October | 2023 | Quantum Frontiers

The origin of life appears to share little with quantum computation, apart from the difficulty of achieving it and its potential for clickbait. Yet similar notions of complexity have recently garnered attention in both fields. Each topic’s researchers expect only special systems to generate high values of such complexity, or complexity at high rates: organisms, in one community, and quantum computers (and perhaps black holes), in the other.

Each community appears fairly unaware of its counterpart. This article is intended to introduce the two. Below, I review assembly theory from origin-of-life studies, followed by quantum complexity. I’ll then compare and contrast the two concepts. Finally, I’ll suggest that origin-of-life scientists can quantize assembly theory using quantum complexity. The idea is a bit crazy, but, well, so what?

Assembly theory in origin-of-life studies

Imagine discovering evidence of extraterrestrial life. How could you tell that you’d found it? You’d have detected a bunch of matter—a bunch of particles, perhaps molecules. What about those particles could evidence life?

This question motivated Sara Imari Walker and Lee Cronin to develop assembly theory. (Most of my assembly-theory knowledge comes from Sara, about whom I wrote this blog post years ago and with whom I share a mentor.) Assembly theory governs physical objects, from proteins to self-driving cars.

Imagine assembling a protein from its constituent atoms. First, you’d bind two atoms together. Then, you might bind another two atoms together. Eventually, you’d bind two pairs together. Your sequence of steps would form an algorithm for assembling the protein. Many algorithms can generate the same protein. One algorithm has the least number of steps. That number is called the protein’s assembly number.

Different natural processes tend to create objects that have different assembly numbers. Stars form low-assembly-number objects by fusing two hydrogen atoms together into helium. Similarly, random processes have high probabilities of forming low-assembly-number objects. For example, geological upheavals can bring a shard of iron near a lodestone. The iron will stick to the magnetized stone, forming a two-component object.

My laptop has an enormous assembly number. Why can such an object exist? Because of information, Sara and Lee emphasize. Human beings amassed information about materials science, Boolean logic, the principles of engineering, and more. That information—which exists only because organisms exists—helped engender my laptop.

If any object has a high enough assembly number, Sara and Lee posit, that object evidences life. Absent life, natural processes have too low a probability of randomly throwing together molecules into the shape of a computer. How high is “high enough”? Approximately fifteen, experiments by Lee’s group suggest. (Why do those experiments point to the number fifteen? Sara’s group is working on a theory for predicting the number.)

In summary, assembly number quantifies complexity in origin-of-life studies, according to Sara and Lee. The researchers propose that only living beings create high-assembly-number objects.

Quantum complexity in quantum computation

Quantum complexity defines a stage in the equilibration of many-particle quantum systems. Consider a clump of $N$ quantum particles isolated from its environment. The clump will be in a pure quantum state $| \psi(0) \rangle$ at a time $t = 0$ . The particles will interact, evolving the clump’s state as a function $| \psi(t) \rangle$ .

Quantum many-body equilibration is more complicated than the equilibration undergone by your afternoon pick-me-up as it cools.

The interactions will equilibrate the clump internally. One stage of equilibration centers on local observables $O$ . They’ll come to have expectation values $\langle \psi(t) | O | \psi(t) \rangle$ approximately equal to thermal expectation values ${\rm Tr} ( O \, \rho_{\rm th} )$ , for a thermal state $\rho_{\rm th}$ of the clump. During another stage of equilibration, the particles correlate through many-body entanglement.

The longest known stage centers on the quantum complexity of $| \psi(t) \rangle$ . The quantum complexity is the minimal number of basic operations needed to prepare $| \psi(t) \rangle$ from a simple initial state. We can define “basic operations” in many ways. Examples include quantum logic gates that act on two particles. Another example is an evolution for one time step under a Hamiltonian that couples together at most $k$ particles, for some $k$ independent of $N$ . Similarly, we can define “a simple initial state” in many ways. We could count as simple only the $N$ -fold tensor product $| 0 \rangle^{\otimes N}$ of our favorite single-particle state $| 0 \rangle$ . Or we could call any $N$ -fold tensor product simple, or any state that contains at-most-two-body entanglement, and so on. These choices don’t affect the quantum complexity’s qualitative behavior, according to string theorists Adam Brown and Lenny Susskind.

How quickly can the quantum complexity of $| \psi(t) \rangle$ grow? Fast growth stems from many-body interactions, long-range interactions, and random coherent evolutions. (Random unitary circuits exemplify random coherent evolutions: each gate is chosen according to the Haar measure, which we can view roughly as uniformly random.) At most, quantum complexity can grow linearly in time. Random unitary circuits achieve this rate. Black holes may; they scramble information quickly. The greatest possible complexity of any $N$ -particle state scales exponentially in $N$ , according to a counting argument.

