Doctrine of the (measurement) mean

Don’t invite me to dinner the night before an academic year begins.

You’ll find me in an armchair or sitting on my bed, laptop on my lap, journaling. I initiated the tradition the night before beginning college. I take stock of the past year, my present state, and hopes for the coming year.

Much of the exercise fosters what my high-school physics teacher called “an attitude of gratitude”: I reflect on cities I’ve visited, projects firing me up, family events attended, and subfields sampled. Other paragraphs, I want off my chest: Have I pushed this collaborator too hard or that project too little? Miscommunicated or misunderstood? Strayed too far into heuristics or into mathematical formalisms?

If only the “too much” errors, I end up thinking, could cancel the “too little.”

In one quantum-information context, they can.

Imagine that you’ve fabricated the material that will topple steel and graphene; let’s call it a supermetatopoconsulator. How, you wonder, do charge, energy, and particles move through this material? You’ll learn by measuring correlators.

A correlator signals how much, if you poke this piece here, that piece there responds. At least, a two-point correlator does: $\langle A(0) B(\tau) \rangle$. $A(0)$ represents the poke, which occurs at time $t = 0$. $B(\tau)$ represents the observable measured there at $t = \tau$. The $\langle . \rangle$ encapsulates which state $\rho$ the system started in.

Condensed-matter, quantum-optics, and particle experimentalists have measured two-point correlators for years. But consider the three-point correlator $\langle A(0) B(\tau) C (\tau' ) \rangle$, or a $k$-point $\langle \underbrace{ A(0) \ldots M (\tau^{(k)}) }_k \rangle$, for any $k \geq 2$. Higher-point correlators relate more-complicated relationships amongst events. Four-pointcorrelators associated with multiple times signal quantum chaos and information scrambling. Quantum information scrambles upon spreading across a system through many-body entanglement. Could you measure arbitrary-point, arbitrary-time correlators?

Supermetatopoconsulator (artist’s conception)

Yes, collaborators and I have written, using weak measurements. Weak measurements barely disturb the system being measured. But they extract little information about the measured system. So, to measure a correlator, you’d have to perform many trials. Moreover, your postdocs and students might have little experience with weak measurements. They might not want to learn the techniques required, to recalibrate their detectors, etc. Could you measure these correlators easily?

Yes, if the material consists of qubits,2 according to a paper I published with Justin Dressel, José Raúl González Alsonso, and Mordecai Waegell this summer. You could build such a system from, e.g., superconducting circuits, trapped ions, or quantum dots.

You can measure $\langle \underbrace{ A(0) B (\tau') C (\tau'') \ldots M (\tau^{(k)}) }_k \rangle$, we show, by measuring $A$ at $t = 0$, waiting until $t = \tau'$, measuring $B$, and so on until measuring $M$ at $t = \tau^{(k)}$. The $t$-values needn’t increase sequentially: $\tau''$ could be less than $\tau'$, for instance. You’d have to effectively reverse the flow of time experienced by the qubits. Experimentalists can do so by, for example, flipping magnetic fields upside-down.

Each measurement requires an ancilla, or helper qubit. The ancilla acts as a detector that records the measurement’s outcome. Suppose that $A$ is an observable of qubit #1 of the system of interest. You bring an ancilla to qubit 1, entangle the qubits (force them to interact), and look at the ancilla. (Experts: You perform a controlled rotation on the ancilla, conditioning on the system qubit.)

Each trial yields $k$ measurement outcomes. They form a sequence $S$, such as $(1, 1, 1, -1, -1, \ldots)$. You should compute a number $\alpha$, according to a formula we provide, from each measurement outcome and from the measurement’s settings. These numbers form a new sequence $S' = \mathbf{(} \alpha_S(1), \alpha_S(1), \ldots \mathbf{)}$. Why bother? So that you can force errors to cancel.

Multiply the $\alpha$’s together, $\alpha_S(1) \times \alpha_S(1) \times \ldots$, and average the product over the possible sequences $S$. This average equals the correlator $\langle \underbrace{ A(0) \ldots M (\tau^{(k)}) }_k \rangle$. Congratulations; you’ve characterized transport in your supermetatopoconsulator.

When measuring, you can couple the ancillas to the system weakly or strongly, disturbing the system a little or a lot. Wouldn’t strong measurements perturb the state $\rho$ whose properties you hope to measure? Wouldn’t the perturbations by measurements one through $\ell$ throw off measurement $\ell + 1$?

Yes. But the errors introduced by those perturbations cancel in the average. The reason stems from how we construct $\alpha$’s: Our formula makes some products positive and some negative. The positive and negative terms sum to zero.

The cancellation offers hope for my journal assessment: Errors can come out in the wash. Not of their own accord, not without forethought. But errors can cancel out in the wash—if you soap your $\alpha$’s with care.

1and six-point, eight-point, etc.

2Rather, each measured observable must square to the identity, e.g., $A^2 = 1$. Qubit Pauli operators satisfy this requirement.

With apologies to Aristotle.

I get knocked down…

“You’ll have to have a thick skin.”

Marcelo Gleiser, a college mentor of mine, emailed the warning. I’d sent a list of physics PhD programs and requested advice about which to attend. Marcelo’s and my department had fostered encouragement and consideration.

Suit up, Marcelo was saying.

