Mark Srednicki doesn’t look like a high priest. He’s a professor of physics at the University of California, Santa Barbara (UCSB); and you’ll sooner find him in khakis than in sacred vestments. Humor suits his round face better than channeling divine wrath would; and I’ve never heard him speak in tongues—although, when an idea excites him, his hands rise to shoulder height of their own accord, as though halfway toward a priestly blessing. Mark belongs less on a ziggurat than in front of a chalkboard. Nevertheless, he called himself a high priest.
Specifically, Mark jokingly called himself a high priest of the eigenstate thermalization hypothesis, a framework for understanding how quantum many-body systems thermalize internally. The eigenstate thermalization hypothesis has an unfortunate number of syllables, so I’ll call it the ETH. The ETH illuminates closed quantum many-body systems, such as a clump of ultracold atoms. The clump can begin in a pure product state , then evolve under a chaotic1 Hamiltonian . The time- state will remain pure; its von Neumann entropy will always vanish. Yet entropy grows according to the second law of thermodynamics. Breaking the second law amounts almost to a enacting a miracle, according to physicists. Does the clump of atoms deserve consideration for sainthood?
No—although the clump’s state remains pure, a small subsystem’s state does not. A subsystem consists of, for example, a few atoms. They’ll entangle with the other atoms, which serve as an effective environment. The entanglement will mix the few atoms’ state, whose von Neumann entropy will grow.
The ETH predicts this growth. The ETH is an ansatz about and an operator —say, an observable of the few-atom subsystem. We can represent as a matrix relative to the energy eigenbasis. The matrix elements have a certain structure, if and satisfy the ETH. Suppose that the operators do and that lacks degeneracies—that no two energy eigenvalues equal each other. We can prove that thermalizes: Imagine measuring the expectation value at each of many instants . Averaging over instants produces the time-averaged expectation value .
Another average is the thermal average—the expectation value of in the appropriate thermal state. If conserves just itself,2 the appropriate thermal state is the canonical state, . The average energy defines the inverse temperature , and normalizes the state. Hence the thermal average is .
The time average approximately equals the thermal average, according to the ETH: . The correction is small in the total number of atoms. Through the lens of , the atoms thermalize internally. Local observables tend to satisfy the ETH, and we can easily observe only local observables. We therefore usually observe thermalization, consistently with the second law of thermodynamics.
I agree that Mark Srednicki deserves the title high priest of the ETH. He and Joshua Deutsch independently dreamed up the ETH in 1994 and 1991. Since numericists reexamined it in 2008, studies and applications of the ETH have exploded like a desert religion. Yet Mark had never encountered the question I posed about it in 2021. Next month’s blog post will share the good news about that question.
1Nonintegrable.
2Apart from trivial quantities, such as projectors onto eigenspaces of .
Terry Rudolph, PsiQuantum & Imperial College London
During a recent visit to the wild western town of Pasadena I got into a shootout at high-noon trying to explain the nuances of this question to a colleague. Here is a more thorough (and less risky) attempt to recover!
tl;dr Photonic quantum computers can perform a useful computation orders of magnitude faster than a superconducting qubit machine. Surprisingly, this would still be true even if every physical timescale of the photonic machine was an order of magnitude longer (i.e. slower) than those of the superconducting one. But they won’t be.
SUMMARY
There is a misconception that the slow rate of entangled photon production from many current (“postselected”) experiments is somehow relevant to the logical speed of a photonic quantum computer. It isn’t, because those experiments don’t use an optical switch.
If we care about how fast we can solve useful problems then photonic quantum computers will eventually win that race. Not only because in principle their components can run faster, but because of fundamental architectural flexibilities which mean they need to do fewer things.
Unlike most quantum systems for which relevant physical timescales are determined by “constants of nature” like interaction strengths, the relevant photonic timescales are determined by “classical speeds” (optical switch speeds, electronic signal latencies etc). Surprisingly, even if these were slower – which there is no reason for them to be – the photonic machine can still compute faster.
In a simple world the speed of a photonic quantum computer would just be the speed at which it’s possible to make small (fixed sized) entangled states. GHz rates for such are plausible and correspond to the much slower MHz code-cycle rates of a superconducting machine. But we want to leverage two unique photonic features: Availability of long delays (e.g. optical fiber) and ease of nonlocal operations, and as such the overall story is much less simple.
If what floats your boat are really slow things, like cold atoms, ions etc., then the hybrid photonic/matter architecture outlined here is the way you can build a quantum computer with a faster logical gate speed than (say) a superconducting qubit machine. You should be all over it.
Magnifying the number of logical qubits in a photonic quantum computer by 100 could be done simply by making optical fiber 100 times less lossy. There are reasons to believe that such fiber is possible (though not easy!). This is just one example of the “photonics is different, photonics is different, ” mantra we should all chant every morning as we stagger out of bed.
The flexibility of photonic architectures means there is much more unexplored territory in quantum algorithms, compiling, error correction/fault tolerance, system architectural design and much more. If you’re a student you’d be mad to work on anything else!
Sorry, I realize that’s kind of an in-your-face list, some of which is obviously just my opinion! Lets see if I can make it yours too 🙂
I am not going to reiterate all the standard stuff about how photonics is great because of how manufacturable it is, its high temperature operation, easy networking modularity blah blah blah. That story has been told many times elsewhere. But there are subtleties to understanding the eventual computational speed of a photonic quantum computer which have not been explained carefully before. This post is going to slowly lead you through them.
I will only be talking about useful, large-scale quantum computing – by which I mean machines capable of, at a minimum, implementing billions of logical quantum gates on hundreds of logical qubits.
PHYSICAL TIMESCALES
In a quantum computer built from matter – say superconducting qubits, ions, cold atoms, nuclear/electronic spins and so on, there is always at least one natural and inescapable timescale to point to. This typically manifests as some discrete energy levels in the system, the levels that make the two states of the qubit. Related timescales are determined by the interaction strengths of a qubit with its neighbors, or with external fields used to control it. One of the most important timescales is that of measurement – how fast can we determine the state of the qubit? This generally means interacting with the qubit via a sequence of electromagnetic fields and electronic amplification methods to turn quantum information classical. Of course, measurements in quantum theory are a pernicious philosophical pit – some people claim they are instantaneous, others that they don’t even happen! Whatever. What we care about is: How long does it take for a readout signal to get to a computer that records the measurement outcome as classical bits, processes them, and potentially changes some future action (control field) interacting with the computer?
For building a quantum computer from optical frequency photons there are no energy levels to point to. The fundamental qubit states correspond to a single photon being either “here” or “there”, but we cannot trap and hold them at fixed locations, so unlike, say, trapped atoms these aren’t discrete energy eigenstates. The frequency of the photons does, in principle, set some kind of timescale (by energy-time uncertainty), but it is far too small to be constraining. The most basic relevant timescales are set by how fast we can produce, control (switch) or detect the photons. While these depend on the bandwidth of the photons used – itself a very flexible design choice – typical components operate in GHz regimes. Another relevant timescale is that we can store photons in a standard optical fiber for tens of microseconds before its probability of getting lost exceeds (say) 10%.
There is a long chain of things that need to be strung together to get from component-level physical timescales to the computational speed of a quantum computer built from them. The first step on the journey is to delve a little more into the world of fault tolerance.
TIMESCALES RELEVANT FOR FAULT TOLERANCE
The timescales of measurement are important because they determine the rate at which entropy can be removed from the system. All practical schemes for fault tolerance rely on performing repeated measurements during the computation to combat noise and imperfection. (Here I will only discuss surface-code fault tolerance, much of what I say though remains true more generally.) In fact, although at a high level one might think a quantum computer is doing some nice unitary logic gates, microscopically the machine is overwhelmingly just a device for performing repeated measurements on small subsets of qubits.
In matter-based quantum computers the overall story is relatively simple. There is a parameter , the “code distance”, dependent primarily on the quality of your hardware, which is somewhere in the range of 20-40. It takes qubits to make up a logical qubit, so let’s say 1000 of them per logical qubit. (We need to make use of an equivalent number of ancillary qubits as well). Very roughly speaking, we repeat twice the following: each physical qubit gets involved in a small number (say 4-8) of two-qubit gates with neighboring qubits, and then some subset of qubits undergo a single-qubit measurement. Most of these gates can happen simultaneously, so (again, roughly!) the time for this whole process is the time for a handful of two-qubit gates plus a measurement. It is known as a code cycle and the time it takes we denote . For example, in superconducting qubits this timescale is expected to be about 1 microsecond, for ion-trap qubits about 1 millisecond. Although variations exist, lets stick to considering a basic architecture which requires repeating this whole process on the order of times in order to complete one logical operation (i.e., a logical gate). So, the time for a logical gate would be , this sets the effective logical gate speed.
If you zoom out, each code cycle for a single logical qubit is therefore built up in a modular fashion from copies of the same simple quantum process – a process that involves a handful of physical qubits and gates over a handful of time steps, and which outputs a classical bit of information – a measurement outcome. I have ignored the issue of what happens to those measurement outcomes. Some of them will be sent to a classical computer and processed (decoded) then fed back to control systems and so on. That sets another relevant timescale (the reaction time) which can be of concern in some approaches, but early generations of photonic machines – for reasons outlined later – will use long delay lines, and it is not going to be constraining.
