The question of how to formulate a quantum theory of gravity is a long-standing open problem in theoretical physics. Somewhat recently, an idea that has gained a lot of traction (and that Spiros has blogged about before) is emergence. This is the idea that space and time may emerge from some more fine-grained quantum objects and their interactions. If we could understand how classical spacetime emerges from an underlying quantum system, then it’s not too much of a stretch to hope that this understanding would give us insight into the full quantum nature of spacetime.

One type of emergence is exhibited in holography, which is the idea that certain (*D*+1)-dimensional systems with gravity are exactly equivalent to *D*-dimensional quantum theories without gravity. (Note that we’re calling time a dimension here. For example, you would say that on a day-to-day basis we experience *D *= 4 dimensions.) In this case, that extra +1 dimension and the concomitant gravitational dynamics are emergent phenomena.

A nice aspect of holography is that it is explicitly realized by the AdS/CFT correspondence. This correspondence proposes that a particular class of spacetimes—ones that asymptotically look like anti-de Sitter space, or AdS—are equivalent to states of a particular type of quantum system—a conformal field theory, or CFT. A convenient visualization is to draw the AdS spacetime as a cylinder, where time marches forward as you move up the cylinder and different slices of the cylinder correspond to snapshots of space at different instants of time. Conveniently, in this picture you can think of the corresponding CFT as living on the boundary of the cylinder, which, you should note, has one less dimension than the “bulk” inside the cylinder.

Even within this nice picture of holography that we get from the AdS/CFT correspondence, there is a question of how exactly do CFT, or boundary quantities map onto quantities in the AdS bulk. This is where a certain tool from quantum information theory called tensor networks has recently shown a lot of promise.

A tensor network is a way to efficiently represent certain states of a quantum system. Moreover, they have nice graphical representations which look something like this:

Beni discussed one type of tensor network in his post on holographic codes. In this post, let’s discuss the tensor network shown above, which is known as the Multiscale Entanglement Renormalization Ansatz, or MERA.

The MERA was initially developed by Guifre Vidal and Glen Evenbly as an efficient approximation to the ground state of a CFT. Roughly speaking, in the picture of a MERA above, one starts with a simple state at the centre, and as you move outward through the network, the MERA tells you how to build up a CFT state which lives on the legs at the boundary. The MERA caught the eye of Brian Swingle, who noticed that it looks an awfully lot like a discretization of a slice of the AdS cylinder shown above. As such, it wasn’t a preposterously big leap to suggest a possible “AdS/MERA correspondence.” Namely, perhaps it’s more than a simple coincidence that a MERA both encodes a CFT state and resembles a slice of AdS. Perhaps the MERA gives us the tools that are required to construct a map between the boundary and the bulk!

So, how seriously should one take the possibility of an AdS/MERA correspondence? That’s the question that my colleagues and I addressed in a recent paper. Essentially, there are several properties that a consistent holographic theory should satisfy in both the bulk and the boundary. We asked whether these properties are still simultaneously satisfied in a correspondence where the bulk and boundary are related by a MERA.

What we found was that you invariably run into inconsistencies between bulk and boundary physics, at least in the simplest construals of what an AdS/MERA correspondence might be. This doesn’t mean that there is no hope for an AdS/MERA correspondence. Rather, it says that the simplest approach will not work. For a good correspondence, you would need to augment the MERA with some additional structure, or perhaps consider different tensor networks altogether. For instance, the holographic code features a tensor network which hints at a possible bulk/boundary correspondence, and the consistency conditions that we proposed are a good list of checks for Beni and company as they work out the extent to which the code can describe holographic CFTs. Indeed, a good way to summarize how our work fits into the picture of quantum gravity alongside holography and tensors networks is by saying that it’s nice to have good signposts on the road when you don’t have a map.

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I’ve asked in seminars, in lectures, in offices, and at group meetings. I’ve asked about physical conjectures, about theorems, and about mathematical properties.

“I don’t know.” Lecturers have shrugged. “It’s just a name.”

This spring, I asked about master equations. I thought of them as tools used in statistical mechanics, the study of vast numbers of particles. We can’t measure vast numbers of particles, so we can’t learn about stat-mech systems everything one might want to know. The magma beneath Santorini, for example, consists of about 10^{24} molecules. Good luck measuring every one.

Imagine, as another example, using a quantum computer to solve a problem. We load information by initializing the computer to a certain *state*: We orient the computer’s particles in certain directions. We run a program, then read out the output.

Suppose the computer sits on a tabletop, exposed to the air like leftover casserole no one wants to save for tomorrow. Air molecules bounce off the computer, becoming entangled with the hardware. This entanglement, or quantum correlation, alters the computer’s state, just as flies alter a casserole.^{*} To understand the computer’s output—which depends on the state, which depends on the air—we must have a description of the air. But we can’t measure all those air molecules, just as we can’t measure all the molecules in Santorini’s magma.

