Posted by Kevin Miao and Matt McEwen, Research Scientists, Quantum AI Team

The qubits that make up Google quantum devices are delicate and noisy, so it’s necessary to incorporate error correction procedures that identify and account for qubit errors on the way to building a useful quantum computer. Two of the most prevalent error mechanisms are bit-flip errors (where the energy state of the qubit changes) and phase-flip errors (where the phase of the encoded quantum information changes). Quantum error correction (QEC) promises to address and mitigate these two prominent errors. However, there is an assortment of other error mechanisms that challenges the effectiveness of QEC.

While we want qubits to behave as ideal two-level systems with no loss mechanisms, this is not the case in reality. We use the lowest two energy levels of our qubit (which form the computational basis) to carry out computations. These two levels correspond to the absence (computational ground state) or presence (computational excited state) of an excitation in the qubit, and are labeled |0⟩ (“ket zero”) and |1⟩ (“ket one”), respectively. However, our qubits also host many higher levels called leakage states, which can become occupied. Following the convention of labeling the level by indicating how many excitations are in the qubit, we specify them as |2⟩, |3⟩, |4⟩, and so on.

In “Overcoming leakage in quantum error correction”, published in Nature Physics, we identify when and how our qubits leak energy to higher states, and show that the leaked states can corrupt nearby qubits through our two-qubit gates. We then identify and implement a strategy that can remove leakage and convert it to an error that QEC can efficiently fix. Finally, we show that these operations lead to notably improved performance and stability of the QEC process. This last result is particularly critical, since additional operations take time, usually leading to more errors.

Working with imperfect qubits

Our quantum processors are built from superconducting qubits called transmons. Unlike an ideal qubit, which only has two computational levels — a computational ground state and a computational excited state — transmon qubits have many additional states with higher energy than the computational excited state. These higher leakage states are useful for particular operations that generate entanglement, a necessary resource in quantum algorithms, and also keep transmons from becoming too non-linear and difficult to operate. However, the transmon can also be inadvertently excited into these leakage states through a variety of processes, including imperfections in the control pulses we apply to perform operations or from the small amount of stray heat leftover in our cryogenic refrigerator. These processes are collectively referred to as leakage, which describes the transition of the qubit from computational states to leakage states.

Consider a particular two-qubit operation that is used extensively in our QEC experiments: the CZ gate. This gate operates on two qubits, and when both qubits are in their |1⟩ level, an interaction causes the two individual excitations to briefly “bunch” together in one of the qubits to form |2⟩, while the other qubit becomes |0⟩, before returning to the original configuration where each qubit is in |1⟩. This bunching underlies the entangling power of the CZ gate. However, with a small probability, the gate can encounter an error and the excitations do not return to their original configuration, causing the operation to leave a qubit in |2⟩, a leakage state. When we execute hundreds or more of these CZ gates, this small leakage error probability accumulates.

Transmon qubits support many leakage states (|2⟩, |3⟩, |4⟩, …) beyond the computational basis (|0⟩ and |1⟩). While we typically only use the computational basis to represent quantum information, sometimes the qubit enters these leakage states, and disrupts the normal operation of our qubits.

A single leakage event is especially damaging to normal qubit operation because it induces many individual errors. When one qubit starts in a leaked state, the CZ gate no longer correctly entangles the qubits, preventing the algorithm from executing correctly. Not only that, but CZ gates applied to one qubit in leaked states can cause the other qubit to leak as well, spreading leakage through the device. Our work includes extensive characterization of how leakage is caused and how it interacts with the various operations we use in our quantum processor.

Once the qubit enters a leakage state, it can remain in that state for many operations before relaxing back to the computational states. This means that a single leakage event interferes with many operations on that qubit, creating operational errors that are bunched together in time (time-correlated errors). The ability for leakage to spread between the different qubits in our device through the CZ gates means we also concurrently see bunches of errors on neighboring qubits (space-correlated errors). The fact that leakage induces patterns of space- and time-correlated errors makes it especially hard to diagnose and correct from the perspective of QEC algorithms.

The effect of leakage in QEC

We aim to mitigate qubit errors by implementing surface code QEC, a set of operations applied to a collection of imperfect physical qubits to form a logical qubit, which has properties much closer to an ideal qubit. In a nutshell, we use a set of qubits called data qubits to hold the quantum information, while another set of measure qubits check up on the data qubits, reporting on whether they have suffered any errors, without destroying the delicate quantum state of the data qubits. One of the key underlying assumptions of QEC is that errors occur independently for each operation, but leakage can persist over many operations and cause a correlated pattern of multiple errors. The performance of our QEC strategies is significantly limited when leakage causes this assumption to be violated.

Once leakage manifests in our surface code transmon grid, it persists for a long time relative to a single surface code QEC cycle. To make matters worse, leakage on one qubit can cause its neighbors to leak as well.

Our previous work has shown that we can remove leakage from measure qubits using an operation called multi-level reset (MLR). This is possible because once we perform a measurement on measure qubits, they no longer hold any important quantum information. At this point, we can interact the qubit with a very lossy frequency band, causing whichever state the qubit was in (including leakage states) to decay to the computational ground state |0⟩. If we picture a Jenga tower representing the excitations in the qubit, we tumble the entire stack over. Removing just one brick, however, is much more challenging. Likewise, MLR doesn’t work with data qubits because they always hold important quantum information, so we need a new leakage removal approach that minimally disturbs the computational basis states.