In a photonic quantum computer we also build up a single logical qubit code cycle from copies of some quantum stuff. In this case it is from copies of an entangled state of photons that we call a resource state. The number of entangled photons comprising one resource state depends a lot on how nice and clean they are, lets fix it and say we need a 20-photon entangled state. (The noisier the method for preparing resource states the larger they will need to be). No sequence of gates is performed on these photons. Rather, photons from adjacent resource states get interfered at a beamsplitter and immediately detected – a process we call fusion. You can see a toy version in this animation:
Highly schematic depiction of photonic fusion based quantum computing. An array of 25 resource state generators each repeatedly create resource states of 6 entangled photons, depicted as a hexagonal ring. Some of the photons in each ring are immediately fused (the yellow flashes) with photons from adjacent resource states, the fusion measurement outputs classical bits of information. One photon from each ring gets delayed for one clock cycle and fused with a photon from the next clock cycle.
Measurements destroy photons, so to ensure continuity from one time step to the next some photons in a resource state get delayed by one time step to fuse with a photon from the subsequent resource state – you can see the delayed photons depicted as lit up single blobs if you look carefully in the animation.
The upshot is that the zoomed out view of the photonic quantum computer is very similar to that of the matter-based one, we have just replaced the handful of physical qubits/gates of the latter with a 20-photon entangled state. (And in case it wasn’t obvious – building a bigger computer to do a larger computation means generating more of the resource states, it doesn’t mean using larger and larger resource states.)
If that was the end of the story it would be easy to compare the logical gate speeds for matter-based and photonic approaches. We would only need to answer the question “how fast can you spit out and measure resource states?”. Whatever the time for resource state generation, , the time for a logical gate would be and the photonic equivalent of would simply be . (Measurements on photons are fast and so the fusion time becomes effectively negligible compared to .) An easy argument could then be made that resource state generation at GHz rates is possible, therefore photonic machines are going to be orders of magnitude faster, and this article would be done! And while I personally do think its obvious that one day this is where the story will end, in the present day and age….
… there are two distinct ways in which this picture is far too simple.
FUNKY FEATURES OF PHOTONICS, PART I
The first over-simplification is based on facing up to the fact that building the hardware to generate a photonic resource state is difficult and expensive. We cannot afford to construct one resource state generator per resource state required at each time step. However, in photonics we are very fortunate that it is possible to store/delay photons in long lengths of optical fiber with very low error rates. This lets us use many resource states all produced by a single resource state generator in such a way that they can all be involved in the same code-cycle. So, for example, all resource states required for a single code cycle may come from a single resource state generator:
Here the 25 resource state generators of the previous figure are replaced by a single generator that “plays fusion games with itself” by sending some of its output photons into either a delay of length 5 or one of length 25 times the basic clock cycle. We achieve a massive amplification of photonic entanglement simply by increasing the length of optical fiber used. By mildly increasing the complexity of the switching network a photon goes through when it exits the delay, we can also utilize small amounts of (logarithmic) nonlocal connectivity in the network of fusions performed (not depicted), which is critical to doing active volume compiling (discussed later).
You can see an animation of how this works in the figure – a single resource state generator spits out resource states (depicted again as a 6-qubit hexagonal ring), and you can see a kind of spacetime 3d-printing of entanglement being performed. We call this game interleaving. In the toy example of the figure we see some of the qubits get measured (fused) immediately, some go into a delay of length and some go into a delay of length .
So now we have brought another timescale into the photonics picture, the length of time that some photons spend in the longest interleaving delay line. We would like to make this as long as possible, but the maximum time is limited by the loss in the delay (typically optical fiber) and the maximum loss our error correcting code can tolerate. A number to have in mind for this (in early machines) is a handful of microseconds – corresponding to a few Km of fiber.
The upshot is that ultimately the temporal quantity that matters most to us in photonic quantum computing is:
What is the total number of resource states produced per second?
It’s important to appreciate we care only about the total rate of resource state production across the whole machine – so, if we take the total number of resource state generators we have built, and divide by , we get this total rate of resource state generation that we denote . Note that this rate is distinct from any physical clock rate, as, e.g., 100 resource state generators running at 100MHz, or 10 resource state generators running at 1GHz, or 1 resource state generator running at 10GHz all yield the same total rate of resource state production
The second most important temporal quantity is , the time of the longest low-loss delay we can use.
We then have that the total number of logical qubits in the machine is:
You can see this is proportional to which is effectively the total number of resource states “alive” in the machine at any given instant of time, including all the ones stacked up in long delay lines. This is how we leverage optical fiber delays for a massive amplification of the entanglement our hardware has available to compute with.
The time it takes to perform a logical gate is determined both by and by the total number of resource states that we need to consume for every logical qubit to undergo a gate. Even logical qubits that appear to not be part of a gate in that time step do, in fact, undergo a gate – the identity gate – because they need to be kept error free while they “idle”. As such the total number of resource states consumed in a logical time step is just and the logical gate time of the machine is
.
Because is expected to be about the same as for superconducting qubits (microseconds), the logical gate speeds are comparable.
At least they are, until…………
FUNKY FEATURES OF PHOTONICS, PART II
But wait! There’s more.
The second way in which unique features of photonics play havoc with the simple comparison to matter-based systems is in the exciting possibility of what we call an active-volume architecture.
A few moments ago I said:
Even logical qubits that seem to not be part of a gate in that time step undergo a gate – the identity gate – because they need to be kept error free while they “idle”. As such the total number of resource states consumed is just
and that was true. Until recently.
It turns out that there is a way of eliminating the majority of consumption of resources expended on idling qubits! This is done by some clever tricks that make use of the possibility of performing a limited number of non-nearest neighbor fusions between photons. It’s possible because photons are not anyway stuck in one place, and they can be passed around readily without interacting with other photons. (Their quantum crosstalk is exactly zero, they do really seem to despise each other.)
What previously was a large volume of resource states being consumed for “thumb-twiddling”, can instead all be put to good use doing non-trivial computational gates. Here is a simple quantum circuit with what we mean by the active volume highlighted:
Now, for any given computation the amount of active volume will depend very much on what you are computing. There are always many different circuits decomposing a given computation, some will use more active volume than others. This makes it impossible to talk about “what is the logical gate speed” completely independent of considerations about the computation actually being performed.
In this recent paper https://arxiv.org/abs/2306.08585 Daniel Litinski considers breaking elliptic curve cryptosystems on a quantum computer. In particular, he considers what it would take to run the relevant version of Shor’s algorithm on a superconducting qubit architecture with a microsecond code cycle – the answer is roughly that with 10 million physical superconducting qubits it would take about 4 hours (with an equivalent ion trap computer the time balloons to more than 5 months).
He then compares solving the same problem on a machine with an active volume architecture. Here is a subset of his results:
Recall that is the photonics parameter which is roughly equivalent to the code cycle time. Thus taking 1 microsecond compares to the expected for superconducting qubits. Imagine we can produce resource states at . This could be 6000 resource state generators each producing resource states at or 3500 generators producing them at 1GHz for example. Then the same computation would take 58 seconds, instead of four hours, a speedup by a factor of more than 200!
Now, this whole blog post is basically about addressing confusions out there regarding physical versus computational timescales. So, for the sake of illustration, let me push a purely theoretical envelope: What if we can’t do everything as fast as in the example just stated? What if our rate of total resource state generation was 10 times slower, i.e. ? And what if our longest delay is ten times longer, i.e. microseconds (so as to be much slower than )? Furthermore, for the sake of illustration, lets consider a ridiculously slow machine that achieves by building 350 billion resource state generators that can each produce resource states at only 1Hz. Yes, you read that right.
The fastest device in this ridiculous machine would only need to be a (very large!) slow optical switch operating at 100KHz (due to the chosen ). And yet this ridiculous machine could still solve the problem that takes a superconducting qubit machine four hours, in less than 10 minutes.
To reiterate:
Despite all the “physical stuff going on” in this (hypothetical, active-volume) photonic machine running much slower than all the “physical stuff going on” in the (hypothetical, non-active-volume) superconducting qubit machine, we see the photonic machine can still do the desired computation 25 times faster!
Hopefully the fundamental murkiness of the titular question “what is the logical gate speed of a photonic quantum computer” is now clear! Put simply: Even if it did “fundamentally run slower” (it won’t), it would still be faster. Because it has less stuff to do. It’s worth noting that the 25x increase in speed is clearly not based on physical timescales, but rather on the efficient parallelization achieved through long-range connections in the photonic active-volume device. If we were to scale up the hypothetical 10-million-superconducting-qubit device by a factor of 25, it could potentially also complete computations 25 times faster. However, this would require a staggering 250 million physical qubits or more. Ultimately, the absolute speed limit of quantum computations is set by the reaction time, which refers to the time it takes to perform a layer of single-qubit measurements and some classical processing. Early-generation machines will not be limited by this reaction time, although eventually it will dictate the maximum speed of a quantum computation. But even in this distant-future scenario, the photonic approach remains advantageous. As classical computation and communication speed up beyond the microsecond range, slower physical measurements of matter-based qubits will hinder the reaction time, while fast single-photon detectors won’t face the same bottleneck.
In the standard photonic architecture we saw that would scale proportionally with – that is, adding long delays would slow the logical gate speed (while giving us more logical qubits). But remarkably the active-volume architecture allows us to exploit the extra logical qubits without incurring a big negative tradeoff. I still find this unintuitive and miraculous, it just seems to so massively violate Conservation of Trouble.