Criticism fuels science, as Oxford physicist David Deutsch has written. We have choices about how we criticize. Some criticism styles reflect consideration for the criticized work’s creator. Tufts University philosopher Daniel Dennett has devised guidelines for “criticizing with kindness”:1

1. You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, “Thanks, I wish I’d thought of putting it that way.

2. You should list any points of agreement (especially if they are not matters of general or widespread agreement).

3. You should mention anything you have learned from your target.

4. Only then are you permitted to say so much as a word of rebuttal or criticism.

Scientists skip to step four often—when refereeing papers submitted to journals, when posing questions during seminars, when emailing collaborators, when colleagues sketch ideas at a blackboard. Why? Listening and criticizing require time, thought, and effort—three of a scientist’s most valuable resources. Should any scientist spend those resources on an idea of mine, s/he deserves my gratitude. Spending empathy atop time, thought, and effort can feel supererogatory. Nor do all scientists prioritize empathy and kindness. Others of us prioritize empathy but—as I have over the past five years—grown so used to its latency, I forget to demonstrate it.

Doing science requires facing not only criticism, but also “That doesn’t make sense,” “Who cares?” “Of course not,” and other morale boosters.

Doing science requires resilience.

So do measurements of quantum information (QI) scrambling. Scrambling is a subtle, late, quantum stage of equilibration2 in many-body systems. Example systems include chains of spins,3 such as in ultracold atoms, that interact with each other strongly. Exotic examples include black holes in anti-de Sitter space.4

Imagine whacking one side of a chain of interacting spins. Information about the whack will disseminate throughout the chain via entanglement.5 After a long interval (the scrambling time, $t_*$), spins across the systems will share many-body entanglement. No measurement of any few, close-together spins can disclose much about the whack. Information will have scrambled across the system.

QI scrambling has the subtlety of an assassin treading a Persian carpet at midnight. Can we observe scrambling?

A Stanford team proposed a scheme for detecting scrambling using interferometry.6 Justin Dressel, Brian Swingle, and I proposed a scheme based on weak measurements, which refrain from disturbing the measured system much. Other teams have proposed alternatives.

Many schemes rely on effective time reversal: The experimentalist must perform the quantum analog of inverting particles’ momenta. One must negate the Hamiltonian $\hat{H}$, the observable that governs how the system evolves: $\hat{H} \mapsto - \hat{H}$.

At least, the experimentalist must try. The experimentalist will likely map $\hat{H}$ to $- \hat{H} + \varepsilon$. The small error $\varepsilon$ could wreak havoc: QI scrambling relates to chaos, exemplified by the butterfly effect. Tiny perturbations, such as the flap of a butterfly’s wings, can snowball in chaotic systems, as by generating tornadoes. Will the $\varepsilon$ snowball, obscuring observations of scrambling?

It needn’t, Brian and I wrote in a recent paper. You can divide out much of the error until $t_*$.

You can detect scrambling by measuring an out-of-time-ordered correlator (OTOC), an object I’ve effused about elsewhere. Let’s denote the time-$t$ correlator by $F(t)$. You can infer an approximation $\tilde{F}(t)$ to $F(t)$ upon implementing an $\varepsilon$-ridden interferometry or weak-measurement protocol. Remove some steps from that protocol, Brian and I say. Infer a simpler, easier-to-measure object $\tilde{F}_{\rm simple}(t)$. Divide the two measurement outcomes to approximate the OTOC:

$F(t) \approx \frac{ \tilde{F}(t) }{ \tilde{F}_{\rm simple}(t) }$.

OTOC measurements exhibit resilience to error.

Physicists need resilience. Brian criticizes with such grace, he could serve as the poster child for Daniel Dennett’s guidelines. But not every scientist could. How can we withstand kindness-lite criticism?

By drawing confidence from what we’ve achieved, with help from mentors like Marcelo. I couldn’t tell what about me—if anything—could serve as a rock on which to plant a foot, as an undergrad. Mentors identified what I had too little experience to appreciate. You question what you don’t understand, they said. You assimilate perspectives from textbooks, lectures, practice problems, and past experiences. You scrutinize details while keeping an eye on the big picture. So don’t let so-and-so intimidate you.

I still lack my mentors’ experience, but I’ve imbibed a drop of their insight. I savor calculations that I nail, congratulate myself upon nullifying referees’ concerns, and celebrate the theorems I prove.

I’ve also created an email folder entitled “Nice messages.” In go “I loved your new paper; combining those topics was creative,” “Well done on the seminar; I’m now thinking of exploring that field,” and other rarities. The folder affords an umbrella when physics clouds gather.

Finally, I try to express appreciation of others’ work.7 Science thrives on criticism, but scientists do science. And scientists are human—undergrads, postdocs, senior researchers, and everyone else.

Doing science—and attempting to negate Hamiltonians—we get knocked down. But we can get up again.

Around the time Brian and I released “Resilience” two other groups proposed related renormalizations. Check out their schemes here and here.

1Thanks to Sean Carroll for alerting me to this gem of Dennett’s.

2A system equilibrates as its large-scale properties, like energy, flatline.

3Angular-momentum-like quantum properties

4Certain space-times different from ours

5Correlations, shareable by quantum systems, stronger than any achievable by classical systems

6The cancellation (as by a crest of one wave and a trough of another) of components of a quantum state, or the addition of components (as two waves’ crests)

7Appreciation of specific qualities. “Nice job” can reflect a speaker’s belief but often reflects a desire to buoy a receiver whose work has few merits to elaborate on. I applaud that desire and recommend reinvesting it. “Nice job” carries little content, which evaporates under repetition. Specificity provides content: “Your idea is alluringly simple but could reverberate across multiple fields” has gristle.