Gently removing leakage

We introduce a new quantum operation called data qubit leakage removal (DQLR), which targets leakage states in a data qubit and converts them into computational states in the data qubit and a neighboring measure qubit. DQLR consists of a two-qubit gate (dubbed Leakage iSWAP — an iSWAP operation with leakage states) inspired by and similar to our CZ gate, followed by a rapid reset of the measure qubit to further remove errors. The Leakage iSWAP gate is very efficient and greatly benefits from our extensive characterization and calibration of CZ gates within the surface code experiment.

Recall that a CZ gate takes two single excitations on two different qubits and briefly brings them to one qubit, before returning them to their respective qubits. A Leakage iSWAP gate operates similarly, but almost in reverse, so that it takes a single qubit with two excitations (otherwise known as |2⟩) and splits them into |1⟩ on two qubits. The Leakage iSWAP gate (and for that matter, the CZ gate) is particularly effective because it does not operate on the qubits if there are fewer than two excitations present. We are precisely removing the |2⟩ Jenga brick without toppling the entire tower.

By carefully measuring the population of leakage states on our transmon grid, we find that DQLR can reduce average leakage state populations over all qubits to about 0.1%, compared to nearly 1% without it. Importantly, we no longer observe a gradual rise in the amount of leakage on the data qubits, which was always present to some extent prior to using DQLR.

This outcome, however, is only half of the puzzle. As mentioned earlier, an operation such as MLR could be used to effectively remove leakage on the data qubits, but it would also completely erase the stored quantum state. We also need to demonstrate that DQLR is compatible with the preservation of a logical quantum state.

The second half of the puzzle comes from executing the QEC experiment with this operation interleaved at the end of each QEC cycle, and observing the logical performance. Here, we use a metric called detection probability to gauge how well we are executing QEC. In the presence of leakage, time- and space-correlated errors will cause a gradual rise in detection probabilities as more and more qubits enter and stay in leakage states. This is most evident when we perform no reset at all, which rapidly leads to a transmon grid plagued by leakage, and it becomes inoperable for the purposes of QEC.

The prior state-of-the-art in our QEC experiments was to use MLR on the measure qubits to remove leakage. While this kept leakage population on the measure qubits (green circles) sufficiently low, data qubit leakage population (green squares) would grow and saturate to a few percent. With DQLR, leakage population on both the measure (blue circles) and data qubits (blue squares) remain acceptably low and stable.

With MLR, the large reduction in leakage population on the measure qubits drastically decreases detection probabilities and mitigates a considerable degree of the gradual rise. This reduction in detection probability happens even though we spend more time dedicated to the MLR gate, when other errors can potentially occur. Put another way, the correlated errors that leakage causes on the grid can be much more damaging than the uncorrelated errors from the qubits waiting idle, and it is well worth it for us to trade the former for the latter.

When only using MLR, we observed a small but persistent residual rise in detection probabilities. We ascribed this residual increase in detection probability to leakage accumulating on the data qubits, and found that it disappeared when we implemented DQLR. And again, the observation that the detection probabilities end up lower compared to only using MLR indicates that our added operation has removed a damaging error mechanism while minimally introducing uncorrelated errors.

Leakage manifests during surface code operation as increased errors (shown as error detection probabilities) over the number of cycles. With DQLR, we no longer see a notable rise in detection probability over more surface code cycles.

Prospects for QEC scale-up

Given these promising results, we are eager to implement DQLR in future QEC experiments, where we expect error mechanisms outside of leakage to be greatly improved, and sensitivity to leakage to be enhanced as we work with larger and larger transmon grids. In particular, our simulations indicate that scale-up of our surface code will almost certainly require a large reduction in leakage generation rates, or an active leakage removal technique over all qubits, such as DQLR.

Having laid the groundwork by understanding where leakage is generated, capturing the dynamics of leakage after it presents itself in a transmon grid, and showing that we have an effective mitigation strategy in DQLR, we believe that leakage and its associated errors no longer pose an existential threat to the prospects of executing a surface code QEC protocol on a large grid of transmon qubits. With one fewer challenge standing in the way of demonstrating working QEC, the pathway to a useful quantum computer has never been more promising.

Acknowledgements

This work would not have been possible without the contributions of the entire Google Quantum AI Team.

Posted by Jesse Hoke, Student Researcher, and Pedram Roushan, Senior Research Scientist, Quantum AI Team

Quantum mechanics allows many phenomena that are classically impossible: a quantum particle can exist in a superposition of two states simultaneously or be entangled with another particle, such that anything you do to one seems to instantaneously also affect the other, regardless of the space between them. But perhaps no aspect of quantum theory is as striking as the act of measurement. In classical mechanics, a measurement need not affect the system being studied. But a measurement on a quantum system can profoundly influence its behavior. For example, when a quantum bit of information, called a qubit, that is in a superposition of both “0” and “1” is measured, its state will suddenly collapse to one of the two classically allowed states: it will be either “0” or “1,” but not both. This transition from the quantum to classical worlds seems to be facilitated by the act of measurement. How exactly it occurs is one of the fundamental unanswered questions in physics.