We can package our knowledge about the computer’s state into a mathematical object, called a *density operator*, labeled by ρ(t). A *quantum master equation *describes how ρ(t) changes. I had no idea, till this spring, why we call master equations “master equations.” Had someone named “John Master” invented them? Had the inspiration for the Russell Crowe movie *Master and Commander*? Or the Igor who lisps, “Yeth, mathter” in adaptations of *Frankenstein*?

Jenia Mozgunov, a fellow student and Preskillite, proposed an answer: Using master equations, we can calculate how averages of observable properties change. Imagine describing a laser, a cavity that spews out light. A master equation reveals how the average number of photons (particles of light) in the cavity changes. We want to predict these averages because experimentalists measure them. Because master equations spawn many predictions—many equations—they merit the label “master.”

Jenia’s hypothesis appealed to me, but I wanted certainty. I wanted Truth. I opened my laptop and navigated to Facebook.

“Does anyone know,” I wrote in my status, “why master equations are called ‘master equations’?”

Ian Durham, a physicist at St. Anselm College, cited Tom Moore’s *Six Ideas that Shaped Physics. *Most physics problems, Ian wrote, involve “some overarching principle.” Example principles include energy conservation and invariance under discrete translations (the system looks the same after you step in some direction). A master equation encapsulates this principle.

Ian’s explanation sounded sensible. But fewer people “liked” his reply on Facebook than “liked” a quip by a college friend: Master equations deserve their name because “[t]hey didn’t complete all the requirements for the doctorate.”

My advisor, John Preskill, dug through two to three books, one set of lecture notes, one German Wikipedia page, one to two articles, and Google Scholar. He concluded that Nordsieck, Lamb, and Uhlenbeck coined “master equation.” According to a 1940 paper of theirs,^{**} “When the probabilities of the elementary processes are known, one can write down a continuity equation for *W* [a set of probabilities], from which all other equations can be derived and which we will call therefore the ‘master’ equation.”

“Are you sure you were meant to be a physicist,” I asked John, “rather than a historian?”

“Procrastination is a powerful motivator,” he replied.

Lecturers have shrugged at questions about names. Then they’ve paused, pondered, and begun, “I guess because…” Theorems and identities derive their names from symmetries, proof techniques, geometric illustrations, and applications to problems I’d thought unrelated. A name taught me about uses for master equations. Names reveal physics I wouldn’t learn without asking about names. Names aren’t just names. They’re lamps and guides.

Pity about the origin of “master equation,” though. I wish an Igor had invented them.

^{*}Apologies if I’ve spoiled your appetite.

^{**}A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I,” *Physica* **7**, 344-60 (1940), p. 353.

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fig has total # uses of qubit in arxiv (divided by 10) per month, and total # docs per month: an impressive 669394 total in 29587 docs. the graph starts at 9412 (dec '94), but that is illusory since qubit only shows up in v2 of hep-th/9412048, posted in 2004. the actual first was quant-ph/9503016 by bennett/divicenzo/shor et al (posted 23 Mar '95) where they carefully attribute the term to schumacher ("PRA, to appear '95") and jozsa/schumacher ("J. Mod Optics '94"), followed immediately by quant-ph/9503017 by deutsch/jozsa et al (which no longer finds it necessary to attribute term) [neither of schumacher's first two articles is on arxiv, but otherwise probably have on arxiv near 100% coverage of its usage and growth, so permits a viral epidemic analysis along the lines of kaiser's "drawing theories apart" of use of Feynman diagrams in post wwII period]. ever late to the party, the first use by j.preskill was quant-ph/9602016, posted 21 Feb 1996 #articles by primary subject area as follows (hep-th is surprisingly low given the firewall connection...): quant-ph 22096 cond-mat.mes-hall 3350 cond-mat.supr-con 880 cond-mat.str-el 376 cond-mat.mtrl-sci 250 math-ph 244 hep-th 228 physics.atom-ph 224 cond-mat.stat-mech 213 cond-mat.other 200 physics.optics 177 cond-mat.quant-gas 152 physics.gen-ph 120 gr-qc 105 cond-mat 91 cs.CC 85 cs.IT 67 cond-mat.dis-nn 55 cs.LO 49 cs.CR 43 physics.chem-ph 33 cs.ET 25 physics.ins-det 21 math.CO,nlin.CD 20 physics.hist-ph,physics.bio-ph,math.OC 19 hep-ph 18 cond-mat.soft,cs.DS,math.OA 17 cs.NE,cs.PL,math.QA 13 cs.AR,cs.OH 12 physics.comp-ph 11 math.LO 10 physics.soc-ph,physics.ed-ph,cs.AI 9 math.ST,physics.pop-ph,cs.GT 8 nlin.AO,astro-ph,cs.DC,cs.FL,q-bio.GN 7 nlin.PS,math.FA,cs.NI,math.PR,q-bio.NC,physics.class-ph,math.GM, physics.data-an 6 nlin.SI,math.CT,q-fin.GN,cs.LG,q-bio.BM,cs.DM,math.GT 5 math.DS,physics.atm-clus,q-bio.PE 4 math.DG,math.CA,nucl-th,q-bio.MN,math.HO,stat.ME,cs.MS,q-bio.QM, math.RA,math.AG,astro-ph.IM,q-bio.OT 3 stat.AP,cs.CV,math.SG,cs.SI,cs.SE,cs.SC,cs.DB,stat.ML,physics.med-ph, math.RT 2 cs.CL,cs.CE,q-fin.RM,chao-dyn,astro-ph.CO,q-fin.ST,math.NA, cs.SY,math.MG,physics.plasm-ph,hep-lat,math.GR,cs.MM,cs.PF,math.AC, nucl-ex 1