The Curious Behavior of Topological Insulators

IQIM hosts a Summer Research Institute that invites high school Physics teachers to work directly with staff, students, and researchers in the lab.  Last summer I worked with Marcus Teague, a highly intelligent and very patient Caltech Staff Scientist in the Yeh Group, to help set up an experiment for studying exotic material samples under circularly polarized light.  I had researched, ordered, and assembled parts for the optics and vacuum chamber.  As I returned to Caltech this summer, I was eager to learn how the Yeh Group had proceeded with the study.

Yeh group (2017): I am the one on the front-left of the picture, next to Dr. Yeh and in front of Kyle Chen. Benjamin Fackrell, another physics teacher interning at the Yeh lab, is all the way to the right.

The optics equipment I had researched, ordered, and helped to set up last summer is being used currently to study topological insulator (TI) samples that Kyle Chien-Chang Chen, a doctoral candidate, has worked on in the Yeh Lab.  Yes, a high school Physics teacher played a small role in their real research! It is exciting and humbling to have a connection to real-time research.

Quartz quarter-wave plates are important elements in many experiments involving light. They convert linearly polarized light to circularly polarized light.

Kyle receives a variety of TI samples from UCLA; the current sample up for review is Bismuth Antimony Telluride $\mathrm{(BiSb)}_2\mathrm{Te}_3$.  Depending on the particular sample and the type of testing, Kyle has a variety of procedures to prep the samples for study.  And this summer, Kyle has help from visiting Canadian student Adrian Llanos. Below are figures of some of the monolayer and bilayer structures for topological insulators studied in the lab.

Pictures of samples from UCLA

Under normal conditions, a topological insulator (TI) is only conductive on the surface. The center of a TI sample is an insulator. But when the surface states open an energy gap, the surface of the TI becomes insulating. The energy gap is the amount of energy necessary to remove an electron from the top valence band to become free to move about.  This gap is the result of the interaction between the conduction band and valence band surface states from the opposing surfaces of a thin film. The resistance of the conducting surface actually increases. The Yeh group is hoping that the circularly polarized light can help align the spin of the Chromium electrons, part of the bilayer of the TI.  At the same time, light has other effects, like photo-doping, which excites more electrons into the conduction bands and thus reduces the resistance. The conductivity of the surface of the TI changes as the preferentially chosen spin up or spin down is manipulated by the circularly polarized light or by the changing magnetic field.

A physical property measurement system.

This interesting experiment on TI samples is taking place within a device called a Physical Property Measurement System (PPMS).  The PPMS is able to house the TI sample and the optics equipment to generate circularly polarized light, while allowing the researchers to vary the temperature and magnetic field.  The Yeh Group is able to artificially turn up the magnetic field or the circularly polarized light in order to control the resistance and current signal within the sample.  The properties of surface conductivity are studied up to 8 Tesla (over one-hundred thousand times the Earth’s magnetic field), and from room temperature (just under 300 Kelvin) to just below 2 Kelvin (colder than outer space).

Right-Hand-Rule used to determine the direction of the magnetic (Lorentz) force.

In the presence of a magnetic field, when a current is applied to a conductor, the electrons will experience a force at a right angle to the magnetic field, following the right-hand rule (or the Physics gang sign, as we affectionately call it in my classroom).  This causes the electrons to curve perpendicular to their original path and perpendicular to the magnetic field. The build up of electrons on one end of the conductor creates a potential difference. This potential difference perpendicular to the original current is known as the ordinary Hall Effect.  The ratio of the induced voltage to the applied current is known as the Hall Resistance.

Under very low temperatures, the Quantum Hall Effect is observed. As the magnetic field is changed, the Hall Voltage increases in set quantum amounts, as opposed to gradually. Likewise, the Hall Resistance is quantized.  It is a such an interesting phenomenon!

For a transport measurement of the TI samples, Kyle usually uses a Hall Bar Geometry in order to measure the Hall Effect accurately. Since the sample is sufficiently large, he can simply solder it for measurement.

Transport Measurements of TI Samples follow the same setup as Quantum Hall measurements on graphene: Current runs through electrodes attached to the North/South ends of the sample, while electron flow is measured longitudinally, as well as along the East/West ends (Hall conductance).

What is really curious is that the Bismuth Antimony Telluride samples are exhibiting the Hall Effect even when no external magnetic field is applied!  When the sample is measured, there is a Hall Resistance despite no external magnetic field. Hence the sample itself must be magnetic.  This phenomenon is called the Anomalous Hall Effect.

According to Kyle, there is no fancy way to measure the magnetization directly; it is only a matter of measuring a sample’s Hall Resistance. The Hall Resistance should be zero when there is no Anomalous Hall Effect, and when there is ferromagnetism (spins want to align in the direction of their neighbors), you see a non-zero value.  What is really interesting is that they assume ferromagnetism would break the time-reversal symmetry and thus open a gap at the surface states.  A very strange behavior that is also observed is that the longitudinal resistance increases gradually.

Running PPMS

Typically the quantum Hall Resistance increases in quantum increments.  Even if the surface gap is open, the sample is not insulating because the gap is small (<0.3 eV); hence, under these conditions this TI is behaving much more like a semiconductor!

Next, the group will examine these samples using the Scanning Tunneling Microscope (STM).  The STM will be able to provide local topological information by examining 1 micron by 1 micron areas.  In comparison, the PPMS research with these samples is telling the story of the global behavior of the sample.  The combination of information from the PPMS and STM research will provide a more holistic story of the behavior of these unique samples.

