Monthly Archives: July 2018

So long, and thanks for all the Fourier transforms

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The air conditioning in Caltech’s Annenberg Center for Information Science and Technology broke this July. Pasadena reached 87°F on the fourth, but my office missed the memo. The thermostat read 62°.

Hyperactive air conditioning suits a thermodynamicist’s office as jittery wifi suits an electrical-engineering building. Thermodynamicists call air conditioners “heat pumps.” A heat pump funnels heat—the energy of random motion—from cooler bodies to hotter. Heat flows spontaneously only from hot to cold on average, according to the Second Law of Thermodynamics. Pumping heat against its inclination costs work, organized energy drawn from a reliable source.

Reliable sources include batteries, coiled springs, and ACME anvils hoisted into the air. Batteries have chemical energy that power electric fans. ACME anvils have gravitational potential energy that splat coyotes.

Thermostat

I hoisted binder after binder onto my desk this July. The binders felt like understudies for ACME anvils, bulging with papers. They contained notes I’d written, and articles I’d read, for research throughout the past five years. My Caltech sojourn was switching off its lights and drawing its shutters. A control theorist was inheriting my desk. I had to move my possessions to an office downstairs, where I’d moonlight until quitting town.

Quitting town.

I hadn’t expected to feel at home in southern California, after stints in New and old England. But research and researchers drew me to California and then hooked me. Caltech’s Institute for Quantum Information and Matter (IQIM) has provided an intellectual home, colleagues-cum-friends, and a base from which to branch out to other scholars and institutions.

The IQIM has provided also the liberty to deck out my research program as a college dorm room with posters—according to my tastes, values, and exuberances. My thesis demanded the title “Quantum steampunk: Quantum information, thermodynamics, their intersection, and applications thereof across physics.” I began developing the concept of quantum steampunk on this blog. Writing a manifesto for the concept, in the thesis’s introduction, proved a delight:

The steampunk movement has invaded literature, film, and art over the past three decades. Futuristic technologies mingle, in steampunk works, with Victorian and wild-west settings. Top hats, nascent factories, and grimy cities counterbalance time machines, airships, and automata. The genre arguably originated in 1895, with the H.G. Wells novel The Time Machine. Recent steampunk books include the best-selling The Invention of Hugo Cabret; films include the major motion picture Wild Wild West; and artwork ranges from painting to jewelry to sculpture.

Steampunk captures the romanticism of fusing the old with the cutting-edge. Technologies proliferated during the Victorian era: locomotives, Charles Babbage’s analytical engine, factories, and more. Innovation facilitated exploration. Add time machines, and the spirit of adventure sweeps you away. Little wonder that fans flock to steampunk conventions, decked out in overcoats, cravats, and goggles.

What steampunk fans dream, quantum-information thermodynamicists live.

Thermodynamics budded during the late 1800s, when steam engines drove the Industrial Revolution. Sadi Carnot, Ludwig Boltzmann, and other thinkers wondered how efficiently engines could operate. Their practical questions led to fundamental insights—about why time flows; how much one can know about a physical system; and how simple macroscopic properties, like temperature, can capture complex behaviors, like collisions by steam particles. An idealization of steam—the classical ideal gas—exemplifies the conventional thermodynamic system. Such systems contain many particles, behave classically, and are often assumed to remain in equilibrium.

But thermodynamic concepts—such as heat, work, and equilibrium—characterize small scales, quantum systems, and out-of-equilibrium processes. Today’s experimentalists probe these settings, stretching single DNA strands with optical tweezers [4], cooling superconducting qubits to build quantum computers [5, 6], and extracting work from single-electron boxes [7]. These settings demand reconciliation with 19th-century thermodynamics. We need a toolkit for fusing the old with the new.

Quantum information (QI) theory provides such a toolkit. Quantum phenomena serve as resources for processing information in ways impossible with classical systems. Quantum computers can solve certain computationally difficult problems quickly; quantum teleportation transmits information as telephones cannot; quantum cryptography secures messages; and quantum metrology centers on high- precision measurements. These applications rely on entanglement (strong correlations between quantum systems), disturbances by measurements, quantum uncertainty, and discreteness.

Technological promise has driven fundamental insights, as in thermodynamics. QI theory has blossomed into a mathematical toolkit that includes entropies, uncertainty relations, and resource theories. These tools are reshaping fundamental science, in applications across physics, computer science, and chemistry.

QI is being used to update thermodynamics, in the field of quantum thermodynamics (QT) [8, 9]. QT features entropies suited to small scales; quantum engines; the roles of coherence in thermalization and transport; and the transduction of information into work, à la Maxwell’s demon [10].