In a large system comprising many qubits, the effect of measurements can cause new phases of quantum information to emerge. Similar to how changing parameters such as temperature and pressure can cause a phase transition in water from liquid to solid, tuning the strength of measurements can induce a phase transition in the entanglement of qubits.

Today in “Measurement-induced entanglement and teleportation on a noisy quantum processor”, published in Nature, we describe experimental observations of measurement-induced effects in a system of 70 qubits on our Sycamore quantum processor. This is, by far, the largest system in which such a phase transition has been observed. Additionally, we detected "quantum teleportation" — when a quantum state is transferred from one set of qubits to another, detectable even if the details of that state are unknown — which emerged from measurements of a random circuit. We achieved this breakthrough by implementing a few clever “tricks” to more readily see the signatures of measurement-induced effects in the system.

Background: Measurement-induced entanglement

Consider a system of qubits that start out independent and unentangled with one another. If they interact with one another , they will become entangled. You can imagine this as a web, where the strands represent the entanglement between qubits. As time progresses, this web grows larger and more intricate, connecting increasingly disparate points together.

A full measurement of the system completely destroys this web, since every entangled superposition of qubits collapses when it’s measured. But what happens when we make a measurement on only a few of the qubits? Or if we wait a long time between measurements? During the intervening time, entanglement continues to grow. The web’s strands may not extend as vastly as before, but there are still patterns in the web.

There is a balancing point between the strength of interactions and measurements, which compete to affect the intricacy of the web. When interactions are strong and measurements are weak, entanglement remains robust and the web’s strands extend farther, but when measurements begin to dominate, the entanglement web is destroyed. We call the crossover between these two extremes the measurement-induced phase transition.

In our quantum processor, we observe this measurement-induced phase transition by varying the relative strengths between interactions and measurement. We induce interactions by performing entangling operations on pairs of qubits. But to actually see this web of entanglement in an experiment is notoriously challenging. First, we can never actually look at the strands connecting the qubits — we can only infer their existence by seeing statistical correlations between the measurement outcomes of the qubits. So, we need to repeat the same experiment many times to infer the pattern of the web. But there’s another complication: the web pattern is different for each possible measurement outcome. Simply averaging all of the experiments together without regard for their measurement outcomes would wash out the webs’ patterns. To address this, some previous experiments used “post-selection,” where only data with a particular measurement outcome is used and the rest is thrown away. This, however, causes an exponentially decaying bottleneck in the amount of “usable” data you can acquire. In addition, there are also practical challenges related to the difficulty of mid-circuit measurements with superconducting qubits and the presence of noise in the system.

How we did it

To address these challenges, we introduced three novel tricks to the experiment that enabled us to observe measurement-induced dynamics in a system of up to 70 qubits.

Trick 1: Space and time are interchangeable

As counterintuitive as it may seem, interchanging the roles of space and time dramatically reduces the technical challenges of the experiment. Before this “space-time duality” transformation, we would have had to interleave measurements with other entangling operations, frequently checking the state of selected qubits. Instead, after the transformation, we can postpone all measurements until after all other operations, which greatly simplifies the experiment. As implemented here, this transformation turns the original 1-spatial-dimensional circuit we were interested in studying into a 2-dimensional one. Additionally, since all measurements are now at the end of the circuit, the relative strength of measurements and entangling interactions is tuned by varying the number of entangling operations performed in the circuit.

Exchanging space and time. To avoid the complication of interleaving measurements into our experiment (shown as gauges in the left panel), we utilize a space-time duality mapping to exchange the roles of space and time. This mapping transforms the 1D circuit (left) into a 2D circuit (right), where the circuit depth (T) now tunes the effective measurement rate.

Trick 2: Overcoming the post-selection bottleneck

Since each combination of measurement outcomes on all of the qubits results in a unique web pattern of entanglement, researchers often use post-selection to examine the details of a particular web. However, because this method is very inefficient, we developed a new “decoding” protocol that compares each instance of the real “web” of entanglement to the same instance in a classical simulation. This avoids post-selection and is sensitive to features that are common to all of the webs. This common feature manifests itself into a combined classical–quantum “order parameter”, akin to the cross-entropy benchmark used in the random circuit sampling used in our beyond-classical demonstration.

This order parameter is calculated by selecting one of the qubits in the system as the “probe” qubit, measuring it, and then using the measurement record of the nearby qubits to classically “decode” what the state of the probe qubit should be. By cross-correlating the measured state of the probe with this “decoded” prediction, we can obtain the entanglement between the probe qubit and the rest of the (unmeasured) qubits. This serves as an order parameter, which is a proxy for determining the entanglement characteristics of the entire web.

In the decoding procedure we choose a “probe” qubit (pink) and classically compute its expected value, conditional on the measurement record of the surrounding qubits (yellow). The order parameter is then calculated by the cross correlation between the measured probe bit and the classically computed value.

