February 2019 Archive

This is an archive of what I found useful, interesting, or simply enjoyable from February 2019.

Research

Quantum state verification

• Why are amplitudes complex?, Shtetl-Optimized (An update on Scott’s personal conundrum)

  In the quantum foundations community, it is popular to derive the complex amplitudes (as opposed to real or quaternionic) from the “local tomography” principle ¹ which Scott has never found intuitive. Now he stumbles upon the following question:

  suppose we have a two-outcome measurement $E$ (let’s say, on $n$ qubits), and suppose it accepts every product state with the same probability $p$. Must $E$ then accept every entangled state with probability $p$ as well? Or, a closely-related question: suppose we know $E$’s acceptance probabilities on every product state. Is that enough to determine its acceptance probabilities on all $n$-qubit states?

  The answer is that, yes, the acceptance probabilities are determined and, in fact, constant for all states. This follows from a result in the Braunstein et al 1999, specifically that there is always a ball of constant size around the maximally mixed state, all states in which are separable. This is in fact true only if amplitudes are complex.

  However, to disprove the statement “the acceptance probabilities are constant for all states” when it is false requires knowing the acceptance probabilities on product states to exponential accuracy. A non-example of the statement to illustrate this: the measurement projecting an $n$-qubit state onto $n/2$ Bell pairs accepts every product state with exponentially small probability (but the probabilities are different for different product states) even though it accepts $n$ Bell pairs with probability 1.

  Quotable: “quantum mechanics over the quaternions is a flaming garbage fire”.

• Single-copy entanglement detection, Dimić and Dakić 2018 (The idea of a test in which the acceptance probability for product states falls off exponentially can be used to detect entanglement.)

• Optimal verification of entangled states with local measurements, Pallister, Linden, and Montanaro 2018

  Verification “is the state $\ket{\psi}$?” is easier than tomography. Given a state $\sigma$, we want to know if it is “closed” to $\ket{\psi}$ so that we can accept or otherwise reject it. This can also be phrased as a promise problem: either the given state $\sigma$ is the true state $\ket{\psi}$ or $\av{\psi

  \sigma

  \psi} \le 1 - \epsilon$. The quantity $1 - \epsilon$ is the (test-independent) worst case fidelity that we’re willing to accept.

  The authors consider a binary hypothesis testing that disallows type I error (rejecting the true state) while having a minimum type-II error probability $\delta_n$, possibly depending on $n$. Their argument goes as follows: suppose that the test has the probability $p$ of accepting the true state and $p-\Delta_{\epsilon}$ of accepting a wrong state. Then our task is to distinguish between Bernoulli trials drawn from i.i.d. distributions ${p,1-p}$ and ${p-\Delta_{\epsilon}, 1-(p-\Delta_{\epsilon})}$. The Chernoff-Stein lemma says that \(\lim_{n\to\infty} \frac{\delta_n}{n} = - D(p||p-\Delta_{\epsilon}),\) where $D(Q||P)$ is the relative entropy \(D(Q||P) = \sum_x Q(x) \log \frac{Q(x)}{P(x)}.\) Rearranging the equation, we get a bound on the number of samples \(n \ge \frac{\ln \delta^{-1}}{D(p||p-\Delta_{\epsilon})}.\)

  They argue that the “worst case” scaling of $D(p

  p-\Delta_{\epsilon})$ is quadratic in $\Delta_{\epsilon}$ because infinitesimally the relative entropy is just the Fisher information metric, which scales as the square of the distance $\sum_x \Delta^2{\epsilon}/4p$ 2, whereas setting $p=1$ gives you a linear scaling across all values of $\Delta{\epsilon}$. (Small $\Delta_{\epsilon}$ makes it hard to distinguish the two distributions but we probably expect it to be small in practice, both in adversarial and non-adversarial scenarios.)

