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To begin discussing the theory behind Quantum Computing, it is important to have a well understood foundation of Quantum Mechanics. I will be presuming that the reader has some level of understanding in Calculus and Linear Algebra in order for us to have some meaningful progress in understanding this field.

Firstly, we must begin our discussion with asking “what even is a classical state”? Well, a classical state is a configuration that the specific system of interest can exist in. Let’s imagine the system of interest as the outcome of a coin toss. We know intuitively that the two states of this system can be said to be “heads (H)” or “tails (T)”. Another state of the coin could be its temperature - if it was found outside, it would be in a colder state than if it was found in our wallet.

A Quantum state is essentially the same thing, but it adds another layer of complexity - the ability for a given state of the system to be described as a linear superposition of the pure states of the system. For those who are unfamiliar, a linear superposition is the weighted sum of two or more terms. For example, take a vector V⃗ V→ in R2R2 space,

V⃗ =αx^+βy^V→=αx^+βy^

A pure state is simply a state that cannot be expressed as a superposition of other states. Typically, these Quantum states are expressed with Dirac ‘brakets’ - |State⟩|State⟩. Now let’s revisit our coin toss and pretend we have a Quantum coin in our hands. The most generic state of this system can be described as the following,

|Quantum Coin⟩=α|T⟩+β|H⟩|Quantum Coin⟩=α|T⟩+β|H⟩

Similar to our vector analogy, each state can be thought of as a vector in a more general Hilbert Space with each pure state as the unit vectors. The weights αα or ββ are coefficients of the vectors which change its orientation in this Hilbert Space. We can naturally extend this to any system, let’s imagine that |ψi⟩|ψi⟩ (where ii is just a natural number) are the pure states of the system at hand. We can express a Quantum state of the system as a sum over all of the pure states with their own weights cici,


In the next blog post we will be discussing the mathematics involved with the two-state system we have introduced with the Quantum coin.

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Now that we’ve established the definition of the Quantum state as well as the Axioms of Quantum Mechanics, we can begin meaningful discussion on the two-state system. Though it may not see like much o

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