Quantum Trick or Treat

NOAA src

Lesley Smith is a climate researcher at the University of Colorado’s Cooperative Institute for Research in Environmental Sciences. She is also associated to the National Oceanic and Atmospheric Administration’s Physical Science Laboratory and has worked on behalf of several other national organizations. Her PhD was in particle physics at the University of Kansas, and she incorporates quantum physics into some of her non-nonfiction writings.

Today we discuss a quantum walk on a simple graph with behavior wild enough to inspire at least the latter kind of writing.

Smith was most recently second author on a paper in the April, 2021 Journal of Climate whose title immediately speaks a vital subject: “Explaining the Spatial Pattern of U.S. Extreme Daily Precipitation Change.” The years-long drought in the US West has critically depleted lakes and reservoirs and rivers. But just one week ago, much of California had its wettest day ever. Both the drought and the pattern of extremes have spread across the country, as many easily-found stories this year attest. The question is how to distinguish standard fluctuations from deep-seated causal factors.

As for her other writings, this is representative:

Amazon page

The first lines are likewise topical: “Despite it being October, the sun felt warm on my skin. Forest-fire smoke from states west of Colorado gave the air a three-dimensional quality…” She has also written novels titled Quantum Murder, The Quantum Cop, and Quantum Juneteenth, plus just now a short story, “Lucky Halloween,” about a quantum computer scientist. She is writing a series called Kat Cubed—well we will raise Schrödinger cat-walks to powers well beyond cubed below.

Climate and Dynamism

The climate paper’s opening paragraphs review the the general prediction of greater precipitation from the thermal component of climate change and the effects of changes in atmospheric currents. Then it observes:

Yet, the regional pattern of observed heavy precipitation trends across the United States departs from this theoretical expectation, being remarkable for its heterogeneity rather than the more uniform structure that thermodynamic effects alone would predict. … The regionally diverse trends in extreme precipitation are likely related to dynamical factors.

The paper speaks to care needed to distinguish causes and draws two conclusions:

First, the absence of appreciable increases [in annual maximum single day of rain] over the West is attributed to a dynamical effect of the observed centennial-scale trends in [sea-surface temperature], sea ice, and atmospheric composition. … A second factor is sampling variability […from…] a combination of moderate forced increases comingled with a strong articulation of internal atmospheric variability.

The nub of the latter from my perspective is distinguishing what we can infer on the basis of increased dynamism alone. In our post seven years ago on global warming, I opined that greater energy and dynamism should be treated as more-primary effects than temperature. The challenge of inferring further effects from dynamism is recognized by another excerpt from the conclusions:

Recognizing the potency of unforced atmospheric dynamics, [one cannot discount] a scenario in which internal variability could mute if not reverse the observed upward trend in eastern U.S. [extreme rains].

In the rest of this post, we’ll offer an example of wild dynamics from a small quantum system that may speak to both her métiers.

Classical Random and Quantum Walks

Consider the following graph . It looks like a Q-tip. The two self-loops depart from usual examples of undirected graphs.

A classical random walk on starts at some node—say node —and at each step flips a coin to determine the next node. The “walker” stays on node if the result is heads () but goes to node if tails (). The walker cannot go from to in one step because there is no edge. The probabilistic transition matrix is given at right. The vector

gives the probabilities for each node after two steps. (Although multiplying row vectors on the left is commonly used for classical walks, we will use column vectors for consistency with quantum usage; that is symmetric moots the difference anyway.) For any , can be called the “classical state” of the walk after steps.

The state whose entries are given by the degree of node divided by the sum of the degrees makes , and thus gives a stable distribution for the walk. A central theorem states that for every that is not bipartite, the powers of converge entrywise in every column to a unique stable distribution. When is regular, as here, this is simply the uniform distribution. The convergence in our case is quite quick, as signified by the exact values

The quantum case needs to bind the graph and the “coin” into one system, so its domain has six members, which we list in order

In quantum notation, this is the tensor product of the “node space” and the “coin space” . The meaning of is that the walker just arrived at node via the coin outcome . It is incumbent that every node can be reached on either outcome. Our – labeling above complies. Then we get a permutation on from the action

where is the vertex reached from by the outcome from the classical walk, and is kept the same. The penultimate trick is to expand this to a mapping by

whereby the previous coin outcome is “forgotten” while the current flip is preserved in the state. We can represent this by the directed graph shown next.

Incidentally, is planar (switch nodes and to see). The last trick is to label its edges by quantum amplitudes for the coin outcomes.

