## Quantum State Transfer

This page contains a technical description relating to perfect quantum state transfer on networks. For a more introductory article on the setting and motivations, see here.

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?

- 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.

- 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.

- 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 $O(2^d)$.

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

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

- 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 - A. Kay and P. J. Pemberton-Ross,
*Computation on Spin Chains with Limited Access*, Phys. Rev. A 81, 010301(R) (2010) arXiv - P. J. Pemberton-Ross and A. Kay,
*Perfect Quantum Routing in Regular Spin Networks*, Phys. Rev. Lett. 106, 020503 (2011) arXiv