# Energy Landscapes

Imagine you are viewing your surroundings.  You look around and notice **MOUNTAINS** and *valleys*.  In front of you, there's a flat plain with a *BUMP* in the distance.  As you move closer, the **BUMP** takes on a shape of its own with small, imperceptible *HILLS* and valleys in its own local vicinity.

Now, instead of viewing your surroundings nearby, take a step back, and see it from a broader frame of reference.  These *hidden hills* and valleys in local areas are a **GLOBAL PATCHWORK** of the same phenomena.

And if you imagine this **landscape** as a potential *ENERGY* function, with the elevation representing the **ENERGY** at a particular point on an xy-plane, then the valleys and **HILLS**, i.e. the z-axis, are the local minima and local *MAXIMA* of the function.

# The Adiabatic Theorem

Broadly speaking, the [*adiabatic theorem*](https://en.wikipedia.org/wiki/Adiabatic_theorem) states that if you have an **ENERGY** function and if you *slowly* and continuously *perturb* it by **ADDING** or removing ENERGY, the *nested* HILLS and **sloping** valleys of the *ENERGY LANDSCAPE* will also change in in the **SAME**, *SLOW*, continuous manner.

This theorem is the **CRUX** of quantum computation.

This **physical** process is the so-called *'black box'* algorithm behind **ALL** of quantum adiabatic computation.  Most of the time, the (quantum) physicists I encountered would wave their arms erratically (similar to mathematicians when they're teaching **ALGEBRAIC GEOMETRY** to grad students) and say that this adiabatic process *SEEMS TO WORK.* 

If questioned further about this process, most would readily admit that they didn't know *HOW* or **WHY** it worked.

As a mathematician, my **curiosity** was not satiated.   To simply explain away and state this process *worked* without any explanation seem suspicious to me.  It seemed to me some **DEEPER** mathematics could explain this whole **adiabatic business**.

# Slowly Perturbing the Hamiltonian

A *potential energy function*, also known as a **Hamiltonian** (named after [Irish Mathematician William Rowan Hamilton](https://en.wikipedia.org/wiki/William_Rowan_Hamilton)), can be described by its **energy** states by considering the corresponding square matrix that *represents* the Hamiltonian, where the so-called [eigenspace](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) of the matrix is intimately related to the energy states of the Hamiltonian.

For **quantum computation**, the interesting quantity that **is important** is the *lowest energy* or **ground** state of the Hamiltonian.  When considering the **potential ENERGY** in its *matrix form*, the *smallest* eigenvalue plays the role of the ground state of the Hamiltonian. 

A *STANDARD* tool of **MATHEMATICS** is to consider the **same** object in as many *different* (and **equivalent**) representations as possible.  When an object **changes** its role by putting on a different *mask*, the object itself does not *change*, and **ALL** of its properties, regardless of the mask that it wears, can lead us to develop a *DEEPER* understanding about the object itself or the **PHYSICAL** process that the object models.  

Similarly, one can **change** one object into another via a **transformation**.  This is what happens with *Adiabatic Quantum Computation*.  A **Hamiltonian** whose *ground state* gives us the **solution to a problem**, say, for instance, [Boolean Satisfiability](https://en.wikipedia.org/wiki/Boolean_satisfiability_problem) is constructed.  We will call this Hamiltonian **H_F**.  

Another Hamiltonian that is *easily constructible*, called **H_0**, is created as a *starting* *POTENTIAL* **energy** **FUNCTION** and *slowly* **perturbed** to the final Hamiltonian under the *smooth* transformation:  **H(t) = (1-t)H_0 + tH_F**.

H(t) represents the *Hamiltonian* **at any time** between **0 and 1**.  Note that **H(0) = H_0 and H1(1) = H_F**.

Now, just because this final Hamiltonian was constructed by us does not mean **WE ALREADY KNOW** its ground state -- otherwise we would already have **THE SOLUTION** to our problem!

In order to *arrive at our destination*, the **adiabatic** process is applied and the *ground state* **H_0** is followed all the way to find (**statistically**) the **lowest energy** state of **H_F**.  

Remember when **WE** said that *if* a *perturbation* happens *slowly* enough, the *energy* minima and *maxima* won't change quickly?  That's what's happening with *quantum computation*!  The lowest *ENERGY* state of the initial Hamiltonian moves *slowly* as the underlying Hamiltonian is perturbed and **CAN BE TRACKED** all the way to its final ground state. 

# But ... 

What happens *IF YOU* have the ground state and the next highest energy state *SWAP POSITIONS* during the adiabatic process?  Then, you're no longer **following** the ground state all the way to the final Hamiltonian, **and you miss the solution** to the problem that you were originally ***trying to solve***.  

