# Peeking into the world of quantum intelligence

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

We, as intelligent beings, receive, process, and store classical information. The information comes from vision, hearing, smell, and tactile sensing. The data is encoded as analog classical information through the electrical pulses sending through our nerve fibers. Our brain processes this information classically through neural circuits (at least that is our current understanding, but one should check out this blogpost). We then store this processed classical information in our hippocampus that allows us to retrieve it later to combine it with future information that we obtain. Finally, we use the stored classical information to make predictions about the future (imagine/predict the future outcomes if we perform certain action) and choose the action that would most likely be in our favor.

Such abilities have enabled us to make remarkable accomplishments: soaring in the sky by constructing accurate models of how air flows around objects, or building weak forms of intelligent beings capable of performing basic conversations and play different board games. Instead of receiving/processing/storing classical information, one could imagine some form of quantum intelligence that deals with quantum information instead of classical information. These quantum beings can receive quantum information through quantum sensors built up from tiny photons and atoms. They would then process this quantum information with quantum mechanical evolutions (such as 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 be like. While we have never encountered such a strange creature in the real world (yet), the mathematics of quantum mechanics, machine learning, and information theory allow us to peek into what such a fantastic world would be like. The physical world we live in is intrinsically quantum. So one may imagine that a quantum being is capable of making more powerful predictions than a classical being. Maybe he/she/they could better predict events that happened further away, such as tell us how a distant black hole was engulfing another? Or perhaps he/she/they could improve our lives, for example by presenting us with an entirely new approach for capturing energy from sunlight?

One may be skeptical about finding quantum intelligent beings in nature (and rightfully so). But it may not be so absurd to synthesize a weak form of quantum (artificial) intelligence in an experimental lab, or enhance our classical human intelligence with quantum devices to approximate a quantum-mechanical being. Many famous companies, like Google, IBM, Microsoft, and Amazon, as well as many academic labs and startups have been building better quantum machines/computers day by day. By combining the concepts of machine learning on classical computers with these quantum machines, the future of us interacting with some form of quantum (artificial) intelligence may not be so distant.

Before the day comes, could we peek into the world of quantum intelligence? And could one better understand how much more powerful they could be over classical intelligence?

In a recent publication [1], my advisor John Preskill, my good friend Richard Kueng, and I made some progress toward these questions. We consider a quantum mechanical world where classical beings could obtain classical information by measuring the world (performing POVM measurement). In contrast, quantum beings could retrieve quantum information through quantum sensors and store the data in a quantum memory. We study how much better quantum over classical beings could learn from the physical world to accurately predict the outcomes of unseen events (with the focus on the number of interactions with the physical world instead of computation time). We cast these problems in a rigorous mathematical framework and utilize high-dimensional probability and quantum information theory to understand their respective prediction power. Rigorously, one refers to a classical/quantum being as a classical/quantum model, algorithm, protocol, or procedure. This is because the actions of these classical/quantum beings are the center of the mathematical analysis.

Formally, we consider the task of learning an unknown physical evolution described by a CPTP map $\mathcal{E}$ that takes in $n$-qubit state and maps to $m$-qubit state. The classical model can select an arbitrary classical input to the CPTP map and measure the output state of the CPTP map with some POVM measurement. The quantum model can access the CPTP map coherently and obtain quantum data from each access, which is equivalent to composing multiple CPTP maps with quantum computations to learn about the CPTP map. The task is to predict a 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 classical input $x \in \{0, 1\}^n$. And the goal is to achieve the task while accessing $\mathcal{E}$ as few times as possible (i.e., fewer interactions or experiments in the physical world). We denote the number of interactions needed by classical and quantum models 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 facilitate better information synthesis with information obtained from previous experiments. Nevertheless, in [1], we show that there is a fundamental limit to how much more efficient quantum models can be. In order 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 the speed-up $N_{\mathrm{C}} / N_{\mathrm{Q}}$ is upper bounded by $m / \epsilon$, where $m > 0$ is the number of qubits each experiment provides (the output number of qubits in the CPTP map $\mathcal{E}$), and $\epsilon > 0$ is the desired prediction error (smaller $\epsilon$ means we want to predict more accurately).

In contrast, when we want to accurately predict all unseen events, we prove that quantum models could use exponentially fewer experiments than classical models. We give a construction for predicting properties of quantum systems showing that quantum models could substantially outperform classical models. These rigorous results show that quantum intelligence shines when we seek stronger prediction performance.

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

Further information:

• A classical model that can be used to accurately predict properties of quantum systems is the classical shadow formalism [2] that we proposed a year ago. In many tasks, this model can be shown to be one of the strongest rivals that quantum models have to surpass.
• Even if a quantum model only receives and stores classical data, the ability to process the data using a quantum-mechanical evolution can still be advantageous [3]. However, obtaining large advantage will be harder in this case as the computational power in data can slightly boost classical machines/intelligence [3].
• Another nice paper by Dorit Aharonov, Jordan Cotler, and Xiao-Liang Qi [4] also proved advantages of quantum models over classical one in some classification tasks.

References:

[1] Huang, Hsin-Yuan, Richard Kueng, and John Preskill. “Information-Theoretic Bounds on 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. “Predicting many properties of a quantum system from very few measurements.” Nature 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 communications 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 preprint arXiv:2101.04634 (2021).
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