Perfect quantum transport in arbitrary spin networks

Spin chains have been proposed as wires to transport information between distributed registers in a quantum information processor. Unfortunately, the challenges in manufacturing linear chains with engineered couplings has hindered experimental implementations. Here we present strategies to achieve perfect quantum information transport in arbitrary spin networks. Our proposal is based on the weak coupling limit for pure state transport, where information is transferred between two end spins that are only weakly coupled to the rest of the network. This regime allows ignoring the complex, internal dynamics of the bulk network and relying on virtual transitions or on the coupling to a single bulk eigenmode. We further introduce control methods capable of tuning the transport process and achieve perfect fidelity with limited resources, involving only manipulation of the end qubits. These strategies could be thus applied not only to engineered systems with relaxed fabrication precision, but also to naturally occurring networks; specifically, we discuss the practical implementation of quantum state transfer between two separated nitrogen vacancy (NV) centers through a network of nitrogen substitutional impurities.

Figure 1

Example of spin network, consisting of NV centers (blue spheres) and P1 centers (red) in a diamond lattice. The network is given by randomly positioned P1 centers in the diamond lattice with a concentration of $0.2$ ppm and a $5\%$ conversion efficiency to NV. The proposed strategies enable perfect quantum state transfer between the two NV spins in this naturally occurring topology of P1 centers.[20, 21]

Figure 2

(a) On-resonance balancing sequence applied to the end spins for perfect quantum transport. Yellow blocks indicate microwave irradiation that brings the end spins on resonance with the bulk network. The resulting $\Lambda$ network is in general unbalanced, but appropriately placed $\pi$ pulses (orange) on the end spin with higher mode overlap (here spin 1) can balance the overlaps $O_{(1,N)}$.[25] (b) The $\pi$ pulses invert the sign of the coupling of spin at node 1 to the bulk mode in the toggling frame such that $O_{1,N}$ become equal on average. For the network of Fig. 1, $|O_1|>|O_N|$ and $r=1/2(1+|O_N/O_1\left|\right)=0.5501$. The sequence is symmetrized[26, 27] and repeated for $L$ cycles.

Figure 3

(a) Off-resonance transport fidelity for the network in Fig. 1 for $\gamma =25$. Perfect transport occurs, but on a slow time scale, an order of magnitude longer than the on-resonance balanced case (Fig. 4). (b) Increasing $\gamma$ improves transport fidelity (red circles) but also increases the time required for perfect transport (blue diamonds).

Figure 4

(a) On-resonance transport fidelity for the network of Fig. 1, with (red) and without (blue, dotted) balancing, for $\gamma =1$ and $L=20$. Almost perfect transport occurs in the balanced case, obtained by the control sequence in Fig. 2. (b) Increasing the number of cycles $L$ improves the Trotter approximation yielding enhanced fidelity. The symmetrized sequence performs better than the sequence without symmetrization.

Figure 5

(a) The control sequence for robust quantum transport under dephasing noise consists of $L$ cycles of a balanced sequence similar to Fig. 2, but constructed out of decoupling blocks $D\left(\tau \right)$. Here ${\tau }_1=r{t}_m/2L$ and ${\tau }_2=(1-r)t_m/2L$. Panel (b) details the decoupling block $D\left(\tau \right)$, consisting of $M$ cycles. The ${\pi }_x$ pulses are applied to both the end spins, and the bulk network. They are accompanied by a toggling of the phase of the microwave irradiation used to bring the end spins in resonance with the bulk mode. The sequence filters out any dephasing noise field that has a correlation time greater than $\tau /M$; and by appropriately choosing $M$, the transport can be made immune to noise.

Figure 6

Time for optimal transport in dipolar coupled spin networks of P1 centers in diamond. The optimal time was calculated from the average over 5000 random lattice realizations of P1 centers of density 10 ppm; we considered transport between two NV centers located at increasing distance. The red line (circles) is the time required to achieve a transport fidelity of at least $99\%$ via resonant balancing. The black line (diamonds) is the time at which optimal fidelity is achieved for on-resonant transport without balancing. The fidelity is however quite low ($\sim 15\%$) in this case. The blue dashed line shows the number of P1 centers in the network, $N-2$.

Figure 7

A general detuned $\Lambda$ network with multiple $\Lambda$ paths between the end spins. Each leg of any of these $\Lambda$ paths may have an arbitrary detuning. For the representative case of the $\Lambda$ path going through node $j$ (red, thick), these detunings are ${\delta }_{1j}$ and ${\delta }_{jN}$.

Figure 8

Positive frequencies of transport fidelity for a $\Lambda$ network (Fig. 7) consisting of two $\Lambda$ paths, $1\to j\to N$ and $1\to k\to N$. The frequencies are plotted as a function of the relative detuning between the paths, ${\Delta }_{jk}^2=({\delta }_{1j}{\delta }_{kN}-{\delta }_{jN}{\delta }_{1k})$. The actual transport also contains symmetric negative frequencies and a dc (zero frequency) component. In general there are four frequencies of transport, derived in Eq. (B6). Note that when ${\delta }_{jN}/{\delta }_{1j}$ is a constant for both paths, there are only two frequencies, $S$ and $2S$, that carry the transport.

Figure 9

Figure shows the scaling with network size $N$ of the matrix norms and largest eigenmodes of the adjacency matrices corresponding to (a) a random network and (b) a dipolar random network. The solid lines are average values obtained from 100 manifestations of the networks. The dashed lines are theoretical results. For the dipolar network, the largest eigenmode $E_{\text{max}}$ approaches a constant $1.6/d^3$ (dashed magenta line).