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.

Arm 2

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 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.bit_commit_cartoon

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.

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.


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.

A poem for Stephen Hawking

Everyone is talking
About Stephen Hawking.
My good friend
Explained how time can end.
And clued us in
On how time can begin.

Always droll,
He spoke about a hole:
“Now, wait a minute, Jack,
A black hole ain’t so black!”

Those immortal words he said,
Which millions now have duly read,
Hit physics like a ton of bricks.
Well, that’s how Stephen got his kicks.

Always grinning through his glasses,
He brought science to the masses,
Displayed a rare capacity
For humor and audacity.

And that’s why, on this somber day,
With relish we can gladly say:
“Thanks, Stephen, for the things you’ve done.
And most of all, thanks for the fun!”

And though there’s more to say, my friend,
This poem, too, must, sadly, end.

Techs in flux & Rock & Roll

Each year, 10000 physicists descend on one of America’s finest inner cities in a ritual known as the American Physical Society’s March Meeting. If you are thinking that this is going to be one big nerd fest, you’re about right. From my experience, the backpacks, poster tubes, non-brand clothing, and distracted looks will be very easy to distinguish among the inhabitants of downtown LA (this year’s location) come next week.

However, with that many physicists, you will find a few trying to make science cool, or at least having fun while they try. One relatively untapped market in my opinion is montages. Take the Imagine Dragons song Believer, whose music video has lead signer Dan Reynolds mostly getting his ass kicked by veteran brawler Dolph Lundgren. Who says that training montages can’t also be for mental training? Sub out Dan for a young graduate student, replace Dolph with an imposing physicist, and substitute boxing with drama about writing equations on paper or a blackboard. Don’t believe it can be cool? I don’t blame you, but science montages have been done before, playing to science’s mystical side. And with sufficient experience, creativity, and money, I believe the sky is the limit.

But back to more concrete things. Having fun while trying to promote science is the main goal of the March Meeting Rock ‘n Roll Physics Sing-Along — a social and outreach event where a band of musicians, mostly scientists attending the meeting, plays well-known songs whose lyrics are substituted for science-themed prose. The audience then sings the new technically oriented lyrics along with the performers. Below is an example with the Smashmouth song I’m a Believer, but we play all kinds of genres, from power ballads to Britney Spears.

This year, we have an especially exciting line-up as we are joined by professional science entertainer, Einstein’s girl Gia Mora! Some of you may remember Gia from her performance with John Preskill at One Entangled Evening. She will join us to perform, among other hits, the funky E=mc^2:

The sing-along is run by the curator of all things related to physics songs, singer and songwriter Prof. Walter F. Smith of Haverford College. Adept at using songs to help teach physics, Walter has carefully collected a database of such songs dating back to the early 20th century; he believes that James Clerk Maxwell may have been the first song parody-er with his version of the lyrics to the Scotch Air Comin’ Thro’ the Rye. You can see James jamming alongside Emmy Noether, Paul Dirac, and Satyendra Bose below to questionable lyrics. The most well-known US physics song pioneer is Harvard grad Tom Lehrer, who recorded his first album in the 50s. Contrary to the general nature of scientists to be constantly worried about preserving their neutral academic self-image, Lehrer tackled edgy topics with creativity and humor.

poster art medium.jpg

The sing-along started in 2006, where the only accompaniment was a guitar and bongo, growing into a full rock band later on. The drums were first played by a Soviet-born physicist named Victor, and that has yet to change today despite it being a different person. The rest of the band this year consists of Walter, his wife Marian McKenzie on the flute, Lev Krayzman from Yale on the guitar, Prof. Esa Räsänen from Tampere University of Technology on the bass, Lenny Campanello from the University of Maryland on the keyboard, and of course the talented Gia Mora on voice. We hope that you can join us next week, as this year’s sing-along is sure to be one for the books!

March Meeting Rock-n-Roll Physics Sing-along
Wednesday, March 7, 2018
9:00 PM–10:30 PM
J.W. Marriott Room: Platinum D

See you there!

Glass beads and weak-measurement schemes

Richard Feynman fiddled with electronics in a home laboratory, growing up. I fiddled with arts and crafts.1 I glued popsicle sticks, painted plaques, braided yarn, and designed greeting cards. Of the supplies in my family’s crafts box, I adored the beads most. Of the beads, I favored the glass ones.