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But the event that moved me to write a blog post is the 1995 appearance of the word “qubit” in an American Physical Society journal. When I was a boy, two-level quantum systems were called “two-level quantum systems.” Which is a descriptive name, but a mouthful and far from euphonious. Think of all the time I’ve saved in the past 20 years by saying “qubit” instead of “two-level quantum system.” And saying “qubit” not only saves time, it also conveys the powerful insight that a quantum state encodes a novel type of information. (Alas, the spelling was bound to stir controversy, with the estimable David Mermin a passionate holdout for “qbit”. Give it up, David, you lost.)

For the word “qubit” we know whom to thank: Ben Schumacher. He introduced the word in his paper “Quantum Coding” which appeared in the April 1995 issue of Physical Review A. (History is complicated, and in this case the paper was actually submitted in 1993, which allowed another paper by Jozsa and Schumacher to be published earlier even though it was written and submitted later. But I’m celebrating the 20th anniversary of the qubit now, because otherwise how will I justify this blog post?)

In the acknowledgments of the paper, Ben provided some helpful background on the origin of the word:

The term “qubit” was coined in jest during one of the author’s many intriguing and valuable conversations with W. K. Wootters, and became the initial impetus for this work.

I met Ben (and other luminaries of quantum information theory) for the first time at a summer school in Torino, Italy in 1996. After reading his papers my expectations were high, all the more so after Sam Braunstein warned me that I would be impressed: “Ben’s a good talker,” Sam assured me. I was not disappointed.

(I also met Asher Peres at that Torino meeting. When I introduced myself Asher asked, “Isn’t there someone with a similar name in particle theory?” I had no choice but to come clean. I particularly remember that conversation because Asher told me his secret motivation for studying quantum entanglement: it might be important in quantum gravity!)

A few years later Ben spent his sabbatical year at Caltech, which gave me an opportunity to compose a poem for the introduction to Ben’s (characteristically brilliant) talk at our physics colloquium. This poem does homage to that famous 1995 paper in which Ben not only introduced the word “qubit” but also explained how to compress a quantum state to the minimal number of qubits from which the original state can be recovered with a negligible loss of fidelity, thus formulating and proving the quantum version of Shannon’s famous source coding theorem, and laying the foundation for many subsequent developments in quantum information theory.

Sometimes when I recite a poem I can sense the audience’s appreciation. But in this case there were only a few nervous titters. I was going for edgy but might have crossed the line into bizarre.. Since then I’ve (usually) tried to be more careful.

(For reading the poem, it helps to know that the quantum state appears to be random when it has been compressed as much as possible.)

**On Quantum Compression (in honor of Ben Schumacher)**

Ben.

He rocks.

I remember

When

He showed me how to fit

A qubit

In a small box.I wonder how it feels

To be compressed.

And then to pass

A fidelity test.Or does it feel

At all, and if it does

Would I squeal

Or be just as I was?If not undone

I’d become as I’d begun

And write a memorandum

On being random.

Had it felt like a belt

Of rum?And might it be predicted

That I’d become addicted,

Longing for my session

Of compression?I’d crawl

To Ben again.

And call,

“Put down your pen!

Don’t stall!

Make me small!”

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I was standing in a hallway at the University of Maryland. On one side stood quantum-information theorists. On the other side stood statistical-mechanics scientists.^{*} The groups eyed each other, like Jets and Sharks in *West Side Story*, except without fighting or dancing.

