The quantum brachistochrone problem for two spins-$\frac{1}{2}$ with anisotropic Heisenberg interaction

Abstract

We study the quantum brachistochrone evolution for a system of two spins-1 2 described by an anisotropic Heisenberg Hamiltonian without zx, zy interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by | ↑↑ , | ↓↓ and | ↑↓ , | ↓↑ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.

Measuring distance between quantum states on a quantum computer

“…Using expression (9) we show in Fig. 6 the protocol for measuring this value on a quantum a computer.…”

“…Each of the terms in the evolution operator can be performed on a quantum computer using two controlled-NOT operator and one R z (−χ/2) operator. Based on expression (9) in Fig. 8 we represent a quantum circuit which allows measuring the square of the modulus of the scalar product between initial state (23) and state which is achieved during the evolution (26).…”

“…The concept of a distance between quantum states in Hilbert space [1][2][3] has found its application in different fields of quantum mechanics related to the evolution of quantum systems [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], quantum entanglement [8,[20][21][22][23][24][25][26][27][28][29][30], quantum computations [31][32][33][34][35], etc. It was shown that the distance, which the quantum system passes during the evolution in the Hilbert space, is related to the integral of the uncertainty of energy that in turn defines the speed of evolution [4].…”

Geometry and speed of evolution for a spin-s system with long-range zz-type Ising interaction

“…where √ gdA is the element of area of the manifold M, g is the determinant of the metric tensor, ∆ defines the contribution of angular defects and X(M) is the Euler characteristic of the manifold. Using the fact that θ ∈ [0, π], χ ∈ [0, χ max ] the integral in formula (21) takes the value 4χ max (N − 1)s. The angular defects are located very close to the point θ = 0 and π. This fact allows to rewrite the metric in these areas as follows…”

Preparation of quantum states of two spin- particles in the form of the Schmidt decomposition

“…We seth = 1 and it means that the coupling parameter A is measured in frequency units. Note that the Hamiltonian (1) is a special case of the Hamiltonian considered in our previous paper where the brachistochrone problem for two spin particles was examined [37]. Let us consider evolution of the system of two spins having started from the initial state | ↑↓ .…”