Looking back at the history of computers, it’s hard to overestimate the rate at which computing power has scaled in the course of just a single human lifetime. But yet, existing classical computers have fundamental limits. If quantum computers are successfully built and eventually fully come online, they will be able to tackle certain classes of problems that elude classical computers. And they may be the computational tool needed to fully understand and simulate the brain.

## How fast and powerful are existing computers?

As of this writing, the fastest supercomputer in the world is Japan’s Fugaku supercomputer, developed jointly by Riken and Fujitsu. It can perform 442 peta-floating-point operations per second.

Let’s break that number down in order to arrive at an intuitive (as much as possible) grasp of what it means.

A floating-point number is a way to express, or write down, a real number - ‘real’ in a mathematical sense - with a fixed amount of precision. Real numbers are all the continuous numbers from the number line. 5, -23, 7/8, and numbers like pi (3.1415926 ...) that go on forever are all real numbers. The problem is a computer, which is digital, has a hard time internally representing continuous numbers. So one way around this is to specify a limited number of digits, and then specify how big or small the actual number is by some base power. For example, the number 234 can be written as 2.34 x 10^{2}, because 2.34 x 100 equals 234. Floating point numbers specify a fixed number of significant digits the computer must store in its memory. It fixes the accuracy of the number. This is important because if you do any mathematical operation (e.g. addition, subtraction, division or multiplication) with the fixed accuracy version of a real number, small errors in your results will be generated that propagate (and can grow) throughout other calculations. But as long as the errors remain small it’s okay.

A floating point operation then, is any arithmetic operation between two floating-point numbers (abbreviated as FLOP). Computer scientists and engineers use the number of FLOP per second - or FLOPS - as a benchmark to compare the speed and computing power of different computers.

One petaFLOP is equivalent to 1,000,000,000,000,000 - or one quadrillion - mathematical operations. A supercomputer with a computing speed of one petaFLOPS is therefore performing one quadrillion operations per second! The Fugaku supercomputer is 442 times faster than that.

## Yet, classical computers are fundamentally limited

For many types of important scientific and technological problems however, even the fastest supercomputer isn’t fast enough. In fact, they never will be. This is because for certain classes of problems, the number of possible combinations of solutions that need to be checked grow so fast, compared to the number of things that need to be ordered, that it becomes essentially impossible to compute and check them all.

Here’s a version of a classic example. Say you have a group of people with differing political views, and you want to seat them around a table in order to maximize constructive dialogue while minimizing potential conflict. The ‘rules’ you decide to use don’t matter here, just that some set of rules exist. For example, maybe you always want to seat a moderate between a conservative and a liberal in order to act as a bit of a buffer.

This is what scientists and engineers call an optimization problem. How many possible combinations of seating arrangements are there? Well, if you only have two people, there are only two possible arrangements. One individual on each side of a table, and then the reverse, where the two individuals change seats. But if you have five people, the number of possible combinations jumps to 120. Ten people? Well, now you’re looking at 3,628,800 different combinations. And that’s just for ten people, or more generally, any ten objects. If you had 100 objects, the number of combinations is so huge that it’s a number with 158 digits (roughly, 9 x 10^{157}). By comparison, there are ‘only’ about 10^{21} stars in the observable universe.

Imagine now if you were trying to do a biophysics simulation of a protein in order to develop a new drug that had millions or billions of individual molecules interacting with each other. The number of possible combinations that would need to be computed and checked far exceed the capability of any computer that exists today. Because of how they’re designed, even the fastest supercomputer is forced to check each combination sequentially - one after another. No matter how fast a classical computer is or can be, given the literally greater than astronomical sizes of the number of combinations, many of these problems would take a practical infinity to solve. It just becomes impossible.

Related, the other problem classical computers face is it’s impossible to build one with sufficient memory to store each of the combinations, even if all the combinations could be computed.

## Quantum computers should be able overcome these limitations

The details of how a quantum computer and quantum computing algorithms work is well beyond the scope or intent of this article, but we can briefly introduce one of the key ideas in order to understand how they can overcome the combinatorial limitations of classical computers.

Classical computers represent information - all information - as numbers. And all numbers can be represented as absolute binary combinations of ‘1’s’ and ‘0’s’. The ‘1’ and ‘0’ each represent a bit of information, the fundamental unit of classical information. Or put another way, information is represented by combinations of two possible states. For example, the number ‘24’ in binary notation is ‘11000’. The number ‘13’ is ‘1101’. You can also do all arithmetic in binary as well. This is convenient, because physically, at the very heart of classical computers is the transistor, which is just an on-off electrical switch. When it’s on it encodes a ‘1’, and when it’s off it encodes a ‘0’. Computers do all their math by combining billions of tiny transistors that very quickly switch back and forth as needed. Yet, as fast as this can occur, it still takes finite amounts of time, and all calculations need to be done in an appropriate ordered sequence. If the number of necessary calculations become big enough, as is the case with the combinatorial problems discussed above, you run into an unfeasible computational wall.

