One-Spin Quantum Logic Gates from Exchange Interactions and a Global Magnetic Field

Abstract

It has been widely assumed that one-qubit gates in spin-based quantum computers suffer from severe technical difficulties. We show that one-qubit gates can in fact be generated using only modest and presently feasible technological requirements. Our solution uses only global magnetic fields and controllable Heisenberg exchange interactions, thus circumventing the need for single-spin addressing.PACS numbers: 03.67.Lx, 03.67.HkThe Heisenberg exchange interaction features prominently in several of the promising solid state proposals for quantum computation (QC) [1,2,3,4,5,6,7]. For this reason extensive studies have been made of the problem of constructing a universal set of quantum logic gates [8] that employs this interaction. The difficulty is that as long as each qubit is represented by a single spin it appears necessary to supplement the Heisenberg interaction with single-spin addressing, thus requiring either strongly localized magnetic fields [9] or g-factor engineering [10], both of which are technically highly demanding. This problem can be alleviated by encoding a logical qubit into the state of at least three spins, in which case the Heisenberg interaction by itself can be shown to be universal [11,12,13,14], or by using a combination of Heisenberg interactions with inhomogeneous, delocalized magnetic fields, and an encoding into at least two spins [15,16,17]. Another option is to supplement the Heisenberg interaction with certain two-spin measurements [18]. These alternatives to single-spin addressing each come with space and time overhead [11,12,13,14,15,16,17], or a probabilistic protocol that needs to be repeated several times until it converges [18].Here we show that in fact it is not necessary to introduce any encoding overhead when using the Heisenberg interaction in conjunction with inhomogeneous, delocalized, time-dependent magnetic fields. Furthermore, unlike previous proposals we do not require spin-resonance techniques [16] (which involve the difficult demand of modulating the interaction strength at high frequency), and do not need the recoupling assumption [15] that the exchange coupling can be made much greater than the difference in Zeeman energies. These considerations overall amount to a significant simplification compared to various aspects of previously proposed methods [11,12,13,14,15,16,17]. Instead, one can generate the single-spin gates needed for universality by pulsing the Heisenberg interaction together with global magnetic fields, under relaxed controllability assumptions.Interactions.-Let us now define the basic interactions involved in our method. We consider N spins interacting via a controllable Heisenberg exchange interaction J ij (t)S i · S j , that generates a unitary evolutionHereare the spin-1/2 matrices acting on qubit i. In addition we assume the presence of a global (i.e., delocalized) magnetic field that can be switched on/off fast on the timescale of spin decoherence. Recent theoretical estimates [19] of dephasing times for electron spins in GaAs quantum dots ar...

Decoherence due to contacts in ballistic nanostructures

“…While the physics must be the same regardless of the projection operator used, the opacity of the equations obtained certainly varies. Equation (14) shows explicitly how the subdynamics looks for P ; by generalizing the proof in Appendix A, one can write the subdynamics for any other projection operator instead of P . The reason we are using P instead of the projection operator P induced by the initial environmental statistical operator is that, as stated previously, P is the only projection operator that has an orthonormal eigenbasis in which it is represented by a diagonal form (4).…”

“…These are given by The other decomposition is into R(P ), the range of the projector P induced by the initial environmental statistical operator ρE, and N (P † ), the null space of P † and orthocomplement to R(P ). These two decompositions enable us to construct the mapping M used in the reduced dynamics (14).…”

“…We will therefore proceed with deriving the Markovian approximation to the exact non-Markovian equation (14) in the presence of an environment that loses memory on a timescale τ , presumed much shorter that the response time of the open system, and we will derive the relationships that the coarse graining time τ must satisfy for the approximation to be consistent. Then, in Sec.…”

“…The description and manipulation of the contact-induced decoherence are presently of great importance not only in quantum transport studies, 4,5,6,7,8,9,10,11,12 but also in the theory of measurement 13 and quantum information. 14 The purpose of this paper is to provide a simple description of the nonunitary evolution of a ballistic nanostructure's active region due to the injection of carriers from the contacts. Carrier injection from the contacts into the active region is traditionally described by either an explicit source term, such as in the single-particle density matrix, 15,16,17,18,19,20 Wigner function 2,3,21,22,23,24,25,26,27,28,29 and Pauli equation 30,31 transport formalisms, or via a special self-energy term in the ubiquitous nonequilibrium Green's function formalism.…”

Quantum Computing with Spins in Solids

“…Alternative single-qubit-rotation schemes may require global magnetic field gradients (Wu et al 2004, Tokura et al 2006, ESR (see Figure 2) or, in the presence of spin-orbit interaction, electric-dipole spin resonance (EDSR) techniques. EDSR has been analyzed in great detail for two-dimensional systems in theory Efros 2003, Duckheim and and experiment (Kato et al 2004), and can also be applied to lower-dimensional systems (quantum wires and quantum dots) (Levitov andRashba 2003, Golovach et al 2006), with the advantage that single-qubit operations could then be performed using fast all-electrical control.…”