Physicists find a brighter way to diagnose quantum states

A two-dimensional quantum object is called a qubit in analogy with classical bits, but the same concept can be generalized to a higher number of dimensions. The resulting higher-dimensional object is usually called a qudit, and it can be equivalent to many entangled qubits. “Most quantum systems are actually higher dimensional, and we just ignore, or sometimes actively empty, most of our higher-dimensional system for it to become a qubit, ” says Markus Rambach, a researcher at the University of Queensland, Australia and lead author of the study, which is published in Physical Review Letters.

Rambach’s team showed that these higher-dimensional quantum objects could, in principle, be used to characterize the quantum state of large entangled systems in an efficient way. The team created its qudits using devices called spatial light modulators (SLMS), which function like a transparency on an old-fashioned overhead projector: they only allow a certain shape of light – a quantum state – to pass through perfectly. While one SLM prepares the initial quantum state, creating superpositions of ring-like patterns of light, another SLM performs the analysis measurement and sends the light through an optical fibre to a detector. Then, the researchers apply the self-guided tomography method to the second SLM, iteratively tweaking the “transparency” until the two SLMs perfectly align, providing them with the correct quantum state.

A more practical way to uncover quantum states

To understand how self-guided tomography works, picture a quantum state as an intricate cathedral steeple in the middle of a barren landscape, standing out as a flash of colour above the monotony. To capture the beauty of that building using conventional quantum state tomography, you would need to paint a picture of the entire landscape, spending just as much effort on the rocks and sky as you do on the Gothic architecture. This would require a lot of time and paint, and moving to an ever higher dimensional space would be even worse – like having a larger landscape to consider while the tower stays the same size.

A schematic representation of self-guided tomography for a qudit with arbitrary dimensions.

In contrast, self-guided tomography directs your attention more rapidly towards the church. Taking into consideration just a few portions of the broader landscape, your gaze quickly settles on the space near the building, bringing it into sharp focus. Then you create a detailed sketch of only the region of interest – the bell tower, the stainedglass windows, and the carefully planted garden. This method is similar to the gradient descent methods popular in machine learning. However, while gradient descent narrowly follows the path of maximal change, and can therefore get stuck in local extrema, self-guided tomography only moves in the optimal direction on average. By using repeated averages, it therefore converges to the global maximum, representing the desired quantum state.

In a quantum landscape, how close you are to that ideal quantum state is quantified by the fidelity, which ranges from 0% for no overlap to 100% for two identical states. The self-guided tomography method shows that increasing the number of iterations of the algorithm makes it possible to reach ever higher fidelities, even for high-dimensional qudits. “To get to the same fidelity as in standard tomography, we need less resources or less copies of the unknown state,” Rambach says.

Performing measurements with the power of light

Using photons as quantum systems enabled the team to perform measurements very rapidly, collecting hundreds of thousands of samples per second. However, since not all quantum systems are capable of quickly preparing many copies of the unknown state, the researchers also performed a separate low photon count experiment, measuring in a regime with high statistical noise. Although the absolute number of iterations required did increase in this noisier region, the trend of achieving continuously increasing fidelities with more iterations remained the same. This means that other quantum systems can take advantage of the same computational speedup.

As a member of the Australian Research Council Centre of Excellence for Engineering Quantum Systems, Rambach foresees many other research groups within the centre using self-guided tomography in all types of quantum architectures. “We are currently reaching out to experimentalists in other systems to see who is keen to pick up our method, to actually use it in their experiments,” he says.

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