Quantum computers are fundamentally different. They overcome the classical limitations by being able to represent information internally not just as a function of two discrete states, but as a continuous probabilistic ‘mixing’ of states. This allows quantum bits, or qubits, to have many more possible states they can represent at once, and so many more possible combinations of arrangements of objects at once. Put another way, the state space and computational space that a quantum computer has access too is much larger than that of a classical computer. And because of the wave nature of quantum mechanics and superposition (concepts we will not explore here), the internal mixing and probabilistic representation of states and information eventually ‘converge’ to one dominant solution that the computer outputs. You can’t actually observe that internal mixing, but you can observe the final computed output. In essence, as the number of qubits in the quantum computer increase, you can exponentially do more calculations in parallel.

The key concept here is not that quantum computers will necessarily be able to solve new and exotic classes of problems that classical computers can’t - although computer scientists have discovered a theoretical class of problem that only quantum computers can solve - but rather that they will be able to solve classes of problems that are - and always will be - beyond the reach of classical computers.

And this isn’t to say that quantum computers will replace classical computers. That is not likely to happen anytime in the foreseeable future. For most classes of computational problems classical computers will still work just fine and probably continue being the tool of choice. But for certain classes of problems, quantum computers will far exceed anything possible today.

## So why does this all matter for doing simulations of the brain?

Well, it depends on the scale at which the dynamics of the brain is being simulated. For sure, there has been much work within the field of computational neuroscience over many decades successfully carrying out computer simulations of the brain and brain activity. But it’s important to understand the scale at which any given simulation is done.

The brain is exceedingly structurally and functionally hierarchical - from genes, to molecules, cells, network of cells and networks of brain regions. Any simulation of the brain needs to begin with an appropriate mathematical model, a set of equations that capture the chosen scale being modeled that then specify a set of ‘rules’ to simulate on a computer. It’s like a map of a city. The mapmaker needs to make a decision about the scale of the map - how much detail to include and how much to ignore. Why? Because the structural and computational complexity of the brain is so vast and huge that it’s impossible given existing classical computers to carry out simulations that cut across the many scales with any significant amount of detail.

Even though a wide range of mathematical models about the molecular and cell biology and physiology exist across this huge structural and computational landscape, it is impossible to simulate with any accuracy because of the sheer size of the combinatorial space this landscape presents. It is the same class of problem as that of optimizing people with different political views around a table. But on a much larger scale.

## How computationally complex is the human brain, and how large is its combinatorial space?

Once again, it in part depends on how you choose to look at it. There is an exquisite amount of detail and structure to the brain across many scales of organization. Here’s a more in depth article on this topic.

But if you just consider the number of cells that make up the brain and the number of connections between them as a proxy for the computational complexity - the combinatorial space - of the brain, then it is staggeringly large. In fact, it defies any intuitive grasp.

The brain is a massive network of densely interconnected cells consisting of about 171 trillion brain cells - 86 billion neurons, the main class of brain cell involved in information processing, and another 85 billion non-neuronal cells. There are approximately 10 quadrillion connections between neurons — that is a ‘1’ followed by 16 zeros. And of the 85 billion other non-neuronal cells in the brain, one major type of cell called astrocyte glial cells have the ability to both listen in and modulate neuronal signaling and information processing. Astrocytes form a massive network onto themselves, while also cross-talking with the network of neurons. So the brain actually has two distinct networks of cells. Each carrying out different physiological and communication functions, but at the same time overlapping and interacting with each other.

On top of all that structure, there are billions upon billions upon billions of discrete electrical impulses, called action potentials, that act as messages between connected neurons. Astrocytes, unlike neurons, don’t use electrical signals. They rely on a different form of biochemical signaling to communicate with each other and with neurons. So there is an entire other molecularly-based information signaling mechanism at play in the brain.

Somehow, in ways neuroscientists still do not fully understand, the interactions of all these electrical and chemical signals carry out all the computations that produce everything the brain is capable of.

Now pause for a moment, and think about the uncountable number of dynamic and ever changing combinations that the state of the brain can take on given this incredible complexity. Yet, it is this combinatorial space, the computations produced by trillions of signals and billions of cells in a hierarchy of networks, that result in everything your brain is capable of doing, learning, experiencing, and perceiving.

