Steve Girvin - Introduction to Quantum Error Correction

hello my name is Steven Garvin I'm a professor of physics and professor of Applied Physics at Yale University and a member of the Yale quantum Institute I'm going to talk to you today about quantum computing about circuit QED which is the primary architecture for building a quantum computer using an all-electronic system of superconducting quantum bits it was developed here at Yale with my colleagues Michele Devery and Rob schulhoff and I'm going to introduce you to that topic and then talk about quantum error correction how you can try to do nearly perfect calculations on a quantum computer built from imperfect parts the first quantum revolution was in the first half of the 20th century it brought us the transistor the laser the atomic clock the global positioning system and the inventors of quantum mechanics didn't were doing basic research they did not foresee these remarkable quantum devices the inventors of these quantum devices did not foresee their amazing applications that brought us the technological revolution of the 20th century that the first makers of individual handmade transistors could not imagine integrated circuits and the fact that today in the world approximately 60 trillion transistors are produced every second these quantum devices however from the 20th century do not actually take advantage of the full power of quantum mechanics there's now in the 21st century a second quantum revolution which is building devices to take full advantage of the full power of quantum mechanics and one of the powers has to do with computing processing information in completely new ways quantum computers use the principle of superposition that a quantum bit or qubit can exist in in an infinite number of states that are intermediate between 0 and 1 there is some very crude sense in which quantum bits can be both 0 and 1 at the same time a more subtle concept is that of entanglement an entanglement is another kind of superposition that allows there to be non classical extra-strong correlations amongst different bits that are widely separated from each other something that I'm stained objected to strongly because he understood the implications of this new quantum theory he called it spooky action at a distance and did not like it these two effects however give us a kind of quantum parallelism that lets a quantum computer do many calculations in parallel and at a rate which is impossible on any conventional computer the irony is that as part of a daily routine engineering calibration test of our quantum computers we carry out these spooky operations that Einstein felt had to be impossible and we use those to verify that our quantum computer really is a quantum computer and not a classical computer so now that you are experts on quantum computing here's a quiz but worry I'm going to give you the answers the first question in the quiz is is quantum information carried by waves or particles and the answer of course is yes second question is quantum information analog or digital and again the answer is yes quantum information can be stored in a continuous analog fashion in intermediate superpositions between 0 and 1 and yet when we measure the quantum bit we always get the result 0 or 1 so now let's talk about circuit quantum electrodynamics the the idea of quantum electrical circuits quantum integrated chips the experiments I'm going to talk about are carried out by my colleagues Michelle Duggar a Luigi friends erm show off and there are many graduate students and postdocs and both the theory and experiment side that have contributed to the ideas that I'm going to tell you about so what is quantum electrodynamics that's the study of atoms and electrons coupled to photons to particles of light and the fact that the electromagnetic field is quantized means that it has zero point fluctuations or vacuum fluctuations quantum uncertainty and how big the electric and magnetic fields are in the VAT even in a vacuum when there's no light supposedly and these fluctuations modify the spectrum of atoms of the energy levels of atoms so for example if I have a hydrogen atom the first thing you learn is there are certain quantized energy levels which are stationary States that live forever but if you look more closely if you include the fact that this atom is sitting in the electromagnetic vacuum which has fluctuating those fluctuations will destabilized the two-piece state and caused the electron to fall down to the ground state the 1s state irreversibly by spontaneously emitting a photon into the vacuum in a time typical time of about 1.6 nanoseconds other effects of vacuum fluctuations in which an electron can virtually emit and then quickly reabsorb a photon leads to an effect called the Lamb shift which causes the 2s and 2p states to not quite be at exactly the same energy the goal of cavity quantum electrodynamics is to modify the available photon States by for example trapping atoms between two highly reflecting mirrors that reflect optical light that can interact with the atoms and make instead of a continuous range of photon frequencies available to the atom only discrete allowed resonances based on the distance between the two mirrors in the cavity I'm going to talk to you about microwave cavity QED or in circuit quantum electrodynamics in which we have some sort of superconducting box or other resonator in an electrical circuit which completely surrounds a region and can trap microwave photons inside so it's like having almost perfectly reflecting mirrors completely surrounding an artificial atom which I'll tell you about and unlike