Anisotropic Spin Exchange in Pulsed Quantum Gates

Abstract

We show how to eliminate the first-order effects of the spin-orbit interaction in the performance of a two-qubit quantum gate. Our procedure involves tailoring the time dependence of the coupling between neighboring spins. We derive an effective Hamiltonian which permits a systematic analysis of this tailoring. Time-symmetric pulsing of the coupling automatically eliminates several undesirable terms in this Hamiltonian. Well chosen pulse shapes can produce an effectively isotropic exchange gate, which can be used in universal quantum computation with appropriate coding. PACS: 03.67.Lx, 71.70.Ej, 85.35.Be The exchange interaction between spins is a promising physical resource for constructing two-qubit quantum gates in quantum computers [1][2][3][4][5]. In the idealized case of vanishing spin-orbit coupling, this interaction is isotropic, and any Hamiltonian describing time-dependent exchange between two spin-1/2 qubits, H 0 (t) = J(t)S 1 · S 2 , commutes with itself at different times. Thus, the resulting quantum gate depends on J(t) only through its time integral -a convenient simplification, particularly because, when carrying out quantum gates, the exchange interaction should be pulsed adiabatically on time scales longer thanh/∆E, where ∆E is a typical level spacing associated with the internal degrees of freedom of the qubits [3]. In addition, isotropic exchange alone has been shown to be sufficient for universal quantum computation, provided the logical qubits of the computer are properly encoded [6,7].Given the potential advantages of isotropic exchange for quantum gates, it is important to understand the effect of the inevitable anisotropic corrections due to spinorbit coupling. When these corrections are included, the Hamiltonian describing time-dependent exchange iswhereHere β(t) is the Dzyaloshinski-Moriya vector, which is first order in spin-orbit coupling, and IΓ(t) is a symmetric tensor which is second order in spin-orbit coupling [8]. Although these corrections may be small, they will, in general, not be zero unless forbidden by symmetry. For example, Kavokin has recently estimated that β(t) can be as large as 0.01 for coupled quantum dots in GaAs [9]. In this Letter we construct the quantum gates produced by pulsing H(t). This is nontrivial because H(t) typically does not commute with itself at different times. We represent the resulting gates using an effective Hamiltonian H(t), which we derive perturbatively in powers of the spin-orbit coupling. H(t) is simple to work with because it does commute with itself at different times. As an application of this effective Hamiltonian, we use it to tailor pulse forms that effectively eliminate any firstorder anisotropic corrections.The quantum gate obtained by pulsing a particular H(t) is found by solving the time-dependent Schrödinger equation i d dt |Ψ(t) = H(t)|Ψ(t) where |Ψ(t) is the state vector describing the two spin-1/2 qubits (here, and in what follows,h = 1). In general this problem cannot be solved analytically. However, since we expect spin...

Prospects for Spin-Based Quantum Computing in Quantum Dots

“…Fortunately, strategies have been developed with which the SOI-induced gate errors can be strongly suppressed [39][40][41]. In general, gate errors can be reduced from first to second (or higher) order in SOI when the coupling strength J(t) is varied symmetrically in time, followed by additional qubit rotations [39,41].…”

Quantum Computing with Spins in Solids

“…The √ swap operation has now been successfully implemented in experiments involving two electrons confined to two neighboring quantum dots (as in Figure 1) (Petta et al 2005a. Errors during the √ swap operation have been investigated due to nonadiabatic transitions to higher orbital states (Schliemann et al 2001, Requist et al 2005, spin-orbit-interaction (Bonesteel et al 2001, Stepanenko et al 2003, and hyperfine coupling to surrounding nuclear spins (Petta et al 2005a, Klauser et al 2005, Taylor et al 2006. The isotropic form of the exchange interaction given in Equation (1) is not always valid.…”

“…The isotropic form of the exchange interaction given in Equation (1) is not always valid. In realistic systems, a finite spin-orbit interaction leads to anisotropic terms which may cause additional errors, but could also be used to perform universal quantum computing with two-spin encoded qubits, in the absence of single-spin rotations (Bonesteel et al 2001, Stepanenko and Bonesteel 2004, Chutia et al 2006) (see also Section 4.4 below).…”

Quantum gates between capacitively coupled double quantum dot two-spin qubits

“…The relative strength of the anisotropic interaction in the quantum dot systems in GaAs is estimated to be |β| ∼ 0.1 − 0.01 [21,22]. The influence of the anisotropic corrections can be reduced in specific implementations of the quantum gates [23,24]. In our study of a two-qubit gate operation, we will only consider the case of isotropic interaction and neglect the weak anisotropy.…”