Quantum tinkerer (Posts by Anton Akhmerov)

How to estimate the number of false positives

Anton Akhmerov — Wed, 10 Aug 2022 08:00:00 GMT

Prelude: the topological gap protocol preprint¶

In arXiv:2207.02472 Microsoft Quantum reported validating and then applying the topological gap protocol. I summarized my overall opinion about it in a commentary. However, even after finishing the commentary, I was confused about how the manuscript estimates the chance that they found false positives. Because I am not trained in statistics, I've used this opportunity to learn how to make such an analysis properly. Here's what I found.

I need your data, your code, and your DOI

Anton Akhmerov — Sat, 05 Jan 2019 14:00:00 GMT

I love open science. Since you are reading a scientific blog, I believe it is likely that you also support many of open science ideas. Indeed, easy access to publications, code, and research data makes research easier to reuse, while also ensuring transparency of the process and better quality control. Unfortunately the academic community is extremely conservative and it just takes forever for new standards to become commonplace.

There is however one easy way to help … (3 min remaining to read)

In the footsteps of Einstein

Anton Akhmerov — Sat, 14 Apr 2018 19:04:18 GMT

I teach the undergraduate solid state physics course, where we just switched to a shiny new book "Oxford Solid State Basics" by Steve Simon.

Steve's story of condensed matter physics starts with the heat capacity of solid materials. It's a great way to dive into how quantum mechanics combines with lucky guesses to improve our understanding of what is happening. It is also what we do in our course.

A great source of experimental data showing the problem is Einstein's original work, and Steve's book reproduces the plot from Einstein. (See also the English translation) Unfortunately that plot belongs to the current publisher of Annalen der Physik and cannot be republished under a free license. So in order to provide this data in the lecture notes and to make it available to whoever wants, I decided to take the original data Einstein has and repeat the exercise. Because we are living in an enlightened age, I also wanted to see if the more advanced Debye model would be any better for Einstein's data.

Time to get hands dirty … (5 min remaining to read)

Bad professional advice

Anton Akhmerov — Thu, 29 Jun 2017 18:00:00 GMT

I wrote a column for the newsletter of our institute. Since I liked the result, I'm also reposting it below.

As a child I had a book "Bad advice" that contained nothing but poems suggesting you to do what you should really never do. So here is my bad professional advice (except that I won't risk making poetry):

Always remember: … (2 min remaining to read)

Connecting the dots

Anton Akhmerov — Wed, 31 May 2017 14:20:12 GMT

Why do spectrum plots look ugly?¶

Very often when we compute the spectrum of a Hamiltonian over a finite grid of parameter values, we cannot resolve whether crossings are avoided or not. Further if we only compute a part of the spectrum using e.g. a sparse diagonalization routine, we fail to find a proper sequence of levels.

Let us illustrate these two failure modes.

# Just some initialization

%matplotlib inline
import numpy as np
from scipy import linalg
from scipy.optimize import linear_sum_assignment
import matplotlib
from matplotlib import pyplot

matplotlib.rcParams['figure.figsize'] = (8, 6)

def ham(n):
    """A random matrix from a Gaussian Unitary Ensemble."""
    h = np.random.randn(n, n) + 1j*np.random.randn(n, n)
    h += h.T.conj()
    return h

def bad_ham(x, alpha1=.2, alpha2=.0001, n=10, seed=0):
    """A messy Hamiltonian with a bunch of crossings."""
    np.random.seed(seed)
    h1, h2, h3 = ham(n), ham(n), ham(n)
    a1, a2 = alpha1 * ham(2*n), alpha2 * ham(3*n) * (1 + 0.1*x)
    a2[:2*n, :2*n] += a1
    a2[:n, :n] += h1 * (1 - x)
    a2[n:2*n, n:2*n] += h2 * x
    a2[-n:, -n:] += h3 * (x - .5)
    return a2

xvals = np.linspace(0, 1)
data = [linalg.eigvalsh(bad_ham(x)) for x in xvals]
pyplot.plot(data)
pyplot.ylim(-2.5, 2.5);

This is mock data produced by a random Hamiltonian with a bunch of crossings. We know that some of these apparent avoided crossings are too tiny to resolve, and should instead be classified as real crossings.

Let's now simulate what would happen if we also use sparse diagonalization to obtain some number of eigenvalues closest to 0.

truncated = [sorted(i[np.argsort(abs(i))[:13]]) for i in data]
pyplot.plot(truncated);

The ugly jumps are not real, they appear merely because some levels exit our window and new ones enter.

A desperate person who needs results right now at this point replots the data using a scatterplot.

pyplot.plot(truncated, '.');

This is OK, but at the points where the lines are dense our eye identifies vertical lines, making the plot harder to interpret.

Can we do better? … (5 min remaining to read)