I am thrilled to see how the group has used what we started with last summer to find interesting new results.  I am fascinated to see what they learn in the coming months with the different samples and STM testing. And I am quite excited to share these applications with my students in the upcoming new school year.  Another summer packed with learning!

The Quantum Wave in Computing

Summer is a great time for academics. Imagine: three full months off! Hit the beach. Tune that golf pitch. Hike the sierras. Go on a cruise. Watch soccer with the brazilenos (there’s been better years for that one). Catch the sunset by the Sydney opera house. Take a nap.

A visiting researcher taking full advantage of the Simons Institute’s world-class relaxation facilities. And yes, I bet you he’s proving a theorem at the same time.

Think that’s outrageous? We have it even better. Not only do we get to travel the globe worry-free, but we prove theorems while doing it. For some of us summer is the only time of year when we manage to prove theorems. Ideas accumulate during the year, blossom during the conferences and workshops that mark the start of the summer, and hatch during the few weeks that many of us set aside as “quiet time” to finally “wrap things up”.

I recently had the pleasure of contributing to the general well-being of my academic colleagues by helping to co-organize (with Andrew Childs, Ignacio Cirac, and Umesh Vazirani) a 2-month long program on “Challenges in Quantum Computation” at the Simons Institute in Berkeley. In this post I report on the program and describe one of the highlights discussed during it: Mahadev’s very recent breakthrough on classical verification of quantum computation.

Challenges in Quantum Computation

The Simons Institute has been in place on the UC Berkeley campus since the Fall of 2013, and in fact one of their first programs was on “Quantum Hamiltonian Complexity”, in Spring 2014 (see my account of one of the semester’s workshops here). Since then the institute has been hosting a pair of semester-long programs at a time, in all areas of theoretical computer science and neighboring fields. Our “summer cluster” had a slightly different flavor: shorter, smaller, it doubled up as the prelude to a full semester-long program scheduled for Spring 2020 (provisional title: The Quantum Wave in Computing, a title inspired from Umesh Vazirani’s recent tutorial at STOC’18 in Los Angeles) — (my interpretation of) the idea being that the ongoing surge in experimental capabilities supports a much broader overhaul of some of the central questions of computer science, from the more applied (such as, programming languages and compilers), to the most theoretical (such as, what complexity classes play the most central role).

This summer’s program hosted a couple dozen participants at a time. Some stayed for the full 2 months, while others visited for shorter times. The Simons Institute is a fantastic place for collaborative research. The three-story building is entirely devoted to us. There are pleasant yet not-too-comfortable shared offices, but the highlight is the two large communal rooms meant for organized and spontaneous discussion. Filled with whiteboards, bright daylight, comfy couches, a constant supply of tea, coffee, and cookies, and eager theorists!

After a couple weeks of settling down the program kicked off with an invigorating workshop. Our goal for the workshop was to frame the theoretical questions raised by the sudden jump in the capabilities of experimental quantum devices that we are all witnessing. There were talks describing progress in experiments (superconducting qubits, ion traps, and cold atoms were represented), suggesting applications for the new devices (from quantum simulation & quantum chemistry to quantum optimization and machine learning through “quantum supremacy” and randomness generation), and laying the theoretical framework for trustworthy interaction with the quantum devices (interactive proofs, testing, and verifiable delegation). We had an outstanding line-up of speakers. All talks (except the panel discussions, unfortunately) were recorded, and you can watch them here.

The workshop was followed by five additional weeks of “residency”, that allowed long-term participants to digest and develop the ideas presented during the workshop. In my experience these few additional weeks, right after the workshop, make all the difference. It is the same difference as between a quick conference call and a leisurely afternoon at the whiteboard: while the former may feel productive and bring the adrenaline levels up, the latter is more suited to in-depth exploration and unexpected discoveries.

There would be much to say about the ideas discussed during the workshop and following weeks. I will describe a single one of these ideas — in my opinion, one of the most outstanding ideas to have emerged at the interface of quantum computing and theoretical computer science in recent years! The result, “Classical Verification of Quantum Computations”, is by Urmila Mahadev, a Ph.D.~student at UC Berkeley (I think she just graduated). Urmila gave a wonderful talk on her result at the workshop, and I highly recommend watching the recorded video. In the remainder of this post I’ll provide an overview of the result. I also wrote a slightly more technical introduction that eager readers will find here.

A cryptographic leash on quantum systems

Mahadev’s result is already famous: announced on the blog of Scott Aaronson, it has earned her a long-standing 25\$ prize, awarded for “solving the problem of proving the results of an arbitrary quantum computation to a classical skeptic”. Or, in complexity-theoretic linguo, for showing that “every language in the class BQP admits an interactive protocol where the prover is in BQP and the verifier is in BPP”. What does this mean?

Verifying quantum computations in the high complexity regime

On his blog Scott Aaronson traces the question back to a talk given by Daniel Gottesman in 2004. An eloquent formulation appears in a subsequent paper by Dorit Aharonov and Umesh Vazirani, aptly titled “Is Quantum Mechanics Falsifiable? A computational perspective on the foundations of Quantum Mechanics”.

