Connecting superconducting quantum computers with light

Entanglement is a peculiar feature of quantum systems that makes them behave as if they were sitting directly next to each other even if they are kilometres away. Such behaviour does not occur in classical physics. Classical particles can affect each other through fields — such as the gravitational or electromagnetic field — but these fields propagate with the speed of light; the interaction between entangled particles, however, is instantaneous. Simply as if the distance between them did not exist.

Classical particles do not affect each other when they are far apart. In quantum physics, entangled particles do influence one another; it is as if they were always close to each other.
Classical particles do not affect each other when they are far apart. In quantum physics, entangled particles do influence one another; it is as if they were always close to each other.

Entanglement started to interest scientists in the early days of quantum physics (it was first mentioned by Erwin Schrödinger in 1935) but only in the 1990s, it was realised that entanglement can also be used as a resource for quantum communication. After some 20 years of intensive research, not much has been achieved in creating entanglement (over long distances as is needed for quantum communication) in the laboratory. The main problem is that the only system that we can send over such distances is light; if a system does not interact with light, it cannot be entangled with another system that is sitting far away.

One particular example of such a system is a superconducting circuit. These typically work at energies corresponding to microwave fields and microwaves cannot be transmitted as easily as light (at least in the quantum regime; the world is too hot for them and the signal they carry does not survive). But superconducting systems seem to be very well suited for quantum computing. And having a (quantum) computer which cannot communicate with other computers over (quantum) internet… well, what’s the point?

A superconducting system with a bus resonator (black line) and two quantum bits (in red squares). © Schoelkopf lab, Yale University
A superconducting system with a bus resonator (black line) and two quantum bits (in red squares). © Schoelkopf lab, Yale University.

Naturally, scientists started looking for a way to connect superconducting circuits and light using a third system that can interact with both. There are several candidates for such an interface — sort of a quantum network card, if you like — and one of the most promising options is to use a mechanical oscillator for the task. Those can be relatively easily manufactures, well controlled, and they can strongly interact with both light and superconducting circuits.

A scheme for entangling superconducting qubits that uses their interaction with mechanical oscillators and optomechanical measurement.
A scheme for entangling superconducting qubits that uses their interaction with mechanical oscillators and optomechanical measurement.

What can we do with all that? Let us start with the simplest possible task — entangling two quantum bits formed by superconducting circuits1 and connected by light. There are many ways this can be done; we will use an approach where the light is used to measure the two qubits. A well-chosen measurement which reveals some joint property of the qubits can result in an entangled state; furthermore, it has the advantage that the right measurement outcome signals that the entangled state has been successfully created. (This is also a reminder of the importance measurements have in quantum physics that I wrote about before.)

Suppose we now start by preparing the qubits in such a state that each qubit has values 0 and 1 at the same time. Now, we let each qubit interact with a mechanical oscillator and the oscillators interact with a beam of light that we measure. If we build the system the right way, the measurement of light will tell us how many qubits have the value 1. It can happen that both or none have this value, which is uninteresting. But if exactly one of the qubits has the value 1 (the other, naturally, has the value 0), they are entangled because we cannot tell which qubit has which value. No matter how far apart they are, if we now measure one of the qubits to be in the state 0, the other will immediately end up having the value 1 and vice versa.

There is are many things one can do once the qubits are entangled — transfer quantum states using quantum teleportation, send encrypted messages using quantum key distribution, or try to confirm quantum mechanics by violating Bell’s inequality, for example.

Ultimately, people are interested in creating entanglement in more complicated systems; with superconducting circuits, it would be interesting to have many qubits entangled. There is one very practical reason for that: Superconducting quantum computers need to work at very low temperatures (only about 0.01 °C above absolute zero) and it is very difficult to cool things to such low temperatures. As a result, only small things can be successfully cooled. Future superconducting quantum computers therefore cannot be very large; to have one large, powerful quantum computer, it is then necessary to connect several such computers using entanglement. Then, the many small computers will behave as if they all were in the same large fridge, forming parts of a large quantum computer.

This post aims to summarise the main results of a paper I wrote with my PhD advisor on the topic of generating entanglement of superconducting qubits using optomechanical systems. A free preprint can be found at arXiv.

