Why Build a Quantum Transducer?

The paper reporting the main result of my postdoctoral research was finally published on Tuesday, nearly a year after my departure from Boulder. I wrote a twitter thread about the paper when we posted it online in preprint form six months ago, but I thought I’d take this occasion to rouse the blog from its regular slumber and write something longer.

This result is part of an ongoing research effort that started before I got to Boulder. The overarching goal is to develop a device called a quantum microwave-to-optical transducer that could someday be used to network quantum computers. A microwave-to-optical transducer is a box with one input and one output: put in some information encoded in an electric current that oscillates at a microwave frequency, and out will come the same information encoded in visible or near-infrared light.¹Etymology notwithstanding, “optical physics” includes the study of infrared light as well as visible light, in large part because the key experimental tools (lasers, mirrors, lenses, detectors, etc.) are rather similar. Throughout this post I’ll use “optical” in this broad sense. I’ll have more to say about “microwave” further down. Such transducers already exist and are widely used in the communications infrastructure we rely on every day. But it’s much harder to make a device that can faithfully convert quantum information between these two encodings.

There’s a lot of buzz about quantum computing these days, some of it justified. It’s widely accepted that sufficiently powerful quantum computers will be able to perform certain useful tasks, like simulating complex molecules and factoring large numbers, much faster than conventional computers. “Certain” is the keyword here — there are only a few classes of problems for which a quantum computational advantage is known to exist, so you should take claims that quantum computing will “revolutionize everything” with a hefty grain of salt.²For further reading about quantum computing, I recommend this article by Scott Aaronson.

But let’s back up. What does it mean for information to be encoded in electricity or light in the first place? Why should we care about microwave and optical encodings specifically, and why do we want to transfer information from one to the other? And how does the story change when I insist on repeating the word “quantum” twice in every sentence?

These are the questions I will address in this post — I’ll say nothing at all about what goes on inside a microwave-to-optical transducer, quantum or otherwise, and thus completely ignore what I actually did as a postdoc. I plan to explore the inner workings of transducers and what our latest result means in a follow-up post more analogous to my post about the experiment I worked on as a grad student.³I’ve learned better than to any make promises about when I’ll get around to writing this second post.

This post consists of four sections of gradually increasing complexity: even if you don’t follow the discussion of quantum communication through to its conclusion, I hope you still come away with a renewed appreciation for all the aspects of communication we so often take for granted in everyday life.

I’ll begin with some simple examples that establish how I’ll use familiar terms like “information” and “communication” throughout the post, then move on in the second section to a discussion of how modern communication networks use both electricity and light to distribute information.

In the third section, the focus will shift to quantum information science. I’ll start with a brief introduction to quantum bits (or ‘qubits’), then describe a specific type of qubit based on the electric current flowing through a superconducting circuit, and finally discuss how individual superconducting qubits use microwaves to communicate. The fourth section will explain why quantum microwave-to-optical transducers will be critical to the success of superconducting quantum computing on a large scale.

Code Switching

Let’s begin with a principle that’s indispensable to all communication: information is independent of the medium in which it’s encoded. Stop for a second and think about how cool this is. Let’s say I have some thoughts I want to share with the world. By tapping away on a keyboard for a while, I command my computer to represent those thoughts in some configuration of digital electronic components. I don’t have to know anything about how these components work, because the computer presents my thoughts back at me as a pattern of black and white pixels on a screen.

When I’ve arranged this pattern of pixels to my satisfaction, I command my computer to relay the information to a web server in Utah. By opening a web browser and entering your favorite URL, you command this computer to relay the information to you, whereupon you digest it by looking at a pattern of black and white pixels resembling the one I generated using my keyboard. You then decide to tell a friend about this great blog you’ve been reading, by making your vocal cords vibrate in a particular way, generating pressure waves in the air that make tiny bones in your friend’s ears vibrate. The net result is that your friend now shares my enthusiasm for microwave-to-optical transducers.