I would pour them on the carpet, some weekend afternoons. I’d inherited a hodgepodge: The beads’ sizes, colors, shapes, trimmings, and craftsmanship varied. No property divided the beads into families whose members looked like they belonged together. But divide the beads I tried. I might classify them by color, then subdivide classes by shape. The color and shape groupings precluded me from grouping by size. But, by loosening my original classification and combining members from two classes, I might incorporate trimmings into the categorization. I’d push my classification scheme as far as I could. Then, I’d rake the beads together and reorganize them according to different principles.

Why have I pursued theoretical physics? many people ask. I have many answers. They include “Because I adored organizing craft supplies at age eight.” I craft and organize ideas.


I’ve blogged about the out-of-time-ordered correlator (OTOC), a signature of how quantum information spreads throughout a many-particle system. Experimentalists want to measure the OTOC, to learn how information spreads. But measuring the OTOC requires tight control over many quantum particles.

I proposed a scheme for measuring the OTOC, with help from Chapman University physicist Justin Dressel. The scheme involves weak measurements. Weak measurements barely disturb the systems measured. (Most measurements of quantum systems disturb the measured systems. So intuited Werner Heisenberg when formulating his uncertainty principle.)

I had little hope for the weak-measurement scheme’s practicality. Consider the stereotypical experimentalist’s response to a stereotypical experimental proposal by a theorist: Oh, sure, we can implement that—in thirty years. Maybe. If the pace of technological development doubles. I expected to file the weak-measurement proposal in the “unfeasible” category.

But experimentalists started collaring me. The scheme sounds reasonable, they said. How many trials would one have to perform? Did the proposal require ancillas, helper systems used to control the measured system? Must each ancilla influence the whole measured system, or could an ancilla interact with just one particle? How did this proposal compare with alternatives?

I met with a cavity-QED2 experimentalist and a cold-atoms expert. I talked with postdocs over skype, with heads of labs at Caltech, with grad students in Taiwan, and with John Preskill in his office. I questioned an NMR3 experimentalist over lunch and fielded superconducting-qubit4 questions in the sunshine. I read papers, reread papers, and powwowed with Justin.

I wouldn’t have managed half so well without Justin and without Brian Swingle. Brian and coauthors proposed the first OTOC-measurement scheme. He reached out after finding my first OTOC paper.

According to that paper, the OTOC is a moment of a quasiprobability.5 How does that quasiprobability look, we wondered? How does it behave? What properties does it have? Our answers appear in a paper we released with Justin this month. We calculate the quasiprobability in two examples, prove properties of the quasiprobability, and argue that the OTOC motivates generalizations of quasiprobability theory. We also enhance the weak-measurement scheme and analyze it.

Amidst that analysis, in a 10 x 6 table, we classify glass beads.


We inventoried our experimental conversations and distilled them. We culled measurement-scheme features analogous to bead size, color, and shape. Each property labels a row in the table. Each measurement scheme labels a column. Each scheme has, I learned, gold flecks and dents, hues and mottling, an angle at which it catches the light.

I’ve kept most of the glass beads that fascinated me at age eight. Some of the beads have dispersed to necklaces, picture frames, and eyeglass leashes. I moved the remnants, a few years ago, to a compartmentalized box. Doesn’t it resemble the table?


That’s why I work at the IQIM.


1I fiddled in a home laboratory, too, in a garage. But I lived across the street from that garage. I lived two rooms from an arts-and-crafts box.

2Cavity QED consists of light interacting with atoms in a box.

3Lots of nuclei manipulated with magnetic fields. “NMR” stands for “nuclear magnetic resonance.” MRI machines, used to scan brains, rely on NMR.

4Superconducting circuits are tiny, cold quantum circuits.

5A quasiprobability resembles a probability but behaves more oddly: Probabilities range between zero and one; quasiprobabilities can dip below zero. Think of a moment as like an average.

With thanks to all who questioned me; to all who answered questions of mine; to my wonderful coauthors; and to my parents, who stocked the crafts box.

Here’s one way to get out of a black hole!