1 Superconducting qubits work similar as classical bits in a computer — there is a current running through a circuit and the value of this current determines the value of the bit. The only principal difference is that a classical bit has a value of either 0 or 1 whereas a quantum bit can also be in their superposition, having values 0 and 1 simultaneously.


Simplifying quantum systems


I already talked about using measurements and feedback in quantum physics and how these tools can be used to prepare interesting quantum states. But it is not an easy task — experimental realisations require ultrafast electronics to apply feedback in real time. And theoretical analyses? Those are not easy either.

When monitoring the outside field of an optical cavity with an atom inside, the exact dynamics of the field is often irrelevant.
When monitoring the outside field of an optical cavity with an atom inside, the exact dynamics of the field is often irrelevant.

Take a simple example — an atom that is placed inside an optical cavity. We measure what leaves the cavity and want to use the information we get to control the state of the atom. The first thing a theoretical physicist will do is write the equation that describes the time evolution of the whole system (i.e., the atom and the cavity field). But we do not really care what happens with the field. We only want to know what the field can tell us about the atom. If only there was a way to get an equation that describes only the dynamics of the atom…

There actually is a whole bunch of methods that can help us do just that — they are generally known under the name adiabatic elimination. We eliminate the uninteresting part of the system (the cavity mode), leaving an equation just for the relevant part (the atom). And why is it called adiabatic? Because all the methods assume that the uninteresting part evolves much faster than the interesting one — the cavity mode will thus quickly reach a steady state (i.e., a state that does not further evolve in time) and, as the atom slowly evolves, the steady state of the cavity will follow it. And physicists call such following of one system by another adiabatic.

All these methods generally suffer from two problems. Firstly, they work only if the cavity field is in a so-called pure state. These are some rather special quantum states that you can get if there is no thermal noise (i.e., the system is cold or uses high — typically optical — frequencies). You can imagine thermal noise as if you were shining inside the cavity with a regular light bulb. Its light is very chaotic (much more than that of a laser) so the state of the field inside the cavity will be chaotic as well. And that is more difficult to deal with than when the light entering the cavity a nice coherent laser beam.

Secondly, adiabatic elimination methods can work well if you need to eliminate a single field.  If you have a more complicated system that you need to get rid of, it is not that simple. You can, in principle, eliminate more fields one by one but that takes a long time. And the order in which you eliminate imposes additional conditions on the system. (You start by eliminating the fastest of the fields, then the second fastest, and so on.)

Complex systems -- a pair of qubits coupled to optomechanical transducers, for instance -- are too large and cannot be numerically simulated.
Complex systems — a pair of qubits coupled to optomechanical transducers, for instance — are too large and cannot be numerically simulated.

Imagine now that you want to work with a more complicated system — you want to entangle two superconducting qubits coupled to optomechanical transducers (like I do). You have the transducers — consisting of a microwave cavity, a mechanical oscillator, and an optical cavity — that you do not really care about and the qubits that are the important part of the system. So if you now want to eliminate the transducers, you have a problem because you have many fields (three for each transducer) and mechanical oscillators which will have thermal noise.

Here, the adiabatic elimination becomes more crucial than with a single atom and a single optical cavity. Whereas it is just a matter of convenience for the simple system, the two qubits with two optomechanical transducers cannot be numerically simulated exactly. You would need several terabytes of memory to store state of such a large system in a single point in time. And what should the feedback applied to the qubits look like? You cannot guess that well with such a complicated system.

In order to be able to deal with such big and complicated systems, we had to develop a brand new method of adiabatic elimination and we had to take a completely different approach than people usually take. We made a different assumption than the usual purity — instead, we assume that the eliminated system is Gaussian. This means that there are some quantities in the system (our optomechanical transducer) that behave as a classical Gaussian probability distribution. That is true for a large class of systems (including our optomechanical transducers) and makes it possible to describe the transducers using parameters of these Gaussian distributions which is much easier than using a full quantum state.

The applications of this method are much broader than this particular system. Measurements and feedback are often used in superconducting systems which typically interact with microwave fields. As a result, thermal noise can be present and standard methods of adiabatic elimination do not work. The way around this problem is to assume that the noise is so small that we can safely neglect it and apply standard methods of adiabatic elimination. This assumption usually works relatively well but our new method works even better (with the almost nonexistent noise!) and is not much more complicated to deal with.