Trick 3: Using noise to our advantage

A key feature of the so-called “disentangling phase” — where measurements dominate and entanglement is less widespread — is its insensitivity to noise. We can therefore look at how the probe qubit is affected by noise in the system and use that to differentiate between the two phases. In the disentangling phase, the probe will be sensitive only to local noise that occurs within a particular area near the probe. On the other hand, in the entangling phase, any noise in the system can affect the probe qubit. In this way, we are turning something that is normally seen as a nuisance in experiments into a unique probe of the system.

What we saw

We first studied how the order parameter was affected by noise in each of the two phases. Since each of the qubits is noisy, adding more qubits to the system adds more noise. Remarkably, we indeed found that in the disentangling phase the order parameter is unaffected by adding more qubits to the system. This is because, in this phase, the strands of the web are very short, so the probe qubit is only sensitive to the noise of its nearest qubits. In contrast, we found that in the entangling phase, where the strands of the entanglement web stretch longer, the order parameter is very sensitive to the size of the system, or equivalently, the amount of noise in the system. The transition between these two sharply contrasting behaviors indicates a transition in the entanglement character of the system as the “strength” of measurement is increased.

Order parameter vs. gate density (number of entangling operations) for different numbers of qubits. When the number of entangling operations is low, measurements play a larger role in limiting the entanglement across the system. When the number of entangling operations is high, entanglement is widespread, which results in the dependence of the order parameter on system size (inset).

In our experiment, we also demonstrated a novel form of quantum teleportation that arises in the entangling phase. Typically, a specific set of operations are necessary to implement quantum teleportation, but here, the teleportation emerges from the randomness of the non-unitary dynamics. When all qubits, except the probe and another system of far away qubits, are measured, the remaining two systems are strongly entangled with each other. Without measurement, these two systems of qubits would be too far away from each other to know about the existence of each other. With measurements, however, entanglement can be generated faster than the limits typically imposed by locality and causality. This “measurement-induced entanglement” between the qubits (that must also be aided with a classical communications channel) is what allows for quantum teleportation to occur.

Proxy entropy vs. gate density for two far separated subsystems (pink and black qubits) when all other qubits are measured. There is a finite-size crossing at ~0.9. Above this gate density, the probe qubit is entangled with qubits on the opposite side of the system and is a signature of the teleporting phase.

Conclusion

Our experiments demonstrate the effect of measurements on a quantum circuit. We show that by tuning the strength of measurements, we can induce transitions to new phases of quantum entanglement within the system and even generate an emergent form of quantum teleportation. This work could potentially have relevance to quantum computing schemes, where entanglement and measurements both play a role.

Acknowledgements

This work was done while Jesse Hoke was interning at Google from Stanford University. We would like to thank Katie McCormick, our Quantum Science Communicator, for helping to write this blog post.

Posted by Ted White and Ofer Naaman, Staff Research Scientists, Google Quantum AI

The Google Quantum AI team is building quantum computers with superconducting microwave circuits, but much like a classical computer the superconducting processor at the heart of these computers is only part of the story. An entire technology stack of peripheral hardware is required to make the quantum computer work properly. In many cases these parts must be custom designed, requiring extensive research and development to reach the highest levels of performance.

In this post, we highlight one aspect of this supplemental hardware: our superconducting microwave amplifiers. In “Readout of a Quantum Processor with High Dynamic Range Josephson Parametric Amplifiers”, published in Applied Physics Letters, we describe how we increased the maximum output power of our superconducting microwave amplifiers by a factor of over 100x. We discuss how this work can pave the way for the operation of larger quantum processor chips with improved performance.

Why microwave amplifiers?

One of the challenges of operating a superconducting quantum processor is measuring the state of a qubit without disturbing its operation. Fundamentally, this comes down to a microwave engineering problem, where we need to be able to measure the energy inside the qubit resonator without exposing it to noisy or lossy wiring. This can be accomplished by adding an additional microwave resonator to the system that is coupled to the qubit, but far from the qubit’s resonance frequency. The resonator acts as a filter that isolates the qubit from the control lines but also picks up a state-dependent frequency shift from the qubit. Just like in the binary phase shift keying (BPSK) encoding technique, the digital state of the qubit (0 or 1) is translated into a phase for a probe tone (microwave signal) reflecting off of this auxiliary resonator. Measuring the phase of this probe tone allows us to infer the state of the qubit without directly interfacing with the qubit itself.

While this sounds simple, the qubit actually imposes a severe cap on how much power can be used for this probe tone. In normal operation, a qubit should be in the 0 state or the 1 state or some superposition of the two. A measurement pulse should collapse the qubit into one of these two states, but using too much power can push it into a higher excited state and corrupt the computation. A safe measurement power is typically around -125 dBm, which amounts to only a handful of microwave photons interacting with the processor during the measurement. Typically, small signals are measured using microwave amplifiers, which increase the signal level, but also add their own noise. How much noise is acceptable? If the measurement process takes too long, the qubit state can change due to energy loss in the circuit. This means that these very small signals must be measured in just a few hundred nanoseconds with very high (>99%) fidelity. We therefore cannot afford to average the signal over a longer time to reduce the noise. Unfortunately, even the best semiconductor low-noise amplifiers are still almost a factor of 10 too noisy.