Here is the problem. As readers of this blog are well aware, Feynman’s idea of a quantum computer, and the subsequent formalization by Bernstein and Vazirani of the Quantum Turing Machine, layed the theoretical foundation for the construction of computing devices whose inner functioning is based on the laws of quantum physics. Most readers also probably realize that we currently believe that these quantum devices will have the ability to efficiently solve computational problems (the class of which is denoted BQP) that are thought to be beyond the reach of classical computers (represented by the class BPP). A prominent example is factoring, but there are many others. The most elementary example is arguably Feynman’s original proposal: a quantum computer can be used to simulate the evolution of any quantum mechanical system “in real time”. In contrast, the best classical simulations available can take exponential time to converge even on concrete examples of practical interest. This places a computational impediment to scientific progress: the work of many physicists, chemists, and biologists, would be greatly sped up if only they could perform simulations at will.

So this hypothetical quantum device claims (or will likely claim) that it has the ability to efficiently solve computational problems for which there is no known efficient classical algorithm. Not only this but, as is widely believed in complexity-theoretic circles (a belief recently strenghtened by the proof of an oracle separation between BQP and PH by Tal and Raz), for some of these problems, even given the answer, there does not exist a classical proof that the answer is correct. The quantum device’s claim cannot be verified! This seems to place the future of science at the mercy of an ingenuous charlatan, with good enough design & marketing skills, that would convince us that it is providing the solution to exponentially complex problems by throwing stardust in our eyes. (Wait, did this happen already?)

Today is the most exciting time in quantum computing since the discovery of Shor’s algorithm for factoring: while we’re not quite ready to run that particular algorithm yet, experimental capabilities have ramped up to the point where we are just about to probe the “high-complexity” regime of quantum mechanics, by making predictions that cannot be emulated, or even verified, using the most powerful classical supercomputers available. What confidence will we have that the predictions have been obtained correctly? Note that this question is different from the question of testing the validity of the theory of quantum mechanics itself. The result presented here assumes the validity of quantum mechanics. What it offers is a method to test, assuming the correctness of quantum mechanics, that a device performs the calculation that it claims to have performed. If the device has supra-quantum powers, all bets are off. Even assuming the correctness of quantum mechanics, however, the device may, intentionally or not (e.g. due to faulty hardware), mislead the experimentalist. This is the scenario that Mahadev’s result aims to counter.

Interactive proofs

The first key idea is to use the power of interaction. The situation can be framed as follows: given a certain computation, such that a device (henceforth called “prover”) has the ability to perform the computation, but another entity, the classical physicist (henceforth called “verifier”) does not, is there a way for the verifier to extract the right answer from the prover with high confidence — given that the prover may not be trusted, and may attempt to use its superior computing power to mislead the verifier instead of performing the required computation?

The simplest scenario would be one where the verifier can execute the computation herself, and check the prover’s outcome. The second simplest scenario is one where the verifier cannot execute the computation, but there is a short proof that the prover can provide that allows her to fully certify the outcome. These two scenario correspond to problems in BPP and NP respectively; an example of the latter is factoring. As argued earlier, not all quantum computations (BQP) are believed to fall within these two classes. Both direct computation and proof verification are ruled out. What can we do? Use interaction!

The framework of interactive proofs originates in complexity theory in the 1990s. An interactive proof is a protocol through which a verifier (typically a computationally bounded entity, such as the physicist and her classical laptop) interacts with a more powerful, but generally untrusted, prover (such as the experimental quantum device). The goal of the protocol is for the verifier to certify the validity of a certain computational statement.

Here is a classical example (the expert — or impatient — reader may safely skip this). The example is for a problem that lies in co-NP, but is not believed to lie in NP. Suppose that both the verifier and prover have access to two graphs, ${G}$ and ${H}$, such that the verifier wishes to raise an “ACCEPT” flag if and only if the two graphs are not isomorphic. In general this is a hard decision to make, because the verifier would have to check all possible mappings from one graph to the other, of which there are exponentially many. Here is how the verifier can extract the correct answer by interacting with a powerful, untrusted prover. First, the verifier flips a fair coin. If the coin comes up heads, she selects a random relabeling of the vertices of ${G}$. If the coin comes up tail, she selects a random relabeling of the vertices of ${H}$. The verifier then sends the relabeled graph to the prover, and asks the prover to guess which graph the verifier has hidden. If the prover provides the correct answer (easy to check), the verifier concludes that the graphs were not isomorphic. Otherwise, she concludes that they were isomorphic. It is not hard to see that, if the graphs are indeed not isomorphic, the prover always has a means to correctly identify the hidden graph, and convince the verifier to make the right decision. But if the graphs are isomorphic, then there is no way for the prover to distinguish the random relabelings (since the distributions obtained by randomly relabeling each graph are identical), and so the verifier makes the right decision with probability 1/2. Repeating the protocol a few times, with a different choice of relabeling each time, quickly drives the probability of making an error to ${0}$.

A deep result from the 1990s exactly charaterizes the class of computational problems (languages) that a classical polynomial-time verifier can decide in this model: IP = PSPACE. In words, any problem whose solution can be found in polynomial space has an interactive proof in which the verifier only needs polynomial time. Now observe that PSPACE contains NP, and much more: in fact PSPACE contains BQP as well (and even QMA)! (See this nice recent article in Quanta for a gentle introduction to these complexity classes, and more.) Thus any problem that can be decided on a quantum computer can also be decided without a quantum computer, by interacting with a powerful entity, the prover, that can convince the verifier of the right answer without being able to induce her in error (in spite of the prover’s greater power).

Are we not done? We’re not! The problem is that the result PSPACE = IP, even when specialized to BQP, requires (for all we know) a prover whose power matches that of PSPACE (almost: see e.g. this recent result for a slighlty more efficient prover). And as much as our experimental quantum device inches towards the power of BQP, we certainly wouldn’t dare ask it to perform a PSPACE-hard computation. So even though in principle there do exist interactive proofs for BQP-complete languages, these interactive proofs require a prover whose computational power goes much beyond what we believe is physically achievable. But that’s useless (for us): back to square zero.