With all this in mind it is also worth noting as an aside that optical fibers made from (expensive!) exotic glasses or with funky core structures are theoretically calculated to be possible with up to 100 times less loss than conventional fiber – therefore allowing for an equivalent scaling of . How many approaches to quantum computing can claim that perhaps one day, by simply swapping out some strands of glass, they could instantaneously multiply the number of logical qubits in the machine from (say) 100 to 10000? Even a (more realistic) factor of 10 would be incredible.
Obviously for pedagogical reasons the above discussion is based around the simplest approaches to logic in both standard and active-volume architectures, but more detailed analysis shows that conclusions regarding total computational time speedup persist even after known optimizations for both approaches.
Now the reason I called the example above a “ridiculous machine” is that even I am not cruel enough to ask our engineers to assemble 350 billion resource state generators. Fewer resource state generators running faster is desirable from the perspective of both sweat and dollars.
We have arrived then at a simple conclusion: what we really need to know is “how fast and at what scale can we generate resource states, with as large a machine as we can afford to build”.
HOW FAST COULD/SHOULD WE AIM TO DO RESOURCE STATE GENERATION?
In the world of classical photonics – such as that used for telecoms, LIDAR and so on – very high speeds are often thrown around: pulsed lasers and optical switches readily run at 100’s of GHz for example. On the quantum side, if we produce single photons via a probabilistic parametric process then similarly high repetition rates have been achieved. (This is because in such a process there are no timescale constraints set by atomic energy levels etc.) Off-the-shelf single photon avalanche photodiode detectors can count photons at multiple GHz.
Seems like we should be aiming to generate resource states at 10’s of GHz right?
Well, yes, one day – one of the main reasons I believe the long-term future of quantum computing is ultimately photonic is because of the obvious attainability of such timescales. [Two others: it’s the only sensible route to a large-scale room temperature machine; eventually there is only so much you can fit in a single cryostat, so ultimately any approach will converge to being a network of photonically linked machines].
In the real world of quantum engineering there are a couple of reasons to slow things down: (i) It relaxes hardware tolerances, since it makes it easier to get things like path lengths aligned, synchronization working, electronics operating in easy regimes etc (ii) in a similar way to how we use interleaving during a computation to drastically reduce the number of resource state generators we need to build, we can also use (shorter than length) delays to reduce the amount of hardware required to assemble the resource states in the first place and (iii) We want to use multiplexing.
Multiplexing is often misunderstood. The way we produce the requisite photonic entanglement is probabilistic. Producing the whole 20-photon resource state in a single step, while possible, would have very low probability. The way to obviate this is to cascade a couple of higher probability, intermediate, steps – selecting out successes (more on this in the appendix). While it has been known since the seminal work of Knill, Laflamme and Milburn two decades ago that this is a sensible thing to do, the obstacle has always been the need for a high performance (fast, low loss) optical switch. Multiplexing introduces a new physical “timescale of convenience” – basically dictated by latencies of electronic processing and signal transmission.
The brief summary therefore is: Yeah, everything internal to making resource states can be done at GHz rates, but multiple design flexibilities mean the rate of resource state generation is itself a parameter that should be tuned/optimized in the context of the whole machine, it is not constrained by fundamental quantum things like interaction energies, rather it is constrained by the speeds of a bunch of purely classical stuff.
I do not want to leave the impression that generation of entangled photons can only be done via the multistage probabilistic method just outlined. Using quantum dots, for example, people can already demonstrate generation of small photonic entangled states at GHz rates (see e.g. https://www.nature.com/articles/s41566-022-01152-2). Eventually, direct generation of photonic entanglement from matter-based systems will be how photonic quantum computers are built, and I should emphasize that its perfectly possible to use small resource states (say, 4 entangled photons) instead of the 20 proposed above, as long as they are extremely clean and pure. In fact, as the discussion above has hopefully made clear: for quantum computing approaches based on fundamentally slow things like atoms and ions, transduction of matter-based entanglement into photonic entanglement allows – by simply scaling to more systems – evasion of the extremely slow logical gate speeds they will face if they do not do so.
Right now, however, approaches based on converting the entanglement of matter qubits into photonic entanglement are not nearly clean enough, nor manufacturable at large enough scales, to be compatible with utility-scale quantum computing. And our present method of state generation by multiplexing has the added benefit of decorrelating many error mechanisms that might otherwise be correlated if many photons originate from the same device.
So where does all this leave us?
I want to build a useful machine. Lets back-of-the-envelope what that means photonically. Consider we target a machine comprising (say) at least 100 logical qubits capable of billions of logical gates. (From thinking about active volume architectures I learn that what I really want is to produce as many “logical blocks” as possible, which can then be divvied up into computational/memory/processing units in funky ways, so here I’m really just spitballing an estimate to give you an idea).
Staring at
and presuming and is going to be about 10 microseconds, we need to be producing resource states at a total rate of at least . As I hope is clear by now, as a pure theoretician, I don’t give a damn if that means 10000 resource state generators running at 1MHz, 100 resource state generators running at 100MHz, or 10 resource state generators running at 1GHz. However, the fact this flexibility exists is very useful to my engineering colleagues – who, of course, aim to build the smallest and fastest possible machine they can, thereby shortening the time until we let them head off for a nice long vacation sipping mezcal margaritas on a warm tropical beach.
None of these numbers should seem fundamentally indigestible, though I do not want to understate the challenge: all never-before-done large-scale engineering is extremely hard.
But regardless of the regime we operate in, logical gate speeds are not going to be the issue upon which photonics will be found wanting.
REAL-WORLD QUANTUM COMPUTING DESIGN
Now, I know this blog is read by lots of quantum physics students. If you want to impact the world, working in quantum computing really is a great way to do it. The foundation of everything round you in the modern world was laid in the 40’s and 50’s when early mathematicians, computer scientists, physicists and engineers figured out how we can compute classically. Today you have a unique opportunity to be part of laying the foundation of humanity’s quantum computing future. Of course, I want the best of you to work on a photonic approach specifically (I’m also very happy to suggest places for the worst of you to go work). Please appreciate, therefore, that these final few paragraphs are my very biased – though fortunately totally correct – personal perspective!
The broad features of the photonic machine described above – it’s a network of stuff to make resource states, stuff to fuse them, and some interleaving modules, has been fixed now for several years (see the references).
Once we go down even just one level of detail, a myriad of very-much-not-independent questions arise: What is the best resource state? What series of procedures is optimal for creating that state? What is the best underlying topological code to target? What fusion network can build that code? What other things (like active volume) can exploit the ability for photons to be easily nonlocally connected? What types of encoding of quantum information into photonic states is best? What interferometers generate the most robust small entangled states? What procedures for systematically growing resource states from smaller entangled states are most robust or use the least amount of hardware? How can we best use measurements and classical feedforward/control to mitigate error accumulation?
Those sorts of questions cannot be meaningfully addressed without going down to another level of detail, one in which we do considerable modelling of the imperfect devices from which everything will be built – modelling that starts by detailed parameterization of about 40 component specifications (ranging over things like roughness of silicon photonic waveguide walls, stability of integrated voltage drivers, precision of optical fiber cutting robots,….. Well, the list goes on and on). We then model errors of subsystems built from those components, verify against data, and proceed.
The upshot is none of these questions have unique answers! There just isn’t “one obviously best code” etc. In fact the answers can change significantly with even small variations in performance of the hardware. This opens a very rich design space, where we can establish tradeoffs and choose solutions that optimize a wide variety of practical hardware metrics.
In photonics there is also considerably more flexibility and opportunity than with most approaches on the “quantum side” of things. That is, the quantum aspects of the sources, the quantum states we use for encoding even single qubits, the quantum states we should target for the most robust entanglement, the topological quantum logical states we target and so on, are all “on the table” so to speak.
Exploring the parameter space of possible machines to assemble, while staying fully connected to component level hardware performance, involves both having a very detailed simulation stack, and having smart people to help find new and better schemes to test in the simulations. It seems to me there are far more interesting avenues for impactful research than more established approaches can claim. Right now, on this planet, there are only around 30 people engaged seriously in that enterprise. It’s fun. Perhaps you should join in?
REFERENCES
A surface code quantum computer in siliconhttps://www.science.org/doi/10.1126/sciadv.1500707. Figure 4 is a clear depiction of the circuits for performing a code cycle appropriate to a generic 2d matter-based architecture.
Here is a common misconception: Current methods of producing ~20 photon entangled states succeed only a few times per second, so generating resource states for fusion-based quantum computing is many orders of magnitude away from where it needs to be.
This misconception arises from considering experiments which produce photonic entangled states via single-shot spontaneous processes and extrapolating them incorrectly as having relevance to how resource states for photonic quantum computing are assembled.
Such single-shot experiments are hit by a “double whammy”. The first whammy is that the experiments produce some very large and messy state that only has a tiny amplitude in the component of the desired entangled state. Thus, on each shot, even in ideal circumstances, the probability of getting the desired state is very, very small. Because billions of attempts can be made each second (as mentioned, running these devices at GHz speeds is easy) it does occasionally occur. But only a small number of times per second.
The second whammy is that if you are trying to produce a 20-photon state, but each photon gets lost with probability 20%, then the probability of you detecting all the photons – even if you live in a branch of the multiverse where they have been produced – is reduced by a factor of . Loss reduces the rate of production considerably.