Two weeks ago I attended an exciting workshop at Stanford, organized by the It from Qubit collaboration, which I covered enthusiastically on Twitter. Many of the talks at the workshop provided fodder for possible blog posts, but one in particular especially struck my fancy. In explaining how to recover information that has fallen into a black hole (under just the right conditions), Juan Maldacena offered a new perspective on a problem that has worried me for many years. I am eagerly awaiting Juan’s paper, with Douglas Stanford and Zhenbin Yang, which will provide more details.


My cell-phone photo of Juan Maldacena lecturing at Stanford, 22 March 2017.

Almost 10 years ago I visited the Perimeter Institute to attend a conference, and by chance was assigned an office shared with Patrick Hayden. Patrick was a professor at McGill at that time, but I knew him well from his years at Caltech as a Sherman Fairchild Prize Fellow, and deeply respected him. Our proximity that week ignited a collaboration which turned out to be one of the most satisfying of my career.

To my surprise, Patrick revealed he had been thinking about  black holes, a long-time passion of mine but not previously a research interest of his, and that he had already arrived at a startling insight which would be central to the paper we later wrote together. Patrick wondered what would happen if Alice possessed a black hole which happened to be highly entangled with a quantum computer held by Bob. He imagined Alice throwing a qubit into the black hole, after which Bob would collect the black hole’s Hawking radiation and feed it into his quantum computer for processing. Drawing on his knowledge about quantum communication through noisy channels, Patrick argued that  Bob would only need to grab a few qubits from the radiation in order to salvage Alice’s qubit successfully by doing an appropriate quantum computation.


Alice tosses a qubit into a black hole, which is entangled with Bob’s quantum computer. Bob grabs some Hawking radiation, then does a quantum computation to decode Alice’s qubit.

This idea got my adrenaline pumping, stirring a vigorous dialogue. Patrick had initially assumed that the subsystem of the black hole ejected in the Hawking radiation had been randomly chosen, but we eventually decided (based on a simple picture of the quantum computation performed by the black hole) that it should take a time scaling like M log M (where M is the black hole mass expressed in Planck units) for Alice’s qubit to get scrambled up with the rest of her black hole. Only after this scrambling time would her qubit leak out in the Hawking radiation. This time is actually shockingly short, about a millisecond for a solar mass black hole. The best previous estimate for how long it would take for Alice’s qubit to emerge (scaling like M3), had been about 1067 years.

This short time scale aroused memories of discussions with Lenny Susskind back in 1993, vividly recreated in Lenny’s engaging book The Black Hole War. Because of the black hole’s peculiar geometry, it seemed conceivable that Bob could distill a copy of Alice’s qubit from the Hawking radiation and then leap into the black hole, joining Alice, who could then toss her copy of the qubit to Bob. It disturbed me that Bob would then hold two perfect copies of Alice’s qubit; I was a quantum information novice at the time, but I knew enough to realize that making a perfect clone of a qubit would violate the rules of quantum mechanics. I proposed to Lenny a possible resolution of this “cloning puzzle”: If Bob has to wait outside the black hole for too long in order to distill Alice’s qubit, then when he finally jumps in it may be too late for Alice’s qubit to catch up to Bob inside the black hole before Bob is destroyed by the powerful gravitational forces inside. Revisiting that scenario, I realized that the scrambling time M log M, though short, was just barely long enough for the story to be self-consistent. It was gratifying that things seemed to fit together so nicely, as though a deep truth were being affirmed.


If Bob decodes the Hawking radiation and then jumps into the black hole, can he acquire two identical copies of Alice’s qubit?

Patrick and I viewed our paper as a welcome opportunity to draw the quantum information and quantum gravity communities closer together, and we wrote it with both audiences in mind. We had fun writing it, adding rhetorical flourishes which we hoped would draw in readers who might otherwise be put off by unfamiliar ideas and terminology.

In their recent work, Juan and his collaborators propose a different way to think about the problem. They stripped down our Hawking radiation decoding scenario to a model so simple that it can be analyzed quite explicitly, yielding a pleasing result. What had worried me so much was that there seemed to be two copies of the same qubit, one carried into the black hole by Alice and the other residing outside the black hole in the Hawking radiation. I was alarmed by the prospect of a rendezvous of the two copies. Maldacena et al. argue that my concern was based on a misconception. There is just one copy, either inside the black hole or outside, but not both. In effect, as Bob extracts his copy of the qubit on the outside, he destroys Alice’s copy on the inside!