Interactive proofs with quantum provers

Prior to Mahadev’s result, a sequence of beautiful results in the late 2000’s introduced a clever extension of the model of interactive proofs by allowing the verifier to make use of a very limited quantum computer. For example, the verifier may have the ability to prepare single qubits in two possible bases of her choice, one qubit at a time, and send them to the prover. Or the verifier may have the ability to receive single qubits from the prover, one at a time, and measure them in one of two bases of her choice. In both cases it was shown that the verifier could combine such limited quantum capacity with the possibility to interact with a quantum polynomial-time prover to verify arbitrary polynomial-time quantum computation. The idea for the protocols crucially relied on the ability of the verifier to prepare qubits in a way that any deviation by the prover from the presecribed honest behavior would be detected (e.g. by encoding information in mutually unbiased bases unknown to the prover). For a decade the question remained open: can a completely classical verifier certify the computation performed by a quantum prover?

Mahadev’s result brings a positive resolution to this question. Mahadev describes a protocol with the following properties. First, as expected, for any quantum computation, there is a quantum prover that will convince the classical verifier of the right outcome for the computation. This property is called completeness of the protocol. Second, no prover can convince the classical verifier to accept a wrong outcome. This property is called soundness of the protocol. In Mahadev’s result the latter property comes with a twist: soundness holds provided the prover cannot break post-quantum cryptography. In contrast, the earlier results mentioned in the previous paragraph obtained protocols that were sound against an arbitrarily powerful prover. The additional cryptographic assumption gives Mahadev’s result a “win-win” flavor: either the protocol is sound, or someone in the quantum cloud has figured out how to break an increasingly standard cryptographic assumption (namely, post-quantum security of the Learning With Errors problem) — in all cases, a verified quantum feat!

In the remaining of the post I will give a high-level overview of Mahadev’s protocol and its analysis. For more detail, see the accompanying blog post.

The protocol is constructed in two steps. The first step builds on insights from works preceding this one. This step reduces the problem of verifying the outcome of an arbitrary quantum computation to a seemingly much simpler problem, that nevertheless encapsulates all the subtlety of the verification task. The problem is the following — in keeping with the terminology employed by Mahadev, I’ll call it the qubit commitment problem. Suppose that a prover claims to have prepared a single-qubit state of its choice; call it ${| \psi \rangle}$ (${| \psi \rangle}$ is not known to the verifier). Suppose the verifier challenges the prover for the outcome of a measurement performed on ${| \psi \rangle}$, either in the computational basis (the eigenbasis of the Pauli Z), or in the Hadamard basis (the eigenbasis of the Pauli X). Which basis to use is the verifier’s choice, but of course only one basis can be asked. Does there exist a protocol that guarantees that, at the end of the protocol, the verifier will be able to produce a bit that matches the true outcome of a measurement of ${| \psi \rangle}$ in the chosen basis? (More precisely, it should be that the verifier’s final bit has the same distribution as the outcome of a measurement of ${| \psi \rangle}$ in the chosen basis.)

The reduction that accomplishes this first step combines Kitaev’s circuit-to-Hamiltonian construction with some gadgetry from perturbation theory, and I will not describe it here. An important property of the reduction is that it is ultimately sufficient that the verifier has the guarantee that the measurement outcomes she obtains in either case, computational or Hadamard, are consistent with measurement outcomes for the correct measurements performed on some quantum state. In principle the state does not need to be related to anything the prover does (though of course the analysis will eventually define that state from the prover), it only needs to exist. Specifically, we wish to rule out situations where e.g. the prover claims that both outcomes are deterministically “0”, a claim that would violate the uncertainty principle. (For the sake of the argument, let’s ignore that in the case of a single qubit the space of outcomes allowed by quantum mechanics can be explicitly mapped out — in the actual protocol, the prover commits to ${n}$ qubits, not just one.)

Committing to a qubit

The second step of the protocol construction introduces a key idea. In order to accomplish the sought-after commitment, the verifier is going to engage in an initial commitment phase with the prover. In this phase, the prover is required to provide classical information to the verifier, that “commits” it to a specific qubit. This committed qubit is the state on which the prover will later perform the measurement asked by the verifier. The classical information obtained in the commitment phase will bind the prover to reporting the correct outcome, for both of the verifier’s basis choice — or risk being caught cheating.

How does this work? Commitments to bits, or even qubits, are an old story in cryptography. The standard method for committing to a bit ${b}$ is based on the use of a one-way permutation ${f}$, together with a hardcore predicate ${h}$ for ${f}$ (i.e.~an efficiently computable function ${h: \{0,1\}^n\rightarrow \{0,1\}}$ such that given ${f(x)}$, it is hard to predict ${h(x)}$). The construction goes as follows. The committer selects a uniformly random string ${r}$ and sends ${(y,m)=(f(r),h(r)\oplus b)}$. To unveil the commitment ${b}$, it is enough to reveal a string ${r}$ such that ${f(r)=y}$; the receiver can then compute ${h(r)}$ and decode ${b=h(r)\oplus m}$. The point is that since ${f}$ is a permutation, the value ${y}$ uniquely “commits” the sender to an ${r}$, and thus to a ${b}$; however, given ${y=f(r)}$ for an unknown ${r}$ the hardcore predicate ${h(r)}$ looks uniformly random, thus ${(y,m)}$ reveals no information about ${b}$ to the receiver.