  Given all that, the task of finding an optimal verification scheme is reduced to minimizing the probability of accepting the “worst” wrong state (easiest to have us fooled) \(\min_{\Omega} \max_{\substack{\sigma \\ \av{\psi|\sigma|\psi} \le 1 - \epsilon}} \Tr (\Omega \sigma) := 1-\Delta_{\epsilon}\) over the “test operator” $\Omega = \sum_j \mu_j P_j$, a convex sum of all the tests $P_j$ with eigenvalue 1 for the target state. The (test-dependent) $\delta$ is no worse than the worst case error $(1-\Delta_{\epsilon})^n$, thus giving \(n \ge \frac{\ln \delta^{-1}}{\ln (1-\Delta_{\epsilon})^{-1}} \approx \frac{\ln \delta^{-1}}{\Delta_{\epsilon}},\) where we have assumed $\Delta_{\epsilon}$ to be small again.

  The following paper shows that $\Delta_{\epsilon}$, which determines the efficiency of the test, is linearly related to the worst case fidelity $\epsilon$, the proportionality constant being the gap between two largest eigenvalues of $\Omega$. The paper also explores the adversarial scenario, in which case all eigenvalues matter, not just the second largest one. All in all, I need to read more of it.

• Efficient verification of pure quantum states with applications to hypergraph states, Zhu and Hayashi 1806.05565

  The Pallister paper gives the optimal local measurement for all two-qubit pure states and optimal stabilizer measurements for all $n$-qubit stabilizer states: $\Delta_{\epsilon}\approx \epsilon/2$ for measuring all stabilizer operators and $\epsilon/n$ for measuring only a set of stabilizer generators.

• Validating and certifying stabilizer states, Kalev and Kyrillidis 1808.10786

  Direct fidelity estimation is great because it doesn’t matter how many qubits you want to verify, the number of samples stay the same. However, it is also not great by the same reason; even though all we have now are on the order of 10-100 qubits, you still need the same number of samples as if you have a thousand of qubits.

  The problem considered by the current paper is in a sense opposite to that in the Pallister paper. in the Pallister paper, we are given the worst case fidelity $\epsilon$ and are asked to find an optimal measurement, whereas in this paper we are given a (stabilizer) measurement and are asked for the worst case fidelity $1-\epsilon$ to a desired stabilizer state. The latter can be done with $n^3\ln(2\delta^{-1})/2r^2$ samples, where $r$ is the radius away from $1-\epsilon$.

Representation theory

• Displacement operator by beam splitter, Paris 1996

  How to implement a displacement operator (a Heisenberg group element) \(D(\alpha) = \exp(\alpha b\dgg - \alpha^* b)\) on an arbitrary quantum state using an auxillary laser mode and a beam splitter (a symplectic group element)? The beam splitter unitary \(\mathrm{BS} = \exp \left[ i \tan^{-1} \sqrt{\frac{1-\tau}{\tau}} (ab\dgg + a\dgg b) \right]\) looks almost like a displacement operator, if only we could replace $a$ by $\alpha$ and flip the sign. Here comes a cool trick: notice that $ab\dgg$ and $a\dgg b$ are respectively the raising and lowering operator in $\sl(2)$ made out of Schwinger’s bosons. ($a\dgg a - b\dgg b$ is $Z$.) So I can do the Gauss decomposition to move the exponentials of $ab\dgg$ and $a\dgg b$ to the left and the right, where $\bra{z}$ and $\ket{z}$ are waiting to turn $b$ into $z$. This gives (barred some mistake) \(\av{z|\mathrm{BS}|z} = \exp (\beta a\dgg a/2) \otimes \exp (\xi z^* b) \exp(-\beta b\dgg b/2) \exp (-\xi^* zb\dgg)\) Then rotate the linear term by the quadratic Hamiltonian $\beta b\dgg b/4$ before taking the appropriate limits to obtain the result.