Now, photonic fusion-based quantum computing could not be based on this type of entangled photon generation anyway, because the production of the resource states needs to be heralded, while these experiments only postselect onto the very tiny part of the total wavefunction with the desired entanglement. But let us put that aside, because the two whammy’s could, in principle, be showstoppers for production of heralded resource states, and it is useful to understand why they are not.
Imagine you can toss coins, and you need to generate 20 coins showing Heads. If you repeatedly toss all 20 coins simultaneously until they all come up heads you’d typically have to do so millions of times before you succeed. This is even more true if each coin also has a 20% chance of rolling off the table (akin to photon loss). But if you can toss 20 coins, set aside (switch out!) the ones that came up heads and re-toss the others, then after only a small number of steps you will have 20 coins all showing heads. This large gap is fundamentally why the first whammy is not relevant: To generate a large photonic entangled state we begin by probabilistically attempting to generate a bunch of small ones. We then select out the success (multiplexing) and combine successes to (again, probabilistically) generate a slightly larger entangled state. We repeat a few steps of this. This possibility has been appreciated for more than twenty years, but hasn’t been done at scale yet because nobody has had a good enough optical switch until now.
The second whammy is taken care of the fact that for fault tolerant photonic fusion-based quantum computing there never is any need to make the resource state such that all photons are guaranteed to be there! The per-photon loss rate can be high (in principle 10’s of percent) – in fact the larger the resource state being built the higher it is allowed to be.
The upshot is that comparing this method of entangled photon generation with the methods which are actually employed is somewhat like a creation scientist claiming monkeys cannot have evolved from bacteria, because it is all so unlikely for suitable mutations to have happened simultaneously!
Acknowledgements
Very grateful to Mercedes Gimeno-Segovia, Daniel Litinski, Naomi Nickerson, Mike Nielsen and Pete Shadbolt for help and feedback.
I first heard the song “Fireflies,” by Owl City, shortly after my junior year of college. During the refrain, singer Adam Young almost whispers, “I’d like to make myself believe / that planet Earth turns slowly.” Goosebumps prickled along my neck. Yes, I thought, I’ve studied Foucault’s pendulum.
Léon Foucault practiced physics in France during the mid-1800s. During one of his best-known experiments, he hung a pendulum from high up in a building. Imagine drawing a wide circle on the floor, around the pendulum’s bob.1
Pendulum bob and encompassing circle, as viewed from above.
Imagine pulling the bob out to a point above the circle, then releasing the pendulum. The bob will swing back and forth, tracing out a straight line across the circle.
You might expect the bob to keep swinging back and forth along that line, and to do nothing more, forever (or until the pendulum has spent all its energy on pushing air molecules out of its way). After all, the only forces acting on the bob seem to be gravity and the tension in the pendulum’s wire. But the line rotates; its two tips trace out the circle.
How long the tips take to trace the circle depends on your latitude. At the North and South Poles, the tips take one day.
Why does the line rotate? Because the pendulum dangles from a building on the Earth’s surface. As the Earth rotates, so does the building, which pushes the pendulum. You’ve experienced such a pushing if you’ve ridden in a car. Suppose that the car is zipping along at a constant speed, in an unchanging direction, on a smooth road. With your eyes closed, you won’t feel like you’re moving. The only forces you can sense are gravity and the car seat’s preventing you from sinking into the ground (analogous to the wire tension that prevents the pendulum bob from crashing into the floor). If the car turns a bend, it pushes you sidewise in your seat. This push is called a centrifugal force. The pendulum feels a centrifugal force because the Earth’s rotation is an acceleration like the car’s. The pendulum also feels another force—a Coriolis force—because it’s not merely sitting, but moving on the rotating Earth.
We can predict the rotation of Foucault’s pendulum by assuming that the Earth rotates, then calculating the centrifugal and Coriolis forces induced, and then calculating how those forces will influence the pendulum’s motion. The pendulum evidences the Earth’s rotation as nothing else had before debuting in 1851. You can imagine the stir created by the pendulum when Foucault demonstrated it at the Observatoire de Paris and at the Panthéon monument. Copycat pendulums popped up across the world. One ended up next to my college’s physics building, as shown in this video. I reveled in understanding that pendulum’s motion, junior year.
My professor alluded to a grander Foucault pendulum in Paris. It hangs in what sounded like a temple to the Enlightenment—beautiful in form, steeped in history, and rich in scientific significance. I’m a romantic about the Enlightenment; I adore the idea of creating the first large-scale organizational system for knowledge. So I hungered to make a pilgrimage to Paris.
I made the pilgrimage this spring. I was attending a quantum-chaos workshop at the Institut Pascal, an interdisciplinary institute in a suburb of Paris. One quiet Saturday morning, I rode a train into the city center. The city houses a former priory—a gorgeous, 11th-century, white-stone affair of the sort for which I envy European cities. For over 200 years, the former priory has housed the Musée des Arts et Métiers, a museum of industry and technology. In the priory’s chapel hangs Foucault’s pendulum.2
A pendulum of Foucault’s own—the one he exhibited at the Panthéon—used to hang in the chapel. That pendulum broke in 2010; but still, the pendulum swinging today is all but a holy relic of scientific history. Foucault’s pendulum! Demonstrating that the Earth rotates! And in a jewel of a setting—flooded with light from stained-glass windows and surrounded by Gothic arches below a painted ceiling. I flitted around the little chapel like a pollen-happy bee for maybe 15 minutes, watching the pendulum swing, looking at other artifacts of Foucault’s, wending my way around the carved columns.
Almost alone. A handful of visitors trickled in and out. They contrasted with my visit, the previous weekend, to the Louvre. There, I’d witnessed a Disney World–esque line of tourists waiting for a glimpse of the Mona Lisa, camera phones held high. Nobody was queueing up in the musée’s chapel. But this was Foucault’s pendulum! Demonstrating that the Earth rotates!
I confess to capitalizing on the lack of visitors to take a photo with Foucault’s pendulum and Foucault’s Pendulum, though.
Shortly before I’d left for Paris, a librarian friend had recommended Umberto Eco’s novel Foucault’s Pendulum. It occupied me during many a train ride to or from the center of Paris.
The rest of the museum could model in an advertisement for steampunk. I found automata, models of the steam engines that triggered the Industrial Revolution, and a phonograph of Thomas Edison’s. The gadgets, many formed from brass and dark wood, contrast with the priory’s light-toned majesty. Yet the priory shares its elegance with the inventions, many of which gleam and curve in decorative flutes.
The grand finale at the Musée des Arts et Métiers.
I tore myself away from the Musée des Arts et Métiers after several hours. I returned home a week later and heard the song “Fireflies” again not long afterward. The goosebumps returned worse. Thanks to Foucault, I can make myself believe that planet Earth turns.
With thanks to Kristina Lynch for tolerating my many, many, many questions throughout her classical-mechanics course.
This story’s title refers to a translation of Goethe’s Faust. In the translation, the demon Mephistopheles tells the title character, “You let the great world spin and riot; / we’ll nest contented in our quiet” (to within punctuational and other minor errors, as I no longer have the text with me). A prize-winning 2009 novel is called Let the Great World Spin; I’ve long wondered whether Faust inspired its title.
1Why isn’t the bottom of the pendulum called the alice?
2After visiting the musée, I learned that my classical-mechanics professor had been referring to the Foucault pendulum that hangs in the Panthéon, rather than to the pendulum in the musée. The musée still contains the pendulum used by Foucault in 1851, whereas the Panthéon has only a copy, so I’m content. Still, I wouldn’t mind making a pilgrimage to the Panthéon. Let me know if more thermodynamic workshops take place in Paris!
By guest blogger Clarice D. Aiello, faculty at UCLA
Imagine using your cellphone to control the activity of your own cells to treat injuries and disease. It sounds like something from the imagination of an overly optimistic science fiction writer. But this may one day be a possibility through the emerging field of quantum biology.
Over the past few decades, scientists have made incredible progress in understanding and manipulating biological systems at increasingly small scales, from protein folding to genetic engineering. And yet, the extent to which quantum effects influence living systems remains barely understood.
Quantum effects are phenomena that occur between atoms and molecules that can’t be explained by classical physics. It has been known for more than a century that the rules of classical mechanics, like Newton’s laws of motion, break down at atomic scales. Instead, tiny objects behave according to a different set of laws known as quantum mechanics.
For humans, who can only perceive the macroscopic world, or what’s visible to the naked eye, quantum mechanics can seem counterintuitive and somewhat magical. Things you might not expect happen in the quantum world, like electrons “tunneling” through tiny energy barriers and appearing on the other side unscathed, or being in two different places at the same time in a phenomenon called superposition.
I am trained as a quantum engineer. Research in quantum mechanics is usually geared toward technology. However, and somewhat surprisingly, there is increasing evidence that nature – an engineer with billions of years of practice – has learned how to use quantum mechanics to function optimally. If this is indeed true, it means that our understanding of biology is radically incomplete. It also means that we could possibly control physiological processes by using the quantum properties of biological matter.
Quantumness in biology is probably real
Researchers can manipulate quantum phenomena to build better technology. In fact, you already live in a quantum-powered world: from laser pointers to GPS, magnetic resonance imaging and the transistors in your computer – all these technologies rely on quantum effects.