What is new in Mahadev’s scheme is not only that the commitment is to a qubit, instead of a bit, but even more importabtly that the commitment is provided by classical information, which is necessary to obtain a classical protocol. (Commitments to qubits, using qubits, can be obtained by combining the quantum one-time pad with the commitment scheme described above.) To explain how this is achieved we’ll need a slightly more advanced crypographic primitive: a pair of injective trapdoor one-way functions ${f_0,f_1:\{0,1\}^n\rightarrow\{0,1\}^n}$. This means that it is easy to evaluate both functions on any input, but that given a value ${y}$ in their common range, it is hard to find a preimage of ${y}$ under either function — except if one is given the trapdoor information. (Note that this is an over-simplification of the actual primitive used by Mahadev, which has additional properties, including that of being “claw-free”.)

The commitment phase of the protocol works as follows. Starting from a state ${| \psi \rangle=\alpha| 0 \rangle+\beta| 1 \rangle}$ of its choice, the prover is supposed to perform the following steps. First, the prover creates a uniform superposition over the common domain of ${f_0}$ and ${f_1}$. Then it evaluates either function, ${f_0}$ or ${f_1}$, in an additional register, by controlling on the qubit of ${| \psi \rangle}$. Finally, the prover measures the register that contains the image of ${f_0}$ or ${f_1}$. This achieves the following sequence of transformations:

$\displaystyle \begin{array}{rcl} \alpha| 0 \rangle+\beta| 1 \rangle &\mapsto& (\alpha| 0 \rangle + \beta| 1 \rangle) \otimes \Big(2^{-n/2} \sum_{x\in\{0,1\}^n} | x \rangle\Big) \\ &\mapsto & 2^{-n/2} \sum_x \alpha | 0 \rangle| x \rangle| f_0(x) \rangle + \beta | 1 \rangle| f_1(x) \rangle\\ &\mapsto & \big(\alpha| 0 \rangle| x_0 \rangle+\beta| 1 \rangle| x_1 \rangle\big)| y \rangle\;, \end{array}$

where ${y\in\{0,1\}^n}$ is the measured image. The string ${y}$ is called the prover’s commitment string. It is required to report it to the verifier.

In what sense is ${y}$ a commitment to the state ${| \psi \rangle}$? The key point is that, once it has measured ${y}$, the prover has “lost control” over its qubit — it has effectively handed over that control to the verifier. For example, the prover no longer has the ability to perform an arbitrary rotation on its qubit. Why? The prover knows ${y}$ (it had to report it to the verifier) but not ${x_0}$ and ${x_1}$ (this is the claw-free assumption on the pair ${(f_0,f_1)}$). What this means — though of course it has to be shown — is that the prover can no longer recover the state ${| \psi \rangle}$! It does not have the ability to “uncompute” ${x_0}$ and ${x_1}$. Thus the qubit has been “set in cryptographic stone”. In contrast, the verifier can use the trapdoor information to recover ${x_0}$ and ${x_1}$. This gives her extra leverage on the prover. This asymmetry, introduced by cryptography, is what eventually allows the verifier to extract a truthful measurement outcome from the prover (or detect lying).

It is such a wonderful idea! It stuns me every time Urmila explains it. Proving it is of course rather delicate. In this post I make an attempt at going into the idea in a little more depth. The best resource remains Urmila’s paper, as well as her talk at the Simons Institute.

Open questions

What is great about this result is not that it closes a decades-old open question, but that by introducing a truly novel idea it opens up a whole new field of investigation. Some of the ideas that led to the result were already fleshed out by Mahadev in her work on homomorphic encryption for quantum circuits, and I expect many more results to continue building on these ideas.

An obvious outstanding question is whether the cryptography is needed at all: could there be a scheme achieving the same result as Mahadev’s, but without computational assumptions on the prover? It is known that if such a scheme exists, it is unlikely to have the property of being blind, meaning that the prover learns nothing about the computation that the verifier wishes it to execute (aside from an upper bound on its length); see this paper for “implausibility” results in this direction. Mahadev’s protocol relies on “post-hoc” verification, and is not blind. Urmila points out that it is likely the protocol could be made blind by composing it with her protocol for homomorphic encryption. Could there be a different protocol, that would not go through post-hoc verification, but instead directly guide the prover through the evaluation of a universal circuit on an encrypted input, gate by gate, as did some previous works?

So long, and thanks for all the Fourier transforms

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.

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.

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.

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.

The World Cup from a Quantum Perspective

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)

What’s the worst that could happen?

The archaeologist Howard Carter discovered Tutankhamun’s burial site in 1922. No other Egyptian pharaoh’s tomb had survived mostly intact until the modern era. Gold and glass and faience, statues and pendants and chariots, had evaded looting. The discovery would revolutionize the world’s understanding of, and enthusiasm for, ancient Egypt.

First, the artifacts had to leave the tomb.

Tutankhamun lay in three coffins nested like matryoshka dolls. Carter describes the nesting in his book The Tomb of Tutankhamen. Lifting the middle coffin from the outer coffin raised his blood pressure:

Everything may seem to be going well until suddenly, in the crisis of the process, you hear a crack—little pieces of surface ornament fall. Your nerves are at an almost painful tension. What is happening? All available room in the narrow space is crowded by your men. What action is needed to avert a catastrophe?

In other words, “What’s the worst that could happen?”