The solution is to design our own custom amplifiers based on the same circuit elements as the qubits themselves. These amplifiers typically consist of Josephson junctions to provide a tunable inductance wired into a superconducting resonant circuit. By constructing a resonant circuit out of these elements, you can create a parametric amplifier where amplification is achieved by modulating the tunable inductance at twice the frequency you want to amplify. Additionally, because all of the wiring is made of lossless superconductors, these devices operate near the quantum limit of added noise, where the only noise in the signal is coming from amplification of the zero point quantum voltage fluctuations.

The one downside to these devices is that the Josephson junctions constrain the power of the signals we can measure. If the signal is too large, the drive current can approach the junction critical current and degrade the amplifier performance. Even if this limit was sufficient to measure a single qubit, our goal was to increase efficiency by measuring up to six qubits at a time using the same amplifier. Some groups get around this limit by making traveling wave amplifiers, where the signals are distributed across thousands of junctions. This increases the saturation power, but the amplifiers get very complicated to produce and take up a lot of space on the chip. Our goal was to create an amplifier that could handle as much power as a traveling wave amplifier but with the same simple and compact design we were used to.

Results

The critical current of each Josephson junction limits our amplifier’s power handling. However, increasing this critical current also changes the inductance and, thus, the operating frequency of the amplifier. To avoid these constraints, we replaced a standard 2-junction DC SQUID with a nonlinear tunable inductor made up of two RF-SQUID arrays in parallel, which we call a snake inductor. Each RF-SQUID consists of a Josephson junction and geometric inductances L1 and L2, and each array contains 20 RF-SQUIDs. In this case, each junction of a standard DC SQUID is replaced by one of these RF-SQUID arrays. While the critical current of each RF-SQUID is much higher, we chain them together to keep the inductance and operating frequency the same. While this is a relatively modest increase in device complexity, it enables us to increase the power handling of each amplifier by roughly a factor of 100x. It is also fully compatible with existing designs that use impedance matching circuits to provide large measurement bandwidth.

Circuit diagram of our superconducting microwave amplifier. A split bias coil allows both DC and RF modulation of the snake inductor, while a shunt capacitor sets the frequency range. The flow of current is illustrated in the animation where an applied current (blue) on the bias line causes a circulating current (red) in the snake. A tapered impedance transformer lowers the loaded Q of the device. Since the Q is defined as frequency divided by bandwidth, lowering the Q with a constant frequency increases the bandwidth of the amplifier. Example circuit parameters used for a real device are Cs=6.0 pF, L1=2.6 pH, L2=8.0 pH, Lb=30 pH, M=50 pH, Z0 = 50 Ohms, and Zfinal = 18 ohms. The device operation is illustrated with a small signal (magenta) reflecting off the input of the amplifier. When the large pump tone (blue) is applied to the bias port, it generates amplified versions of the signal (gold) and a secondary tone known as an idler (also gold).

Microscope image of the nonlinear resonator showing the resonant circuit that consists of a large parallel plate capacitor, nonlinear snake inductor, and a current bias transformer to tune the inductance.

We measure this performance improvement by measuring the saturation power of the amplifier, or the point at which the gain is compressed by 1 dB. We also measure this power value vs. frequency to see how it scales with amplifier gain and distance from the center of the amplifier bandwidth. Since the amplifier gain is symmetric about its center frequency we measure this in terms of absolute detuning, which is just the absolute value of the difference between the center frequency of the amplifier and the probe tone frequency.

Input and output saturation power (1-dB gain compression point), calibrated using a superconducting quantum processor vs. absolute detuning from the amplifier center frequency.

Conclusion and future directions

The new microwave amplifiers represent a big step forward for our qubit measurement system. They will allow us to measure more qubits using a single device, and enable techniques that require higher power for each measurement tone. However, there are still quite a few areas we would like to explore. For example, we are currently investigating the application of snake inductors in amplifiers with advanced impedance matching techniques, directional amplifiers, and non-reciprocal devices like microwave circulators.

Acknowledgements

We would like to thank the Quantum AI team for the infrastructure and support that enabled the creation and measurement of our microwave amplifier devices. Thanks to our cohort of talented Google Research Interns that contributed to the future work mentioned above: Andrea Iorio for developing algorithms that automatically tune amplifiers and provide a snapshot of the local parameter space, Ryan Kaufman for measuring a new class of amplifiers using multi-pole impedance matching networks, and Randy Kwende for designing and testing a range of parametric devices based on snake inductors. With their contributions, we are gaining a better understanding of our amplifiers and designing the next generation of parametrically-driven devices.

Posted by Jarrod McClean, Staff Research Scientist, Google Quantum AI, and Hsin-Yuan Huang, Graduate Student, Caltech

In efforts to learn about the quantum world, scientists face a big obstacle: their classical experience of the world. Whenever a quantum system is measured, the act of this measurement destroys the “quantumness” of the state. For example, if the quantum state is in a superposition of two locations, where it can seem to be in two places at the same time, once it is measured, it will randomly appear either ”here” or “there”, but not both. We only ever see the classical shadows cast by this strange quantum world.

