Intelligent beings have the ability to receive, process, store information and, based on the processed information, predict what will happen in the future and act accordingly.

As intelligent beings, we receive, process and store classic information (0/1 bit). The information comes from sight, hearing, smell and sense of touch. The data is encoded in the form of 0/1 bits by the electrical impulses sent through our nerve fibers. Our brain processes this information classically through neural circuits (at least that’s our current understanding, but you should take a look at this blog post). We then store this processed classic information in our hippocampus, which enables us to retrieve it later to combine it with future information we receive. Finally, we use the stored classical information to make predictions about the future (to imagine / predict the future outcomes if we take a certain action) and choose the action that would most likely be in our favor.

Such skills have allowed us to achieve remarkable feats: soaring to the sky by constructing accurate models of airflow around objects or building weak forms of intelligent beings capable of basic conversations and playing various board games. Instead of receiving / processing / storing classical information, one could imagine a form of quantum intelligence that deals with quantum information instead of classical information. These quantum beings can receive quantum information (represented in qubits) through quantum sensors made up of tiny photons and atoms. They would then process this quantum information with quantum mechanical evolutions (like quantum computers) and store the processed qubits in a quantum memory (protected with a surface code or toric code).

It is natural to wonder what a world of quantum intelligence would look like. While we have never (yet) encountered such a strange creature in the real world, the mathematics of quantum mechanics, machine learning and information theory allow us a glimpse into such a fantastic world. The physical world in which we live is quantum by nature. So one can imagine that a quantum being is able to make stronger predictions than a classical being. Perhaps he / she / she could better predict events that happened further away, such as telling us how one black hole far away devoured another? Or could he / she / she improve our lives, for example by presenting us with a whole new approach to generating energy from sunlight?

One can be skeptical about finding quantum intelligent beings in nature (and rightly so). But it may not be so absurd to (artificially) synthesize a weak form of quantum intelligence in an experimental laboratory or to improve our classical human intelligence with quantum devices in order to approximate a quantum mechanical being. Many famous companies like Google, IBM, Microsoft and Amazon as well as many academic laboratories and startups are building better quantum machines / computers day in and day out. By combining the concepts of machine learning on classic computers with these quantum machines, the future of our interaction with some form of quantum intelligence (artificial) may not be that far off.

Before the day comes, could we take a look into the world of quantum intelligence? And could one better understand how much more powerful they could be compared to classical intelligence?

In a recent publication [1] , my advisor John Preskill, my good friend Richard Kueng and I have made some progress on these issues. We consider a quantum mechanical world in which classical beings could receive classical information by measuring the world (performing a POVM measurement). In contrast, quantum beings could retrieve quantum information through quantum sensors and store the data in a quantum memory. We investigate how much better quanta than classical beings could learn from the physical world to accurately predict the outcomes of invisible events (with a focus on the number of interactions with the physical world instead of computing time). We put these problems into a rigorous mathematical framework and use high-dimensional probability and quantum information theory to understand their predictive power. Strictly speaking, a classical / quantum being is referred to as a classical / quantum model, algorithm, protocol or procedure. This is because the actions of these classical / quantum beings are at the center of mathematical analysis.

Formally we consider the task of learning an unknown physical evolution described by a CPTP map $mathcal {E}$ that takes a $No$-Qubit state and assignment to $I$-Qubit state. The classic model can select any classic input for the CPTP card and measure the output state of the CPTP card with some POVM measurements. The quantum model can coherently access the CPTP card and get quantum data from anyone access, which is the same as creating multiple CPTP cards with quantum calculations to learn more about the CPTP card. The task is to predict some property of the output state $mathcal {E} ( lvert x rangle ! langle x rvert)$, given by $mathrm {Tr} (O mathcal {E} ( lvert x rangle ! langle x rvert))$, for a new classic entrance $x in {0, 1 } ^ n$. And the goal is to do the job in accessing it $mathcal {E}$ as few times as possible (i.e. fewer interactions or experiments in the physical world). We refer to the number of interactions that classical and quantum models require as $N _ { mathrm {C}}, N _ { mathrm {Q}}$.

In general, quantum models could learn from fewer interactions with the physical world (or experiments in the physical world) than classical models. This is because coherent quantum information can enable better information synthesis with information from previous experiments. Still in [1] , we show that there is a fundamental limit to how much more efficient quantum models can be. To achieve a prediction error

$mathbb {E} _ {x sim mathcal {D}} | h (x) - mathrm {Tr} (O mathcal {E} ( lvert x rangle ! langle x rvert)) | leq mathcal {O} ( epsilon),$

Where $h (x)$ is the hypothesis learned from the classical / quantum model and $mathcal {D}$ is an arbitrary distribution over the input space ${0, 1 } ^ n$, we found that acceleration $N _ { mathrm {C}} / N _ { mathrm {Q}}$ is limited by $m / epsilon$, Where $m> 0″ class=”latex”/> is the number of qubits that each experiment delivers (the output number of qubits in the CPTP map ), and $epsilon> 0″ class=”latex”/> is the desired prediction error (smaller means that we want to predict more accurately).

In contrast, when we want to predict accurately all We prove to unseen events that quantum models could use exponentially fewer experiments than classical models. We give a construction for predicting properties of quantum systems, which shows that quantum models could clearly outperform classical models. These rigorous results show that quantum intelligence shines when we are looking for more predictive power.

We have only scratched the surface of what is possible with quantum intelligence. As the future progresses, I hope that we will discover more of what can only be achieved through quantum intelligence, mathematical analysis, rigorous numerical studies, and physical experiments.

• A classic model that can be used to accurately predict the properties of quantum systems is classic shadow formalism [2] that we proposed a year ago. In many tasks, this model can be shown as one of the strongest competitors that quantum models have to surpass.
• Even if a quantum model only receives and stores classical data, the possibility of processing the data by means of a quantum mechanical evolution can still be an advantage [3] . However, in this case it will be more difficult to achieve a great advantage, since the computing power in the data can easily increase classic machines / intelligence [3] .
• Another nice article by Dorit Aharonov, Jordan Cotler and Xiao-Liang Qi [4] also demonstrated advantages of quantum models compared to classical models in some classification tasks.

References:

[1] Huang, Hsin-Yuan, Richard Kueng, and John Preskill. “Information-theoretical limits for the quantum advantage in machine learning.”Physical review letters 126: 190505 (2021). https://doi.org/10.1103/PhysRevLett.126.190505

[2] Huang, Hsin-Yuan, Richard Kueng, and John Preskill. “Predict many properties of a quantum system from very few measurements.”Natural physics 16: 1050-1057 (2020). https://doi.org/10.1038/s41567-020-0932-7

[3] Huang, Hsin-Yuan, et al. “Power of data in quantum machine learning.”Nature communication 12.1 (2021): 1-9. https://doi.org/10.1038/s41467-021-22539-9

[4] Aharonov, Dorit, Jordan Cotler and Xiao-Liang Qi. “Quantum Algorithmic Measurement.”arXiv form arXiv: 2101.04634 (2021).