This March, the groups were generous enough to host me for a visit. I parked first at QuICS, the Joint Center for Quantum Information and Computer Science. Established in October 2014, QuICS had moved into renovated offices the previous month. QuICSland boasts bright colors, sprawling armchairs, and the scent of novelty. So recently had QuICS arrived that the restroom had not acquired toilet paper (as I learned later than I’d have preferred).

From QuICS, I yoyo-ed to the chemistry building, where Chris Jarzynski’s group studies fluctuation relations. *Fluctuation relations*, introduced elsewhere on this blog, describe out-of-equilibrium systems. A system is *out of equilibrium* if large-scale properties of it change. Many systems operate out of equilibrium—boiling soup, combustion engines, hurricanes, and living creatures, for instance. Physicists want to describe nonequilibrium processes but have trouble: Living creatures are complicated. Hence the buzz about fluctuation relations.

My first Friday in Maryland, I presented a seminar about quantum voting for QuICS. The next Tuesday, I was to present about one-shot information theory for stat-mech enthusiasts. Each week, the stat-mech crowd invites its speaker to lunch. Chris Jarzynski recommended I invite QuICS. Hence the Jets-and-Sharks tableau.

“Have you interacted before?” I asked the hallway.

“No,” said a voice. QuICS hadn’t existed till last fall, and some QuICSers hadn’t had offices till the previous month.**

Silence.

“We’re QuICS,” volunteered Stephen Jordan, a quantum-computation theorist, “the Joint Center for Quantum Information and Computer Science.”

So began the mingling. It continued at lunch, which we shared at three circular tables we’d dragged into a chain. The mingling continued during the seminar, as QuICSers sat with chemists, materials scientists, and control theorists. The mingling continued the next day, when QuICSer Alexey Gorshkov joined my discussion with the Jarzynski group. Back and forth we yoyo-ed, between buildings and topics.

“Mingled,” said Yigit Subasi. Yigit, a postdoc of Chris’s, specialized in quantum physics as a PhD student. I’d asked how he thinks about quantum fluctuation relations. Since Chris and colleagues ignited fluctuation-relation research, theorems have proliferated like vines in a jungle. Everyone and his aunty seems to have invented a fluctuation theorem. I canvassed Marylanders for bushwhacking tips.

Imagine, said Yigit, a system whose state you know. Imagine a gas, whose temperature you’ve measured, at equilibrium in a box. Or imagine a trapped ion. Begin with a state about which you have information.

Imagine performing work on the system “violently.” Compress the gas quickly, so the particles roil. Shine light on the ion. The system will leave equilibrium. “The information,” said Yigit, “gets mingled.”

Imagine halting the compression. Imagine switching off the light. Combine your information about the initial state with assumptions and physical laws.^{***} Manipulate equations in the right way, and the information might “unmingle.” You might capture properties of the violence in a fluctuation relation.

I’m grateful to have exchanged information in Maryland, to have yoyo-ed between groups. We have work to perform together. I have transformations to undergo.^{****} Let the unmingling begin.

*With gratitude to Alexey Gorshkov and QuICS, and to Chris Jarzynski and the University of Maryland Department of Chemistry, for their hospitality, conversation, and camaraderie.*

^{*}*Statistical mechanics* is the study of systems that contain vast numbers of particles, like the air we breathe and white dwarf stars. I harp on about statistical mechanics often.

**Before QuICS’s birth, a future QuICSer had collaborated with a postdoc of Chris’s on combining quantum information with fluctuation relations.

^{***}Yes, physical laws are assumptions. But they’re glorified assumptions.

^{****}Hopefully nonviolent transformations.

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I tacked Dirac’s quote onto the bulletin board above my desk, the summer before senior year of high school. I’d picked quotes by T.S. Elliot and Einstein, Catullus and Hatshepsut.^{*} In a closet, I’d found amber-, peach-, and scarlet-colored paper. I’d printed the quotes and arranged them, starting senior year with inspiration that looked like a sunrise.

Not that I knew who Paul Dirac was. Nor did I evaluate his opinion. But I’d enrolled in Advanced Placement Physics C and taken the helm of my school’s literary magazine. The confluence of two passions of mine—science and literature—in Dirac’s quote tickled me.

A fiery lecturer began to alleviate my ignorance in college. Dirac, I learned, had co-invented quantum theory. The “*Dee-*rac Equa-shun,” my lecturer trilled in her Italian accent, describes relativistic quantum systems—tiny particles associated with high speeds. I developed a taste for *spin*, a quantum phenomenon encoded in Dirac’s equation. Spin serves quantum-information scientists as two-by-fours serve carpenters: Experimentalists have tried to build quantum computers from particles that have spins. Theorists keep the idea of electron spins in a mental car trunk, to tote out when illustrating abstract ideas with examples.

