We consider the problem of steering control via an input electro-magnetic field for a system of two interacting spin 1/2 particles. This model is of interest in applications because it is used to perform logic operations in quantum computing that involve two quantum bits. The describing model is a bilinear system whose state varies on the Lie group of special unitary matrices of dimension 4, SU(4). By using decompositions of the latter Lie group, the problem can be decomposed into a number of subproblems for a system whose state varies on the (smaller) Lie group of 4/spl times/4 proper orthogonal matrices, SO(4). We tackle the time optimal control problem for this system and show that the extremals can be computed explicitly and they are the superposition of a constant field and a sinusoidal one.
In this paper we provide an explicit parameterization of arbitrary unitary transformation acting on n qubits, in terms of one and two qubit quantum gates. The construction is based on successive Cartan decompositions of the semi-simple Lie group, SU(2^n). The decomposition highlights the geometric aspects of building an arbitrary unitary transformation out of quantum gates and makes explicit the choice of pulse sequences for the implementation of arbitrary unitary transformation on $n coupled spins. Finally we make observations on the optimality of the design procedure.
In quantum control theory, a question of fundamental and practical interest is how an arbitrary unitary transformation can be decomposed into a minimum number of elementary rotations for implementation, subject to various physical constraints. Examples include the singlet-triplet (ST) and exchange-only (EO) qubits in quantum-dot systems, and gate construction in the Solovay--Kitaev algorithm. For two important scenarios, we present complete solutions to the problems of optimal decomposition of single-qubit unitary gates with non-orthogonal rotations. For each unitary gate, the criteria for determining the minimal number of pieces is given, the explicit gate construction procedure, as well as a computer code for practical uses. Our results include an analytic explanation to the four-gate decomposition of EO qubits, previously determined numerically by Divincenzo et al. [Nature (London) 408, 339 (2000)]. Furthermore, compared with the approaches of Ramon sequence and its variant [Phys. Rev. Lett. 118, 216802 (2017)], our method can reduce about 50% of gate time for ST qubits. Finally, our approach can be extended to solve the problem of optimal control of topological qubits, where gate construction is achieved through the braiding operations.
Given a compact, connected Lie group G with Lie algebra � . We discuss time-optimal control of bilinear systems of the form (I) where H d , H j ∈ � , U ∈ G , and the v j act as control variables. The case G = SU (2 n ) has found interesting appli- cations to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n -particle system with fixed drift Hamiltonian H d , which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v j . The question of interest here is to transfer the sys- tem from a given initial state U 0 to a prescribed final state U 1 in least possible time. Denote bythe Lie algebra spanned by H 1 , …, H m , and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G / K we discuss in detail time-optimal control strategies for system (I) in the case where G / K carries the structure of a Riemannian sym- metric space.
We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (cnot) and swap gates explicitly. Our approach is independent of the standard cnot implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with cnot gates, achieving an asymptotic cnot cost of
16214n for
n qubits.
Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called
KP-problem, where target unitaries belong to a semi-simple Lie group manifold
G whose Lie algebra admits a
g=k⊕p decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in
p. In this paper, we propose a new method utilising global Cartan decompositions
G=KAK of symmetric spaces
G/K for generating time-optimal unitaries for targets
−iX∈[p,p]⊂k with controls
−iH(t)∈p. Target unitaries are parametrised as
U=kac where
k,c∈K and
a=eiΘ with
Θ∈a. We show that the assumption of
dΘ=0 equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by
p and local rotations are generated by
[p,p].