In general, quantum effects only manifest at very small length and mass scales, or when temperatures approach absolute zero. This is because quantum objects like atoms and molecules lose their “quantumness” when they uncontrollably interact with each other and their environment. In other words, a macroscopic collection of quantum objects is better described by the laws of classical mechanics. Everything that starts quantum dies classical. For example, an electron can be manipulated to be in two places at the same time, but it will end up in only one place after a short while – exactly what would be expected classically.
In a complicated, noisy biological system, it is thus expected that most quantum effects will rapidly disappear, washed out in what the physicist Erwin Schrödinger called the “warm, wet environment of the cell.” To most physicists, the fact that the living world operates at elevated temperatures and in complex environments implies that biology can be adequately and fully described by classical physics: no funky barrier crossing, no being in multiple locations simultaneously.
Chemists, however, have for a long time begged to differ. Research on basic chemical reactions at room temperature unambiguously shows that processes occurring within biomolecules like proteins and genetic material are the result of quantum effects. Importantly, such nanoscopic, short-lived quantum effects are consistent with driving some macroscopic physiological processes that biologists have measured in living cells and organisms. Research suggests that quantum effects influence biological functions, including regulating enzyme activity, sensing magnetic fields, cell metabolism and electron transport in biomolecules.
How to study quantum biology
The tantalizing possibility that subtle quantum effects can tweak biological processes presents both an exciting frontier and a challenge to scientists. Studying quantum mechanical effects in biology requires tools that can measure the short time scales, small length scales and subtle differences in quantum states that give rise to physiological changes – all integrated within a traditional wet lab environment.
In my work, I build instruments to study and control the quantum properties of small things like electrons. In the same way that electrons have mass and charge, they also have a quantum property called spin. Spin defines how the electrons interact with a magnetic field, in the same way that charge defines how electrons interact with an electric field. The quantum experiments I have been building since graduate school, and now in my own lab, aim to apply tailored magnetic fields to change the spins of particular electrons.
Research has demonstrated that many physiological processes are influenced by weak magnetic fields. These processes include stem cell development and maturation, cell proliferation rates, genetic material repair and countless others. These physiological responses to magnetic fields are consistent with chemical reactions that depend on the spin of particular electrons within molecules. Applying a weak magnetic field to change electron spins can thus effectively control a chemical reaction’s final products, with important physiological consequences.
Currently, a lack of understanding of how such processes work at the nanoscale level prevents researchers from determining exactly what strength and frequency of magnetic fields cause specific chemical reactions in cells. Current cellphone, wearable and miniaturization technologies are already sufficient to produce tailored, weak magnetic fields that change physiology, both for good and for bad. The missing piece of the puzzle is, hence, a “deterministic codebook” of how to map quantum causes to physiological outcomes.
In the future, fine-tuning nature’s quantum properties could enable researchers to develop therapeutic devices that are noninvasive, remotely controlled and accessible with a mobile phone. Electromagnetic treatments could potentially be used to prevent and treat disease, such as brain tumors, as well as in biomanufacturing, such as increasing lab-grown meat production.
A whole new way of doing science
Quantum biology is one of the most interdisciplinary fields to ever emerge. How do you build community and train scientists to work in this area?
Since the pandemic, my lab at the University of California, Los Angeles and the University of Surrey’s Quantum Biology Doctoral Training Centre have organized Big Quantum Biology meetings to provide an informal weekly forum for researchers to meet and share their expertise in fields like mainstream quantum physics, biophysics, medicine, chemistry and biology.
Research with potentially transformative implications for biology, medicine and the physical sciences will require working within an equally transformative model of collaboration. Working in one unified lab would allow scientists from disciplines that take very different approaches to research to conduct experiments that meet the breadth of quantum biology from the quantum to the molecular, the cellular and the organismal.
The existence of quantum biology as a discipline implies that traditional understanding of life processes is incomplete. Further research will lead to new insights into the age-old question of what life is, how it can be controlled and how to learn with nature to build better quantum technologies.
Clarice D. Aiello is a quantum engineer interested in how quantum physics informs biology at the nanoscale. She is an expert on nanosensors that harness room-temperature quantum effects in noisy environments. Aiello received a bachelor’s in physics from the Ecole Polytechnique, France; a master’s degree in physics from the University of Cambridge, Trinity College, UK; and a PhD in electrical engineering from the Massachusetts Institute of Technology. She held postdoctoral appointments in bioengineering at Stanford University and in chemistry at the University of California, Berkeley. Two months before the pandemic, she joined the University of California, Los Angeles, where she leads the Quantum Biology Tech (QuBiT) Lab.
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The author thanks Nicole Yunger Halpern and Spyridon Michalakis for the opportunity to talk about quantum biology to the physics audience of this wonderful blog!
During the past seven months, I’ve steamed across the Atlantic, sailed in a flying castle, teleported across the globe, and shuttled forward and backward in time. Literarily, not literally—the Quantum-Steampunk Short-Story Contest began welcoming submissions in October 2022. We challenged everybody aged 13 and over to write a steampunk narrative that involves a real or imagined quantum technology. One hundred sixty-seven entries arrived from 29 countries. Professional writers submitted stories, as did 13-year-olds. Tenured physics professors, librarians, English and math teachers, undergraduates, physicians, graduate students, and a United States Senate staffer entered. Thanks to their creativity, I now have a folder full of other worlds.
I’m over the moon (in a steam-powered ship) to announce the winners. David Wakeham received the $1,500 grand prize for the story The Creature of Ashen House. First runner-up Gerard McCaul won $1,000 for Doctor Up and Mister Down, and second runner-up Paulo Barreto won $500 for Eikonal. The People’s Choice Award ($500) went to Cristina Legarda for Pursuit, also nominated by two judges for a “Please Turn This into a Novel” award. Thanks to the 261 of you who voted in the People’s Choice competition!
In addition to traditional awards, we created four idiosyncratic ones, each entailing $250. We recognized Jeff Provine’s Stealing Buttons for its badass steampunk heroine; Matt King’s Three Imperiled Scientists for its wit and (relatedly) its portrayal of academia; Rick Searle’s The Recurrence Machine for its steampunk atmosphere; and Claudia Clarke’s Looking Forward, Looking Back, for its heart-capturing automaton. You can read all the finalist stories here.
Quantum-steampunk graphic by contest entrant Kayla Phan, who used YouChat Imagine
Sending our judges the finalists in March, I felt not only exhilaration (and relief, as whittling down 167 entries entails no little hand wringing), but also anxiety. Would the stories measure up? So I must have glowed when the first judge submitted his evaluations: Speculative-fiction author Ken Liu enthused, “The entries were so fun to read.” Similar reactions followed from across the panel, which featured experts in mathematics, philosophy, creative writing, experimental quantum physics, and history: “I had a very good time reading these stories,” another panelist wrote. “This was fun and some excellent spring break airplane (no dirigibles, I’m afraid) reading,” said another. Many thanks to our judges and short-listing committee for their input. University of Maryland undergraduates Hannah Cho and Jade Leschack led the team of students who narrowed down the candidates. I couldn’t resist treating the committee to a Victorian-inspired thank-you upon announcing the winners.
Thank-yous baked by Panera, not me
Although this year’s contest has ended, quantum-steampunk literature has just shipped out from its berth. Two contest entrants have posted their stories on their own online domains: You can read the mystery by Duke physics professor Ken Brown here and the adventure by quantum-algorithm designer Brian Siegelwax here. All other entrants, please feel free to post your stories and to submit them to other literary contests. Drop me a line, and leave a link in the chat below, when your story is published. I’d love to hear how your journey continues.
Also, stay tuned for v2.0 of the Quantum-Steampunk Short-Story Contest. An organization has expressed interest in a reboot during the 2024–2025 academic year. AI-collaboration category, anyone? Bonus points if you use a quantum neural network. Please email me if you’d like to support the effort!
Quantum-steampunk graphic by contest entrant Necklace Devkota
The opportunity to helm this contest has been a privilege and a dream. Many thanks to our writers, readers, funder (the John Templeton Foundation), staff (especially webmaster Anıl Zenginoğlu), judges, and shortlisting committee. Keep writing, and keep experimenting.
pon receiving my speaking assignments for the Tucson Festival of Books, I mentally raised my eyebrows. I’d be participating in a panel discussion with Mike Evans, the founder of Grubhub? But I hadn’t created an app that’s a household name. I hadn’t transformed 30 million people’s eating habits. I’m a theoretical physicist; I build universes in my head for a living. I could spend all day trying to prove a theorem and failing, and no stocks would tumble as a result.
Once the wave of incredulity had crested, I noticed that the panel was entitled “The Future of Tech.” Grubhub has transformed technology, I reasoned, and quantum computing is in the process of doing so. Fair enough.
Besides, my husband pointed out, the food industry requires fridges. Physicists building quantum computers from superconductors need fridges. The latter fridges require temperatures ten million times lower than restaurateurs do, but we still share an interest.
Very well, I thought. Game on.
Tucson hosts the third-largest book festival in the United States. And why shouldn’t it, as the festival takes place in early March, when much of the country is shivering and eyeing Arizona’s T-shirt temperatures with envy? If I had to visit any institution in the winter, I couldn’t object to the festival’s home, the University of Arizona.