Realm of the Coin

Our quantum coin can be “flipped” by applying any chosen matrix to the current state of the coin, provided is a unitary matrix. In general, if is a -regular graph and we have a suitable labeling for results of a -sided die, then can be a matrix acting on a single qudit, or we can use qubits to encode the coin. But here we just need one qubit. Before fixing the choice of coin matrix, we can represent it abstractly as

To apply the coin on , we form the matrix , where is the identity matrix. To move the walker after the coin result, we apply the matrix of the permutation . Since we use column vectors for states, the walk matrix is given by

We have drawn lines to show how the coin acts between each pair of nodes. Notice that is no longer symmetric, so specifying column-vector inputs matters. For instance, the in the top row comes in from the column at and exits at . This means the previous coin result was tails and took the walker (from node 2) to node ; now the coin gives heads so the walker stays at node . We can diagram this action also on the expanded graph , so that the goes on the arrow from to , and so on:

The first step has no “previous coin result” but we still have to specify one to initialize the coin state as well as the walker’s state. For starting on node there is still an infinitude of combinations of the basis states and . If we fix the start as , this is like saying the coin was heads-up before the initial act of flipping it.

Walking Into Chaos

A natural and popular choice of coin is the Hadamard matrix

We need only mark the edges with , keeping the normalizing in mind. Here are and the three steps of the quantum walk after the first:

To visualize the -step walk, say starting from , we take each possible path of length and multiply the numbers on its edges. Over all paths that end at the same we add those products to give the amplitude , which in our column-first notation equals the entry .

For example, there are two paths of steps from back to itself. One takes the loop at three times and contributes the phase value , but the other does and picks up the value . Adding those values cancels them, leaving the red shown at upper left in . If our walker is a Schrödinger cat, that part of it is not just “dead” but completely annihilated.

It is still possible for the cat to end on node after being cubed, because the path is not canceled. That the cat ends with phase does not matter, because to get the probabilities—given we started at —we take the squared magnitude of each entry in column and add the adjacent pairs denoting the same vertex of the original graph . Thus the cat has probability only

of being still on node after three time steps. Given the ease of looping at node , this is counterintuitive. After nine and ten steps, we have:

The corresponding node probabilities are and . The probability of the first node is the same between every odd and even step, echoing how starting at presumes a previous coin flip of heads, which kept the cat on node at time . At the cat has almost come back fully to node , indeed to the initial quantum state , but not quite. The swing is even closer at :

with state and node probabilities . The next probabilities, however, are , which is not close to the first-step distribution .

Most treatments of quantum walks, including a seminal paper by Dorit Aharonov, Andris Ambainis, Julia Kempe, and Umesh Vazirani, and an influential 2003 survey by Kempe, emphasize the dispersion time being linear rather than quadratic as in classical random walks on undirected graphs. Here, the quadratic nature effects quick divergence between close states like and , which is the signature of dynamical chaos.

The coin is reputed to make quantum walks smoother. That works to some extent for our three-node graph, but there appears to be no periodic stability, let alone convergence as in the classical case. Simple Python code for trying other coins is available here.

Real or Fiction?

Our main question is simple:

Are these crazy walks real?

The answer is yes if we can engineer a qutrit+qubit system that evolves by repeating . Simulating such a system throws up a second issue: No one would say that the classical computer executing my Python code is behaving chaotically. We can program via basic gates available to run on real hardware at IBM Quantum or some other services.

Is quantum hardware that applies powers of behaving chaotically? Does that pose a concretely physical impediment to maintaining its coherence?

On expecting a yes answer to the walks’ physical existence in the microworld, our questions next try to connect to Smith’s paper:

Does the mathematical existence of chaotic quantum systems on such small scales have macroscopic effects in our world? Such as on the chaotic dynamics of our atmosphere?

Again, the answer should be yes at the level of connecting Brownian motion to quantum processes, or of quantum-walk interpretations of physical systems such as the quantum harmonic oscillator (QHO). So the most specific form of our question is:

Does our (not-so-)simple three-node quantum walk appear as a non-negligible term in a Feynman-style summation needed to understand a macroscopic physical system?

Open Problems

Can the quantum chaos of the Q-tip graph be tamed? See this for success on other graphs including the 3-cycle. There are are affinities between the 3-cycle and Q-tip walks, especially at even timesteps, and the 3-cycle is an option in the Python code.

Smith also has many activities promoting STEM, including her Phyiscs Is Fun website and a book on particle physics. She maintains a blog of her writing and reading, plus much else on her author site. Dick and I were already wondering about women in the community of math/theory bloggers, such as Tanya Khovanova, Sophia Wood, Vi Hart, and Anna Weltman.

[fixed link for Python code, some word changes, fixed entry of J matrix]