the optical QED cavity QED case the atom when it tries to emit a photon in any direction it's always reflected back and is not able to escape and as a result of that the lifetime force Ponte mission of the excited state of our artificial atom can be enhanced by a factor of a thousand if the cavity does not have a resonance frequency that matches that of the atom and this enhancement of the lifetime is very very helpful in trying to store and manipulate quantum information in superposition states of our artificial atoms so in circuit QED which is an analogue of cavity QED as I said the microwave photons are inside superconducting circuits and are we don't use real atoms we use artificial atoms Josephson Junction quantum bits or qubits and these artificial atoms are very large and couple very strongly to photons and give us the ability to do what's called nonlinear quantum optics in a completely new regime of individual photons so here you see a cartoon of a hydrogen atom obviously not to scale the diameter of the boar orbit is about 10 to the minus 10 meters the frequency of transition between the 1s and 2p state that I talked about is about two and a half pedda Hertz 10 to the 15 Hertz the lifetime of the excited state before it spontaneously decays as I mentioned about 1.6 nanoseconds the cue of this oscillation the quality factor which is the product of the frequency and the lifetime is about four million and the transition dipole moment the ability of this atom to couple to electromagnetic fields because there is a charge moving around is captured by this so-called dipole moment which is about 1/2 by or about one electron charge times the Bohr radius of this but let's compare this to an artificial atom made of superconducting aluminum evaporated on an insulating substrate and containing a joke Josephson Junction which I'll explain momentarily the size of this atom is about a millimeter it's seven orders of magnitude larger than this atom it has a transition frequency six orders of magnitude smaller about seven gigahertz which is in the domain of cell phone signals microwave signals a lifetime of the excited state is about 300 microseconds much longer than the hydrogen atom but the Q the quality factor is about the same two million but what's really different is the transition dipole moment which is sort of a measure of how much charge is moving over what distance that's several pairs of electrons moving a millimeter it's about 30 million times larger than for the hydrogen atom and this enormous transition dipole moment allows us to do quantum mechanics and quantum electrodynamics in a completely new regime so this object this jozin Junction object can be an artificial atom it can be viewed as a quantum bit or qubit it has to be operated near absolute zero so that the material the metal is superconducting but you can build superconducting integrated circuits out of this by the same technology that's used to make semiconducting integrated circuits in today's classical computers the transistor of quantum computing the heart of our artificial atom is this thing called the Josephson Junction which I mentioned before it consists of two pieces of aluminum which are superconducting separated by a very Phin one nanometer thick insulating layer of aluminum oxide here you see an electron micrograph of it it's about two thousand items wide and two thousand atoms high and a few atoms thick this object provides the anharmonic energy structure of an LC oscillator it acts like the artificial atom has a capacitor and it has some nonlinear inductance caused by this Joe's injunction which allows pairs so-called Cooper pairs of electrons in the superconductor to tunnel through this thin barrier and slosh back and forth between the two capacitor plates on this artificial atom and it acts like an inductor whose inductance is not constant but varies with how much current is flowing in the circuit and makes this into an an harmonic oscillator the the potential seen by this oscillator is a cosine of a certain angle a certain phase difference of the superconducting order parameter across the two sides of the junction here's a different design the so-called transman qubit this is like an artificial atom with atomic number one trillion it has ten to the twelve mobile electrons in these aluminium films evaporated on this insulating substrate it's about a it's like an antenna a millimeter long and the two halves of the antenna are connected by this Josephson tunnel Junction the soup you might think if I had an atom with atomic number 10 to the 12 it would be impossible to solve for the spectrum of that atom it would be impossibly complex but superconductivity gaps out all of the single particle excitation zin the electron gas in the aluminum and the only low energy excitations are pairs of electrons sloshing back and forth between the two antennae pads and so the quantized energy levels of this object are simply those of an an harmonic oscillator the ground to first excited state transition is at perhaps five gigahertz and the first excited - second excited transition is at perhaps four point nine gigahertz slightly lower because the cosine potential is softer than the parabola of a harmonic oscillator and so it has negative and harmonicity the quality factor of this is I mentioned about the same as a hydrogen 1s 2p but the enormous transition dipole moment caused by the charges moving back and forth a millimeter means that this atom which has its own antenna built in can talk very very strongly couple very strongly to microwave photons in a cavity which is surrounding this artificial atom so this is the the