The day before the festival, I presented a colloquium at the university, for the Arizona Quantum Alliance. The talk took place in the Wyant College of Optical Sciences, the home of an optical-instruments museum. Many of the instruments date to the 1800s and, built from brass and wood, smack of steampunk. I approved. Outside the optics building, workers were setting up tents to house the festival’s science activities.
The next day—a Saturday—dawned clear and bright. Late in the morning, I met Mike and our panel’s moderator, Bob Griffin, another startup veteran. We sat down at a table in the back of a broad tent, the tent filled up with listeners, and the conversation began.
I relished the conversation as I’d relished an early-morning ramble along the trails by my hotel at the base of the Santa Catalina Mountains. I joined theoretical physics for the love of ideas, and this exchange of ideas offered an intellectual workout. One of Mike’s points resonated with me most: Grubhub didn’t advance technology much. He shifted consumers from ordering pizza via phone call to ordering pizza via computer, then to ordering pizza via apps on phones. Yet these small changes, accumulated across a population and encouraged by a pandemic, changed society. Food-delivery services exploded and helped establish the gig economy (despite Mike’s concerns about worker security). One small step for technology, adopted by tens of millions, can constitute one giant leap for commerce.
To me, Grubhub offered a foil for quantum computing, which offers a giant leap in technology: The physical laws best-suited to describing today’s computers can’t describe quantum computers. Some sources portray this advance as bound to transform all our lives in countless ways. This portrayal strikes some quantum scientists as hype that can endanger quality work.
Quantum computers will transform cybersecurity, being able to break the safeguards that secure our credit-card information when we order food via Grubhub. Yet most consumers don’t know what safeguards are protecting us. We simply trust that safeguards exist. How they look under the hood will change by the time large-scale quantum computers exist—will metamorphose perhaps as dramatically as did Gregor Samsa before he woke up as an insect. But consumers’ lives might not metamorphose.
Quantum scientists hope and anticipate that quantum computers will enable discoveries in chemistry, materials science, and pharmacology. Molecules are quantum, and many materials exhibit quantum properties. Simulating quantum systems takes classical (everyday) computers copious amounts of time and memory—in some cases, so much that a classical computer the size of the universe would take ages. Quantum computers will be able to simulate quantum subjects naturally. But how these simulations will impact everyday life remains a question.
For example, consider my favorite potential application of quantum computers: fertilizer production, as envisioned by Microsoft’s quantum team. Humanity spends about 3% of the world’s energy on producing fertilizer, using a technique developed in 1909. Bacteria accomplish the same goal far more efficiently. But those bacteria use a molecule—nitrogenase—too complicated for us to understand using classical computers. Being quantum, the molecule invites quantum computation. Quantum computers may crack the molecule’s secrets and transform fertilizer production and energy use. The planet and humanity would benefit. We might reduce famines or avert human-driven natural disasters. But would the quantum computation change my neighbor’s behavior as Grubhub has? I can’t say.
Finally, evidence suggests that quantum computers can assist with optimization problems. Imagine a company that needs to transport supplies to various places at various times. How can the company optimize this process—implement it most efficiently? Quantum computers seem likely to be able to help. The evidence isn’t watertight, however, and quantum computers might not solve optimization problems exactly. If the evidence winds up correct, industries will benefit. But would this advance change Jane Doe’s everyday habits? Or will she only receive pizza deliveries a few minutes more quickly?
Don’t get me wrong; quantum technology has transformed our lives. It’s enabled the most accurate, most precise clocks in the world, which form the infrastructure behind GPS. Quantum physics has awed us, enabling the detection of gravitational waves—ripples, predicted by Einstein, in spacetime. But large-scale quantum computers—the holy grail of quantum technology—don’t suit all problems, such as totting up the miles I traveled en route to Tucson; and consumers might not notice quantum computers’ transformation of cybersecurity. I expect quantum computing to change the world, but let’s think twice about whether quantum computing will change everyone’s life like a blockbuster app.
I’ve no idea how many people have made this pun about Mike’s work, but the panel discussion left me with food for thought. He earned his undergraduate degree at MIT, by the way; so scientifically inclined Quantum Frontiers readers might enjoy his memoir, Hangry. It conveys a strong voice and dishes on data and diligence through stories. (For the best predictor of whether you’ll enjoy a burrito, ignore the starred reviews. Check how many people have reordered the burrito.)
The festival made my week. After the panel, I signed books; participated in a discussion about why “The Future Is Quantum!” with law professor Jane Bambauer; and narrowly missed a talk by Lois Lowry, a Newbury Award winner who wrote novels that I read as a child. (The auditorium filled up before I reached the door, but I’m glad that it did; Lois Lowry deserves a packed house and then some.) I learned—as I’d wondered—that yes, there’s something magical to being an author at a book festival. And I learned about how the future of tech depends on more than tech.
“If I had a nickel for every unsolicited and very personal health question I’ve gotten at parties, I’d have paid off my medical school loans by now,” my doctor friend complained. As a physicist, I can somewhat relate. I occasionally find myself nodding along politely to people’s eccentric theories about the universe. A gentleman once explained to me how twin telepathy (the phenomenon where, for example, one twin feels the other’s pain despite being in separate countries) comes from twins’ brains being entangled in the womb. Entanglement is a nonclassical correlation that can exist between spatially separated systems. If two objects are entangled, it’s possible to know everything about both of them together but nothing about either one. Entangling two particles (let alone full brains) over tens of kilometres (let alone full countries) is incredibly challenging. “Using twins to study entanglement, that’ll be the day,” I thought. Well, my last paper did something like that.
In theory, a twin study consists of two people that are as identical as possible in every way except for one. What that allows you to do is isolate the effect of that one thing on something else. Aleksander Lasek (postdoc at QuICS), David Huse (professor of physics at Princeton), Nicole Yunger Halpern (NIST physicist and Quantum Frontiers blogger), and I were interested in isolating the effects of quantities’ noncommutation (explained below) on entanglement. To do so, we first built a pair of twins and then compared them.
Consider a well-insulated thermos filled with soup. The heat and the number of “soup particles” inside the thermos are conserved. So the energy and the number of “soup particles” are conserved quantities. In classical physics, conserved quantities commute. This means that we can simultaneously measure the amount of each conserved quantity in our system, like the energy and number of soup particles. However, in quantum mechanics, this needn’t be true. Measuring one property of a quantum system can change another measurement’s outcome.
Conserved quantities’ noncommutation in thermodynamics has led to some interesting results. For example, it’s been shown that conserved quantities’ noncommutation can decrease the rate of entropy production. For the purposes of this post, entropy production is something that limits engine efficiency—how well engines can convert fuel to useful work. For example, if your car engine had zero entropy production (which is impossible), it would convert 100% of the energy in your car’s fuel into work that moved your car along the road. Current car engines can convert about 30% of this energy, so it’s no wonder that people are excited about the prospective application of decreasing entropy production. Other results (like this one and that one) have connected noncommutation to potentially hindering thermalization—the phenomenon where systems interact until they have similar properties, like when a cup of coffee cools. Thermalization limits memory storage and battery lifetimes. Thus, learning how to resist thermalization could also potentially lead to better technologies, such as longer-lasting batteries.
One can measure the amount of entanglement within a system, and as quantum particles thermalize, they entangle. Given the above results about thermalization, we might expect that noncommutation would decrease entanglement. Testing this expectation is where the twins come in.
Say we built a pair of twins that were identical in every way except for one. Nancy, the noncommuting twin, has some features that don’t commute, say, her hair colour and height. This means that if we measure her height, we’ll have no idea what her hair colour is. For Connor, the commuting twin, his hair colour and height commute, so we can determine them both simultaneously. Which twin has more entanglement? It turns out it’s Nancy.
Disclaimer: This paragraph is written for an expert audience. Our actual models consist of 1D chains of pairs of qubits. Each model has three conserved quantities (“charges”), which are sums over local charges on the sites. In the noncommuting model, the three local charges are tensor products of Pauli matrices with the identity (XI, YI, ZI). In the commuting model, the three local charges are tensor products of the Pauli matrices with themselves (XX, YY, ZZ). The paper explains in what sense these models are similar. We compared these models numerically and analytically in different settings suggested by conventional and quantum thermodynamics. In every comparison, the noncommuting model had more entanglement on average.
Our result thus suggests that noncommutation increases entanglement. So does charges’ noncommutation promote or hinder thermalization? Frankly, I’m not sure. But I’d bet the answer won’t be in the next eccentric theory I hear at a party.
My best friend—who’s held the title of best friend since kindergarten—calls me the keeper of her childhood memories. I recall which toys we played with, the first time I visited her house,1 and which beverages our classmates drank during snack time in kindergarten.2 She wouldn’t be surprised to learn that the first workshop I’ve co-organized centered on memory.
Memory—and the loss of memory—stars in thermodynamics. As an example, take what my husband will probably do this evening: bake tomorrow’s breakfast. I don’t know whether he’ll bake fruit-and-oat cookies, banana muffins, pear muffins, or pumpkin muffins. Whichever he chooses, his baking will create a scent. That scent will waft across the apartment, seep into air vents, and escape into the corridor—will disperse into the environment. By tomorrow evening, nobody will be able to tell by sniffing what my husband will have baked.
That is, the kitchen’s environment lacks a memory. This lack contributes to our experience of time’s arrow: We sense that time passes partially by smelling less and less of breakfast. Physicists call memoryless systems and processes Markovian.