We can sort the steps outlined above into three broad classes — processing, storage, and communication. Writing is a form of information processing: as far as my computer is concerned, my creative process amounts to a series of operations that transform bits, turning 0s into 1s or vice versa. Those 0s and 1s must be encoded in a form that is stable — we don’t want them flipping back and forth at random — but easy to manipulate. In practice computers represent these bits using voltages across transistors.

When I’m satisfied with the draft, I want the bits representing it to remain unchanged after I direct my computer’s processing power elsewhere. For long-term storage, my computer encodes the pattern of 0s and 1s in the orientations of atoms in a magnetic film. At a higher level, written language and human memory are also forms of information storage.

In this post I’ll be chiefly concerned with communication: the transmission of information between two systems. During storage and processing, information is encoded in physical systems that are spatially localized, but for communication it must hitch a ride on something that travels from sender to receiver. This something is fittingly called a carrier. In the context of communication, a transducer is anything that swaps information from one carrier to another.

Let’s look at an example. I arrange for a speaker to play a tone that cuts in and out according to a specific pattern, having decided in advance that each second the tone is on will represent a 1 and each second it’s off will represent a 0.⁴This is not an ideal encoding: a stretch of silence due to equipment malfunction would be misinterpreted as a series of 0s, and it takes one second to transmit a single bit. I could use shorter time intervals for each bit to speed things up, but at some point I’d run into the limits of the human auditory system. This is a contrived example of a limit on the bandwidth of a communication channel. More sophisticated encodings with much higher bandwidth are used in practice. This tone is a carrier that I can use to communicate a message in binary to everybody within earshot, albeit at the cost of annoying them so much that they never want to speak to me again. But suppose you forgive me and want to relay my message in real time to somebody farther away. If it’s dark out, you can switch a flashlight on and off in sync with the sound: you’re now communicating using a beam of light as the carrier. The transducer in this example is you: its input is your ear and its output is your thumb on the flashlight switch.

The internet is a series of tubes

I’ll now relieve you of your duties as thought experiment transducer and consider a more realistic example. What happens when I connect to a website that’s hosted in Europe? First, my computer talks to my router using my wifi network — a microwave-frequency electromagnetic field suffusing my apartment — as the carrier. The information encoded in this microwave field is then imprinted on an electric current flowing through a series of cables that terminates somewhere on the New England coast. There, a transducer transfers the information to pulses of infrared laser light that travel through an undersea network of thin strands of glass called optical fibers.⁵For our present purposes, the most important thing about lasers is that they emit light with a very specific wavelength, while other sources emit light with a range of different wavelengths. This matters because the energy loss in optical fibers is different at different wavelengths; it’s lowest at near-infrared wavelengths around 1550 nanometers. On the other side of the ocean, the same process unfolds in reverse.

Why optical fibers? Before the widespread adoption of fiber in the late 1980s, undersea communications cables were made of copper, an excellent electrical conductor. But even copper has some residual electrical resistance that causes signals to lose energy as they travel over long distances, necessitating the use of “repeaters” placed along the line every 10 kilometers or so to boost the signal. State-of-the-art optical fibers are ever so slightly leaky, but much less lossy than even the best electrical conductors, allowing the spacing between repeaters to be increased to more than 100 kilometers.⁶It’s possible to compensate for cable losses to some extent by using very high voltages. Optical fibers also have other advantages, such as much higher bandwidth.

Now what’s so special about microwave frequencies? Thus far, in the interest of not getting bogged down in technical details, I’ve been throwing the word “microwave” around without properly defining it, but now I’ll have to get more specific.

First, we must appreciate that all the light we see — and, with apologies to Anthony Doerr, all the light we cannot see — is made of electromagnetic waves: intertwined oscillating electric and magnetic fields that propagate through space. If “electromagnetic wave” means nothing to you, then this is a profoundly unhelpful definition that just pushes the jargon up one level. For our present purposes, all that matters is that the electric part of an electromagnetic wave tries to pull electrically charged particles in one direction during half of the oscillation cycle, and pull them in the opposite direction during the other half of the cycle.