The next year, I learned that Dirac had predicted the existence of antimatter. Three years later, I learned to represent antimatter mathematically. I memorized the Dirac Equation, forgot it, and re-learned it.

One summer in grad school, visiting my parents, I glanced at my bulletin board.

The sun rises beyond a window across the room from the board. Had the light faded the papers’ colors? If so, I couldn’t tell.

*In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it’s the exact opposite!*

Do poets try to obscure ideas everyone understands? Some poets express ideas that people intuit but feel unable, lack the attention, or don’t realize one should, articulate. Reading and hearing poetry helps me grasp the ideas. Some poets express ideas in forms that others haven’t imagined.

Did Dirac not represent physics in a form that others hadn’t imagined?

Would you have imagined that form? I didn’t imagine it until learning it. Do scientists not express ideas—about gravity, time, energy, and matter—that people feel unable, lack the attention, or don’t realize we should, articulate?

The U.S. and Canada have designated April as National Poetry Month. A hub for cousins of poets,* Quantum Frontiers* salutes. Carry a poem in your pocket this month. Or carry a copy of the Dirac Equation. Or tack either on a bulletin board; I doubt whether their colors will fade.

^{*}“Now my heart turns this way and that, as I think what the people will say. Those who see my monuments in years to come, and who shall speak of what I have done.” I expect to build no such monuments. But here’s to trying.

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In a recent paper with Daniel Harlow, Fernando Pastawski and John Preskill, we have proposed a toy model of the AdS/CFT correspondence based on quantum error-correcting codes. Fernando has already written how this research project started after a fateful visit by Daniel to Caltech and John’s remarkable prediction in 1999. In this post, I hope to write an introduction which may serve as a reader’s guide to our paper, explaining why I’m so fascinated by the beauty of the toy model.

This is certainly a challenging task because I need to make it accessible to everyone while explaining real physics behind the paper. My personal philosophy is that a toy model must be as simple as possible while capturing key properties of the system of interest. In this post, I will try to extract some key features of the AdS/CFT correspondence and construct a toy model which captures these features. This post may be a bit technical compared to other recent posts, but anyway, let me give it a try…

**Bulk locality paradox and quantum error-correction**

The AdS/CFT correspondence says that there is some kind of correspondence between quantum gravity on (d+1)-dimensional asymptotically-AdS space and d-dimensional conformal field theory on its boundary. But how are they related?

The AdS-Rindler reconstruction tells us how to “reconstruct” a bulk operator from boundary operators. Consider a bulk operator and a boundary region A on a hyperbolic space (in other words, a negatively-curved plane). On a fixed time-slice, the causal wedge of A is a bulk region enclosed by the geodesic line of A (a curve with a minimal length). The AdS-Rindler reconstruction says that can be represented by some integral of local boundary operators supported on A if and only if is contained inside the causal wedge of A. Of course, there are multiple regions A,B,C,… whose causal wedges contain , and the reconstruction should work for any such region.

That a bulk operator in the causal wedge can be reconstructed by local boundary operators, however, leads to a rather perplexing paradox in the AdS/CFT correspondence. Consider a bulk operator at the center of a hyperbolic space, and split the boundary into three pieces, A, B, C. Then the geodesic line for the union of BC encloses the bulk operator, that is, is contained inside the causal wedge of BC. So, can be represented by local boundary operators supported on BC. But the same argument applies to AB and CA, implying that the bulk operator corresponds to local boundary operators which are supported inside AB, BC and CA simultaneously. It would seem then that the bulk operator must correspond to an identity operator times a complex phase. In fact, similar arguments apply to any bulk operators, and thus, all the bulk operators must correspond to identity operators on the boundary. Then, the AdS/CFT correspondence seems so boring…

Almheiri, Dong and Harlow have recently proposed an intriguing way of reconciling this paradox with the AdS/CFT correspondence. They proposed that *the AdS/CFT correspondence can be viewed as a quantum error-correcting code*. Their idea is as follows. Instead of corresponding to a single boundary operator, may correspond to different operators in different regions, say , , living in AB, BC, CA respectively. Even though , , are different boundary operators, they may be equivalent inside a certain low energy subspace on the boundary.

This situation resembles the so-called quantum secret-sharing code. The quantum information at the center of the bulk cannot be accessed from any single party A, B or C because does not have representation on A, B, or C. It can be accessed only if multiple parties cooperate and perform joint measurements. It seems that a quantum secret is shared among three parties, and the AdS/CFT correspondence somehow realizes the three-party quantum secret-sharing code!