artificial atom in our circuit quantum electrodynamics what can we do with this well here is an example here is a Hamiltonian you don't need to understand the details but there's a mode in the cavity which is a harmonic oscillator it has an energy which is an integer times Planck's constant times the frequency of the resonator this just counts the number of photons in the microwave cavity we can approximate the two-level atom at the sorry the an harmonic oscillator as a two-level atom and described that mathematically as a pseudo spin 1/2 with spin down being the ground state spin up being the excited state and because the other transition frequencies are different due to the anharmonicity we can ignore them in a certain approximate and when the atom in the cavity have different transition frequencies then the net effect is you are in what's called the dispersive regime and the effective dipole coupling enters in second order and produces this term which causes the transition frequency of the atom to shift by a number Chi each time you add one more photon to the cavity so here you see the spectrum of the cavity I'm sorry of the atom we shine microwaves on the atom and see what frequency causes it to transition from the ground to excited state and we see a series of Peaks corresponding to how many photons are in the cavity 0 1 2 3 4 5 6 7 8 microwave photons in the cavity chi is negative so the frequency goes down there's a quantized light shift associated with how many photons are in the cavity so what you should learn from the fact that there are discrete Peaks here is that microwaves despite their name are particles they are photons they're just as good photons as the optical photons that cause the chemical reactions in your eye to let you see even though they have a hundred thousand or million times less energy there are still particles with quantized lumps of energy and the fact that you now can detect these individual microwave photons is giving us new ways to search for certain types of dark matter called accion particles and so one of the applications of all this is not simply to produce a new kind of quantum computer but also to do quantum sensing to detect extremely weak signals of interest in cosmology and astrophysics so with this Hamiltonian with this strong coupling between the atom and the cavity we can control the state of both we can apply drive signals to the sorry to the cavity shown here we can apply drive signals to this transman two-level system here and produce any quantum state of the combined system for example suppose we wanted to do something that would be considered extremely difficult in ordinary quantum optics take an empty cavity with zero photons and put exactly six photons into that cavity that would be extremely difficult with ordinary optics but is relatively easy here so here you see a plot of some control signals that have been pre computed that we apply to the trans month and another control signal that we apply to the cavity and here is a graph with where the color scale shows you the probability that there are n photons in the cavity as a function of time and you see you start at zero it rises up it dips down it kind of splits into two parts and then at the last second or the last nanosecond 500 nanoseconds into the sequence boom all of the probability lands on N equals six there are now exactly six photons in the cavity and here you see a plot called the Vigna function which verifies for us that we are in the N equals six state so you can think of this Vigna function as a kind of quasi probability distribution in the phase space of the quantum oscillator so think of this axis as the coordinate of the electromagnetic oscillator perhaps the electric field measured at the center of the cavity would be the coordinate and this axis is the momentum or the magnetic field in the electromagnetic oscillator and you see this beautiful bull's eye pattern which is completely circularly symmetric and changes color the colors indicate the sign of the Vigna function so it's not a probability distribution it's a quasi-probability distribution like a quantum amplitude because it can be both positive and negative the fact that it's completely circularly symmetric means that we don't know the phase of the oscillation of the cavity and that's just the number phase uncertainty relation we know that the number of photons is exactly 6 so we cannot assign a phase to that it's a superposition of all possible phases here's another example we can create Schrodinger cat state we can make an oscillator be displaced to the right and to the left to be in two coordinates at the two different places at the same time and here you see the Vigna function for that case you see this blob on the Left represents the coherent state minus alpha we've displaced the oscillator to the left this blob represents the coherent state plus alpha the oscillator is displaced to the right you see that it's a blob not a point as it would be classically and that's vacuum noise that's the fact that the position of the oscillator and the momentum do not commute and so it's impossible to know both of them precisely simultaneously equivalently this is the vacuum noise the electromagnetic noise of the vacuum right before your very eyes in the middle you see what we call the whiskers of the cat these very rapid sine oscillations of the Vigna function and they can be used for quantum sensing if you displace the oscillator very very slightly in the momentum direction you get a very strong change in that signal even though the displacement is much smaller than the zero point uncertainty in the momentum