• Representation Theory and Complex Geometry, Chriss and Ginzburg

• The Borel-Weil theorem in Peter Woit’s lectures (Old lectures 13-16)

  Associated to each weight $\lambda$ under a maximal torus $T$ is a line bundle $L_{\lambda}$ over the base space $G/T$ (the line is the one-dimensional irrep $e^{\lambda(H)}$ of $T$, thinking of the weight $\lambda$ as a linear functional on $\h$) and the space of sections $\Gamma (G/T,L_{\lambda})$.

  The Borel-Weil theorem gives every irrep of a semisimple Lie group (or an algebraic group) over $\C$ a concrete realization as a subspace of $\Gamma (G/T,L_{\lambda})$. As an induced representation, $\Gamma (G/T,L_{\lambda})$ has the isotypic decomposition \(\Gamma(G/T,L_{\lambda}) = \bigoplus_{\mu} V_{\mu} \otimes (V_{\mu}^*)_{-\lambda},\) where $(V^_{\mu})_{-\lambda}$ is the $-\lambda$-weight space of $V_{\mu}^$. (The minus sign comes about because we’re acting on $V^_{\mu}$.) We could single out the irrep $V_{\lambda}$ by putting in by hand the condition that $-\lambda$ has to be a lowest weight, or equivalently, $\lambda$ has to be a highest weight ³. What is beautiful about the Borel-Weil theorem is that this condition has a simple meaning in complex geometry (no pun intended): the sections have to be *holomorphic for the following reason. The tangent space of $G/T$ at the identity is isomorphic to the even-dimensional Lie algebra of negative and positive roots consisting of holomorphic and antiholomorphic vectors: \(\begin{align} E_{\alpha} = \frac{\partial}{\partial z_{\alpha}}, && E_{-\alpha} = \frac{\partial}{\partial \overline{z}_{\alpha}}. \end{align}\) But $-\lambda$ is a lowest weight iff the corresponding section $f$ is annihilated by $E_{-\alpha}$ for all positive roots $\alpha$. In other words, $f$ is a holomorphic function.

  The $\SU(2)$ example is given in Lecture 16, in which case the space of sections are homogeneous polynomials of degree $k$ in two variables, a standard textbook result.

• MO threads on complex and Kählerian flag manifolds (Why are flag manifolds important to physicists? They exhaust all the possibilities of compact homogeneous Kähler manifolds associated with $G$ connected, compact, and semisimple.)
  • Flag Manifolds, Alekseevsky 1996 (11th Yugoslav Geometric Seminar)
• Trying to trace the history of $G/T \cong G^{\C}/B$
  • Kählerian Coset Spaces of Semisimple Lie Groups, Borel 1954
  • Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts, Serre 1954
  • Lie Groups, Duistermaat and Kolk 2000

Media

Links

• A game you play as a paperclip maximizer
• Are we on the road to civilization collapse?, BBC
  • Reconciling taxon senescence with the Red Queen’s hypothesis, Žliobaitė, Fortelius and Stenseth (Random walk with resource constraint often produces the observed unimodal trajectory of civilization growth)
• Dissolving the Fermi paradox, Sandberg, Drexler and Ord [Sandberg himself] [SSC] [LW]
• GPT-2: Language Models are Unsupervised Multitask Learners, Radford et al. 2019 [OpenAI blog]
  • An AI helped us write this article, Vox
  • Do Neural Nets Dream of Electric Hobbits?, SSC
  • GPT-2 as Step Toward General Intelligence, SSC
  • OpenAI Trains Language Model, Mass Hysteria Ensues
• The history of human-cat co-existence, ThoughtCo.
• Manfred Eigen of the ultrafast-chemical-reaction and quasispecies fame passed away.
• All Impossible Foods partner restaurants will have the meat-free burgers 2.0 that taste like real meat by mid-March.
• Nonfiction Writing Advice, SSC

• Thailand’s debate on paraquat ban: [The Momentum] [BIOTHAI] (ภาษาไทย)

  There seems to be an exaggeration regarding paraquat contamination in soil and water. This doesn’t undermine arguments for banning paraquat, but we should be mindful not to oversimplify research findings in order to support our position.

