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In discussions on many quantum algorithms especially related to quantum walks, I have seen the term "perfect state transfer" used to describe some property apparently related to the periodicities of the walk/algorithm, but I cannot quite grasp the significance of the term, or why it would be useful concept to consider.

For example using Grover's algorithm to find a single marked state by applying iterations to a state initially prepared on the uniform superposition over all input vectors will not necessarily reach back to the uniform superposition, thus there may not be perfect state transfer with Grover's iteration. (I think).

Is there a succinct way to understand "perfect state transfer", and why this is an interesting concept to study?

The Wikipedia article is not bad, but I suspect there might be more to be said..

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Perfect state transfer is generally discussed in the context of continuous time evolution. For example, you might be evolving under the influence of a Hamiltonian $H$. Particularly when one is considering some sort of underlying graph structure, you probably prepare an initial state corresponding to a single vertex (perhaps $|1000\ldots 0\rangle$) and you're aiming to reach another vertex (e.g. corresponding with a marked item). Let's call these vertices $u$ and $v$. The perfect transfer condition in this context is then just $$ \exists t:|e^{-iHt}_{v,u}|=1. $$ In other words, if you start at vertex $u$, then after time $t$ you will arrive perfectly at vertex $v$.

Note that if you're doing Grover's in this continuous time case, there aren't different iterations. It's generally just a single Hamiltonian that's switched on, and you can switch it off whenever you want. So, you can get an exact solution for Grover's.

There's a sense in which you might argue that this isn't an interesting thing to study: in real-world devices, there will always be errors. So it's irrelevant if there's a "perfect in theory" solution if, in practice, there are imperfect solutions that are less susceptible to errors (e.g. by being faster).

But where it's useful is precisely for the theoretical studies as the perfectly transferring cases are mathematically much easier, help to shape ones intuition, and set you on the path of being able to deal with other cases.

For my part, I've worked a lot on perfect state transfer in a slightly different context, where it essentially provides a long-range swap built out of nearest-neighbour interactions. (But you can also map the basic structure onto all sorts of other problems.) The motivation (roughly) is that this is the natural "language" to build circuits out of if your device interacts via a nearest-neighbour Hamiltonian, and should therefore give you a better way of doing things compared to the circuit model, for example. Compared to multiple consecutive swap gates, for instance, the perfect state transfer Hamiltonians in 1D can achieve the transfer of a quantum state over long range in half the time, which could make a major difference in real-world devices.

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  • $\begingroup$ Is it correct to say that "perfect state transfer" appears to be a local property of the evolution, e.g. local to a specific pair of vertices/pair of qubits? However "periodicity" may be more of a global property of the graph? Are both properties reflected in the spectral decomposition of $H$? $\endgroup$
    – Mark Spinelli
    Apr 26, 2021 at 21:10
  • $\begingroup$ Periodicity can also be local to a specific vertex. Yes, both are about spectral properties. For periodicity, you just require that all the eigenvectors with support on a given vertex have (up to an overall scale factor)eigenvalues with integer gaps $\endgroup$
    – DaftWullie
    Apr 27, 2021 at 6:41
  • $\begingroup$ Perfect transfer has slightly more detailed conditions. The weights of the eigenvectors on both sites must be equal, meaning the only difference in the amplitudes is the relative phase. The eigenvalues then have to generate that relative phase. Often, the structure of your Hamiltonian might mean that the phase differences are $\pm 1$, in which case your spectrum (roughly speaking) has to be even or odd integers respectively (up to a global offset, and an overall scale factor) for those eigenvectors with support on the two vertices. $\endgroup$
    – DaftWullie
    Apr 27, 2021 at 6:41
  • $\begingroup$ Ahh! In the context of perfect sate transfer, a quantum computer having $n$ qubits with a certain (say nearest-neighbor) layout/topology is thought of as a graph having $n$ vertices, with edges corresponding to the specific layout. I was initially thinking of this as a graph having $\mathcal{O}(2^n)$ vertices, with edges determined by oracle access. $\endgroup$
    – Mark Spinelli
    Apr 27, 2021 at 19:14
  • $\begingroup$ Yes, this is generally the case, typically because in this context one thinks about an "excitation preserving" Hamiltonian, meaning that the graph of all basis elements splits up into (at least) $n+1$ distinct graphs, one of which is the qubit topology. $\endgroup$
    – DaftWullie
    Apr 28, 2021 at 6:42

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