There is more to adiabatic elimination than tractability of numerical simulations (which is still pretty important!). It can give us information about the evolution of the small part of system that we are really interested in. A trained scientist can make a good guess based on the evolution of the whole system (including, for instance, the cavity field) but understanding the exact role of various system parameters (such as the amount of the thermal noise) is not always so easy. Now, we have made an important step in understanding these issues.

This post summarises the main results of a paper I wrote with my colleagues on adiabatic elimination with continuous measurements. A free preprint can be found at arXiv.

Building the quantum internet

Do you remember your first computer? And your first internet connection?  Sure, they were not as powerful as today’s technology but it was something completely new and opened many possibilities. A quantum computer, ideally connected to quantum internet, must then be even more remarkable. Although it is true that algorithms for quantum computers focus on abstract mathematical tasks such as factoring large numbers, everyday life applications will certainly come as well. After all, classical computers were also originally seen solely as calculators.

We have now pretty good idea what the quantum internet could look like. Because quantum systems are very sensitive to disturbances and quantum features do not survive for long, the ideal medium for transmitting quantum signals is light. It travels fast and almost does not interact with the surrounding environment so quantum effects can survive a long-distance transfer.

Quantum computers, on the other hand, can in principle be built in many different ways. Some scientists trap ions in electric fields and use them as the basic building blocks. Others try to build the whole quantum computer from a single molecule and use different parts of this molecule as quantum bits that store information. Some try to use light to perform quantum computations since such quantum computers are then easily connected via quantum internet. There are also those who use superconducting systems.

In a way, superconducting systems are, in their form, most similar to classical computers. You can build a chip from the right material, similarly to an integrated circuit in a normal computer. Then you cool the chip down to temperature of a few Kelvin (around -270 degrees Celsius) and it becomes superconducting — it starts to transmit current without any resistance. Quantum bits can then be represented by superconducting currents of various strength, similar to normal computers.

There is just one problem with superconducting quantum computers — it is not possible to connect them to optical quantum internet. Energy of superconducting qubits is much smaller than that of an optical photon so they do not interact well. Superconducting systems can interact with microwave fields but those cannot be transmitted as easily as light because they require low temperatures (just like superconducting systems) to overcome noise.

The solution is simple: We let the superconducting qubits interact with microwave photons which can then be converted to light using mechanical oscillators. Or we can even skip the microwave field and couple superconducting qubits directly to mechanical oscillators. That is possible because superconducting qubits are built using capacitors and some other elements. If one of the capacitor plates can vibrate, its position will affect the state of the qubit and the state of the qubit, in turn, determines the position of the vibrating plate.

Because we do not have quantum computers just yet, we can start with a smaller task — we can try to entangle two superconducting qubits that sit on two different chips. That would be a first step towards building quantum internet with superconducting systems.

Measurement of the number of excitations
Number of excitations of two qubits can be measured if the signal from the first qubit (the sphere with arrow) is converted using a transducer (blackbox), transmitted and converted back.

The approach I like is based on measurement feedback and there are two options how to use it. The first one uses entanglement swapping where each of the qubits interacts with a microwave field in a way that generates entanglement between them. The microwave field is then converted to light and travels to a detector where both the fields are measured together. In this way, the entangled state is teleported from a microwave field to a qubit and both qubits become entangled.

Entanglement swapping with two qubits
Entangling qubits with their transducers locally and then performing joint measurement on the light fields, one can entangle the two qubits.

Another option is to engineer the system in such a way that we perform a measurement of the number of excitations of the two qubits. Each qubit has two levels — denoted by 0 and 1 and thus showing the number of excitations in the qubit. If we prepare the qubits in a suitable state and the measurement reveals that one qubit is excited but we do not know which one, they become entangled. That is commonly done with superconducting qubits (without coupling to light, though). With the optical link, this can be done in the following way: we let one qubit interact with a microwave field which then gets converted to light. The light gets transmitted to the second qubit where it is converted back to microwave frequency, interacts with the second qubit, and is measured.