  • Brian Clegg, Chemistry World (“environmentally-based decision is anything but black and white”)
  • A Pesticide Information Project, 1993
  • Environmental fate and ecotoxicology of paraquat: a California perspective, Sartori and Vidrio 2018
• Spiral patterns in hallucinations reflect physiological structure in the visual system, PBS math
• This Waifu Does not Exist (The anime version of This Person Does not Exist, a StyleGAN face generator)
• The Yoda of Silicon Valley, NY Times

• Augmenting long-term memory, Nielsen

  Anki isn’t just a tool for memorizing simple facts. It’s a tool for understanding almost anything.

  • You should read 10 most important papers of a field to pick up open problems, questions that you’d personally like answered, tricks, curious observations that seem important, and sometimes even field-wide blind spots.
  • Avoid Ankify intellectually interesting but emotionally dried facts that are so removed from everything else you know (“orphan questions”), which will feel pointless to remember after a while. Find something related to personal projects.

Meditation

• The neuroscience of meditation: classification, phenomenology, correlates, and mechanisms, Brandmeyer, Delorme and Wahbehdf 2019
• On the Relationship Between the Practice of Mindfulness Meditation and Personality—an Exploratory Analysis of the Mediating Role of Mindfulness Skills, van den Hurk et al. 2011

Anime

• A Carnival Phantasm Touhou parody

Physics/quantum

• Ink propulsion engine

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• Linking plasma formation in grapes to microwave resonances of aqueous dimers, Khattak, Bianucci and Slepkov 2019 (Two grapes touching at a point create a plasma ball inside a microwave because wet, grape-sized objects sustains and amplifies standing microwaves, focusing the energy into a single point.)
• Daniel Harlow’s Facebook discussion on the NY Times piece by Sabine Hossenfelder. I agree with one of the comments that there are at least two social consequences of not building a next-generation collider that are tangled together (1) punishing theorists (because science is not self-correcting) for not learning from their mistakes and hyping a next-generation collider up for a wrong reason (2) punishing experimentalists for the mistakes made by theorists (which we don’t want).
• Someone wrote a blog post on what he did to get a job in quantum computing.
• Andy Matuschak and Michael Nielsen recruit alpha testers for spaced-repetition-based “Quantum Computing for the Very Curious”
• Find your ideal quantum interpretation.
• Quantum Game Jam at Helsinki
• QIP 2019 videos are now online.
• Pan Jianwei in MIT Techonology Review: The China Issue

• Quantum cryptanalysis of symmetric, public-key and hash-based cryptographic schemes, Vlad Gheorghiu and Michele Mosca

  Roughly speaking, the physical footprint $nq$ is a space resource and the quantum security parameter $qs$ is a time resource. $nq$ is the number of qubits per CPU parallelizing Grover’s search, the best known quantum attack against symmetric cryptographic protocols and hash functions. $qs$ in this case is the number of surface code cycles, assuming that fault tolerence is achieved via the surface code.

  Breaking RSA and ECC schemes in 24 hours may require at least ten million qubits, $10^{11}$ $T$ gates, and $10^{13}$ cycles. Symmetric schemes may require $>2^{100}$ cycles to break.

Musics

• March of the Spirits, Imaginary Force, Bob Katsionis
• Marble Machine, Wintergatan

Finnish power metal

• Sonata Arctica
  • UnOpened / Destruction Preventer, Ecliptica 1999
  • Weballergy / San Sebastian / Black Sheep / Wolf & Raven, Silence 2001
  • The Cage, Winterheart’s Guild 2003
• Stratovarius
  • Hunting High and Low, Infinite 2000
  • Eagleheart, Elements Pt. 1 2002
  • Father Time, Episode 2010

Similarity/homage

• Rising Force, Yngwie Malmsteen {Rising Scarlet (東方緋想天), THUNDER CONCERTO (C75), IRON ATTACK}
• Back in the Village, Iron Maiden {Suck a Sage, Ishiwatari Daisuke}

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• 2019 7
• 2017 1