A growing number of experiments are implementing machine learning (ML) algorithms to aid in analyzing data, but these have the same limitations as the people they aim to help: They can’t directly access and learn from quantum information. But what if there were a quantum machine learning algorithm that could directly interact with this quantum data?

In “Quantum Advantage in Learning from Experiments”, a collaboration with researchers at Caltech, Harvard, Berkeley, and Microsoft published in Science, we show that a quantum learning agent can perform exponentially better than a classical learning agent at many tasks. Using Google’s quantum computer, Sycamore, we demonstrate the tremendous advantage that a quantum machine learning (QML) algorithm has over the best possible classical algorithm. Unlike previous quantum advantage demonstrations, no advances in classical computing power could overcome this gap. This is the first demonstration of a provable exponential advantage in learning about quantum systems that is robust even on today's noisy hardware.

Quantum Speedup
QML combines the best of both quantum computing and the lesser-known field of quantum sensing.

Quantum computers will likely offer exponential improvements over classical systems for certain problems, but to realize their potential, researchers first need to scale up the number of qubits and to improve quantum error correction. What’s more, the exponential speed-up over classical algorithms promised by quantum computers relies on a big, unproven assumption about so-called “complexity classes” of problems — namely, that the class of problems that can be solved on a quantum computer is larger than those that can be solved on a classical computer.. It seems like a reasonable assumption, and yet, no one has proven it. Until it's proven, every claim of quantum advantage will come with an asterisk: that it can do better than any known classical algorithm.

Quantum sensors, on the other hand, are already being used for some high-precision measurements and offer modest (and proven) advantages over classical sensors. Some quantum sensors work by exploiting quantum correlations between particles to extract more information about a system than it otherwise could have. For example, scientists can use a collection of N atoms to measure aspects of the atoms’ environment like the surrounding magnetic fields. Typically the sensitivity to the field that the atoms can measure scales with the square root of N. But if one uses quantum entanglement to create a complex web of correlations between the atoms, then one can improve the scaling to be proportional to N. But as with most quantum sensing protocols, this quadratic speed-up over classical sensors is the best one can ever do.

Enter QML, a technology that straddles the line between quantum computers and quantum sensors. QML algorithms make computations that are aided by quantum data. Instead of measuring the quantum state, a quantum computer can store quantum data and implement a QML algorithm to process the data without collapsing it. And when this data is limited, a QML algorithm can squeeze exponentially more information out of each piece it receives when considering particular tasks.

Comparison of a classical machine learning algorithm and a quantum machine learning algorithm. The classical machine learning algorithm measures a quantum system, then performs classical computations on the classical data it acquires to learn about the system. The quantum machine learning algorithm, on the other hand, interacts with the quantum states produced by the system, giving it a quantum advantage over the CML.

To see how a QML algorithm works, it’s useful to contrast with a standard quantum experiment. If a scientist wants to learn about a quantum system, they might send in a quantum probe, such as an atom or other quantum object whose state is sensitive to the system of interest, let it interact with the system, then measure the probe. They can then design new experiments or make predictions based on the outcome of the measurements. Classical machine learning (CML) algorithms follow the same process using an ML model, but the operating principle is the same — it’s a classical device processing classical information.

A QML algorithm instead uses an artificial “quantum learner.” After the quantum learner sends in a probe to interact with the system, it can choose to store the quantum state rather than measure it. Herein lies the power of QML. It can collect multiple copies of these quantum probes, then entangle them to learn more about the system faster.

Suppose, for example, the system of interest produces a quantum superposition state probabilistically by sampling from some distribution of possible states. Each state is composed of n quantum bits, or qubits, where each is a superposition of “0” and “1” — all learners are allowed to know the generic form of the state, but must learn its details.

In a standard experiment, where only classical data is accessible, every measurement provides a snapshot of the distribution of quantum states, but since it’s only a sample, it is necessary to measure many copies of the state to reconstruct it. In fact, it will take on the order of 2ⁿ copies.

A QML agent is more clever. By saving a copy of the n-qubit state, then entangling it with the next copy that comes along, it can learn about the global quantum state more quickly, giving a better idea of what the state looks like sooner.

Basic schematic of the QML algorithm. Two copies of a quantum state are saved, then a “Bell measurement” is performed, where each pair is entangled and their correlations measured.

The classical reconstruction is like trying to find an image hiding in a sea of noisy pixels — it could take a very long time to average-out all the noise to know what the image is representing. The quantum reconstruction, on the other hand, uses quantum mechanics to isolate the true image faster by looking for correlations between two different images at once.

Results
To better understand the power of QML, we first looked at three different learning tasks and theoretically proved that in each case, the quantum learning agent would do exponentially better than the classical learning agent. Each task was related to the example given above:

1. Learning about incompatible observables of the quantum state — i.e., observables that cannot be simultaneously known to arbitrary precision due to the Heisenberg uncertainty principle, like position and momentum. But we showed that this limit can be overcome by entangling multiple copies of a state.
2. Learning about the dominant components of the quantum state. When noise is present, it can disturb the quantum state. But typically the “principal component” — the part of the superposition with the highest probability — is robust to this noise, so we can still glean information about the original state by finding this dominant part.
3. Learning about a physical process that acts on a quantum system or probe. Sometimes the state itself is not the object of interest, but a physical process that evolves this state is. We can learn about various fields and interactions by analyzing the evolution of a state over time.