**FORTUNATELY**, this cannot happen when the *adiabatic transformation* is **SLIGHTLY** modified.

In fact, this *SLIGHT* modification is what is used in **PRACTICE**, but *WITHOUT* any **understanding** or **JUSTIFICATION** other than *IT SEEMS TO WORK!*

Indeed, there is some **VERY EARLY** work by Wigner and von Neumann entitled [``On the behaviour of eigenvalues in adiabatic processes``](https://books.google.com/books?id=qsidHRJmUoIC&lpg=PA25&dq=adiabatic%20process%20hermitian%20systems&pg=PA25#v=onepage&q=adiabatic%20process%20hermitian%20systems&f=false)  that says precisely that, i.e. *WE DON'T KNOW WHAT'S HAPPENING*, but **IF WE DO THIS**, things seem to work (**yay***!***?***!?***?!***?***?!**).

What **slight** modification is done you might ask?  Nothing more than adding a **r***A***nD***O***m** Hamiltonian, H_R, to the equation.

Now, H(t) = (1-t)H_0 + tH_F + t(1-t)H_R

What does **adding** this **r**a*N***d**O**M** Hamiltonian do to the *energy* states?   This additional Hamiltonian *DISRUPTS* any degeneracies that could occur when **multiple** eigenvalues are at the same ground state.  This condition occurs when certain algebraic **conditions are satisfied** and is called **degenerate** (although, a better term would be *de-generic*, or *non-generic* -- more on that later).

If a *degenerate space* occurs during the adiabatic process, that is to say, if there is a collapse of **TWO (OR EVEN MULTIPLE**) eigenspaces onto one another **AT ANY TIME**, it is unknown which energy state is the ground state when this degenerate space is perturbed back to its **NORMAL** state.


Throughout the time that I have spent poring through the **AQC** literature and talking with the (*relatively* **few**, I must add!) scientists in this field, no one seems to *UNDERSTAND* the role of **R***a***ND***oM***n***E***Ss**!

# Random Matrix Theory

The role of *R***a**n**D**o*M***N**e**S***S* is *very important to* the adiabatic process as some black-box machinery to be used for *quantum computing*.

The reason that **including** ***a*** **r***A***n***d**O***m** Hamiltonian necessarily causes no degeneracy to occur is because a **generic** or  ***R*** *A* **n** *D* **O** ***m*** matrix has **simple** eigenvalues. In other words, a random matrix has no  eigenvalues of multiplicity bigger than one, i.e. the characteristic polynomial of the matrix has ***NO REPEATED ROOTS***. 

The occurrence where an eigenvalue appears **MULTIPLE TIMES IS** not  **COMMON**.

What Does It All Mean?
====================

The black-box of quantum computation requires **r***A**n*D***O***mn**E*s***S** at its heart to **work** and because **rA**n*D***o**M Hamiltonians are added to the energy state after the **adiabatic** process begins and **before it ends**, one can **apply** all the results from **R**a*N*d***OM*** matrix theory. 

**r**a*N***D**o*M* matrix theory tells us that generic (or random) matrices have simple eigenvalues and, as such, no degeneracies, i.e. no collapsing of multiple eigenspaces.  If no eigenspaces can be collapsed into the same one, then its obvious that the ground state cannot *degenerate* into the **next excited state**.

# The Minimal Distance Between The Lowest Energy State and the Next Excited State

A ***VERY IMPORTANT*** question asked by the quantum physics community concerns the *minimal* distance between the lowest energy state and the **next excited state**.   If the ***GAP***  of the two energy states is too *small*, then one could **accidentally follow** the *next excited* state **to the end** Hamiltonian. 

It turns out that if a **R***a*N**d***O***m** Hamiltonian is used to *perturb* the **energy** states during the *adiabatic process*, then **MORE** knowledge **CAN** still **APPLY**.  In fact, **R**a*ND***oM** matrices have **on average** a well-known gap that is the inverse of the size of the matrix.

For **quantum computing**, this means that as the number of *q-bits* is increased, the corresponding size of the matrix representing the Hamiltonian increases **exponentially**.  

From this, we can now conclude that **ON AVERAGE** the distance between the **ground** state and the next ***EXCITED*** state is *exponentially smaller* as the number of q-bits increases.

# The Future of Quantum Computing

**Quantum computing** is currently a *HOT* topic amongst researchers and lots of funding from ***ALL OVER*** the world is being pored into it, but the crucial role of **R***a*nD**oM***N***e***sS* is being overmissed.

Indeed, new algorithms need to be developed by quantum physicists to use **R***a***ND***o*M***Ne***Ss appropriately so that one can **apply** quantum computation *efficiently.*