Extracting coherence information from random circuits (QuantumCasts)

Key Takeaways

[Music] my name is Julian Kelly I am a quantum Hardware team lead in Google AI quantum and I'm going to be talking about extracting coherence information from random circuits via something we're calling speckle purity benchmarking so I want to start by quickly reviewing cross entry benchmarking which we sometimes call xev so the way this works is we were going to decide on some random circuits these are interleaved layers of randomly chosen single qubit gates and fixed two qubit gates we then take this circuit and we send a copy of it to the quantum computer and then we send a copy of it to a simulator we take a simulator as the ideal distribution of what the quantum evolution should have done and we compare the probabilities to what the quantum computer did the measured probabilities and we expect the ideal probabilities to be sampling from the porter thomas distribution we can then and assign a fidelity for the sequence by comparing the measured in ideal probabilities to figure out how well the quantum computer is done and again I want to emphasize that we are comparing the measured field against the expected unitary evolution of what the circuit should have done so here is what experimental cross entropy benchmarking data looks like so what we're going to do is we're going to take data both over a number of cycles and also random circuit instances and when we look at the raw probabilities we see that it looks pretty random so what we're going to do is we then take the unitary that we expect to get and we can then compute the fidelity and then out of this random looking data we see this nice fidelity decay so the way that this works is that by choosing depolarizing single qubit gates the errors more or less add up and we get this exponential decay we can fit this and extract an error per cycle for our xeb sequence and this is a rare per cycle given some expected given unitary evolution so this is fantastic but we might be interested in understanding where exactly our errors are coming from so this is where a technique known as purity benchmarking comes in and essentially what we're going to do is we're going to take this xeb sequence and we're going to append it with state tomography so in-state tomography we run a collection of experiments to extract the density matrix row we can then figure out the purity which is more or less - trace of Rho squared you can think of this as essentially the length of a vectors squared an n-dimensional block vector so what happens if we then look at the length of this block vector as a function of the number of cycles in the sequence we can look at the decay of it and that tells us something about incoherent errors that we're adding to the system so basically the Bloch vector will shrink if we have noise or decoherence so purity benchmarking allows us to figure out the incoherent error per cycle and the inter coherent error is a decoherence error lower bound it is the best error that we could possibly get so this is great we love this we use this all the time it turns out that predicting incoherent art error is harder than you think and it's very nice to have a measurement Institute in the experiment at you're carrying about to measure it directly so with that I'm going to tell a little bit of a story so last March meeting I was sitting in some of the sessions and I was inspired to try some new type of gate so I decided to run back to my hotel room and code it up and remote in and get things running and I took this data and my experimentalist intuition started tingling and I was thinking that wow the performance of this gate seems really bad so when I was looking at the raw xeb data I noticed that for a low number of cycles the data looked quite speckly and I was pretty happy with the amount of spec leanest I was happening down here but when I went to a high number of cycles I noticed these speckles started to wash out and I found the degree of speckling this to be a little bit disappointing not so speckly so I started thinking about this a bit more and this is pretty interesting because I'm looking at the Adada temps and I'm assessing the quality somehow but I don't even know what unitary I've been actually performing and shouldn't the xeb data just be completely random so I formed an extremely simple hypothesis maybe this speckle contrast is actually telling us something about the purity right so in some sense what I'm saying is that if there's low contrast it's telling us that our system is de cohered in some way so I did something very very simple which was I just took a cut like this and I took the variance of that and I plotted a versus cycle number and I see this nice exponential decay so after this I got excited and I went to talk to some of our awesome Theory friends like Sergio boys oh and Sasha karakov and they helped set me straight by going through some of the math so what we're going to do is we're going to define a rescaled version appeared it looks like this and you say that essentially this is trace of Rho squared and there's some dimension factors they have to do the size of the Hilbert space we define it this way so that pier D equals zero corresponds to a completely decohere state and a purity of one corresponds to a completely pure State we then assume and depolarizing channel model which is to say that the density matrix is some polarization parameter P times the completely pure States I plus 1 minus P times the completely D cohered uniform distribution one over D then if you plug these in to each other we can see that these parameters directly relate the polarization parameter is directly related to the square root of purity which kind of makes sense because the polarization is telling us how much we're in a pure state versus how much we're in a decohere state okay so now let's talk about what it looks like if we're actually going to try and measure this density matrix Rho if we're measuring probabilities from the pure state distribution we'd expect to get a two porter thomas distribution like we talked about before and what we notice is that there's some variance to this distribution that is greater