# Last Frontiers in Quantum Information Science Workshop

** Date:**

It’s been a while since I have last written, hasn’t it? Part of the reason is I attended the “Last Frontiers in Quantum Information Science” (LFQIS) workshop in Juneau, Alaska, last month, and was busy doing research and writing my talk. I took some photos (it’s scenic Alaska, after all!); they are available here. Some people were tweeting about the workshop; you can find those at #lfqis.

Unlike other conferences I have attended, LFQIS was very small - only about 20 participants total. This made for an useful social dynamic - it was possible to meet, and chat with, all the attendees. (Unlike the APS March meeting, where it’s impossible to even just meet all of them!) Because the organizers of the conference wanted the content of the talks to be at the “last frontier” of quantum information, there were lots of really interesting topics presented.

The workshop also had a very distinct schedule - there were only four or five talks per day, with plenty of time in between to chat with people. (Personally, I was really grateful for the two hour lunch!) Having lots of time between talks also helped ensure we could abide by the schedule…which I rather appreciated.

Some of the speakers included:

- Christopher Ferrie - Green Sampling: Using Recycled Randomness to Locate Phase Transitions

In Monte Carlo sampling, we generate a list of random numbers according to some sampling scheme. Suppose we have two different problems we are interested in solving using Monte Carlo. Is it possible to use the

samesample for both? In some cases, yes!Chris discussed how green sampling could be used to more efficiently find phase transitions in some problems. For instance, percolation on a regular lattice - depending on the probability of there being an edge between any two vertices, then the probability that there exists a path from one side of the lattice to another exhibits a phase transition. Another example shows up in computing the probability of finding a satisfying assignment in a \(k\)-SAT problem. Depending on the ratio \(\frac{\text{number of clauses}}{\text{number of variables}}\), there is a phase transition in the probability that

someclause is satisfied.

- Christopher Granade - QInfer: Bayesian Inference for Quantum Information

QInfer is a software library developed by Chris (and Chris Ferrie) to help do Bayesian inference. Chris’ talk emphasized the importance of good software development practices in quantum information, and showed examples of how QInfer could be used “in the wild” on problems related to tomography.

Of note is the fact that the

\[\hat{\theta} = \sum_{j}w_{j}x_{j}\hat{x}_{j}\]particle filteringalgorithm allows for fast and efficient updates of priors. Suppose our best estimate of some parameter of interest iswhere the \(x_{j}\) are the

\[w_{j} \rightarrow w_{j}~\mathrm{Pr}(\text{Data}|x_{j})\]hypotheses, and the \(w_{j}\) are the correspondingweights. The particle filtering algorithm updates the weights asup to a normalization. Also, a

resamplingalgorithm is used to ensure that the hypotheses \(x_{j}\) do not become too concentrated.

- Chris Wood - Quantum Plumbing: Gate Characterization in the Presence of Leakage

Characterizing leakage - when your quantum information goes to a part of some Hilbert space over which you do not have control - is a big problem as we build larger quantum information processors. Chris presented some work on building theoretical models for leakage. A key observation is that leakage comes about from a direct sum structure on the Hilbert space. That is, the total (possibly-infinite dimensional) Hilbert space \(\mathcal{H}\) describing our system should be broken down as \(\mathcal{H} = \mathcal{H}_{\mathrm{controlled}} \oplus \mathcal{H}_\mathrm{uncontrolled}\). Leakage occurs when there are dynamics which populate the \(\mathcal{H}_{\mathrm{uncontrolled}}\) space.

As an example, consider a qutrit \(\rho \in \mathcal{B}(\mathcal{H}_{3})\), where \(H_{3}\) is a three-dimensional Hilbert space. Suppose we embed our qubit in the \(|0\rangle, |1\rangle\) subspace, and we have no control over any population that goes into the \(|2\rangle\) state. Then, a quantum channel which models leakage could be written as

\[\mathcal{E}(\rho) = (1-p)\rho + p|2\rangle\langle 2|\]With probability \(p\), we populate the part of the Hilbert space we have no control over. For this channel, repeated applications actually continue to inject population into \(|2\rangle\langle 2|\):

\[\mathcal{E}^{2}(\rho) = (1-p)^{2}\rho + p(2-p)|2\rangle\langle 2|\]Notice that \(\mathrm{Tr}(\mathcal{E}(\rho)) = \mathrm{Tr}(\rho) = 1\), so \(\mathcal{E}\) is trace-preserving. The line above shows us that \(n\) applications of \(\mathcal{E}\) will simply change the coefficient of \(\rho\) to \((1-p)^{n}\); the fact the channel is trace-preserving implies the coefficient of \(|2 \rangle\langle 2|\) will be \(1-(1-p)^{n}\). Thus, after \(n\) applications, we have

\[\mathcal{E}^{n}(\rho) = (1-p)^{n}\rho + (1-(1-p)^{n})|2\rangle\langle 2|\]For very large \(n\), the channel simply maps our original state to \(|2\rangle\langle 2|\). (In other words, our quantum information has gone someplace where we cannot control it!)

- Gemma de las Cuevas - Simple Universal Models Capture All Classical Spin Physics

There are many families of spin models. Gemma’s work showed how it’s possible to reduce a large number of spin models to one

universalmodel. This involved mapping the spin models to satisfiability problems, and some other tricks.Gemma’s work was featured in several news articles. See, for instance, this one from phys.org.

- Richard Kueng - The Clifford Group Fails Gracefully to be a Unitary 4-Design

A group \(G\) is a \(t\)-design for the \(d\)-dimensional unitary group \(U_{d}\) if it’s possible to approximate the \(t^{th}\) moment of random unitaries using the average over the group elements:

\[\frac{1}{|G|}\sum_{X\in G}X^{\otimes t} \otimes (X^{\dagger})^{\otimes t} = \int_{U_{d}}U^{\otimes t} \otimes (U^{\dagger})^{\otimes t}~dU\]where the measure \(dU\) is the Haar measure. Richard showed us how the Clifford Group (which is a unitary 3-design) is “close to” being a 4-design. For more information about \(t\)-designs in quantum information, see this Wikipedia page.

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