So far it seems that such tasks can be performed with mechanical oscillators that need not be much better that what is available currently. We thus might see the first steps towards quantum networks with superconducting qubits in the near future. But it will still be a long way to go if we want to build quantum computers connected by quantum internet.

This post is loosely based on talk I held at the Spring meeting of the German Physical Society in Heidelberg, March 2015.

Wi-Fi for a quantum computer

The basic picture of an optomechanical system, that even many scientists keep in mind, is that of a cavity with one movable mirror. But that is not the only way to achieve coupling between light and mechanical vibrations. Every time light is strong enough (and the mechanical oscillator light enough), the light can be used to control the vibrational state of the mechanical system.

Optomechanical systems can take on various forms, such as a vibrating mirror inside a cavity or a vibrating microdisk.
Optomechanical systems can take on various forms, such as a vibrating mirror inside a cavity or a vibrating microdisk.

People have studied all sorts of different systems this way. One option is to use a cavity (with both mirrors fixed) and put a vibrating membrane inside. Other scientists work with microdisks where light travels around thanks to total internal reflection; if the disk can vibrate, strong light will excite mechanical vibrations of the disk. And there are optomechanical platforms that are more exotic than these examples.

The beauty of the theoretical description of such systems lies in the fact that they are all described by the same mathematics. This stays true even if we do not use visible light but a microwave field which cannot be trapped in a cavity using two simple mirrors. Instead, microwave cavities have the form of LC circuits — basic electrical circuits with an inductor (basically a coil) and a capacitor (two conducting plates separated by a thin layer of a dielectric material) that have been used in electronics for decades.

Optomechanics can be studied even in microwave systems, where the role of the optical cavity is taken by an LC circuit and vibrating mirror is replaced by an oscillating capacitor plate.
Optomechanics can be studied even in microwave systems, where the role of the optical cavity is taken by an LC circuit and vibrating mirror is replaced by an oscillating capacitor plate.

If such a circuit is to be used in the quantum regime, though, it is not that simple. The circuit has to be built from a superconducting material (and cooled down for the experiments) so that the electrical signals can travel through the circuit many times without being absorbed. If we now make one of the capacitor plates vibrating, usually by making it from a membrane, the following happens:

The microwave field acts as a varying electric field across the capacitor. Since the membrane can freely vibrate, it will move in accordance with the electric field. But that results in varying distance between the capacitor plates which affects the resonance of the LC circuit in a way similar to a moving mirror in an optical cavity. The whole system is then described in the same way as other optomechanical systems — even though we now use a microwave field, instead of visible light!

Imagine that we now take such an LC circuit with a vibrating membrane and put the membrane in an optical cavity (either by making it an end mirror or putting it inside a closed cavity). The microwaves as well as the visible light can now swap state with the vibrating membrane. Using such a system, we can, for example, swap the state of the microwave field and the membrane and then swap the state of the membrane and the visible light. Any signal that was initially encoded in the microwave field has now been converted to light.

Combining microwave and optical cavity with a vibrating membrane, we get a system that is capable of converting microwaves to visible light and vice versa.
Combining microwave and optical cavity with a vibrating membrane, we get a system that is capable of converting microwaves to visible light and vice versa.

Such a conversion is commonly done in the classical world — Wi-Fi uses microwaves to send signals between your computer and router and light is used in optical fibres to transmit these signals over long distances to a server. This is done by the router measuring the microwave signal, transmitting it to a modem via a cable where it is measured again, sent in the form of light to the other end where the process is repeated in reverse. That is something you cannot do in the quantum world where every measurement destroys the quantum nature of the signal. This is why more sophisticated methods — such as swapping the state with a mechanical oscillator — have to be used.

There is one immediate application for these opto-electromechanical systems (i.e., systems comprising an optical cavity, an LC circuit, and a mechanical oscillator). The conversion of microwave signals to visible light can be used to improve detection efficiency of weak microwave fields. That is a task that is very difficult to do. But if you could efficiently convert these signals to light, you would need to measure weak light pulses instead, which is easier. Radio astronomers, for instance, can then use these systems to detect weaker sources of radio waves in the universe. Magnetic resonance imaging can profit by reaching better accuracy than with current detection strategies, which could lead to earlier diagnoses of serious illnesses. But we still have to wait for these applications — there is a long way between a successful experimental demonstration and a practical use of an effect.