furthermore the fact that the central fringe is the same color as the blobs tells us that we have an even cat a plus sign in the superposition of being here and being here and not a random mixture so not an incoherent mixture a true Schrodinger cat with a coherent superposition with a definite sign between coherent displacement to the left and to the right just to give you some icing on the cake here is a kind of Schrodinger cat living in thirty five places at once this is an amazing recent experiment from my colleague Michelle Deborah in which he has made a so-called goddess Minh cat Ioffe press coal state which is very powerful for quantum error correction and you see these blobs which are actually squeezed now they're smaller than the uncertainty in the previous slide and the cat is in 35 different places in phase space at the same time so that just demonstrates for you the level of quantum control that we have over these quantum states okay so now that's the the hardware the background now I want to talk to you about the quantum error correction problem how do I correct errors in a quantum computer that has imperfect parts so here there is the problem in a nutshell I'm going to give you an unknown quantum state if you were to measure it will collapse it will change randomly due to this back action resulting from your measurement that's a problem because if this unknown state develops an error I want you to fix it it seems impossible but miraculously it can be done the fact that in principle quantum error correction is possible is to me much more profound and amazing than the fact that you can do quantum computation in principle with a perfect device you can now even do it with real devices that are imperfect it's still extremely challenging but I will show you experimental progress that we are now able to do this at some level so how does quantum error correction work how can you get around the fact that when you examine a quantum object you change it so what you need to do is store the quantum information not in one physical qubit but in a logical qubit consisting of many physical qubits in this example there are nine and you store the quantum information the superposition of ground and excited state in not in one physical object but in these spread out in some non-local non classical spooky action at a distance kind of correlations among all of these qubits and the non locality of putting the information in there is very important because it you need for it to be true that no single physical qubit can know the state of the logical qubit of the information you're trying to store because if that one qubit develops an error or is measured by the environment which is what decoherence is then you do not want the environment to learn the state of the logical qubit and collapse its state so you need to the information from the environment but then you need a Maxwell demon to correct those errors when they occur and the Maxwell demon which is itself made of imperfect parts has to be so fast and so accurate that it can make certain clever measurements which do not tell it the state of the logical qubit but do tell it what errors may have occurred and on which of the nine physical qubits the error occurred so that it can be corrected and the entropy so to speak pumped into a cold bath to keep this logical state pure well there's a big problem right off the bat that when you have n physical qubits the error rate has suddenly become n times worse because each one of the physical qubits can have independent errors so every quantum error correction protocol starts by taking a giant step backward and making the error rate worse then you have to turn on a very clever Maxwell demon that is so fast and so efficient it's so accurate and doesn't introduce errors of its own that will overcome this factor of N and get you back to the break-even point where the quantum information lives as long as if it had been in one of the single physical qubits and hopefully even better than that you want to be much longer lifetime than just reaching the break-even point so it turns out that all previous attempts to overcome this factor of N and reach the break-even point of quantum error correction have actually failed and the reason is that the Maxwell demon has to figure out all the different errors that could have occurred a bit flip a phase slip a combination of both and which of the N physical qubits suffered that error so it has to make many many clever measurements of so called error syndromes which tell it about what errors may have occurred but do not tell it about the logical state in which the error occurred so that's quite complex and difficult it's been attempted it's sort of works but it never actually makes things better by overcoming that factor of n it always makes things worse so we want to try a different idea which is not to use material objects as quantum bits not those artificial atoms that I showed you but use microwave photon States stored in high quality factor superconducting resonators as our logical qubit and so to compare let's just consider the case of three physical qubits just to keep it simple each is a two-level artificial atom it has a total of eight states in the combination of three the states can be represented by the binary numbers 0 0 0 through 1 1 1 so there are 8 numbers there and each of those 8 quantum states can have some quantum amplitude in the in the entangled correlated state of the logical qubit instead of that which is complicated because again there are many errors that can occur and we have to figure out which physical qubit had the error instead of that