Collaborators and I asked that question in a paper published last month. We had in mind less Egyptology than thermodynamics and information theory. But never mind the distinction; you’re reading Quantum Frontiers! Let’s mix the fields like flour and oil in a Biblical grain offering.

Carter’s team had trouble separating the coffins: Ancient Egyptian priests (presumably) had poured fluid atop the innermost, solid-gold coffin. The fluid had congealed into a brown gunk, gluing the gold coffin to the bottom of the middle coffin. Removing the gold coffin required work—thermodynamic work.

Work consists of “well-ordered” energy usable in tasks like levering coffins out of sarcophagi and motoring artifacts from Egypt’s Valley of the Kings toward Cairo. We can model the gunk as a spring, one end of which was fixed to the gold coffin and one end of which was fixed to the middle coffin. The work $W$ required to stretch a spring depends on the spring’s stiffness (the gunk’s viscosity) and on the distance stretched through.

$W$ depends also on details: How many air molecules struck the gold coffin from above, opposing the team’s effort? How quickly did Carter’s team pull? Had the gunk above Tuankhamun’s nose settled into a hump or spread out? How about the gunk above Tutankhamun’s left eye socket? Such details barely impact the work required to open a 6.15-foot-long coffin. But air molecules would strongly impact $W$ if Tutankhamun measured a few nanometers in length. So imagine Egyptian matryoshka dolls as long as stubs of DNA.

Imagine that Carter found one million sets of these matryoshka dolls. Lifting a given set’s innermost coffin would require an amount $W$ of work that would vary from set of coffins to set of coffins. $W$ would satisfy fluctuation relations, equalities I’ve blogged about many times.

Fluctuation relations resemble the Second Law of Thermodynamics, which illuminates why time flows in just one direction. But fluctuation relations imply more-precise predictions about $W$ than the Second Law does.

Some predictions concern dissipated work: Carter’s team could avoid spending much work by opening the coffin infinitesimally slowly. Speeding up would heat the gunk, roil air molecules, and more. The heating and roiling would cost extra work, called dissipated work, denoted by $W_{\rm diss}$.

Suppose that Carter’s team has chosen a lid-opening speed $v$. Consider the greatest $W_{\rm diss}$ that the team might have to waste on any nanoscale coffin. $W_{\rm diss}^{\rm worst}$ is proportional to each of three information-theoretic quantities, my coauthors and I proved.

For experts: Each information-theoretic quantity is an order-infinity Rényi divergence $D_\infty ( X || Y)$. The Rényi divergences generalize the relative entropy $D ( X || Y )$. $D$ quantifies how efficiently one can distinguish between probability distributions, or quantum states, $X$ and $Y$ on average. The average is over many runs of a guessing game.

Imagine the worst possible run, which offers the lowest odds of guessing correctly. $D_\infty$ quantifies your likelihood of winning. We related $W_{\rm diss}^{\rm worst}$ to a $D_\infty$ between two statistical-mechanical phase-space distributions (when we described classical systems), to a $D_\infty$ between two quantum states (when we described quantum systems), and to a $D_\infty$ between two probability distributions over work quantities $W$ (when we described systems quantum and classical).

The worst case marks an extreme. How do the extremes consistent with physical law look? As though they’ve escaped from a mythologist’s daydream.

In an archaeologist’s worst case, arriving at home in the evening could lead to the following conversation:

“The worst possible.”

“What happened?”

“I accidentally eviscerated a 3.5-thousand-year-old artifact—the most valuable, best-preserved, most information-rich, most lavishly wrought ancient Egyptian coffin that existed yesterday.”

Suppose that the archaeologist lived with a physicist. My group (guided by a high-energy physicist) realized that the conversation could continue as follows:

“Also the worst possible.”

“What happened?”

“I created a black hole.”

General relativity and high-energy physics have begun breeding with quantum information and thermodynamics. The offspring bear extremes like few other systems imaginable. I wonder what our results would have to say about those offspring.

National Geographic reprinted Carter’s The Tomb of Tutankhamen in its “Adventure Classics” series. The series title fits Tomb as a mummy’s bandages fit the mummy. Carter’s narrative stretches from Egypt’s New Kingdom (of 3.5 thousand years ago) through the five-year hunt for the tomb (almost fruitless until the final season), to a water boy’s discovery of steps into the tomb, to the unsealing of the burial chamber, to the confrontation of Tutankhamun’s mummy.

Carter’s book guided me better than any audio guide could have at the California Science Center. The center is hosting the exhibition “King Tut: Treasures of the Golden Pharaoh.” After debuting in Los Angeles, the exhibition will tour the world. The tour showcases 150 artifacts from Tutankhamun’s tomb.

Those artifacts drove me to my desk—to my physics—as soon as I returned home from the museum. Tutankhamun’s tomb, Carter argues in his book, ranks amongst the 20th century’s most important scientific discoveries. I’d seen a smidgeon of the magnificence that Carter’s team— with perseverance, ingenuity, meticulousness, and buckets of sweat shed in Egypt’s heat—had discovered. I don’t expect to discover anything a tenth as magnificent. But how can a young scientist resist trying?

People say, “Prepare for the worst. Hope for the best.” I prefer “Calculate the worst. Hope and strive for a Tutankhamun.”

Postscript: Carter’s team failed to unglue the gold coffin by just “stretching” the gunky “spring.” The team resorted to heat, a thermodynamic quantity alternative to work: The team flipped the middle coffin upside-down above a heat lamp. The lamp raised the temperature to 932°F, melting the goo. The melting, with more work, caused the gold coffin to plop out of the middle coffin.