Whether or not we can see an electromagnetic wave depends on how fast it oscillates, and the range of oscillation frequencies is called the electromagnetic spectrum. Waves that oscillate a few billion times per second are called microwaves, while visible and near-infrared waves oscillate a few hundred trillion times per second.

There’s a big difference between a billion and a hundred trillion. When microwaves pass over a material that conducts electricity, electrons move in response to the oscillating electric field, just as waves on the surface of a body of water cause a buoy to bob up and down. The collective motion of the electrons generates an electric current that oscillates at the same frequency as the passing wave.⁷This is how some radio receivers work. Optical light, on the other hand, oscillates way too fast for electrons to keep up.⁸Fundamentally, this is because electrons have some inertia due to their mass: it takes them a while to start moving, by which point the field has switched directions.

The upshot of all this is that easy to swap information between microwaves and oscillating electric currents with frequencies in the same range, and it’s common to speak of “microwave-frequency signals,” whether the signals are carried by microwaves or oscillating currents in electrical circuits. This versatility is an attractive feature of the microwave region of the electromagnetic spectrum.⁹I still haven’t explained why we’re interested in microwaves specifically, as opposed to radio waves (with lower frequencies) or millimeter waves (with slightly higher frequencies). There’s unfortunately no single simple answer to this question. Suffice it to say here that the technology for transmitting, receiving, and manipulating signals at microwave frequencies is very mature. Microwave electromagnetic fields are used for local wireless communication, and microwave-frequency electrical signals are widely used in information processing — in the central processing units of computers and smartphones, for instance. Optical frequencies aren’t so flexible — they’re great for long-distance communication, but not much use in information processing.

Enter Quantum Physics

We’ve seen that microwave and optical signals play complementary roles in modern communications networks: microwave frequencies are convenient for information processing and local communication, while optical frequencies are ideal for long-distance communication. This divide is even sharper in quantum information science — it’s not just that optical frequencies are more convenient for long-distance quantum communication; rather, long-distance quantum communication with microwaves is altogether hopeless. To see why, we need to understand a few things about the curious nature of quantum information and familiarize ourselves with a specific physical system that can encode quantum information.

Superpositions

Ordinary classical information can be represented by a series of bits, each of which is either 0 or 1. A quantum bit, or qubit, can be thought of as a sphere with 0 at one pole and 1 at the other pole.¹⁰It’s not an actual sphere in three spatial dimensions, just like an ordinary bit is not two points labeled “0” and “1” floating in space. People sometimes say a qubit can be “both 0 and 1 at the same time,” but this isn’t entirely correct. Rather, a qubit has infinitely many states between 0 and 1: any point on the surface of the sphere represents a possible state.¹¹“Infinitely many states between 0 and 1” probably suggests points on a line segment rather than points on the surface of a sphere. The second dimension of “betweenness” here is a consequence of the striking fact that imaginary numbers show in quantum physics: roughly speaking, a position on the surface is specified by a real number analogous to latitude and an imaginary number analogous to longitude. Confusion on this point arises in part because it’s common (though not always necessary) to describe qubit states using their distance from the 0 and 1 states. I could come up with an odd but perfectly sensible way to label the latitude of a point on earth according to which Stockholm, say, would be “mostly north but still a little bit south” in that it’s not all the way at the north pole. That wouldn’t mean it’s in two places at once.

But I’m not being entirely fair to quantum physics popularizers, because there really is something weird going on here. Even though the qubit has infinitely many states, only its 0 and 1 states have well-defined energies.¹²I realize I’m being very vague about what “energy” means in this context, but unfortunately this is a subtle question that would take us very far afield. Just think of energy here as an especially important property of quantum states, while I hang my head in shame for punting on this question. Any other state has what’s called a superposition of these two energies. If you care about the energy of such a state, you must represent it with a mathematical expression involving the sum of two terms: one for the energy associated with its “northness,” and the other for the energy of its “southness.” This is the sense in which the state has “two energies at once;” unfortunately there just isn’t an entirely satisfactory way to phrase this in plain English.¹³My insistence that even though there are “two energies” in this sense there’s only one state may sound like terminological nitpicking. It matters because energy superposition states are only superpositions with respect to energy; other physical quantities associated with these states may have perfectly well-defined values. Conversely, states with well-defined energies are superposition states with respect to other quantities.