This thesis (i) contributes to the theory of QI thermodynamics and (ii) applies the theory, as a toolkit, across physics. Spheres touched on include atomic, molecular, and optical (AMO) physics; nonequilibrium statistical mechanics; condensed matter; chemistry; and high-energy physics. I propose the name quantum steampunk for this program…

Never did I anticipate, in college, that a PhD could reflect my identity and style. I feared losing myself and my perspective in a subproblem of a subproblem of a subproblem. But I found myself blessed with the chance to name the aesthetic that’s guided my work, the scent I’ve unconsciously followed from book to class to research project to conversation, to paper, since…middle school, come to think of it. I’m grateful for that opportunity.

Q. steampunk

Whump, went my quantum-engine binder on my desk. I’d stuck an address label, pointing to Annenberg, to the binder. If the binder walked away, whoever found it would know where it belonged. Scratching at the label with a fingernail failed to budge the sticker. I stuck a label addressed to Cambridge, Massachusetts alongside the Pasadena address.

I’m grateful to be joining Harvard as an ITAMP (Institute for Theoretical Atomic, Molecular, and Optical Physics) Postdoctoral Fellow. You’ll be able to catch me in Harvard’s physics department, in ITAMP, or at MIT, starting this September.

While hunting for a Cambridge apartment, I skyped with potential roommates. I’d inquire about locations, about landlords and landladies, about tidiness, and about heating. The heating system’s pretty old, most tenants would admit. We keep the temperature between 60 and 65 degrees, to keep costs down. I’d nod and extol the layering of sweaters, but I shivered inside.

One tenant surprised me. The heating…works too well, she said. It’s pretty warm, to tell the truth. I thought about heat pumps and quantum engines, about picnics in the Pasadena sunshine, about the Julys I’d enjoyed while the world around me had sweated. Within an hour, I’d committed to sharing the apartment.

Boxes

Some of you have asked whether I’ll continue blogging for Quantum Frontiers. Yes: Extricating me from the IQIM requires more than 3,000 miles.

See you in Cambridge.

 

With apologies to Douglas Adams.

The World Cup from a Quantum Perspective

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Two weeks into the Football World Cup and the group stages are over: 16 teams have gone home, leaving the top 16 teams to contend the knock-out stages. Those fans who enjoy a bet will be poring over the odds in search of a bargain—a mis-calculation on the part of the bookmakers. Is now the time to back Ronaldo for the golden boot, whilst Harry Kane dominates the headlines and sports betting markets? Will the hosts Russia continue to defy lowly pre-tournament expectations and make the semi-finals? Are France about to emerge from an unconvincing start to the tournament and blossom as clear front-runners?

But, whilst for most the sports betting markets may lead to the enhanced enjoyment of the tournament that a bet can bring (as well as the possibility of making a little money), for others they represent a window into the fascinating world of sports statistics. A fascination that can be captured by the simple question: how do they set the odds?

Suppose that a bookmaker has in their possession a list of outcome probabilities for matches between each pair of national football teams in the world and wants to predict the overall winner. There are 32768 possible ways for the tournament knock-out rounds to pan-out—a large, but not insurmountable number of iterations by modern computational standards.

However, if the bookmaker instead considers the tennis grand-slams, with 128 competitors in the first round, then there are a colossal 1.7 × 1038 permutations. Indeed, in a knock-out format there are 2n-1 permutations, where n is the number of entrants. And for those of a certain mindset, this exponentially growing space immediately raises the question of whether a quantum algorithm can yield a speed-up for the related prediction tasks.

A Tiny Cup

The immediate question which we want to answer here is, perhaps, who will win the World Cup. We will walk through the idea on the blackboard first, and then implement it as a quantum algorithm—which, hopefully, will give some insight into how and where quantum computers can outperform classical ones, for this particular way of answering the question.

Let us take a toy setup with four teams A, B, C and D;
the knockout stage starts with a game A vs. B, and C vs. D.
Whoever wins each game will play against each other, so here we have four possible final games: A vs. C, A vs. D, B vs. C, or B vs. D.
Let’s denote by p(X, Y) the probability that X wins when playing against Y.

The likelihood of A winning the cup is then simply given by

p(A, B) × ( p(C, D) × p(A, C) + p(D, C) × p(A, D) ),

i.e. the probability that A wins against B, times the probabilities of A winning against C in case C won against D, plus the probability of A winning against D in case D won.

How can we obtain the same quantity with a quantum algorithm?

First, we set up our Hilbert space so that it can represent all possible Cup scenarios.
Since we have four teams, we need a four-dimensional quantum system as our smallest storage unit—we commonly call those qudits as generalizations of a qubit, which having dimension 2 would be fit to store two teams only (we can always “embed” a qudit into a few qubits of the same dimension.

Remember: k qubits have dimension 2k, so we could also store the qudit as two qubits).
If we write |A\rangle, this simply stands for a qudit representing team A; if we write |A\rangle |B\rangle, then we have a state representing two teams.

To represent a full knockout tree, we follow the same logic: Take four qudits for the initial draw; add two qudits for the winners of the first two matches, and one qudit for the final winner.

