If the universe is as we think it might be,
a sort of closed system surrounded by a horizon,
and that that horizon behaves roughly like the horizon of a black hole.
Any self-contained system, if it satisfies any kind of version of the standard
laws of physics, entropy will increase
until it comes to thermal equilibrium.
Once it comes to thermal equilibrium, um...
not much happens.
Thermal equilibrium is extremely boring.
It really is a dead world.
However, the classical story
of the increase of entropy misses part of the evolution.
Quantum entanglement is still evolving,
things keep happening.
We propose the second law of quantum complexity in direct analogy
to the second law of thermodynamics.
This is a really provocative suggestion.
It struck me as something that did not compute.
They came to the conclusion...
old Lenny used to be a good physicist,
he got into this complexity story, but he didn't know what complexity was.
The standard laws of physics are given
their most rigorous stress tests inside the most exotic of cosmic locales:
black holes.
On top of the much buzzed about work on singularities and event horizons
in the 20th century.
A new enigma has appeared having to do with how big black holes
are on the inside.
About a decade ago, Stanford physicist, Leonard Susskind, found that the interior
spacetime of a black hole is so warped that it can seemingly grow forever.
This appears to conflict with the same laws of physics
that bring a glass of ice water to thermal equilibrium.
Those laws say that entropy should increase toward a maximum value,
at which point things in the universe,
including black holes, should stop changing in any meaningful way.
A fate referred to as a "heat death."
At this point, why would black holes continue to change?
Susskind and his collaborators set out to solve this paradox,
and in doing so, they stumbled upon an unlikely explanation
for the near infinite expansion of a black hole's interior.
My friend, who was my friend and a good friend, Feynman,
he would close his eyes and visualize a system.
And then when he had a visual picture of it,
he would convert that to mathematics.
I don't have the mind of a mathematician,
I have the mind of an auto mechanic.
What works is what's practical,
but sometimes you need some mathematics.
I learned about black holes, and there was something
about black holes that had nothing to do with complexity at that time.
But there was a puzzling feature that I didn't understand.
We had these pictures,
they were called Penrose Diagrams.
It's a kind of map of the whole universe, including a black hole.
The black hole is over here.
Time goes upward.
The interior of the black hole had this strange behavior.
The interior of the black hole came to a smallest size and then expanded.
It bounced and it started to grow.
And I didn't know what that bouncing thing was.
And I gave up, I didn't know what to make out of it.
I would ask my friends, what's growing here?
Black hole experts – and they would....
I don't know...
it's just the volume of the black hole.
Okay.
Physicists started to look towards entropy,
a concept that has a long history in science and engineering.
Entropy was invented by people during the Industrial Revolution
who wished to understand the maximum possible efficiency
of steam engines and other similar machines.
And what they were interested in was the entropy
of the working gas in their engine.
So this was both a important theoretical understanding and also big business.
That is the story, though, of entropy - a statistical claim
that this disorder of the system always increases.
But for Susskind, something didn't fit.
I knew that it wasn't entropy because the entropy
of the black hole comes to thermal equilibrium very quickly.
On the other hand, you can look at the Penrose Diagram
and you could see that expansion lasted for a very, very long time.
So if you start with black coffee and milk
and mix them together, it's not going to take very long before you, in fact,
have at least, as far as the intermixing is concerned, thermodynamic equilibrium.
And in fact, it'll look at least as far as thermodynamic is concerned,
like the arrow of time has stopped,
the thermodynamic arrow of time has ceased.
The system is now in a heat death situation.
There is no longer any ability to generate more entropy.
However, that is
not the end of the story because, even though the system has reached
thermodynamic equilibrium, it is not reached complexity equilibrium.
Complex systems
are characterized by individual parts that interact in a multiplicity of ways,
leading to phenomena like non-linearity, randomness, and emergence.
Sensing that complexity might be at the root
of the explanation for why black holes continue to grow past thermal equilibrium,
Susskind and his collaborators turned
to the mathematics of theoretical computer science.
Feynman taught us that the number of quantum
states of a system is enormously large.
If you have a certain number of qubits,
the number of orthogonal quantum states of that system is exponential in the number of qubits.
What that means, is that for example, a quantum computer or just an ordinary
system will take an enormously long time
to explore the entire space of states.
It will start out at some simple state
and start wandering around in the space of states
much, much longer time than it takes to establish thermal equilibrium.
All of a sudden it would be bang -
the complexity of a quantum state just increases and increases and increases,
long, long past the time
when the system has apparently come to thermal equilibrium
And that this size of the interior of the black hole was a direct measure
of the complexity of the quantum state of the black hole.
