In a computational architecture in which it is necessary to transport quantum states over long distances, but the available inter-qubit interactions are only local, we might try to address how to minimise an experimenter's interaction with their system, allowing for a reduction in the incident errors. Clearly, no interaction with the system is the absolute limit of this, although, in exchange, we must carefully manufacture a specific system in advance. We call this task quantum state transfer.

In general, we consider a graph $G$ where the vertices, $V$, are associated with the qubits of our system and the edges, $E$ describe possible interactions. The distance, $d(A,B)$ between a pair of vertics $A$ and $B$ is the minimum number of edges that one has to traverse to get from one to the other.

The basic question is whether, for a particular pair of vertices $A$ and $B$, there exist initial and final states $\ket{\Psi_{IN}}$ and $\ket{\Psi_{OUT}}$, a time $t$ and a Hamiltonian $$ H=\sum_{(i,j)\in E}h_{ij} $$ such that $$ e^{-iHt}\ket{\psi}_A\ket{\Psi_{IN}}_{V\setminus A}=\ket{\psi}_B\ket{\Psi_{OUT}}_{V\setminus B} $$ independent of the arbitrary single qubit state $\ket{\psi}$. The Hamiltonian will be fixed in time, the idea being that we can put all of our experimental effort into making this with high precision just once, and we can check that it has been correctly implemented before its use in a computation. Other that reading in and out the state to transfer, we don't need to interact with it.

We typically make some simplifying assumptions as to the nature of the Hamiltonian and the initial states. In particular, we fix $$ \ket{\Psi_{IN}}=\ket{0}^{\otimes(N-1)} $$ where $N=|V|$ is the number of qubits in the system. We also choose the Hamiltonian to be excitation preserving, i.e. $$ \left[H,\sum_{n=1}^NZ_n\right]=0, $$ where $Z_n$ is the Pauli $Z$ matrix applied to qubit $n$. Such a choice immediately implies that $\ket{\Psi_{OUT}}=\ket{\Psi_{IN}}$. It also means that the dynamics are entirely determined by $H_1$, the representation of $H$ in the single excitation subspace, which is an $N\times N$ matrix spanned by the basis elements $$ \ket{n}=\ket{0}^{\otimes (n-1)}\ket{1}\ket{0}^{\otimes N-n} $$ rather than the full $2^N\times 2^N$ matrix of $H$. Common choices for the terms $h_{ij}$ are either $$ \frac{1}{2}(X_iX_j+Y_iY_j) $$ or $$ \frac{1}{2}(X_iX_j+Y_iY_j+Z_iZ_j). $$ These are known as the XX and Heisenberg models respectively, and $H_1$ in these cases maps either to the adjacency matrix or laplacian of the graph respectively. Furthermore, weights $J_{ij}$ can be introduced on each term, which correspond to weighting the edges in $H_1$.

There are subsequently a number of properties that you might like to introduce:

• How can one optimize the trade-off between transfer distance, $d(A,B)$, and transfer time, $t$?
• What is the maximum rate of transfer? i.e. Are we limited to sending one state, waiting $t$ for it to arrive, reinitialising the system, and starting again, or can we send states more often?
• Is it possible to route states? i.e. have several different locations at which the state can arrive, and somehow choose which one the state arrives at. If so, what are the trade-offs betwen the number of parties that one can route to and the minimum distance between a sender and any receiver?
• Are there any network topologies that are particularly robust to noise? Studies of circulant graphs for perfect state transfer have been motivated by the fact that these present nice robustness results in the classical regime, but nobody has yet demonstrated any such results in the quantum regime (It has recently been proven that circulant graphs cannot perfectly transfer states over distances greater than 2, so they're probably not that interesting to study.).
• In uniformly coupled graphs, what is the optimal trade-off between transfer distance and the width of the graph?

One further issue that we are just starting to understand is, when moving away from the perfect transfer scenario, what influence noise has on state transfer. While research continues for the best way to maintain coherence even in the presence of noise, it seems that noise can even enhance transfer speeds in some networks, particularly those that would otherwise have symmetries that might prevent transfer. This seems to be especially relevant in describing processes such as energy transport in photosynthesis.