In addition to the theoretical work, we ran some proof-of-principle experiments on the Sycamore quantum processor. We started by implementing a QML algorithm to perform the first task. We fed an unknown quantum mixed state to the algorithm, then asked which of two observables of the state was larger. After training the neural network with simulation data, we found that the quantum learning agent needed exponentially fewer experiments to reach a prediction accuracy of 70% — equating to 10,000 times fewer measurements when the system size was 20 qubits. The total number of qubits used was 40 since two copies were stored at once.

Experimental comparison of QML vs. CML algorithms for predicting a quantum state’s observables. While the number of experiments needed to achieve 70% accuracy with a CML algorithm (“C” above) grows exponentially with the size of the quantum state n, the number of experiments the QML algorithm (“Q”) needs is only linear in n. The dashed line labeled “Rigorous LB (C)” represents the theoretical lower bound (LB) — the best possible performance — of a classical machine learning algorithm.

In a second experiment, relating to the task 3 above, we had the algorithm learn about the symmetry of an operator that evolves the quantum state of their qubits. In particular, if a quantum state might undergo evolution that is either totally random or random but also time-reversal symmetric, it can be difficult for a classical learner to tell the difference. In this task, the QML algorithm can separate the operators into two distinct categories, representing two different symmetry classes, while the CML algorithm fails outright. The QML algorithm was completely unsupervised, so this gives us hope that the approach could be used to discover new phenomena without needing to know the right answer beforehand.

Experimental comparison of QML vs. CML algorithms for predicting the symmetry class of an operator. While QML successfully separates the two symmetry classes, the CML fails to accomplish the task.

Conclusion
This experimental work represents the first demonstrated exponential advantage in quantum machine learning. And, distinct from a computational advantage, when limiting the number of samples from the quantum state, this type of quantum learning advantage cannot be challenged, even by unlimited classical computing resources.

So far, the technique has only been used in a contrived, “proof-of-principle” experiment, where the quantum state is deliberately produced and the researchers pretend not to know what it is. To use these techniques to make quantum-enhanced measurements in a real experiment, we’ll first need to work on current quantum sensor technology and methods to faithfully transfer quantum states to a quantum computer. But the fact that today’s quantum computers can already process this information to squeeze out an exponential advantage in learning bodes well for the future of quantum machine learning.

Acknowledgements
We would like to thank our Quantum Science Communicator Katherine McCormick for writing this blog post. Images reprinted with permission from Huang et al., Science, Vol 376:1182 (2022).

Posted by William J. Huggins, Research Scientist, Google Quantum AI

The intersection between the computational difficulty and practical importance of quantum chemistry challenges run on quantum computers has long been a focus for Google Quantum AI. We’ve experimentally simulated simple models of chemical bonding, high-temperature superconductivity, nanowires, and even exotic phases of matter such as time crystals on our Sycamore quantum processors. We’ve also developed algorithms suitable for the error-corrected quantum computers we aim to build, including the world’s most efficient algorithm for large-scale quantum computations of chemistry (in the usual way of formulating the problem) and a pioneering approach that allows for us to solve the same problem at an extremely high spatial resolution by encoding the position of the electrons differently.

Despite these successes, it is still more effective to use classical algorithms for studying quantum chemistry than the noisy quantum processors we have available today. However, when the laws of quantum mechanics are translated into programs that a classical computer can run, we often find that the amount of time or memory required scales very poorly with the size of the physical system to simulate.

Today, in collaboration with Dr. Joonho Lee and Professor David Reichmann at Colombia, we present the Nature publication “Unbiasing Fermionic Quantum Monte Carlo with a Quantum Computer”, where we propose and experimentally validate a new way of combining classical and quantum computation to study chemistry, which can replace a computationally-expensive subroutine in a powerful classical algorithm with a “cheaper”, noisy, calculation on a small quantum computer. To evaluate the performance of this hybrid quantum-classical approach, we applied this idea to perform the largest quantum computation of chemistry to date, using 16 qubits to study the forces experienced by two carbon atoms in a diamond crystal. Not only was this experiment four qubits larger than our earlier chemistry calculations on Sycamore, we were also able to use a more comprehensive description of the physics that fully incorporated the interactions between electrons.

Google’s Sycamore quantum processor. Photo Credit: Rocco Ceselin.

A New Way of Combining Quantum and Classical
Our starting point was to use a family of Monte Carlo techniques (projector Monte Carlo, more on that below) to give us a useful description of the lowest energy state of a quantum mechanical system (like the two carbon atoms in a crystal mentioned above). However, even just storing a good description of a quantum state (the “wavefunction”) on a classical computer can be prohibitively expensive, let alone calculating one.

Projector Monte Carlo methods provide a way around this difficulty. Instead of writing down a full description of the state, we design a set of rules for generating a large number of oversimplified descriptions of the state (for example, lists of where each electron might be in space) whose average is a good approximation to the real ground state. The “projector” in projector Monte Carlo refers to how we design these rules — by continuously trying to filter out the incorrect answers using a mathematical process called projection, similar to how a silhouette is a projection of a three-dimensional object onto a two-dimensional surface.