For instance, one possible knockout scenario would be

|\text{Game 1}\rangle = \underbrace{|A\rangle |B\rangle |C\rangle |D\rangle}_\text{Initial Draw} \ \underbrace{|A\rangle |D\rangle}_\text{Finals} \ |D\rangle.

The probability associated with Game 1 is then precisely p(A, B) × p(D, C) × p(D, A).

Here is where quantum computing comes in.

Starting from an initial state |A\rangle |B\rangle |C\rangle |D\rangle, we create two new slots in a superposition over all possible match outcomes, weighted by the square-root of their probabilities (which we call q instead of p):

\begin{aligned} |\text{Step 1}\rangle = |A\rangle |B\rangle |C\rangle |D\rangle \big(\ \ &\text{q(A, B)q(C, D)} \,|A\rangle\ |C\rangle +\\ &\text{q(A, B)q(D, C)} \,|A\rangle\ |D\rangle +\\ &\text{q(B, A)q(C, D)} \,|B\rangle\ |C\rangle +\\ &\text{q(B, A)q(D, C)} \,|B\rangle\ |D\rangle\ \big). \end{aligned}

For the final round, we perform the same operation on those two last slots; e.g. we would map |A\rangle |C\rangle to a state |A\rangle |C\rangle ( q(A, C) |A\rangle + q(C, A) |C\rangle ). The final state is thus a superposition over eight possible weighted games (as we would expect).

So you can tell me who wins the World Cup?

Yes. Or well, probably. We find out by measuring the rightmost qudit.
As we know, the probability of obtaining a certain measurement outcome, say A, will then be determined by the square of the weights in front of the measured state; since we put in the square-roots initially we recover the original probabilities. Neat!

And since there are two possible game trees that lead to a victory of A, we have to sum them up—and we get precisely the probability we calculated by hand above. This means the team that is most likely to win will be the most likely measurement outcome.

So what about the World Cup? We have 16 teams; one team can thus be stored in four qubits. The knockout tree has 31 vertices, and a naive implementation can be done on a quantum computer with 124 qubits. Of course we are only a bit naive, so we can simulate this quantum computer on a classical one and obtain the following winning probabilities:

0.194 Brazil
0.168 Spain
0.119 France
0.092 Belgium
0.082 Argentina
0.075 England
0.049 Croatia
0.041 Colombia
0.04 Portugal
0.032 Uruguay
0.031 Russia
0.022 Switzerland
0.019 Denmark
0.018 Sweden
0.012 Mexico
0.006 Japan

It is worth noting that all operations we described can be implemented efficiently with a quantum computer, and the number of required qubits is quite small; for the four teams, we could get away with seven qudits, or fourteen qubits (and we could even save some, by ignoring the first four qudits which are always the same).

So for this particular algorithm there is an exponential speedup over its non-probabilistic classical counterpart: as mentioned, one would have to iterate over all trees; tedious for the World Cup, practically impossible for tennis. However…

Classical vs. Quantum

Does using a quantum algorithm give us a speedup for this task? Here, the answer is no; one could obtain similar results in comparable time using probabilistic methods, for instance, by doing Monte Carlo sampling.

But there are several interesting related questions that we could ask for which there might be a quantum advantage.

For some team A, we can easily create a state that has all game trees in superposition that lead to a victory of A—even weighting them using their respective probabilities.
Given this state as a resource, we can think of questions like “which game tree is most likely, given that we fix A and B as semifinalists”, or “which team should A play in the knockout stages to maximize the probability that B wins the tournament”.

Or, more controversially: can we optimize the winning chances for some team by rearranging the initial draw?

Some questions like these lend themselves to applying Grover search, for which there is a known speedup over classical computers. To inquire deeper into the utility of quantum algorithms, we need to invent the right kind of question to ask of this state.

Let us think of one more toy example. Being part physicists, we assume cows are spheres—so we might as well also assume that if A is likely to win a match against B, it always wins—even if the probability is only 51%. Let’s call this exciting sport “deterministic football”. For a set of teams playing a tournament of deterministic football, does there exist a winning initial draw for every team?

This becomes an especially interesting question in cases where there is a non-trivial cyclic relation between the teams’ abilities, a simple example being: A always beats B, B always beats C, and C always beats A. For example, if this problem turns out to be NP-hard, then it would be reasonable to expect that the quadratic improvement achieved by quantum search is the best we can hope for in using a quantum algorithm for the task of finding a winning initial draw for a chosen team—at least for deterministic football (phew).

To the finals and beyond

World Cup time is an exciting time: whatever the question, we are essentially dealing with binary trees, and making predictions can be translated into finding partitions or cuts that satisfy certain properties defined through a function of the edge weights (here the pairwise winning probabilities). We hope this quantum take on classical bookmaking might point us in the direction of new and interesting applications for quantum algorithms.

Hopefully a good bargain!

(Written with Steven Herbert and Sathyawageeswar Subramanian)