One class of graphs that is particularly well studied are the one dimensional chains. If the chain is uniformly coupled, then only lengths $N=2,3$ can achieve perfect transfer. Beyond that, one has to engineer the coupling strengths so that the following conditions are satisfied:

• The coupling strengths $J_{i,i+1}$ must be symmetric, $J_{i,i+1}^2=J_{N-i,N+1-i}^2$.
• The eigenvalues $\lambda_n$ of $H_1$, when ordered in increasing value must have (up to a scale factor) odd integer gaps.

Moreover, these are not just conditions for checking whether perfect state transfer occurs, but also present a method for finding suitable coupling strengths - if you provide a sequence of suitable eigenvalues, one can solve an inverse eigenvalue problem to find the $J_{i,i+1}$. Obviously, for end-to-end transfer on a chain, this gives us the maximum achievable transfer distance. Moreover,

• an end-to-end transfer chain also transfers between any pair of qubits $i$, $N+1-i$.
• for the coupling scheme $J_{i,i+1}=\sqrt{i(N-i)}$, the transfer is the fastest possible.
• perfect transfer remains possible even if $\ket{\Psi_{IN}}$ is replaced by an arbitrary unknown state.
• routing is impossible without active intervention - if the state $\ket{\psi}$ can never arrive perfectly at any site other than $i$ or $N+1-i$ if it started at $i$ (assuming $J_{i,i+1}\neq 0$).
• No perfect transfer schemes are known with a rate higher than $N^{-1}$, although if perfect transfer schemes are used approximately, rates $N^{-1/2}$ can be achieved.

Much less is known about perfect state transfer on more general network topologies, although some statements can be made:

• If $H_1$ is real, routing is impossible without active intervention.
• The trade-off between number of routing sites and transfer distance can be bounded, but it is not known whether these bounds can be saturated in general.
• The trade-off between the transfer rate and the transfer distance can be bounded, but it is not known whether these bounds can be saturated in general.
• The best example of a uniformly coupled network for perfect transfer (in terms of the graph size/transfer distance relation) is the hypercube. Transfer over a distance $d$ can be achieved for a graph of width and size of $O(2^d)$.

We also know how to recognise a perfect transfer network. We require that the eigenvectors of $H_1$, $\ket{\lambda_n}$ satisfy $$ |\braket{\lambda_n}{A}|=|\braket{\lambda_n}{B}| $$ (if there is degeneracy, there needs to exist a resolution to the degeneracy such that this is satisfied) and, if $$ \theta_n=\text{Arg}\left(\frac{\braket{\lambda_n}{B}}{\braket{\lambda_n}{A}}\right), $$ then we need there to exist a $t$ and $\phi$ such that $$ e^{-i\lambda_nt}=e^{i(\theta_n+\phi)}. $$ Unlike chains, this does not translate back to providing a way to design networks. On the other hand, if one reintroduces a limited amount of control, one can easily design systems that allow high rates of transfer, and routing simulatneously in large, periodic networks.

Most of the details of state transfer in chains can be found in the following review article:

1. A. Kay, A Review of Perfect State Transfer and its Application as a Constructive Tool, Int. J. Quantum Inf. 8, 641 (2010) arXiv

The results alluded to on more general networks (without control) can be found here

2. A. Kay, The Basics of Perfect Communication through Quantum Networks, Phys. Rev. A 84, 022337 (2011) arXiv

If one allows a small amount of control, the range of networks that are useful for state transfer is massively enhanced, and routing also becomes possible.

3. P. J. Pemberton-Ross, A. Kay, and S. G. Schirmer, Quantum Control Theory for State Transformations: Dark States and their Enlightenment , Phys. Rev. A 82, 042322 (2010) arXiv
4. A. Kay and P. J. Pemberton-Ross, Computation on Spin Chains with Limited Access, Phys. Rev. A 81, 010301(R) (2010) arXiv
5. P. J. Pemberton-Ross and A. Kay, Perfect Quantum Routing in Regular Spin Networks, Phys. Rev. Lett. 106, 020503 (2011) arXiv