To reach this conclusion, several ideas are invoked. First, we analyze the problem in the case where we understand quantum gravity best, the case of a negatively curved spacetime called anti-de Sitter space.  In effect, this trick allows us to trap a black hole inside a bottle, which is very advantageous because we can study the physics of the black hole by considering what happens on the walls of the bottle. Second, we envision Bob’s quantum computer as another black hole which is entangled with Alice’s black hole. When two black holes in anti-de Sitter space are entangled, the resulting geometry has a “wormhole” which connects together the interiors of the two black holes. Third, we chose the entangled pair of black holes to be in a very special quantum state, called the “thermofield double” state. This just means that the wormhole connecting the black holes is as short as possible. Fourth, to make the analysis even simpler, we suppose there is just one spatial dimension, which makes it easier to draw a picture of the spacetime. Now each wall of the bottle is just a point in space, with the left wall lying outside Bob’s side of the wormhole, and the right wall lying outside Alice’s side.

An important property of the wormhole is that it is not traversable. That is, when Alice throws her qubit into her black hole and it enters her end of the wormhole, the qubit cannot emerge from the other end. Instead it is stuck inside, unable to get out on either Alice’s side or Bob’s side. Most ways of manipulating the black holes from the outside would just make the wormhole longer and exacerbate the situation, but in a clever recent paper Ping Gao, Daniel Jafferis, and Aron Wall pointed out an exception. We can imagine a quantum wire connecting the left wall and right wall, which simulates a process in which Bob extracts a small amount of Hawking radiation from the right wall (that is, from Alice’s black hole), and carefully deposits it on the left wall (inserting it into Bob’s quantum computer). Gao, Jafferis, and Wall find that this procedure, by altering the trajectories of Alice’s and Bob’s walls, can actually make the wormhole traversable!


(a) A nontraversable wormhole. Alice’s qubit, thrown into the black hole, never reaches Bob. (b) Stealing some Hawking radiation from Alice’s side and inserting it on Bob’s side makes the wormhole traversable. Now Alice’s qubit reaches Bob, who can easily “decode” it.

This picture gives us a beautiful geometric interpretation of the decoding protocol that Patrick and I had described. It is the interaction between Alice’s wall and Bob’s wall that brings Alice’s qubit within Bob’s grasp. By allowing Alice’s qubit to reach Bob at the other end of the wormhole, that interaction suffices to perform Bob’s decoding task, which is especially easy in this case because Bob’s quantum computer was connected to Alice’s black hole by a short wormhole when she threw her qubit inside.


If, after a delay, Bob’s jumps into the black hole, he might find Alice’s qubit inside. But if he does, that qubit cannot be decoded by Bob’s quantum computer. Bob has no way to attain two copies of the qubit.

And what if Bob conducts his daring experiment, in which he decodes Alice’s qubit while still outside the black hole, and then jumps into the black hole to check whether the same qubit is also still inside? The above spacetime diagram contrasts two possible outcomes of Bob’s experiment. After entering the black hole, Alice might throw her qubit toward Bob so he can catch it inside the black hole. But if she does, then the qubit never reaches Bob’s quantum computer, and he won’t be able to decode it from the outside. On the other hand, Alice might allow her qubit to reach Bob’s quantum computer at the other end of the (now traversable) wormhole. But if she does, Bob won’t find the qubit when he enters the black hole. Either way, there is just one copy of the qubit, and no way to clone it. I shouldn’t have been so worried!

Granted, we have only described what happens in an oversimplified model of a black hole, but the lessons learned may be more broadly applicable. The case for broader applicability rests on a highly speculative idea, what Maldacena and Susskind called the ER=EPR conjecture, which I wrote about in this earlier blog post. One consequence of the conjecture is that a black hole highly entangled with a quantum computer is equivalent, after a transformation acting only on the computer, to two black holes connected by a short wormhole (though it might be difficult to actually execute that transformation). The insights of Gao-Jafferis-Wall and Maldacena-Stanford-Yang, together with the ER=EPR viewpoint, indicate that we don’t have to worry about the same quantum information being in two places at once. Quantum mechanics can survive the attack of the clones. Whew!

Thanks to Juan, Douglas, and Lenny for ongoing discussions and correspondence which have helped me to understand their ideas (including a lucid explanation from Douglas at our Caltech group meeting last Wednesday). This story is still unfolding and there will be more to say. These are exciting times!