A highly complex state $| \psi(t) \rangle$ looks simple from one perspective and complicated from another. Human scientists can easily measure only local observables $O$ . Such observables’ expectation values $\langle \psi(t) | O | \psi(t) \rangle$ tend to look thermal in highly complex states, $\langle \psi(t) | O | \psi(t) \rangle \approx {\rm Tr} ( O \, \rho_{\rm th} )$ , as implied above. The thermal state has the greatest von Neumann entropy, $- {\rm Tr} ( \rho \log \rho)$ , of any quantum state $\rho$ that obeys the same linear constraints as $| \psi(t) \rangle$ (such as having the same energy expectation value). Probed through simple, local observables $O$ , highly complex states look highly entropic—highly random—similarly to a flipped coin.

Yet complex states differ from flipped coins significantly, as revealed by subtler analyses. An example underlies the quantum-supremacy experiment published by Google’s quantum-computing group in 2018. Experimentalists initialized 53 qubits (quantum two-level systems) in a tensor product. The state underwent many gates, which prepared a highly complex state. Then, the experimentalists measured the $z$ -component $\sigma_z$ of each qubit’s spin, randomly obtaining a -1 or a 1. One trial yielded a 53-bit string. The experimentalists repeated this process many times, using the same gates in each trial. From all the trials’ bit strings, the group inferred the probability $p(s)$ of obtaining a given string $s$ in the next trial. The distribution $\{ p(s) \}$ resembles the uniformly random distribution…but differs from it subtly, as revealed by a cross-entropy analysis. Classical computers can’t easily generate $\{ p(s) \}$ ; hence the Google group’s claiming to have achieved quantum supremacy/advantage. Quantum complexity differs from simple randomness, that difference is difficult to detect, and the difference can evidence quantum computers’ power.

A fridge that holds one of Google’s quantum computers.

Comparison and contrast

Assembly number and quantum complexity resemble each other as follows:

Each function quantifies the fewest basic operations needed to prepare something.
Only special systems (organisms) can generate high assembly numbers, according to Sara and Lee. Similarly, only special systems (such as quantum computers and perhaps black holes) can generate high complexity quickly, quantum physicists expect.
Assembly number may distinguish products of life from products of abiotic systems. Similarly, quantum complexity helps distinguish quantum computers’ computational power from classical computers’.
High-assembly-number objects are highly structured (think of my laptop). Similarly, high-complexity quantum states are highly structured in the sense of having much many-body entanglement.
Organisms generate high assembly numbers, using information. Similarly, using information, organisms have created quantum computers, which can generate quantum complexity quickly.

Assembly number and quantum complexity differ as follows:

Classical objects have assembly numbers, whereas quantum states have quantum complexities.
In the absence of life, random natural processes have low probabilities of producing high-assembly-number objects. That is, randomness appears to keep assembly numbers low. In contrast, randomness can help quantum complexity grow quickly.
Highly complex quantum states look very random, according to simple, local probes. High-assembly-number objects do not.
Only organisms generate high assembly numbers, according to Sara and Lee. In contrast, abiotic black holes may generate quantum complexity quickly.

Another feature shared by assembly-number studies and quantum computation merits its own paragraph: the importance of robustness. Suppose that multiple copies of a high-assembly-number (or moderate-assembly-number) object exist. Not only does my laptop exist, for example, but so do many other laptops. To Sara, such multiplicity signals the existence of some stable mechanism for creating that object. The multiplicity may provide extra evidence for life (including life that’s discovered manufacturing), as opposed to an unlikely sequence of random forces. Similarly, quantum computing—the preparation of highly complex states—requires stability. Decoherence threatens quantum states, necessitating quantum error correction. Quantum error correction differs from Sara’s stable production mechanism, but both evidence the importance of robustness to their respective fields.

A modest proposal

One can generalize assembly number to quantum states, using quantum complexity. Imagine finding a clump of atoms while searching for extraterrestrial life. The atoms need not have formed molecules, so the clump can have a low classical assembly number. However, the clump can be in a highly complex quantum state. We could detect the state’s complexity only (as far as I know) using many copies of the state, so imagine finding many clumps of atoms. Preparing highly complex quantum states requires special conditions, such as a quantum computer. The clump might therefore evidence organisms who’ve discovered quantum physics. Using quantum complexity, one might extend the assembly number to identify quantum states that may evidence life. However, quantum complexity, or a high rate of complexity generation, alone may not evidence life—for example, if achievable by black holes. Fortunately, a black hole seems unlikely to generate many identical copies of a highly complex quantum state. So we seem to have a low probability of mistakenly attributing a highly complex quantum state, sourced by a black hole, to organisms (atop our low probability of detecting any complex quantum state prepared by anyone other than us).

Would I expect a quantum assembly number to greatly improve humanity’s search for extraterrestrial life? I’m no astrobiology expert (NASA videos notwithstanding), but I’d expect probably not. Still, astrobiology requires chemistry, which requires quantum physics. Quantum complexity seems likely to find applications in the assembly-number sphere. Besides, doesn’t juxtaposing the search for extraterrestrial life and the understanding of life’s origins with quantum computing sound like fun? And a sense of fun distinguishes certain living beings from inanimate matter about as straightforwardly as assembly number does.

With thanks to Jim Al-Khalili, Paul Davies, the From Physics to Life collaboration, and UCLA for hosting me at the workshop that spurred this article.