When a qubit in one of these states interacts with its environment, the interaction tends to mess with the superposition of energies. In this sense quantum information is very fragile, and you can’t do anything useful with quantum information unless you can keep superposition states alive. Experimental physicists working in quantum information science devote a lot of effort to encoding quantum information in clever ways that shield it from the messy world outside.

Superconductors

Thus far I’ve discussed quantum information at a rather abstract level; now it’s time to get specific. Many different physical systems can represent the 0s and 1s of classical information, but for information processing, transistors on silicon chips are unrivaled. Quantum information science is much less mature, and researchers continue to explore the relative merits of many different physical implementations of qubits.

Perhaps the most popular approach is superconducting quantum computing, in which qubits are encoded in microwave-frequency resonant circuits made of superconducting metals. That’s a lot of jargon for one sentence, so let’s break it down. Resonant circuits are electrical circuits in which the electric currents oscillate at particular frequencies — circuits used as qubits are designed to resonate at microwave frequencies in part because microwave electronics technology is so mature. Superconductors are metals whose electrical resistivity drops sharply when the metal is cooled below a certain temperature. The electric current flowing through a superconducting circuit is very well isolated from its environment, and thus can exhibit subtle quantum behavior like superposition that would be smeared out in more conventional circuits.¹⁴This doesn’t mean that any old thing that happens to a superconducting qubit is profoundly quantum.

When I say a qubit is “encoded” in a superconducting circuit, I mean the electric current flowing through the circuit has two well-defined states with slightly different energies that can represent the 0 and 1 states of a qubit. In one common qubit design, for instance, these two states correspond to currents flowing in opposite directions through a superconducting loop. A superposition of clockwise- and counterclockwise-circulating currents is a perfectly valid state of such a qubit!

How qubits communicate

Armed with a basic understanding of superconducting qubits, we can now consider how they talk to each other, starting with the case of local microwave communication between nearby qubits. In the second part of this post, I noted that it’s easy to pass ordinary classical information back and forth between microwaves and electric currents that oscillate at microwave frequencies. This is no less true of quantum information, but the wave-particle duality of quantum physics suggests a radically different perspective on how this information transfer works.

In quantum physics, light sometimes behaves not like a wave but rather like a stream of discrete particles called photons, each with energy determined by the frequency of the associated electromagnetic wave. This relationship between energy and frequency is a feature of all quantum systems, not just light: for example, the difference between the energies of a qubit’s 0 and 1 states bears the same relationship to the qubit’s resonant frequency as a photon’s energy does to its wave frequency.

Let’s take a specific example: a qubit in the higher-energy 1 state can drop down to the 0 state by emitting a microwave photon with energy equal to the difference between the energies of the two states. A second qubit in the 0 state can jump up to the 1 state by absorbing that photon: the photon has mediated an interaction that swaps the states of the two qubits. Superpositions and other complex quantum states can be distributed throughout a system of interconnected qubits in a similar way. Put enough qubits together and you get a quantum computer that might be capable of doing something useful.

Information wants to be freed from its cryogenic prison

It bears repeating that a full-fledged quantum computer remains a distant goal: it will likely require not 100 qubits (roughly the current state of the art) but millions. Scaling up is a challenge for every proposed quantum computing technology, but for superconducting quantum computers specifically, one obvious difficulty is the limited space inside the cryostats that maintain the superconducting qubits at the extremely low temperature at which they must operate.

The problem is not so much the qubits themselves, but rather all the other electronics required to control and measure them. There are scientists working on developing miniature superconducting alternatives to many of these bulky electronic components, but the best route to a large-scale superconducting quantum computer will likely involve local networks of smaller processors in separate cryostats. And just as the internet radically transformed the way we use computers today, global networks of quantum computers may open up entirely new directions in quantum information science.