How to close an open system

When students encounter quantum physics for the first time, it is as simple as it gets — there are no unwanted interactions, no noise, particles do not get lost. In the real world, nothing is so easy, though.

Take a single atom placed in an optical cavity, for instance. (The cavity helps to enhance the interaction between the atom and the electromagnetic field, just like it did with the optomechanical interaction in the last post.) We would like to have just the interaction between the field inside the cavity and the atom but there is a lot more going on. The field can leak out from the cavity or the atom might lose energy. The atom and the cavity field thus represent an open system because they interact with the outside world.

Things would get better if we could somehow keep track of what is happening. We could, for example, place a detector outside the cavity so we can see every photon that leaves. Every time we register a photon, we know that there is one photon less inside the cavity. This approach brings us then more information about the cavity than we would have without the measurement.

This idea was originally developed as a simple numerical tool to solve dynamics of open quantum systems. Because the system fast becomes complex with growing size, only small systems can be analyzed directly. But if we randomly generate many possible measurement results we get from such a system and take the average, we end up with the same result we would get by solving the dynamical equation.

When monitoring the outside field of an optical cavity, we can  undo the effect of losses by conditionally affecting the system inside the cavity.
When monitoring the outside field of an optical cavity, we can undo the effect of losses by conditionally affecting the system inside the cavity.

At first, this was just a useful numerical tool but today experimentalists can indeed watch cavities lose photons in real time. They can do even more — if they see that a photon has been lost, they can inject a new one into the cavity and keep the cavity field at a constant intensity. Moreover, if the cavity field, interacts with an atom, the outgoing photons carry some information about the state of this atom and we can use more complicated feedback on the atom. In this way, the state of the atom and the cavity field can be stabilised and the effect of the losses (at least partly) undone.

Measurement and feedback have become a powerful tool in quantum physics. Apart from protecting quantum systems from losses, they can also be used to bring a system to a desired state. For example, in optomechanics one of the main problems is noise in the mechanical oscillations. Because of low frequencies of mechanical oscillations (usually of the order of megahertz up to a few gigahertz), the mechanical oscillator is full of random vibrations that degrade the interaction with light. Measuring the oscillator position and applying feedback, it is possible damp the random oscillations, leaving the mechanical oscillator in its ground state and ready for a truly quantum interaction with light.

Mixing output of two optical cavities on a beam splitter, it is possible to entangle atoms that sit inside these cavities even though the atoms never interact directly.
Mixing output of two optical cavities on a beam splitter, it is possible to entangle atoms that sit inside these cavities without their direct interaction.

The resulting state can even be more complicated than that. We can take two of the atom-cavity systems and mix the output fields on a beam splitter. (A beam splitter is a partially reflecting mirror, that lets part of the light go through and reflects the other part. It is then possible to send in two different light modes and get their combination at each output.) Using suitable interaction between the atoms and the cavity fields and a proper measurement, one can entangle the two atoms even though they never interact directly. The feedback is then used to ensure that the atoms always end in the same state. This can be important for some tasks because the measurement results are in principle random and the particular state of the atoms is then random as well.

The main advantage of measurement based feedback for preparing desired states lies in combatting losses. If you want to prepare a quantum system in a certain state by well controlled interactions excluding the outside world (i.e., in a closed system setting), any kind of losses will have a negative effect on the state. With measurement and feedback, however, you let losses work to your advantage because you learn information about the system by monitoring what comes out.

All that said, feedback is not all-powerful. There are usually more kinds of losses present and you typically cannot have them all reverted. Even then, detectors never work perfectly so the losses cannot be compensated for completely. It is also not always obvious what form the feedback should take to bring your quantum system to the state you want to reach. Nevertheless, it is a crucial instrument in studying quantum systems and their possible applications.

Of light and springs

Using light, we can achieve more than simply see the world around us. Spectroscopy can be used to find chemical composition of a sample, frequency of light interacting with atoms can be used to measure time. We can even move objects by shining light at them. Such manipulations are far from tractor beams of science fiction, but optical tweezers are commonly used to manipulate small objects in many labs around the world. And there are other ways how to control matter using light.

