Measurement-induced entanglement phase transitions in a quantum circuit

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.