**Entanglement wedge reconstruction?**

Recently, causal wedge reconstruction has been further generalized to the notion of entanglement wedge reconstruction. Imagine we split the boundary into four pieces A,B,C,D such that A,C are larger than B,D. Then the geodesic lines for A and C do not form the geodesic line for the *union* of A and C because we can draw shorter arcs by connecting endpoints of A and C, which form the global geodesic line. The entanglement wedge of AC is a bulk region enclosed by this global geodesic line of AC. And the entanglement wedge reconstruction predicts that can be represented as an integral of local boundary operators on AC if and only if is inside the entanglement wedge of AC [1].

**Building a minimal toy model; the five-qubit code**

Okay, now let’s try to construct a toy model which admits causal and entanglement wedge reconstructions of bulk operators. Because I want a simple toy model, I take a rather bold assumption that *the bulk consists of a single qubit while the boundary consists of five qubits, denoted by A, B, C, D, E*.

What does causal wedge reconstruction teach us in this minimal setup of five and one qubits? First, we split the boundary system into two pieces, ABC and DE and observe that the bulk operator is contained inside the causal wedge of ABC. From the rotational symmetries, we know that the bulk operator must have representations on ABC, BCD, CDE, DEA, EAB. Next, we split the boundary system into four pieces, AB, C, D and E, and observe that the bulk operator is contained inside the entanglement wedge of AB and D. So, the bulk operator must have representations on ABD, BCE, CDA, DEB, EAC. In summary, we have the following:

- The bulk operator must have representations on R if and only if R contains three or more qubits.

This is the property I want my toy model to possess.

What kinds of physical systems have such a property? Luckily, we quantum information theorists know the answer; the five-qubit code. The five-qubit code, proposed here and here, has an ability to encode one logical qubit into five-qubit entangled states and corrects any single qubit error. We can view the five-qubit code as a quantum encoding isometry from one-qubit states to five-qubit states:

where and are the basis for a logical qubit. In quantum coding theory, logical Pauli operators and are Pauli operators which act like Pauli X (bit flip) and Z (phase flip) on a logical qubit spanned by and . In the five-qubit code, for any set of qubits R with volume 3, some representations of logical Pauli X and Z operators, and , can be found on R. While and are different operators for , they act exactly in the same manner on the codeword subspace spanned by and . This is exactly the property I was looking for.

**Holographic quantum error-correcting codes**

We just found possibly the smallest toy model of the AdS/CFT correspondence, the five-qubit code! The remaining task is to construct a larger model. For this goal, we view the encoding isometry of the five-qubit code as a six-leg tensor. The holographic quantum code is a network of such six-leg tensors covering a hyperbolic space where each tensor has one open leg. These open legs on the bulk are interpreted as logical input legs of a quantum error-correcting code while open legs on the boundary are identified as outputs where quantum information is encoded. Then the entire tensor network can be viewed as an encoding isometry.

The six-leg tensor has some nice properties. Imagine we inject some Pauli operator into one of six legs in the tensor. Then, for any given choice of three legs, there always exists a Pauli operator acting on them which counteracts the effect of the injection. An example is shown below:

In other words, if an operator is injected from one tensor leg, one can “push” it into other three tensor legs.

Finally, let’s demonstrate causal wedge reconstruction of bulk logical operators. Pick an arbitrary open tensor leg in the bulk and inject some Pauli operator into it. We can “push” it into three tensor legs, which are then injected into neighboring tensors. By repeatedly pushing operators to the boundary in the network, we eventually have some representation of the operator living on a piece of boundary region A. And the bulk operator is contained inside the causal wedge of A. (Here, the length of the curve can be defined as the number of tensor legs cut by the curve). You can also push operators into the boundary by choosing different tensor legs which lead to different representations of a logical operator. You can even have a rather exotic representation which is supported non-locally over two disjoint pieces of the boundary, realizing entanglement wedge reconstruction.

**What’s next?**

This post is already pretty long and I need to wrap it up…

Shor’s quantum factoring algorithm is a revolutionary invention which opened a whole new research avenue of quantum information science. It is often forgotten, but the first quantum error-correcting code is another important invention by Peter Shor (and independently by Andrew Steane) which enabled a proof that the quantum computation can be performed fault-tolerantly. The theory of quantum error-correcting codes has found interesting applications in studies of condensed matter physics, such as topological phases of matter. Perhaps then, quantum coding theory will also find applications in high energy physics.

Indeed, many interesting open problems are awaiting us. Is entanglement wedge reconstruction a generic feature of tensor networks? How do we describe black holes by quantum error-correcting codes? Can we build a fast scrambler by tensor networks? Is entanglement a wormhole (or maybe a perfect tensor)? Can we resolve the firewall paradox by holographic quantum codes? Can the physics of quantum gravity be described by tensor networks? Or can the theory of quantum gravity provide us with novel constructions of quantum codes?