This is possible because photons, the particles of light, carry momentum (even though they are massless!), and during an interaction between any two objects, momentum has to be conserved. When a photon bounces off a surface — say, a mirror — it changes its momentum because it changes the direction of its movement. This change of momentum has to be compensated by exactly opposite change of momentum of the mirror.

Photon bouncing off a mirror
When photons reflects off a mirror, part of its momentum is transferred to the mirror, causing it to move.

So why don’t we see light moving objects around all the time? Because light’s momentum is tiny. If you turned on a laser pointer (which is the source of the most concentrated light you can get easily) for a single second, the momentum of the light beam would be about a hundred thousand times smaller than that of a flying mosquito.

That does not stop scientists from trying to see this effect, though. Many physics labs have lasers stronger than your average laser pointer. They can also produce tiny mirrors (as small as few hundred micrometers) that are very light and thus much easier to move. When suspended on a spring, such a mirror should start swinging when light is shining on it.

It turns out, however, that this is still not enough to see the mirror jump to motion when laser light reflects off it. But there is a simple trick how to enhance the effect. You can use a second mirror (this one fixed in place) and trap the light between these two mirrors — forming an optical cavity is a well-known way to enhance interactions in quantum optics. The light bounces many times back and forth and while a single kick to the movable mirror is not enough to make it move, kicking it again and again finally sets the mirror in motion. And since there is a cavity and light interacting with a mechanical oscillator, the field studying such systems is called cavity optomechanics.

Standard optomechanical setup
To enhance the interaction between light and a mirror, light is trapped inside a cavity so that each photon interacts with the mirror many times.

There is a very simple and intuitive explanation of what is going on in such a system. This is thanks to the fact that light can survive in the cavity only when the cavity length is a whole number multiple of the light’s half-wavelength. Such a light enters the cavity and starts pushing the moving mirror, lengthening the cavity. But that means that the cavity resonance (i.e., the wavelength or frequency of the light it supports) shifts. The light, no longer supported by the cavity, then begins to leak out which decreases the pressure it exerts on the mirror which thus moves back. The light intensity inside the cavity increases and the cycle start again.

This probably sounds like a neat toy to play with but it might seem that there is not much to do with such an apparatus. But if the system is built very carefully, the light, as well as the vibrating mirror, has to be described quantum mechanically. This means that the forms of vibration are not completely arbitrary but can only have certain discrete values. Like light (electromagnetic vibrations) is build from photons, the vibrations of the mirror come in form of quantum particles — phonons. And that opens up a whole new sea of possibilities.

For example, we can send in light that is not exactly at the cavity resonance but whose frequency is smaller by an amount corresponding to the mechanical frequency. What can happen is that a photon we sent in combines with a phonon and they create a photon at the cavity resonance. Similarly, if we shine both resonant and detuned light, the resonant photon can split into a phonon and a detuned photon. In this way, we can transfer resonant photons to phonons and vice versa, swapping the vibrational state of the cavity and the mirror.

Alternatively, we can also send in light with frequency higher than the resonance frequency. A photon of this frequency can now split into a phonon and a cavity photon. Because the photons and phonons are now created in pairs, the number of excitations in the cavity and the mirror are now correlated. These correlations are stronger than any correlations in classical physics could be and the cavity field and the mirror become entangled. (And trust me, physicists can have loads of fun with that!)

With these tools at hand, there is a lot one can do. Perhaps the most intriguing thing is trying to see how far quantum mechanics can go. If we use larger and heavier mirrors, will we still see quantum behaviour, or will they start behaving classically at some point? Will there be a slow transition from quantum to classical oscillations or will such a change be abrupt? Today, nobody really knows, and it is a question that has bothered physicists for a while.

A more practical option for such systems is to use them for frequency conversion. A single mirror can be coupled to several cavities (or a single cavity with several resonant light modes). Using state swapping as explained above, it is possible to transfer signal from one light field to another by swapping it first from the first light field to the mirror and then to the second light field. It is even possible to convert visible photons to microwave photons or the other way round. This can be used, for instance, to improve detection efficiency of microwave fields from which magnetic resonance imaging could greatly benefit. Since this conversion is something I am working on in my PhD, you will certainly here more about this problem later in more detail.