State transfer in static and dynamic spin chains with disorder David Petrosyan, Georgios M. Nikolopoulos, and P. Lambropoulos Institute of Electronic Structure and Laser, Foundation for Research and Technology - Hellas, 71110 Heraklion, Crete, Greece (Dated: October 31, 2018) We examine the speed and fidelity of several protocols for state or single excitation transfer in finite spin chains subject to diagonal and off-diagonal disorder. We find that, for a given chain length and maximal achievable inter-spin exchange ( XY ) coupling strength, the optimal static spin- coupling protocol, implementing the fastest state transfer between the two ends of the chain, is more susceptible to off-diagonal ( XY coupling) disorder, as compared to a much slower but robust adiabatic transfer protocol with time-dependent coupling strengths. PACS numbers: 03.67.Hk, 75.10.Pq, 73.63.Kv, 03.67.Lx Faithful transfer of quantum states between physical qubits of an integrated quantum register is one of the important prerequisites for scalable quantum computa- tion. Typically, qubit-qubit interactions are short range and implementing quantum logic gates between qubits located at distant sub-registers would involve intercon- necting them via quantum channels, or wires [1], which may consist of arrays of coupled quantum dots [2, 3] or superconducting qubits [4–6], atoms in optical lattices [7, 8], or other realizations of spin chains. Quantum channels of permanently coupled spins would require no dynamical manipulations during the state transfer, but might be susceptible to noise and imper- fections. Conversely, dynamically manipulated networks can be more robust with respect to certain kinds of dis- order, but are more involved requiring time-dependent external control. Here we re-consider critically several protocols for achieving efficient and dependable—ideally perfect—state transfer in disordered spin chains subject to physically constrained maximal achievable inter-spin coupling rate. The present work is an extension of our earlier studies [2, 3] on perfect states transfer to more realistic scenarios with the aim of quantifying and neu- tralizing the influence of static (or slowly changing) noise inevitably present in any imperfect physical realization of the spin chain resulting in diagonal and off-diagonal dis- order. After outlining the model, we examine the speed and reliability of several state transfer protocols first for ideal and then for noisy spin chains, followed by conclusions. The Hamiltonian for a spin chain of length N has a general form [9] H = 1 2 N ∑ j =1 h$_{j}$ ˆ σ z j − 1 2 N − 1 ∑ j =1 J$_{j}$ (ˆ σ x j ˆ σ x j $_{+1}$+ ˆ σ y j ˆ σ y j $_{+1}$+Δˆ σ z $_{j}$ˆ σ z j $_{+1}$) , (1) H = N ∑ j =1 h$_{j}$ ˆ a † $_{j}$ˆ a$_{j}$ − N − 1 ∑ j =1 J$_{j}$ (ˆ a † $_{j}$ˆ a$_{j}$$_{+1}$ + ˆ a † j $_{+1}$ˆ a$_{j}$ ) , (2) where ˆ σ x,y,z j are the Pauli spin operators at position j , h$_{j}$ determines the energy separation between the spin-up and spin-down states playing the role of the local “mag- netic field”, and J$_{j}$ is the nearest-neighbor spin-spin in- teraction which can be static or time-dependent. From now on we set the anisotropy parameter Δ = 0; Eq. (1) reduces then to the Hamiltonian of the XX model, which is isomorphic to the Hubbard Hamiltonian for spinless fermions or hard-core bosons [9], where ˆ a † j (ˆ a$_{j}$ ) is the particle creation (annihilation) op- erator at site j with energy h$_{j}$ and J$_{j}$ now plays the role of tunnel couping between adjacent sites j and j + 1. Our objective here is to transfer an arbitrary single qubit state | ψ 〉 = α | 0 〉 + β | 1 〉 between the two ends of the spin chain. To that end, we assume that all the spins can be prepared in the “ground” state |↓〉 j ≡ | 0 〉 j and at a certain initial time t$_{in}$ = 0 the first site of the chain is initialized to | ψ 〉 $_{1}$. Ideal transfer would imply that at a well-defined final time t$_{out}$ the last site of the chain is in state | ψ 〉 $_{N}$, up to a certain relative phase factor between the amplitudes of states | 0 〉 N and | 1 〉 N (see below). Since the Hamiltonian (1) [or (2)] preserves the number of spin- [or particle-] excitations, we need to consider only the zero | 0 〉 ≡ ∏ N j =1 | 0 〉 j and single excitation | j 〉 ≡ ˆ σ + j | 0 〉 [ˆ a † j 0 ] subspace of the total Hilbert space. Then the system initially in state | Ψ$_{in}$ 〉 = α | 0 〉 + β | 1 〉 evolves in time as | Ψ( t ) 〉 = U ( t ) | Ψ$_{in}$ 〉 = α | 0 〉 + β ∑ N j =1 A$_{j}$ ( t ) | j 〉 , where U ( t ) = T exp [ 1 i ¯ h ∫ t $_{0}$H ( t $^{′}$) dt $^{′}$] is the (time-ordered, T ) evolution operator. Apparently, only the states in the single excitation sub-space { | j 〉} evolve in time with the corresponding amplitudes A$_{j}$ ( t ) ≡ 〈 j | U ( t ) | 1 〉 , while the vacuum (or ground) state | 0 〉 remains unchanged. Thus perfect state transfer would be achieved for the amplitude | A$_{N}$ ( t$_{out}$ ) | = 1, provided its phase φ = arg( A$_{N}$ ) is fixed and known, φ = φ$_{0}$ , and therefore can be amended. We may quantify the performance of the scheme by the transfer fidelity F$_{ψ}$ = 〈 ψ | ρ$_{N}$ | ψ 〉 where ρ$_{N}$ ≡ Tr$_{̸}$$_{N}$( | Ψ 〉〈 Ψ | ) = (1 −| β | $^{2}$| A$_{N}$ | $^{2}$) | 0 〉〈 0 | + | β | $^{2}$| A$_{N}$ | 2 | 1 〉〈 1 | + II. THE MODEL I. INTRODUCTION arXiv:1001.2444v1 [quant-ph] 14 Jan 2010