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arXiv:quant-ph/0511192 · Physical Review A

Geometric phase for N -level systems through unitary integration

Dmitry B. Uskov, A. R. P. Rau

Published Sep 14, 2006• 23 citations
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Abstract

Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin $1∕2$, the Bloch sphere ${S}^{2}$, together with a U(1) phase, provides a complete SU(2) description. We generalize to $N$-level systems and $\mathrm{SU}(N)$ in terms of a $2(N\ensuremath{-}1)$-dimensional base space and reduction to a $(N\ensuremath{-}1)$-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an $N$-level system and gives $(N\ensuremath{-}1)$ geometric phases explicitly. A complete analytical construction of an ${S}^{4}$ Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).

DOI
10.1103/physreva.74.030304

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Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin 1∕21∕21∕2, the Bloch sphere S2{S}^{2}S2, together with a U(1) phase, provides a complete SU(2) description. We generalize to NNN-level systems and SU(N)\mathrm{SU}(N)SU(N) in terms of a 2(N\ensuremath−1)2(N\ensuremath{-}1)2(N\ensuremath−1)-dimensional base space and reduction to a (N\ensuremath−1)(N\ensuremath{-}1)(N\ensuremath−1)-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an NNN-level system and gives (N\ensuremath−1)(N\ensuremath{-}1)(N\ensuremath−1) geometric phases explicitly. A complete analytical construction of an S4{S}^{4}S4 Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).
A two-sphere (``Bloch'' or ``Poincare'') is familiar for describing the dynamics of a spin-1∕21∕21∕2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU(N)\mathrm{SU}(N)SU(N) in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation and entangled states in quantum-information sciences. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various subgroups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitary evolution of a quantal spin pair is thereby expressed as rotations of real, many-dimensional vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) subgroup and an eight-dimensional Grassmannian manifold for the general SU(4). The latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint. This geometrical picture for two spins provides the extension and generalization of the Bloch sphere that has proved invaluable for the understanding of the dynamics of a single spin.
The Bloch sphere is a familiar and useful geometrical picture of the dynamics of a single spin or two-level system's quantum evolution. The analogous geometrical picture for three-level systems is presented, with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds describing the time evolution operator. The first, called the base manifold, is the counterpart of the S^2 Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four-dimensional, it breaks down further into smaller objects depending on alternative representations that we discuss. Geometrical phases are also developed and presented for specific applications. Arbitrary time-dependent couplings between three levels or between two spins (qubits) with SU(3) Hamiltonians can be conveniently handled through these geometrical objects.
Quantum computing leverages fundamental principles of quantum mechanics to perform computations far exceeding classical architectures. At the core of single-qubit quantum state manipulation is the Bloch sphere, a geometric representation mapping the state space of two-level quantum systems onto a three-dimensional unit sphere. This paper presents a comprehensive theoretical framework of qubits, starting from single-qubit geometric state vectors and expanding to rigorous multi-qubit mathematical formulations. Furthermore, it analyzes single-qubit quantum gates as unitary rotations on the Bloch sphere, evaluates multi-qubit system scaling, and explicitly proves state normalization conditions using Born's rule across multi-dimensional Hilbert spaces.
Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.
The existence of Hopf fibrations S^{2N+1}/S^1 = CP^N and S^{4K+3}/S^3 = HP^K allows us to treat the Hilbert space of generic finite-dimensional quantum systems as the total bundle space with respectively U(1)U(1)U(1) and SU(2)SU(2)SU(2) fibers and complex and quaternionic projective spaces as base manifolds. This alternative method of studying quantum states and their evolution reveals the intimate connection between generic quantum mechanical systems and geometrical objects. The exact Abelian and non-Abelian geometric phases, and more generally the geometrical factors for open paths, and their precise correspondence with geometric Kahler and hyper-Kahler connections will be discussed. Explicit physical examples are used to verify and exemplify the formalism.