# One Little Push at a Time: Quantum Trajectories and Weak Measurements

## Tracking the Trajectory of a Qubit’s State in Real-Time

Surely you’ve heard of quantum non-demolition (QND) measurements before: they come strong, or they come weak. Strong measurements fully collapse the wave function in the chosen basis. So, say you use the Bloch sphere representation, a strong measurement looks like collapsing the vector into one of the vertices, e.g., ground |g> or excited state |e>. You have officially killed any quantum superposition there. Among the many examples of this heartless method in use, you should check out the amazing quantum supremacy experiment by Google Quantum AI in 2019 [1].

Weak measurements, on the other hand, allow the quantum state to evolve stochastically, arriving at its final state in a given time scale. Hatridge et al. [2] showed that weak measurements keep the state pure while extracting partial information. It’s more of a gentle nudge rather than a killing blow. Vijay et al. [3] demonstrated analog feedback on the Rabi oscillations of a qubit in a striking demonstration of weak measurements. They showed, amongst other things, that qubits do not need to be mistreated to give out information.

**Sometimes Weak is Better**

In Figure 1, we show how strong and weak measurements look in the IQ plane (or rather on its projection along the line that connects the two blobs under test). Repeated measurements of the system will unveil gaussian distributions for |g> and for the |e> states. The stronger the measurement, i.e., longer or more intense pulses, the more the gaussians overlap. Strong measurements often allow a single shot to clearly distinguish between the states, while individual weak measurements do not show on which bell you ended up. This is how we can understand obtaining partial information without collapsing the wavefunction, as we gather information only insofar the overlap is non-negligible. Still, the same overlap allows for the quantum superposition to live on.

This partial information gathering from weak measurements allows for an estimation of the state and gives access to the deviation of quantum trajectories [4]. So after many, many shots, you can gather statistics and classically calculate the path the system followed, tracking it on its Bloch sphere. Of course, you only know a posteriori the path, and you have no hope of controlling it. Or do you? Can we actually track the trajectory of a qubit’s state in real-time?

It turns out that now we can! We need to perform the classical estimations during continuous weak measurement. This is not an easy feat, but thanks to the OPX+ real-time computational capabilities and its multi-core operation, you can have one core continuously *weakly *measure while another estimates the trajectory and feeds back the information in an ultra-fast feedback loop that takes merely tens of nanoseconds. Have a look at Figure 2 for schematics. This is the first time continuous quantum trajectory estimation becomes truly within reach of every qubit lab. However, we won’t stop here.

**The Marvels of Today**

Now that we have a great new way to track your qubit state *while* it wanders. You can take control by correcting the state path with carefully adjusted real-time pulses. For example, you can build a ** qubit stabilizer**. rack the quantum trajectory and make sure to gently and constantly kick the state in the direction that voids its last move in an endless game of chess. This will allow you to keep a qubit’s state steady, preventing collapse and possibly extending its lifetime. How cool is that?

Many labs have picked up on the possibilities that this capability opens up, and we at Quantum Machines are happy to work with them to exploit the best that OPX+ offers. Imagine having an OPX+ and a bunch of qubits to play with (any technology will do). You have the ability to perform continuous weak measurements, trajectory estimations, and classical computation within quantum coherence times. We made sure to get the controller technology right. Now it is your time to step up and get some crazy physics going – and we’ll be there at your side.

Drop me a line and let’s see what you come up with: lorenzo@quantum-machines.co

**References**

[1] Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

[2] Hatridge, M., Shankar, S., Mirrahimi, M., Schackert, F., Geerlings, K., Brecht, T., … & Devoret, M. H. (2013). Quantum back-action of an individual variable-strength measurement. Science, 339(6116), 178-181.

[3] Vijay, R., Macklin, C., Slichter, D. et al. Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77–80 (2012)

[4] Naghiloo, M. (2019). Introduction to Experimental Quantum Measurement with Superconducting Qubits. ArXiv. https://doi.org/10.48550/arXiv.1904.09291