When that hit me, I suddenly got really,
really interested in the concept of computational complexity.
So a classical system, we basically describe it like,
Oh, I have a bunch of bits and they can be 0101, right?
I can just describe them individually what state they are.
But for a quantum system, that's not enough.
We, we cannot describe the system by starting from the first qubit
and then describe the second cubit because they can be globally entangled
and locally, there might not be information at all, but globally
they might be in some highly non-trivial quantum state.
Entanglement is the key to why complexity can grow
so large for a quantum system.
Working with his Stanford colleague Adam Brown, Susskind developed the idea
that even after classical entropy has reached its maximum,
the interior of a black hole can continue to evolve,
because of the ever increasing complexity of its quantum state.
Meaning that there's life after heat, death for black holes after all.
And to make their case to the scientific community,
they borrowed a longstanding concept from computer science
known as Circuit Complexity.
Originally formulated to measure the complexity of a computer program's output.
Quantum circuit complexity had become the unlikely mathematical language
for a new theory attempting to explain the evolution of black holes, a theory
that wasn't without its critics.
I first heard that there was a connection
being drawn between circuit complexity and black holes
way back when I was in grad school,
when Lenny Susskind and his collaborators made the suggestion
that circuit complexity might be playing a role in understanding quantum gravity.
And when you first hear
this is a computer scientist, this is a really provocative suggestion.
Lenny had formulated this, and I happened to be at that talk.
And what I found hard to believe about it
is that he was equating something that is physical, you know, volume,
with something that, as a complexity theorist, I would say, you know, this is quite
un-physical: Quantum Circuit Complexity.
That's something that's manifestly hard to compute.
I'm not sure we have particular hypothesis, but something felt funny.
To put Susskind and Brown's Quantum Circuit Complexity Theory of Black Holes
to the test.
Adam Bouland and his collaborators brought in tools of modern cryptography
to serve as a testing ground.
The state that Susskind and his collaborators were considering
actually look a lot, like a quantum analog of a block cipher.
A block cipher is a type of algorithm that underpins modern
cryptography, including the technology behind cryptocurrencies.
By scrambling information behind an encryption key-phrase
that must be entered exactly right, to gain access to the information.
For Adam Bouland, the similarity to Susskind's
theory was in the chaotic mixing within a quantum system.
His work drew a comparison between scrambling the information in a cipher
and the rearrangement of qubits,
as a quantum system evolves...
backing up Susskind's approach.
Funilly, after this long period of exploration, our work ended up
in some roundabout way, justifying Susskind and collaborators' conjectures.
That was a really unexpected result of our investigation.
We realized that rather than reinforcing our skepticism about his proposal,
what our results suggested was that, in fact,
that was the only sensible way to resolve the paradox.
So that was a that was a really interesting thing,
you know, that that he had this deep intuition.
It's pretty remarkable!
Faced with a possible explanation for life after heat death,
Susskind and his collaborators decided to propose a new fundamental law of the universe.
Eventually, we just said, let's put forward the idea
that computer scientists had never put it forward...
they hadn't said it.
Let's put forward the idea that there is a second
law of complexity.
That complexity behaves very much like entropy,
and all the ways on the average increases
until it maxes out.
We proposed the second law of quantum complexity
in direct analogy to the second law of thermodynamics.
I would say that it is not a law in the sense that it's been proved.
It is a conjecture that there should exist such a law.
But the analogy with the classic second law of thermodynamics
is sufficiently strong that we felt emboldened to propose such a law.
At the moment, I think it's something which really applies to black holes.
The extent to which it applies to the universe as a whole is not entirely clear.
So this idea of using quantum circuit complexity,
this is treating quantum system in a quantum mechanical way.
Rather than starting from a classical starting point
into some semi classical approximation and
and then bootstrap from there,
that can take us somewhere, but it's not all the way.
So once we accept that the world is quantum and we fully embrace that,
then we can actually understand things much, much better
because we're now standing inside the quantum world
and we're using fundamentally quantum language
to describe it as we should have for a long time.
And that will be a fundamental change
of how we think about these kind of systems.
What the implications of complexity growth are
for the evolution of the universe.
At the moment, this is unexplored territory.
Eventually, if you wait an exponentially long time,
even the quantum complexity achieves its maximum value.
Even the quantum complexity, which is its own analog of heat death....
and then maybe the story begins again.
You know, as people say, if you wait
long enough, everything will happen, including what you started with.