But let’s not get ahead of ourselves. To network superconducting quantum processors, we need a way to extract quantum information from a cryostat without scrambling it. We’ve seen that superconducting qubits communicate locally by exchanging microwave photons. Although optical fiber networks are superior in most respects, microwave carriers can also be used for long-distance classical communication — satellite internet is a prime example. So we’re back to the question that motivated this post. Why are physicists so interested in building a quantum microwave-to-optical transducer? Why not use microwave photons for quantum communication between cryostats?

The fundamental problem is that the energies of microwave photons are too small to faithfully preserve quantum information at room temperature. Whenever you see a sentence like this, your first question should be “small compared to what?” Any claim that a physical quantity is small or large involves an implicit comparison to another quantity. In this case, the correct comparison is to the size of typical energy fluctuations in the environment through which the photon must travel, which are closely related to the temperature of that environment. Large energy fluctuations will screw up any quantum information encoded in superpositions of states with slightly different energies.

We saw earlier that microwave and optical frequencies differ by roughly a factor of 100,000. The quantum link between frequency and energy implies that microwave and optical photon energies differ by the same factor. The energies of individual optical photons are much larger than thermal energy fluctuations at room temperature, making these photons ideal carriers for long-distance quantum communication.¹⁶I explored the deep connections between frequency, energy, and temperature in detail in my blog post about natural units.  In the appendix below (not for the faint of heart!), I’ll use the concepts introduced in that post to make these comparisons between photon energies and thermal fluctuations quantitative.

We’re coming to the end of the post, so let me sum up the main point. If superconducting quantum computers are to ever become viable, we’re going to need some kind of transducer that swaps quantum information from a microwave encoding to an optical encoding inside a cryostat: that information can then be sent up to room temperature with superpositions intact.¹⁷Other proposed quantum computers work at room temperature using ions or atoms as qubits, and quantum microwave-to-optical transduction isn’t necessary. This is one attractive feature of these schemes relative to quantum computing using superconducting circuits or electrons in semiconductors. Of course, we need to ensure that the transducer itself doesn’t scramble the quantum information during the transduction process, which is much easier said than done. This latest result got us one step closer — but that’s a story for another time.

Appendix: Natural units revisited

Warning: heavy jargon ahead! Proceed only if you’re feeling intrepid and are comfortable with the ideas and terminology introduced in my natural units post; I’ll also make use of scientific notation to express some very large or very small numbers.

In that post, I introduced two fundamental constants called \(h\) and \(k_B\) named after physicists Max Planck and Ludwig Boltzmann, respectively. Planck’s constant appears in the quantum relationship \(E = hf\) between frequency and energy, while Boltzmann’s constant relates temperature to an equivalent energy scale: \(E = k_BT\).

Taking \(f_\text{mic} \approx 6~\text{GHz} = 6\times 10^{9}~\text{Hz}\) as a typical microwave frequency, \(f_\text{opt} \approx 300~\text{THz} = 3\times10^{14}~\text{Hz}\) as a typical optical frequency, \(T_\text{c} \approx 30~\text{mK}= 3\times10^{-2}~\text{K}\) as a typical temperature inside a cryostat housing superconducting qubits, and using \(T_\text{r} \approx 300~\text{K}\) for room temperature, we can convert everything into units of energy to obtain \(hf_\text{opt} \approx 1.2~\text{eV}\), \(hf_\text{mic} \approx 2.5\times10^{-5}~\text{eV}\), \(k_BT_\text{r} \approx 2.5\times10^{-2}~\text{eV}\), and \(k_BT_\text{c} \approx  2.5\times10^{-6}~\text{eV}\).

Arranging these energy scales in ascending order, we see that

\[k_BT_\text{c} < hf_\text{mic} \ll k_BT_\text{r} \ll hf_\text{opt}.\]

The second and third inequalities here quantify the points I expressed in words above: quantum information encoded in microwave photons will be scrambled at room temperature, but optical encodings are very robust.

The first inequality, meanwhile, indicates that quantum information encoded in microwave photons is safe at the frigid temperatures inside the cryostat. This is one reason superconducting qubits aren’t designed to resonate at lower frequencies: if quantum information were encoded in radio waves, even 30 millikelvin would be too hot.