than zero however if we're measuring probabilities from the uniform distribution which corresponds ot cohered state we see that there's no variance at all and again this is a ECD effort integrated histogram so this course wants to a delta function with no variance and it turns out if you then go back and actually do the math you can directly relate the purity to the variance of the probability distributions times some dimension factors so the point is that we can directly relate purity to variance experimentally from measured probabilities so is this purity yes turns out so let's talk about what this looks like in an experiment so here we took some very dense xeb data both the number of cycles this way and number of circuit instances that way and so what I'm going to do is I'm going to draw a cut like this I'm going to plot the the EC DF the integrated histogram distribution and we see that it obeys is very nice Porter Thomas line and then if I look over here when the we expect the state have very much D cohered we see that it does look like this d cohered uniform distribution like that so then if we plug all this data into the handy dandy formula that we showed on the previous slide we can then compare it to the purity as measured by tomography and we say these curves line up almost exactly and indeed they provide an error lower bound they're outperforming the x cb fidelity because there can be some for example calibration error so speckle purity agrees it's from our Phee we get it for free from this raw xeb data and we are noticing this by basically observing a general signature of quantum behavior that's what these contrasts mean so here's a simple analogy so if you take some kind of neato coherent laser and you send it through a crazy crystal medium you can see this speckle pattern showing up on for example a wall but if you take a kind of boring flashlight and you pointed at something you just get a very blurred out pattern so this makes sense which is to say that if you have a very decoherence state and you try and perform a complex quantum operation to it it's really not going to tell you anything nothing's gonna happen okay so let's talk about how to actually understand the cost of these different ways of extracting purity in terms of experimental time so typically when we're doing an X to B experiment we take n random circuits times n repetitions per circuit and then we may optionally add some number of additional tomography configurations to figure out how much data we need to take so for example in two qubits who may take twenty right up circus a thousand repetitions per circuit and then for two qubits we have to add an additional nine tomography repetitions and that means we're taking 200 thousand data points to extract a single data point here which is pretty expensive if you look at the order scaling the number of circuits and xeb would typically pick to be about constant the number of repetitions that turns out scales exponentially to the size of the hilbert space but then we also have this additional exponential factor doing full state tomography so what we're taking away from this is that full state tomography really is overkill for extracting the purity we can get it just from the speckles with the same information only we're doing it exponentially cheaper we're actually getting rid of this three to the N factor I do want to point out that the silvus exponential factor that remain is due to having to figure out the full distribution of probabilities so I want to now actually show data for scaling the number of qubits up to larger system sizes so we can for example extend the xeb protocol to many qubits and a pattern that looks something like this and we can measure for three qubits five seven or all the way up to ten cubits to extract a directly extracted purity and we see that this this still works we see that we get nice numbers we get nice decays out of this and we can also then compared to the xep directly I want to point out that going all the way up to here we use kind of standard numbers for the number of sequences a number of stats but then at this point we we started to feel the exponential number of samples that we needed we had to crank up the repetitions up to 20,000 but even that is really not that much for extracting purity for a 10 cubed system okay so now the last thing that we can do is we can actually do some pretty cool error budgeting so we can benchmark the different error processes versus system size using the data that I just showed so if we say the x ebiere is equal to the incoherent error plus the coherent error we directly measured the x cb error and we directly measure the incoherent error the purity that allows us to infer the coherent error so if we look at for example the n cubed x cb verses these error mechanisms we can see that for a low number of qubits we don't have much coherent error we're doing a very good job with calibration but as we scale the system sides we add more cubis we start to see just a little bit of coherent error being added and we'd expect that for example across stock effects to introduce something like this so we can answer these questions now vs system size which is are we getting unexpected incoherent errors we scale and we can also answer are we getting coherent errors for example across talk as we scale and typically these are quite challenging to measure so in conclusion we introduce this notion of speckle purity and it quantifies the decoherence without even having to know the unitary that you did it relies on random circuits and a porter Thomas distribution which is important it comes free with any xeb data that you have so even if you have a historical data set you can go and extract this from it which is pretty neat we can use it for many qubit systems we showed up to ten here and there's exponentially better scaling than doing full state tomography if probes is fundamental behavior of quantum mechanics and it's pretty neat if you spend some time thinking about it I want to point out that we have a discussion of this in the Supplemental of their quantum supremacy publication and I also found this nice paper that talks about a lot of similar concepts from 2012 you