ_{i}} and U_{i} = U(λ(t_{i})) .

Hence, the state |ψ(0)⟩ acquires a geometrical unitary operator given by the holonomy of a connection A as

Γ_{A}(C) := P exp ∮_{C} A , where (A^{λ^{a}})_{αβ} := ⟨α|U^{†}(λ)\frac{∂}{∂λ^{a}} U(λ)|β⟩ .

The states |α⟩ and |β⟩ belong to the same degenerate eigenspace of H_{0} and λ^{a}’s are the real control parameters.

The produced unitary operator is an effect of the non-commutativity of the control transformations which produce effectively a curvature. In the case of the Berry phase produced for example in front of the spin states of an electron when placed in a magnetic field, the non-commutativity is between the U(2) control unitaries which change the direction of the magnetic field in the three dimensional space. What is presented here is the generalization of the Berry phase to the nonabelian case.

The Γ_{A}(C)’s produced by CP^{2} for various loops C, generate the whole U(2),

{Γ_{A}(C); ∀ C ∈ CP^{2}} ≈ U(2) .

In the case of m qubits with their proper interactions, the produced group is U(2^{m}),

{Γ_{A}(C); ∀ C ∈ (CP^{2})^{×m} × (G(4, 2)_{int})^{×(m−1)}} ≈ U(2^{m}) .

4. Quantum Computation

In order to perform quantum computation by using the above constructions we consider the following identifications:

QUANTUM CODE ≡ Degenerate States, |ψ(0)⟩

LOGICAL GATES ≡ Holonomies, Γ_{A}(C)

Let us first investigate the CP^{2} case. The basic question is how we can generate a general U(2) element by moving along a closed path, C. Or in other words, for a specific U ∈ U(2) which loop C is such that Γ_{A}(C) = U. In general, ∀ g ∈ u(2) ∃ loop C ∈ CP^{2} manifold, such that Γ_{A}(C) = exp g, which is the statement of irreducibility of the connection A [Zan]. To answer the above question we perform the following analysis. The loop integral

∮_{C} A = ∮_{C} A_{λ^{1}}dλ^{1} + A_{λ^{2}}dλ^{2} + ⋯

is the main ingredient of the holonomy. Due to the path ordering symbol it is not possible to just calculate it and evaluate its exponential, as in general the connection components do not commute with each other. Still it is possible to consider the following restrictions in the position of the loop. Choose C such that:

• it belongs to one plane (λ^{i}, λ^{j}) = (θ_{i}, ϕ_{j}) or (θ_{i}, θ_{j}), hence only two components of A are involved,

• the position of the plane is such that the connection, A, restricted on it become, A|_{(λ^{i},λ^{j})} = (A^{λ^{i}} = 0, A^{λ^{j}}), that is these two components commute with each other. Still it is important that their related field strength component, F_{ij} = ∂_{i}A_{j} − ∂_{j}A_{i} + [A_{i}, A_{j}], is non-vanishing in order to obtain a non-trivial holonomy. Such a requirement is possible for the CP^{2} model and for a wide class of other models.

On the planes where those conditions are satisfied the evaluation of the holonomy is trivially given by just exponentiating the loop integral of the connection without worrying about the path ordering symbol. Hence,

Γ_{A}(C) = P exp ∮_{C} A = exp(Σg) = 1_{2×2} cos Σ + g sin Σ ,

where Σ represents the area enclosed by the loop C projected on the sphere associated with the compactified CP^{2} manifold. This area may be varied desirably. Furthermore, we are able to obtain a complete set of generators g by choosing C to lie on different planes. In detail we may obtain for g the following forms

−i|α⟩⟨α|  := −iσ_{α}^{3} , α = 1, 2
−i(−i|1⟩⟨2| + i|2⟩⟨1|) := −iσ^{2} ,
−i(|1⟩⟨2| + |2⟩⟨1|) := −iσ^{1} .

The σ_{α}^{3} generators are similar to a Berry phase and they are produced by paths C_{1} on the (θ_{α}, ϕ_{α}) planes. The corresponding holonomy is the exponential of this generator multiplied by the area, Σ_{1}, of the surface the path C_{1} encloses when projected on a sphere S^{2}(2θ_{α}, ϕ_{α}) with spherical coordinates 2θ_{α} and ϕ_{α},

−iσ_{α}^{3} : C_{1} ∈ (θ_{α}, ϕ_{α}) → Γ_{A}(C_{1}) = exp −iΣ_{1}σ_{α}^{3} .