Unfortunately, when it comes to chemistry or materials science, this idea isn’t enough to find the ground state on its own. Electrons belong to a class of particles known as fermions, which have a surprising quantum mechanical quirk to their behavior. When two identical fermions swap places, the quantum mechanical wavefunction (the mathematical description that tells us everything there is to know about them) picks up a minus sign. This minus sign gives rise to the famous Pauli exclusion principle (the fact that two fermions cannot occupy the same state). It can also cause projector Monte Carlo calculations to become inefficient or even break down completely. The usual resolution to this fermion sign problem involves tweaking the Monte Carlo algorithm to include some information from an approximation to the ground state. By using an approximation (even a crude one) to the lowest energy state as a guide, it is usually possible to avoid breakdowns and even obtain accurate estimates of the properties of the true ground state.

Top: An illustration of how the fermion sign problem appears in some cases. Instead of following the blue line curve, our estimates of the energy follow the red curve and become unstable. Bottom: An example of the improvements we might see when we try to fix the sign problem. By using a quantum computer, we hope to improve the initial guess that guides our calculation and obtain a more accurate answer.

For the most challenging problems (such as modeling the breaking of chemical bonds), the computational cost of using an accurate enough initial guess on a classical computer can be too steep to afford, which led our collaborator Dr. Joonho Lee to ask if a quantum computer could help. We had already demonstrated in previous experiments that we can use our quantum computer to approximate the ground state of a quantum system. In these earlier experiments we aimed to measure quantities (such as the energy of the state) that are directly linked to physical properties (like the rate of a chemical reaction). In this new hybrid algorithm, we instead needed to make a very different kind of measurement: quantifying how far the states generated by the Monte Carlo algorithm on our classical computer are from those prepared on the quantum computer. Using some recently developed techniques, we were even able to do all of the measurements on the quantum computer before we ran the Monte Carlo algorithm, separating the quantum computer’s job from the classical computer’s.

A diagram of our calculation. The quantum processor (right) measures information that guides the classical calculation (left). The crosses indicate the qubits, with the ones used for the largest experiment shaded green. The direction of the arrows indicate that the quantum processor doesn’t need any feedback from the classical calculation. The red bars represent the parts of the classical calculation that are filtered out by the data from the quantum computer in order to avoid the fermion sign problem and get a good estimate of properties like the energy of the ground state.

This division of labor between the classical and the quantum computer helped us make good use of both resources. Using our Sycamore quantum processor, we prepared a kind of approximation to the ground state that would be difficult to scale up classically. With a few hours of time on the quantum device, we extracted all of the data we needed to run the Monte Carlo algorithm on the classical computer. Even though the data was noisy (like all present-day quantum computations), it had enough signal that it was able to guide the classical computer towards a very accurate reconstruction of the true ground state (shown in the figure below). In fact, we showed that even when we used a low-resolution approximation to the ground state on the quantum computer (just a few qubits encoding the position of the electrons), the classical computer could efficiently solve a much higher resolution version (with more realism about where the electrons can be).

Top left: a diagram showing the sixteen qubits we used for our largest experiment. Bottom left: an illustration of the carbon atoms in a diamond crystal. Our calculation focused on two atoms (the two that are highlighted in translucent yellow). Right: A plot showing how the error in the total energy (closer to zero is better) changes as we adjust the lattice constant (the spacing between the two carbon atoms). Many properties we might care about, such as the structure of the crystal, can be determined by understanding how the energy varies as we move the atoms around. The calculations we performed using the quantum computer (red points) are comparable in accuracy to two state-of-the-art classical methods (yellow and green triangles) and are extremely close to the numbers we would have gotten if we had a perfect quantum computer rather than a noisy one (black points). The fact that these red and black points are so close tells us that the error in our calculation comes from using an approximate ground state on the quantum computer that was too simple, not from being overwhelmed by noise on the device.

Using our new hybrid quantum algorithm, we performed the largest ever quantum computation of chemistry or materials science. We used sixteen qubits to calculate the energy of two carbon atoms in a diamond crystal. This experiment was four qubits larger than our first chemistry calculations on Sycamore, we obtained more accurate results, and we were able to use a better model of the underlying physics. By guiding a powerful classical Monte Carlo calculation using data from our quantum computer, we performed these calculations in a way that was naturally robust to noise.

We’re optimistic about the promise of this new research direction and excited to tackle the challenge of scaling these kinds of calculations up towards the boundary of what we can do with classical computing, and even to the hard-to-study corners of the universe. We know the road ahead of us is long, but we’re excited to have another tool in our growing toolbox.

Acknowledgements
I’d like to thank my co-authors on the manuscript, Bryan O’Gorman, Nicholas Rubin, David Reichman, Ryan Babbush, and especially Joonho Lee for their many contributions, as well as Charles Neill and Pedram Rousham for their help executing the experiment. I’d also like to thank the larger Google Quantum AI team, who designed, built, programmed, and calibrated the Sycamore processor.