Our kitchen’s environment is Markovian because it’s large and particles churn through it randomly. But not all environments share these characteristics. Metaphorically speaking, a dispersed memory of breakfast may recollect, return to a kitchen, and influence the following week’s baking. For instance, imagine an atom in a quantum computer, rather than a kitchen in an apartment. A few other atoms may form our atom’s environment. Quantum information may leak from our atom into that environment, swish around in the environment for a time, and then return to haunt our atom. We’d call the atom’s evolution and environment non-Markovian.
I had the good fortune to co-organize a workshop about non-Markovianity—about memory—this February. The workshop took place at the Banff International Research Station, abbreviated BIRS, which you pronounce like the plural of what you say when shivering outdoors in Canada. BIRS operates in the Banff Centre for Arts and Creativity, high in the Rocky Mountains. The Banff Centre could accompany a dictionary entry for pristine, to my mind. The air feels crisp, the trees on nearby peaks stand out against the snow like evergreen fringes on white velvet, and the buildings balance a rustic-mountain-lodge style with the avant-garde.
The workshop balanced styles, too, but skewed toward the theoretical and abstract. We learned about why the world behaves classically in our everyday experiences; about information-theoretic measures of the distances between quantum states; and how to simulate, on quantum computers, chemical systems that interact with environments. One talk, though, brought our theory back down to (the snow-dusted) Earth.
Gabriela Schlau-Cohen runs a chemistry lab at MIT. She wants to understand how plants transport energy. Energy arrives at a plant from the sun in the form of light. The light hits a pigment-and-protein complex. If the plant is lucky, the light transforms into a particle-like packet of energy called an exciton. The exciton traverses the receptor complex, then other complexes. Eventually, the exciton finds a spot where it can enable processes such as leaf growth.
A high fraction of the impinging photons—85%—transform into excitons. How do plants convert and transport energy as efficiently as they do?
Gabriela’s group aims to find out—not by testing natural light-harvesting complexes, but by building complexes themselves. The experimentalists mimic the complex’s protein using DNA. You can fold DNA into almost any shape you want, by choosing the DNA’s base pairs (basic units) adroitly and by using “staples” formed from more DNA scraps. The sculpted molecules are called DNA origami.
Gabriela’s group engineers different DNA structures, analogous to complexes’ proteins, to have different properties. For instance, the experimentalists engineer rigid structures and flexible structures. Then, the group assesses how energy moves through each structure. Each structure forms an environment that influences excitons’ behaviors, similarly to how a memory-containing environment influences an atom.
Courtesy of Gabriela Schlau-Cohen
The Banff environment influenced me, stirring up memories like powder displaced by a skier on the slopes above us. I first participated in a BIRS workshop as a PhD student, and then I returned as a postdoc. Now, I was co-organizing a workshop to which I brought a PhD student of my own. Time flows, as we’re reminded while walking down the mountain from the Banff Centre into town: A cemetery borders part of the path. Time flows, but we belong to that thermodynamically remarkable class of systems that retain memories…memories and a few other treasures that resist change, such as friendships held since kindergarten.
1Plushy versions of Simba and Nala from The Lion King. I remain grateful to her for letting me play at being Nala.
2I’d request milk, another kid would request apple juice, and everyone else would request orange juice.
I didn’t fancy the research suggestion emailed by my PhD advisor.
A 2016 email from John Preskill led to my publishing a paper about quantum complexity in 2022, as I explained in last month’s blog post. But I didn’t explain what I thought of his email upon receiving it.
It didn’t float my boat. (Hence my not publishing on it until 2022.)
The suggestion contained ingredients that ordinarily would have caulked any cruise ship of mine: thermodynamics, black-hole-inspired quantum information, and the concept of resources. John had forwarded a paper drafted by Stanford physicists Adam Brown and Lenny Susskind. They act as grand dukes of the community sussing out what happens to information swallowed by black holes.
From Rare-Gallery
We’re not sure how black holes work. However, physicists often model a black hole with a clump of particles squeezed close together and so forced to interact with each other strongly. The interactions entangle the particles. The clump’s quantum state—let’s call it —grows not only complicated with time (), but also complex in a technical sense: Imagine taking a fresh clump of particles and preparing it in the state via a sequence of basic operations, such as quantum gates performable with a quantum computer. The number of basic operations needed is called the complexity of . A black hole’s state has a complexity believed to grow in time—and grow and grow and grow—until plateauing.
This growth echoes the second law of thermodynamics, which helps us understand why time flows in only one direction. According to the second law, every closed, isolated system’s entropy grows until plateauing.1 Adam and Lenny drew parallels between the second law and complexity’s growth.
The less complex a quantum state is, the better it can serve as a resource in quantum computations. Recall, as we did last month, performing calculations in math class. You needed clean scratch paper on which to write the calculations. So does a quantum computer. “Scratch paper,” to a quantum computer, consists of qubits—basic units of quantum information, realized in, for example, atoms or ions. The scratch paper is “clean” if the qubits are in a simple, unentangled quantum state—a low-complexity state. A state’s greatest possible complexity, minus the actual complexity, we can call the state’s uncomplexity. Uncomplexity—a quantum state’s blankness—serves as a resource in quantum computation.
Manny Knill and Ray Laflamme realized this point in 1998, while quantifying the “power of one clean qubit.” Lenny arrived at a similar conclusion while reasoning about black holes and firewalls. For an introduction to firewalls, see this blog post by John. Suppose that someone—let’s call her Audrey—falls into a black hole. If it contains a firewall, she’ll burn up. But suppose that someone tosses a qubit into the black hole before Audrey falls. The qubit kicks the firewall farther away from the event horizon, so Audrey will remain safe for longer. Also, the qubit increases the uncomplexity of the black hole’s quantum state. Uncomplexity serves as a resource also to Audrey.
A resource is something that’s scarce, valuable, and useful for accomplishing tasks. Different things qualify as resources in different settings. For instance, imagine wanting to communicate quantum information to a friend securely. Entanglement will serve as a resource. How can we quantify and manipulate entanglement? How much entanglement do we need to perform a given communicational or computational task? Quantum scientists answer such questions with a resource theory, a simple information-theoretic model. Theorists have defined resource theories for entanglement, randomness, and more. In many a blog post, I’ve eulogized resource theories for thermodynamic settings. Can anyone define, Adam and Lenny asked, a resource theory for quantum uncomplexity?
Resource thinking pervades our world.
By late 2016, I was a quantum thermodynamicist, I was a resource theorist, and I’d just debuted my first black-hole–inspired quantum information theory. Moreover, I’d coauthored a review about the already-extant resource theory that looked closest to what Adam and Lenny sought. Hence John’s email, I expect. Yet that debut had uncovered reams of questions—questions that, as a budding physicist heady with the discovery of discovery, I could own. Why would I answer a question of someone else’s instead?
So I thanked John, read the paper draft, and pondered it for a few days. Then, I built a research program around my questions and waited for someone else to answer Adam and Lenny.
Three and a half years later, I was still waiting. The notion of uncomplexity as a resource had enchanted the black-hole-information community, so I was preparing a resource-theory talk for a quantum-complexity workshop. The preparations set wheels churning in my mind, and inspiration struck during a long walk.2
After watching my workshop talk, Philippe Faist reached out about collaborating. Philippe is a coauthor, a friend, and a fellow quantum thermodynamicist and resource theorist. Caltech’s influence had sucked him, too, into the black-hole community. We Zoomed throughout the pandemic’s first spring, widening our circle to include Teja Kothakonda, Jonas Haferkamp, and Jens Eisert of Freie University Berlin. Then, Anthony Munson joined from my nascent group in Maryland. Physical Review A published our paper, “Resource theory of quantum uncomplexity,” in January.
The next four paragraphs, I’ve geared toward experts. An agent in the resource theory manipulates a set of qubits. The agent can attempt to perform any gate on any two qubits. Noise corrupts every real-world gate implementation, though. Hence the agent effects a gate chosen randomly from near . Such fuzzy gates are free. The agent can’t append or discard any system for free: Appending even a maximally mixed qubit increases the state’s uncomplexity, as Knill and Laflamme showed.
Fuzzy gates’ randomness prevents the agent from mapping complex states to uncomplex states for free (with any considerable probability). Complexity only grows or remains constant under fuzzy operations, under appropriate conditions. This growth echoes the second law of thermodynamics.
We also defined operational tasks—uncomplexity extraction and expenditure analogous to work extraction and expenditure. Then, we bounded the efficiencies with which the agent can perform these tasks. The efficiencies depend on a complexity entropy that we defined—and that’ll star in part trois of this blog-post series.
Now, I want to know what purposes the resource theory of uncomplexity can serve. Can we recast black-hole problems in terms of the resource theory, then leverage resource-theory results to solve the black-hole problem? What about problems in condensed matter? Can our resource theory, which quantifies the difficulty of preparing quantum states, merge with the resource theory of magic, which quantifies that difficulty differently?
Unofficial mascot for fuzzy operations
I don’t regret having declined my PhD advisor’s recommendation six years ago. Doing so led me to explore probability theory and measurement theory, collaborate with two experimental labs, and write ten papers with 21 coauthors whom I esteem. But I take my hat off to Adam and Lenny for their question. And I remain grateful to the advisor who kept my goals and interests in mind while checking his email. I hope to serve Anthony and his fellow advisees as well.