The σ^{2} generator is produced by a path C_{2} along the plane (θ_{1}, θ_{2}) positioned at ϕ_{1} = ϕ_{2} = 0, while σ^{1} is produced by a path, C_{3}, along a parallel plane positioned at ϕ_{1} = \frac{π}{2} and ϕ_{2} = 0. Their corresponding areas are Σ_{2} and Σ_{3}. For example

−iσ^{2} : C_{2} ∈ (θ_{1}, θ_{2})|_{ϕ_{1}=0,ϕ_{2}=0} → Γ_{A}(C_{2}) = exp −iΣ_{2}σ^{2} .

Altogether we have 2^{2} independent generators spanning the Lie algebra of U(2).

For the case of the two qubit interaction the corresponding connection components are given by

A_{θ} = diag(0, 0, 0, 0) , A_{ϕ} = diag(0, 0, 0, −i sin^{2} θ)

which are written in the basis {|13⟩, |14⟩, |23⟩, |24⟩}. A loop C on the (θ, ϕ) plane will produce the following holonomy

Γ_{A}(C) = diag(1, 1, 1, e^{−iΣ}) , Σ = ∫_{D(C)} dθdϕ sin 2θ ,

where Σ can also be interpreted as an area on the sphere S^{2}(2θ, ϕ).

5. One and Two Qubit Logical Gates

By performing appropriate loops we can obtain one qubit phase rotations as well as two qubit gates such as a controlled phase rotation U_{CP H}.

Analytically, by spanning the indicated areas we may obtain

U_{1} = exp [ \begin{matrix}−iΣ_{1} & 0 \\ 0 & 0 \end{matrix} ] , U_{2} = exp [ \begin{matrix}0 & 0 \\ 0 & iΣ_{1} \end{matrix} ] , U_{3} = exp [ \begin{matrix}0 & −Σ_{2} \\ Σ_{2} & 0 \end{matrix} ] .

The combinations

U_{1}U_{2} = exp(−iσ_{3}Σ_{1}) , U_{3} = exp(−iσ_{2}Σ_{2})

can give any U(2) transformation and hence any one qubit rotation.

For the two qubit gates we can construct easily the controlled rotation U_{CP H} = diag(1, 1, 1, exp −iΣ) between any pair of qubits. It is generated by a loop C on the (θ, ϕ) plane. Together with the one qubit rotations they provide a universal set of gates.

6. Epilogue

Apart from the intriguing theoretical formulation of holonomic computation there are several aspects of it, which have appealing technical advantages. Without overlooking the difficulties posed to an experimenter for performing continuous control over a system in order to span a loop, there are several unique characteristics of it, which await for exploitation. For example, robustness of the control procedure in terms of the spanned area, according to errors in the actual form of the performed loop, as well as the isolation of the degenerate states as a calculational space may prove to be advantages worth exploring.

In quantum optics displacing devices, squeezing devices and interferometers acting on laser beams can provide the control parameters for the holonomic computation. Each laser beam is placed in a non-linear Kerr medium with degenerate Hamiltonian H_{0} = n(n−1), where n is the photon numbering operator. The degenerate states |0⟩ and |1⟩ are the basis for encoding one qubit which is manipulated by displacing and squeezing devices. Any two qubit interactions can be implemented by interferometers [Cho].

It is challenging for the experimenters to produced the desired closed paths.

References

[Sha] Geometric Phases in Physics, A. Shapere and F. Wilczek, Eds. World Scientific (1989).

[Wil] F. Wilczek and A. Zee., Phys. Rev. Lett. 52, 2111 (1984).

[Zan] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999), quant-ph/9904011.

[Pac] J. Pachos, P. Zanardi and M. Rasetti, Phys. Rev. A 61 010305(R), quant-ph/9907103.

[Cho] J. Pachos and S. Chountasis, quant-ph/9912093.

(Jiannis Pachos) Institute for Scientific Interchange Foundation, Villa Gualino, Viale Settimio Severo 65, I-10133 Torino, Italy

E-mail address, Jiannis Pachos: pachos@isiosf.isi.it