So any computer simulation of the brain is ultimately going to be very limited. At least on a classical computer.

How big and complete are the biggest simulations of the brain done to date? And how much impact have they had on scientists’ understanding of the brain? The answer critically depends on what’s being simulated. In other words, at what scale - or scales - and with how much detail given the myriad of combinatorial processes. There certainly continue to be impressive attempts from various research groups around the world, but the amount of cells and brain being simulated, the level of detail, and the amount of time being simulated remains rather limited. This is why headlines and claims that tout ground-breaking large scale simulations of the brain can be misleading, sometimes resulting in controversy and backlash.

The challenges of doing large multi-scale simulations of the brain are significant. So in the end, the answer to ‘how big and complete are the biggest simulations of the brain done to date’ and ‘how much impact have they had on scientists’ understanding of the brain’ - is not much.

## Why might large-scale simulations of the brain be the right kind of challenge for quantum computers?

First, by their very nature, given a sufficient number of qubits quantum computers will excel at solving and optimizing very large combinatorial problems. It’s an inherent consequence of the physics of quantum mechanics and the design of the computers.

Second, given the sheer size and computational complexity of the human brain, any attempt at a large multi-scale simulation with sufficient detail will have to contend with the combinatorial space of the problem.

Third, how a potential quantum computer neural simulation is set up might be able to take advantage of the physics the brain is subject to. Despite its computational power, the brain is still a physical object, and so physical constraints could be used to design and guide simulation rules (quantum computing algorithms) that are *inherently* combinatorial and parallelizable, thereby taking advantage of what quantum computers do best.

For example, local rules, such as the computational rules of individual neurons, can be used to calculate aspects of the emergent dynamics of networks of neurons in a decentralized way. Each neuron is doing their own thing and contributing to the larger whole, in this case the functions of the whole brain itself, all acting at the same time, and without realizing what they’re contributing too.

In the end, the goal will be to understand the emergent functions of the brain that give rise to cognitive properties. For example, large scale quantum computer simulations might discover latent (hidden) properties and states that are only observable at the whole brain scale, but not computable without a sufficient level of detail and simulation from the scales below it.

## Beyond neural simulations

If these simulations and research are successful, one can only speculate about what as of yet unknown brain algorithms remain to be discovered and understood. It’s possible that such future discoveries will have a significant impact on related topics such as artificial quantum neural networks, or on specially designed hardware that some day may challenge the boundaries of existing computational systems. For example, just published yesterday, an international team of scientists and engineers announced a computational ‘hardware’ device composed of a molecular-chemical network capable of energy-efficient rapid reconfigurable states, somewhat similar to the reconfigurable nature of biological neurons.

One final comment regarding quantum computers and the brain: This discussion has focused on the potential use of future quantum computers to carry out simulations of the brain that are not currently possible. While some authors and researchers have proposed that neurons *themselves* might be tiny quantum computers, that is completely different and unrelated to the material here.

## But for now, back to ground ...

It may be that quantum computers will usher in a new era for neuroscience and the understanding of the brain. It may even be the only real way forward. But as of now, actually building workable quantum computers with sufficient stable qubits that outperform classical computers at even modest tasks remains a work in progress. While a handful of commercial efforts exist and have claimed various degrees of success, many difficult hardware and technological challenges remain. Some experts argue that quantum computers may in the end never be built due to technical reasons. But there is much research across the world both in academic labs and in industry attempting to overcome these engineering challenges. Neuroscientists will just have to be patient a bit longer.

I am a Professor of Bioengineering and Neurosciences at the University of California San Diego, where I hold a Jacobs Family Scholar in Engineering Endowed Chair. I have

…I am a Professor of Bioengineering and Neurosciences at the University of California San Diego, where I hold a Jacobs Family Scholar in Engineering Endowed Chair. I have been on the faculty at UCSD for the past 18 years. I am also the Founding Director of the Center for Engineered Natural Intelligence, and Associate Director of the Kavli Institute for Brain and Mind. My research aims to understand what the algorithms are that the brain uses, and how technologies can interface with the brain in order to treat neurological disorders. I am also exploring new forms of machine learning derived from neuroscience. My doctorate and post-doctorate fellowship work were at the University of Illinois at Chicago and Northwestern University, respectively, in neuroscience and neural engineering. Prior to that I was at the University of Toronto where I received double Bachelor degrees in human physiology and biophysics, and a Masters degree in neurophysiology. My contributions to Forbes focus on neuroscience and brain research, and their intersections with technology, engineering, and mathematics.