1…en route to obtaining a marriage license. My husband and I married four months after the pandemic throttled government activities. Hours before the relevant office’s calendar filled up, I scored an appointment to obtain our license. Regarding the metro as off-limits, my then-fiancé and I walked from Cambridge, Massachusetts to downtown Boston for our appointment. I thank him for enduring my requests to stop so that I could write notes.
2At least, in the thermodynamic limit—if the system is infinitely large. If the system is finite-size, its entropy grows on average.
Early in the fourth year of my PhD, I received a most John-ish email from John Preskill, my PhD advisor. The title read, “thermodynamics of complexity,” and the message was concise the way that the Amazon River is damp: “Might be an interesting subject for you.”
Below the signature, I found a paper draft by Stanford physicists Adam Brown and Lenny Susskind. Adam is a Brit with an accent and a wit to match his Oxford degree. Lenny, known to the public for his books and lectures, is a New Yorker with an accent that reminds me of my grandfather. Before the physicists posted their paper online, Lenny sought feedback from John, who forwarded me the email.
The paper concerned a confluence of ideas that you’ve probably encountered in the media: string theory, black holes, and quantum information. String theory offers hope for unifying two physical theories: relativity, which describes large systems such as our universe, and quantum theory, which describes small systems such as atoms. A certain type of gravitational system and a certain type of quantum system participate in a duality, or equivalence, known since the 1990s. Our universe isn’t such a gravitational system, but never mind; the duality may still offer a toehold on a theory of quantum gravity. Properties of the gravitational system parallel properties of the quantum system and vice versa. Or so it seemed.
The gravitational system can have two black holes linked by a wormhole. The wormhole’s volume can grow linearly in time for a time exponentially long in the black holes’ entropy. Afterward, the volume hits a ceiling and approximately ceases changing. Which property of the quantum system does the wormhole’s volume parallel?
Envision the quantum system as many particles wedged close together, so that they interact with each other strongly. Initially uncorrelated particles will entangle with each other quickly. A quantum system has properties, such as average particle density, that experimentalists can measure relatively easily. Does such a measurable property—an observable of a small patch of the system—parallel the wormhole volume? No; such observables cease changing much sooner than the wormhole volume does. The same conclusion applies to the entanglement amongst the particles.
What about a more sophisticated property of the particles’ quantum state? Researchers proposed that the state’s complexity parallels the wormhole’s volume. To grasp complexity, imagine a quantum computer performing a computation. When performing computations in math class, you needed blank scratch paper on which to write your calculations. A quantum computer needs the quantum equivalent of blank scratch paper: qubits (basic units of quantum information, realized, for example, as atoms) in a simple, unentangled, “clean” state. The computer performs a sequence of basic operations—quantum logic gates—on the qubits. These operations resemble addition and subtraction but can entangle the qubits. What’s the minimal number of basic operations needed to prepare a desired quantum state (or to “uncompute” a given state to the blank state)? The state’s quantum complexity.1
Quantum complexity has loomed large over multiple fields of physics recently: quantum computing, condensed matter, and quantum gravity. The latter, we established, entails a duality between a gravitational system and a quantum system. The quantum system begins in a simple quantum state that grows complicated as the particles interact. The state’s complexity parallels the volume of a wormhole in the gravitational system, according to a conjecture.2
The conjecture would hold more water if the quantum state’s complexity grew similarly to the wormhole’s volume: linearly in time, for a time exponentially large in the quantum system’s size. Does the complexity grow so? The expectation that it does became the linear-growth conjecture.
Evidence supported the conjecture. For instance, quantum information theorists modeled the quantum particles as interacting randomly, as though undergoing a quantum circuit filled with random quantum gates. Leveraging probability theory,3 the researchers proved that the state’s complexity grows linearly at short times. Also, the complexity grows linearly for long times if each particle can store a great deal of quantum information. But what if the particles are qubits, the smallest and most ubiquitous unit of quantum information? The question lingered for years.
Jonas Haferkamp, a PhD student in Berlin, dreamed up an answer to an important version of the question.4 I had the good fortune to help formalize that answer with him and members of his research group: master’s student Teja Kothakonda, postdoc Philippe Faist, and supervisor Jens Eisert. Our paper, published in Nature Physics last year, marked step one in a research adventure catalyzed by John Preskill’s email 4.5 years earlier.
Imagine, again, qubits undergoing a circuit filled with random quantum gates. That circuit has some architecture, or arrangement of gates. Slotting different gates into the architecture effects different transformations5 on the qubits. Consider the set of all transformations implementable with one architecture. This set has some size, which we defined and analyzed.
What happens to the set’s size if you add more gates to the circuit—let the particles interact for longer? We can bound the size’s growth using the mathematical toolkits of algebraic geometry and differential topology. Upon bounding the size’s growth, we can bound the state’s complexity. The complexity, we concluded, grows linearly in time for a time exponentially long in the number of qubits.
Our result lends weight to the complexity-equals-volume hypothesis. The result also introduces algebraic geometry and differential topology into complexity as helpful mathematical toolkits. Finally, the set size that we bounded emerged as a useful concept that may elucidate circuit analyses and machine learning.
John didn’t have machine learning in mind when forwarding me an email in 2017. He didn’t even have in mind proving the linear-growth conjecture. The proof enables step two of the research adventure catalyzed by that email: thermodynamics of quantum complexity, as the email’s title stated. I’ll cover that thermodynamics in its own blog post. The simplest of messages can spin a complex legacy.
The links provided above scarcely scratch the surface of the quantum-complexity literature; for a more complete list, see our paper’s bibliography. For a seminar about the linear-growth paper, see this video hosted by Nima Lashkari’s research group.
1The term complexity has multiple meanings; forget the rest for the purposes of this article.
2According to another conjecture, the quantum state’s complexity parallels a certain space-time region’s action. (An action, in physics, isn’t a motion or a deed or something that Hamlet keeps avoiding. An action is a mathematical object that determines how a system can and can’t change in time.) The first two conjectures snowballed into a paper entitled “Does complexity equal anything?” Whatever it parallels, complexity plays an important role in the gravitational–quantum duality.
3Experts: Such as unitary -designs.
4Experts: Our work concerns quantum circuits, rather than evolutions under fixed Hamiltonians. Also, our work concerns exact circuit complexity, the minimal number of gates needed to prepare a state exactly. A natural but tricky extension eluded us: approximate circuit complexity, the minimal number of gates needed to approximate the state.
, the “code distance”, dependent primarily on the quality of your hardware, which is somewhere in the range of 20-40. It takes 
qubits to make up a logical qubit, so let’s say 1000 of them per logical qubit. (We need to make use of an equivalent number of ancillary qubits as well). Very roughly speaking, we repeat twice the following: each physical qubit gets involved in a small number (say 4-8) of two-qubit gates with neighboring qubits, and then some subset of qubits undergo a single-qubit measurement. Most of these gates can happen simultaneously, so (again, roughly!) the time for this whole process is the time for a handful of two-qubit gates plus a measurement. It is known as a code cycle and the time it takes we denote 
. For example, in superconducting qubits this timescale is expected to be about 1 microsecond, for ion-trap qubits about 1 millisecond. Although variations exist, lets stick to considering a basic architecture which requires repeating this whole process on the order of 
, this sets the effective logical gate speed.
, the time for a logical gate would be 
and the photonic equivalent of 
and some go into a delay of length 
. 
that some photons spend in the longest interleaving delay line. We would like to make this as long as possible, but the maximum time is limited by the loss in the delay (typically optical fiber) and the maximum loss our error correcting code can tolerate. A number to have in mind for this (in early machines) is a handful of microseconds – corresponding to a few Km of fiber.
. Note that this rate is distinct from any physical clock rate, as, e.g., 100 resource state generators running at 100MHz, or 10 resource state generators running at 1GHz, or 1 resource state generator running at 10GHz all yield the same total rate of resource state production 
which is effectively the total number of resource states “alive” in the machine at any given instant of time, including all the ones stacked up in long delay lines. This is how we leverage optical fiber delays for a massive amplification of the entanglement our hardware has available to compute with.
and the logical gate time of the machine is
.
microsecond code cycle – the answer is roughly that with 10 million physical superconducting qubits it would take about 4 hours (with an equivalent ion trap computer the time balloons to more than 5 months).
1 microsecond compares to the expected 
. This could be 6000 resource state generators each producing resource states at 
or 3500 generators producing them at 1GHz for example. Then the same computation would take 58 seconds, instead of four hours, a speedup by a factor of more than 200!
? And what if our longest delay is ten times longer, i.e. 
microseconds (so as to be much slower than 
by building 350 billion resource state generators that can each produce resource states at only 1Hz. Yes, you read that right.
would scale proportionally with 
and 
. As I hope is clear by now, as a pure theoretician, I don’t give a damn if that means 10000 resource state generators running at 1MHz, 100 resource state generators running at 100MHz, or 10 resource state generators running at 1GHz. However, the fact this flexibility exists is very useful to my engineering colleagues – who, of course, aim to build the smallest and fastest possible machine they can, thereby shortening the time until we let them head off for a nice long vacation sipping mezcal margaritas on a warm tropical beach.
. Loss reduces the rate of production considerably.
qubits. The agent can attempt to perform any gate 
on any two qubits. Noise corrupts every real-world gate implementation, though. Hence the agent effects a gate chosen randomly from near 
Quantum Frontiers 