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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Design of geometric phase gates and controlling the dynamic phase for a two-qubit Ising model in magnetic fields

M. Amniat-Talab, Hossein Rangani Jahromi

Published Mar 1, 2013• 6 citations
Publisher

Abstract

We investigate how to obtain a non-trivial geometric phase gate for a two-qubit spin chain, with Ising interaction in different magnetic fields. Indeed, one of the spins is driven by a time-varying rotating magnetic field, and the other is coupled with a static magnetic field in the direction of the rotation axis. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. It is shown that the static magnetic field does not change the adiabatic states of the system, and it does not affect the geometric phases, whereas it may be used to control the dynamic phases. In addition, by considering the exact two-spin adiabatic geometric phases, we find that a non-trivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with opposite directions of the magnetic fields, opposite signs of the interaction constant and the phase shift of the rotating magnetic field. In addition, in the non-adiabatic case, starting with a certain initial state, a cycle can be achieved and thus the Aharonov–Anandan phase is calculated.

DOI
10.1098/rspa.2012.0743

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We investigate how to obtain a non-trivial geometric phase gate for a two-qubit spin chain, with Ising interaction in different magnetic fields. Indeed, one of the spins is driven by a time-varying rotating magnetic field, and the other is coupled with a static magnetic field in the direction of the rotation axis. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. It is shown that the static magnetic field does not change the adiabatic states of the system, and it does not affect the geometric phases, whereas it may be used to control the dynamic phases. In addition, by considering the exact two-spin adiabatic geometric phases, we find that a non-trivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with opposite directions of the magnetic fields, opposite signs of the interaction constant and the phase shift of the rotating magnetic field. In addition, in the non-adiabatic case, starting with a certain initial state, a cycle can be achieved and thus the Aharonov–Anandan phase is calculated.
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct π/8 and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases, or based on Aharonov-Anandan phases when the gyromagnetic ratios of the two qubits are equal. In implementing a universal set of quantum gates, one may combine geometric π/8 and Hadamard gates and dynamical gate.
Recently, it has been proposed to perform quantum computation through the Berry phase (adiabatic cyclic geometric phase) shift with NMR. This geometric quantum gate will hopefully be fault tolerant to certain types of error because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through the Aharonov-Anandan phase (non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. Moreover, in the scheme we have chosen the zero dynamic phase time evolution loop in obtaining the geometric phase shift; we need not perform any extra operation to cancel the dynamic phase.
While it is often thought that the geometric phase is less sensitive to fluctuations in the control fields, a very general feature of adiabatic Hamiltonians is the unavoidable dynamic phase that accompanies the geometric phase. The effect of control field noise during adiabatic geometric quantum gate operations has not been probed experimentally, especially in the canonical spin qubit system that is of interest for quantum information. We present measurement of the Berry phase and carry out adiabatic geometric phase gate in a single solid-state spin qubit associated with the nitrogen-vacancy center in diamond. We manipulate the spin qubit geometrically by careful application of microwave radiation that creates an effective rotating magnetic field, and observe the resulting Berry phase signal via spin echo interferometry. Our results show that control field noise at frequencies higher than the spin echo clock frequency causes decay of the quantum phase, and degrades the fidelity of the geometric phase gate to the classical threshold after a few (∼10) operations. This occurs inspite of the geometric nature of the state preparation, due to unavoidable dynamic contributions. We have carried out systematic analysis and numerical simulations to study the effects of the control field noise and imperfect driving waveforms on the quantum phase gate.
The geometric phase, in particular the Berry phase, in an entangled state of five spin-1/2 particles is studied. A time-dependent magnetic field is applied to control the time evolution of the cluster. Using the method of algebraic dynamics, we calculate the non-adiabatic geometric phase or Berry phase and the degeneracy energy levels when the magnetic rotates around Z axis. Based on the exact analytical solutions, we show how the Berry phase of the entangled state of this cluster depends on the external magnetic field parameters ω (the angular velocity of the rotating magnetic field) and θ (the angle between the magnetic field and Z axis).
We derive the general formula giving Berry's phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the 'effective' electric and magnetic fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to useful selection rules. Furthermore, the Hamiltonian parameter space coincides with the density matrix space for a spin S = 1. This implies a mathematical identity between Berry's phase and the Aharonov–Anandan phase, which is lost for S > 1. We have found that new physical features of Berry's phases emerge for integer spins ⩾2. We provide explicit numerical results of Berry's phases for S = 2, 3, 4. For any spin, one easily finds well-defined regions of the parameter space where instantaneous eigenstates remain widely separated. We have decided to stay within them and not deal with non-Abelian Berry's phases. We present a thorough and precise analysis of the non-adiabatic corrections with separate treatment for periodic and non-periodic Hamiltonian parameters. In both cases, the accuracy for satisfying the adiabatic approximation within a definite time interval is considerably improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept below the 0.1% level. For a class of quantum cycles, involving as a sole periodic parameter the precession angle of the electric field around the magnetic field, the corrections odd upon the reversal of the associated rotation velocity can be canceled exactly if the quadrupole to dipole coupling is chosen appropriately ('magic values'). The even ones are eliminated by taking the difference of the Berry phases generated by two 'mirror' cycles. We end by indicating a possible application of the theoretical tools developed in this paper. We propose a way to perform a holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian given as a nonlinear function of the total spin operator S , defined as the sum of the N individual spin operators. The basic idea behind this proposal is the mathematical fact that any non-correlated states can be expanded into eigenstates of S 2 and S z . The same eigenvalues appear several times in the decomposition when N > 2 but all these states differ by their symmetry properties under the N -spin permutations. The case N = 4 and S z = 1 is treated explicitly and a maximum entanglement is achieved.