I feel that now is the time for quantum information scientists to jump into the research of black holes. We don’t know if we will be burned by a firewall or not … , but it is worth trying.

1. Whether entanglement wedge reconstruction is possible in the AdS/CFT correspondence or not still remains controversial. In the spirit of the Ryu-Takayanagi formula which relates entanglement entropy to the length of a global geodesic line, entanglement wedge reconstruction seems natural. But that a bulk operator can be reconstructed from boundary operators on two separate pieces A and C non-locally sounds rather exotic. In our paper, we constructed a toy model of tensor networks which allows both causal and entanglement wedge reconstruction in many cases. For details, see our paper.

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Most of you are probably familiar with holograms, these shiny flat films representing a 3D object from essentially any desired angle. I find it quite remarkable how all the information of a 3D object can be printed on an essentially 2D film. True, the colors are not represented as faithfully as in a traditional photograph, but it looks as though we have taken a photograph from every possible angle! The speaker’s main message that day seemed even more provocative than the idea of holography itself. Even if the hologram is broken into pieces, and some of these are lost, we may still use the remaining pieces to recover parts of the 3D image or even the full thing given a sufficiently large portion of the hologram. The 3D object is not only recorded in 2D, it is recorded redundantly!

Half way through Daniel’s exposition, Beni and I exchange a knowing glance. We recognize a familiar pattern from our latest project. A pattern which has gained the moniker of “cleaning lemma” within the quantum information community which can be thought of as a quantitative analog of reconstructing the 3D image from pieces of the hologram. Daniel makes connections using a language that we are familiar with. Beni and I discuss what we have understood and how to make it more concrete as we stride back through campus. We scribble diagrams on the whiteboard and string words such as tensor, encoder, MERA and negative curvature into our discussion. An image from the web gives us some intuition on the latter. We are onto something. We have a model. It is simple. It is new. It is exciting.

Food has not come our way so we head to my apartment as we enthusiastically continue our discussion. I can only provide two avocados and some leftover pasta but that is not important, we are sharing the joy of insight. We arrange a meeting with Daniel to present our progress. By Wednesday Beni and I introduce the holographic pentagon code at the group meeting. A core for a new project is already there, but we need some help to navigate the high-energy waters. Who better to guide us in such an endeavor than our mentor, John Preskill, who recognized the importance of quantum information in Holography as early as 1999 and has repeatedly proven himself a master of both trades.

“I feel that the idea of holography has a strong whiff of entanglement—for we have seen that in a profoundly entangled state the amount of information stored locally in the microscopic degrees of freedom can be far less than we would naively expect. For example, in the case of the quantum error-correcting codes, the encoded information may occupy a small ‘global’ subspace of a much larger Hilbert space. Similarly, the distinct topological phases of a fractional quantum Hall system look alike locally in the bulk, but have distinguishable edge states at the boundary.”

-J. Preskill, 1999

As Beni puts it, the time for using modern quantum information tools in high-energy physics has come. By this he means quantum error correction and maybe tensor networks. First privately, then more openly, we continue to sharpen and shape our project. Through conferences, Skype calls and emails, we further our discussion and progressively shape ideas. Many speculations mature to conjectures and fall victim to counterexamples. Some stand the test of simulations or are even promoted to theorems by virtue of mathematical proofs.

I publicly present the project for the first time at a select quantum information conference in Australia. Two months later, after a particularly intense writing, revising and editing process, the article is almost complete. As we finalize the text and relabel the figures, Daniel and Beni unveil our work to quantum entanglement experts in Puerto Rico. The talks are a hit and it is time to let all our peers read about it.

You are invited to do so and Beni will even be serving a reader’s guide in an upcoming post.

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So began the personal statement in my application to Caltech’s PhD program. I didn’t mention Sir Terry Pratchett, but he belongs in the list. Pratchett wrote over 70 books, blending science fiction with fantasy, humor, and truths about humankind. Pratchett passed away last week, having completed several novels after doctors diagnosed him with early-onset Alzheimer’s. According to the San Francisco Chronicle, Pratchett “parodie[d] everything in sight.” Everything in sight included physics.

Pratchett set many novels on the Discworld, a pancake of a land perched atop four elephants, which balance on the shell of a turtle that swims through space. Discworld wizards quantify magic in units called *thaums*. Units impressed their importance upon me in week one of my first high-school physics class. We define one meter as “the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.” Wizards define one thaum as “the amount of magic needed to create one small white pigeon or three normal-sized billiard balls.”

