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 that takes in -qubit state and maps to -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 , given by , for a new classical input . And the goal is to achieve the task while accessing 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 .

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

where is the hypothesis learned from the classical/quantum model and is an arbitrary distribution over the input space , we found that the speed-up is upper bounded by , where is the number of qubits each experiment provides (the output number of qubits in the CPTP map ), and is the desired prediction error (smaller 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).

“The data is encoded in the form of 0/1 bits through the electrical pulses sending through our nerve fibers.” That seems too strong: is the electrochemical signalling between parts of animal and human bodies 0/1-discrete everywhere in the body? We know that noise can sometimes overwhelm effective functioning (permanently, after death, but also at ill-defined intervals, during strokes, psychosis, …), but it would seem to be difficult to rule out noise playing any role whatsoever in normal operation, in which case purely 0/1 operations would not be a good model for the brain.

I emphasize that I base this comment more on the physics in my article in Annals of Physics 2020, “An algebraic approach to Koopman classical mechanics”, than on a deep knowledge of biological systems (I’m not, for example, referencing the quantum biology literature, which is not always compelling.) The argument in that article is effectively that in classical physics there are hidden observables in classical mechanics that are generated by the Poisson bracket and that make the algebra of measurements noncommutative in the presence of noise. Given a noncommutative algebra of observables we can, I think, say that there is no difference between classical systems and quantum systems: there is just physics with different kinds of noise dominating the local effective dynamics. I apologize that this viewpoint is so contrary to your post —to you it may well be just so much noise— but such, I suppose, is life.

This is a very interesting perspective!

I think it is still debatable whether biological systems maintain some form of quantum coherence (which I hope they do as that would be much more interesting). However, if we believe that these electrical pulses have been completely decohered by noise and become a purely classical signal, i.e., a diagonal density matrix, or approximately so, then such pulses could then be well approximated by a discretization into 0/1 bits. Here, I am referring to the fact that classical digital computers with sufficiently large classical memory and computing time can simulate any analog classical system to high accuracy.

I don’t think the signals and electric fields are really discrete in the body (or in any place we know of). But when they do not have off-diagonal elements in the density matrix, then a further discretization into many 0/1 bits will provide accurate descriptions for the continuous systems (as captured by the extended Church–Turing thesis).

The current understanding of neurons also follows a very digital fashion. When the voltage in the neuron changes drastically in a short amount of interval, the neurons would generate an all-or-nothing pulse (based on the action potential). But that is just how we understand now! Our knowledge about neurons could well change in the future.

FWIW, while it’s true that neurons are either firing or not-firing, my understanding is that the firing state consists of a train of pulses and that the duty cycle and frequency of those pulses can carry information. I’ve seen papers suggesting that even the rise and fall times of the pulses might carry information. Neurons receive and integrate lots of inputs from other neurons. I believe the average number of connections is 7,000. As such, it makes sense the integration has more information then just “yay” or “nay” — the brain is, I think, better seen as an analog signal processing system than a discrete one.

Not that it necessary affects your work. As an analog device, the brain is still classical (as far as we know), and I’m not at all surprised a quantum brain with quantum inputs could outperform a classical one. Superposition and interference seem a powerful platform!

Thank you, Wyrd Smythe, for the additional information!

Yes, the focus is more about classical versus quantum information. Similarly, one could also consider continuous-variable quantum information (common in quantum optics) instead of qubits. Digitization is not my intended emphasis, so I decided to change the wordings in that part of the text 🙂

Sorry; didn’t mean to make extra work for you! 😮

There are those who feel the power of qubits, of QC in general, is from many-worlds, but superposition, to me, make even qubits analog. I think the power of QC comes from the same source as what makes the ocean surface calculate the wave pattern of all the various wind and other influences. Or a guitar calculate its tone different from how a saxophone calculates its tone. Analog wave magic! 🙂

First of all I am in complete agreement

In university we had to design and build our own computerized data aquisition systems

I am very curious if you or anyone else here is familiar with the lithium isotope experiments on rats that observed large differences in behavior compared to regular lithium

Based on asymmetric nuclear weights we are even allowed irrational numbers despite the constant three valence electrons

When you look into a vast, open, sky, you see “space”, but that “space” you see is an image on the retina of your eye. Maybe your sense of “space” is not just an interpretation by your brain. Maybe, you are already effecting quantum field sensing on a subconscious level?¿

Autopoiesis, self-created reality, depends, to a large extent, on the tools inherent in one’s toolkit. Young cats raised in in an environment w/o horizontal lines will walk right into a coffee tabletop because they literally don’t see it. And vice versa for vertical lines.

What you see is what you get, literally, and so to with what you don’t see.