let's think of putting microwave photons inside a cavity and there can be zero photons 1 2 3 4 5 6 or 7 so there's a total of eight states we'll consider and they can have exactly the same quantum amplitudes as are stored in these three physical qubits and it's a continuous variable system it's described by some wave functions I of X is just a harmonic oscillator X is the coordinate of the oscillator but in that continuous variable system we have a discrete basis of photon number States and we can store exactly the same information but it's on one physical object a cavity containing photons and that's simplification using a single object that has many states in its Silbert space instead of many objects with only two in each of their states is the simplification so why why is that an improvement well each this harmonic oscillator has one mode and it's weakly damped very very weakly damped it's a high q-- oscillator and it has only one kind of error it can decay by losing a photon and you don't need to know which of the the modes or which of the physical qubits lost of the energy it's just one single mode with one possible error so if we could think of a thing to measure that would detect that error we only have to measure that one thing and it turns out for us again something that's very difficult to measure in ordinary quantum optics but easy in circuit QED is the photon number parity is the photon number even or odd and if I see the photon number change from even to odd I know that I have lost a photon an error has occurred and I need to correct it so here is the simplest so-called boson ik code or continuous variable quantum error correction code that I know of which we developed a few years ago here yeah this is the binomial code or the Khitan code the it's a whole family of codes but the simplest version is the following the zero logical state is a coherent superposition of zero photons in the resonator plus four photons in the resonator the one logical state is two photons in the resonator so it's very simple if I should lose a which happens when the lowering operator a get supplied to the state then 0 logical becomes has now three photons in the state and one logical in the error state has one Photon and this is very efficient you uses only five photon states or log to five bits it uses fewer states than the corresponding discrete variable error correcting codes because it's has fewer errors and the recovery that I want to apply when I know that the parity has jumped oh I should emphasize that these two code words are both eigenstates of parity they're both even photon numbers and the error words have odd photon numbers so when I'd see the parity jump I have to apply unitary operation that map's 3 back to 0 plus 4 and 1 back to 2 well that's a those are non-trivial tricky operations that would be very difficult in ordinary quantum optics but turn out to be relatively easy with our universal control in circuit QED so I won't go into the details but here are some experimental results from the Luo Yin Sun group in changhua doing quantum error correction on this code and improving the lifetime by applying the error correction and reaching 92% of the break-even point so almost reaching break-even and the first code and so far the only code which is actually exceeded breakeven and was done slightly before that experiment was done here at Yale in the group of my colleague Rob Scholl cough and it used a superposition of two different Schrodinger cat States remember the Schrodinger cat state let's say for the 0 logical is oscillator displaced to the left we proposed with oscilator displaced to the right one logical is the oscillator displaced in positive momentum superposed with displacement to negative momentum and the Vigna functions of those two codewords are shown here and you can produce with our universal control arbitrary superpositions of those states and the properties of these states are very favorable for quantum error correction and by encoding the information in these logical photon States and applying the error correction operation the show-off group for the first time in any technology finally exceeded the break-even point only about 10% but did so and then if you it turns out that this system will raise its hand when there's an error it will Harold errors and in 20% of the cases it raises his hand and says something went wrong if you post select only on the 80% of the data where that didn't happen then you reached 1.75 times break even so this is still not as good as we want to do but it's the first time that breakeven has been robustly reached so we have a great deal a great staircase to climb towards our ultimate goal of fault-tolerant quantum computation and you can see a series of steps here we're somewhere in the middle we've done we've created a logical memory with a slightly longer lifetime than the individual physical parts through quantum error correction we've done operations on single logical qubits we've done entangling gates on pairs of logical qubits but we're still a long way from error correction producing dramatic enhancement a lifetime and being able to do that at large-scale to produce a full fault-tolerant quantum computation but we feel like we're making great progress and doing lots of interesting physics along the way so here thank you very much for listening and here are some references to the experiment and theoretical papers thank you very much

Channel: YaleUniversity

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Keywords: quantum, lecture, education, yale

Id: uD69GCYF9Zg

Length: 40min 35sec (2435 seconds)

Published: Fri Dec 06 2019