Wizards study the thaum in a High-Energy Magic Building reminiscent of Caltech’s Lauritsen-Downs Building. To split the thaum, the wizards built a Thaumatic Resonator. Particle physicists in our world have split atoms into constituent particles called *mesons* and *baryons*. Discworld wizards discovered that the thaum consists of *resons*. Mesons and baryons consist of *quarks*, seemingly elementary particles that we believe cannot be split. Quarks fall into six types, called *flavors*: *up, down, charmed, strange, top* (or *truth*), and *bottom* (or *beauty*). Resons, too, consist of quarks. The Discworld’s quarks have the flavors *up, down, sideways, sex appeal,* and *peppermint*.

Reading about the Discworld since high school, I’ve wanted to grasp Pratchett’s allusions. I’ve wanted to do more than laugh at them. In *Pyramids*, Pratchett describes “ideas that would make even a quantum mechanic give in and hand back his toolbox.” Pratchett’s ideas have given me a hankering for that toolbox. Pratchett nudged me toward training as a quantum mechanic.

Pratchett hasn’t only piqued my curiosity about his allusions. He’s piqued my desire to create as he did, to do physics as he wrote. While reading or writing, we build worlds in our imaginations. We visualize settings; we grow acquainted with characters; we sense a plot’s consistency or the consistency of a system of magic. We build worlds in our imaginations also when doing and studying physics and math. The Standard Model is a system that encapsulates the consistency of our knowledge about particles. We tell stories about electrons’ behaviors in magnetic fields. Theorems’ proofs have logical structures like plots’. Pratchett and other authors trained me to build worlds in my imagination. Little wonder I’m training to build worlds as a physicist.

Around the time I graduated from college, Diana Wynne Jones passed away. So did Brian Jacques (another British novelist) and Madeleine L’Engle. L’Engle wasn’t British, but I forgave her because her *Time Quartet* introduced me to dimensions beyond three. As I completed one stage of intellectual growth, creators who’d led me there left.

Terry Pratchett has joined Jones, Jacques, and L’Engle. I will probably create nothing as valuable as his Discworld, let alone a character in the Standard Model toward which the Discworld steered me.

But, because of Terry Pratchett, I have to try.

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Yet in the not too distant future, a computer may help you become a brilliant detective (or a scheming villain) yourself! The first step is noticing the details, which is known in machine learning as the classification task. Here is a pioneering work that somewhat resembles the above picture, only it’s done by a computer:

The task for the computer here was to produce a verbal description of the image. There are thousands of words in the vocabulary, and a computer has to try them in different combinations to make a sensible sentence. There is no way a computer can be given an exhaustive list of correct sentences with examples of images for each. That kind of list would be a database bigger than the earth (as one can see just by counting the number of combinations). So to train the computer to use language like in a picture above, one only possesses a limited set of examples – maybe a few thousand pictures with descriptions. Yet we as humans are capable of learning from just seeing a few examples, by noticing the repeating patterns. So the computer can do the same! The score next to each word above is an estimate based on those few thousand examples of how relevant is the word “tennis” or “woman” to what’s in the box on the image. The algorithm produces possible sentences, scores them, and then selects the sentence with the highest total score.

Once the classification task is done, one needs to use all the collected information to make a *prediction* – as Sherlock is able to point out the most probable motive in the first picture, we also want to predict a piece of very personal information: we’d like to know how to start up a conversation with that tennis player.

Humans are actually good at *classification* tasks: with luck, we can notice and type in our cellphone all the details the predictor will need, like brand of clothing, hair color, height… though computers recently became better than humans at facial expression recognition, so we don’t have to trust ourselves on that anymore. Finally, when all the data is collected, most humans will still say only generic advice to you on conversation starters. Which means we are very bad at *prediction* tasks. We don’t notice the hidden dependencies between brand of clothes and sense of humor. But such information may not hide from the all-seeing eye of the machine learning algorithm! So expect your cellphones to give you dating advice within 10 years…

Now how do quantum computers come into play? Well if you look at your search results, they are still pretty irrelevant most of the time. Imagine you used them as conversation starters – you’ll embarrass yourself 9 out of 10 times! To make this better, a certain company needs more memory and processing power. Yet most advanced deep learning routines remain out of reach, just because there are exponentially many hidden dependencies one would need to try and reject before the algorithm finds the right predictor. So a certain company turns to us, quantum computing people, as we deal with exponentially hard problems notoriously well! And indeed, quantum algorithms make some of the machine learning routines exponentially faster – see this Quantum Machine Learning article, as well as a talk by Seth Lloyd for technical details. Some anonymous stock trader is already trying to intimidate their fellow quants (quantitative analysts) by calling the top trading system “Quantum machine learning”. I think we should appreciate his sense of humor and invest into his algorithm as soon as Quantiacs.com opens such functionality. Or we could invest in Teagan from Caltech – her code recently won the futures contest on the same website.

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