Already understand the basics and just want to get stuck in? Can't quite remember what the CSWAP matrix looks like? This post is for you.
Here you will find a collection of all the main states/gates/matrices/useful maths bits/etc. covered so far in this blog series. This cheat-sheet will be updated as important new things are introduced in posts 😊
For all posts past and future, please refer to the Hitchhiker's Guide to the Quantum Computing and Q# Blog.
- The Bloch Sphere
- Key Quantum States
- Gates, Matrices and Operations
- Useful Relationships and Equations
The Bloch Sphere
Any unitary transformation we do on |𝜓〉 can be visualised as simply moving the point (marked |𝜓〉) around the Bloch Sphere*. Sadly, this visualisation can only be used for single qubit states, as there is no known (simple) generalisation that applies to multi-qubit systems. You may also see it referred to in places as the Unit Sphere.
*All pure states can be found on the surface of the sphere, whereas mixed states are located within the sphere. Please refer to my other post Quantum Computing Primer: Pure vs. Mixed States for further explanation.
Key Quantum States
Gates, Matrices and Operations
Below is a summary of the key gates as introduced in my previous post about Gates and Circuits. Operations have been included for all single- and two-qubit gates (three plus becomes too big for display). For controlled gates, the identity matrix (𝕀) has been higlighted red and the original gate matrix blue, as seen previously.
|Name(s)||Matrix||Circuit Symbol(s)||Q# Representation||Key Operations|
|Pauli X, X, NOT, bit flip,||X(qubit : Qubit)|
|Pauli Y, Y,||Y(qubit : Qubit)|
|Pauli Z, Z, phase flip,||Z(qubit : Qubit)|
|Hadamard, H||H(qubit : Qubit)|
|Phase,, S||S(qubit : Qubit)|
|, T||T(qubit : Qubit)|
|SWAP||SWAP(qubit1 : Qubit, qubit2 : Qubit)|
|CNOT||CNOT(control : Qubit, target : Qubit)
|CCNOT, Toffoli||CCNOT(control1 : Qubit, control2 : Qubit, target : Qubit)
Useful Relationships and Equations
The Pauli matrices are their own inverse:
The density operator can be defined as follows:
- is the probability of the system being in state at the start.
- The element represents the outer product of the vector with itself, which produces a matrix (this is also known as a projection operator).
- n is the total number of possible states the system could be in (in our example, 3).
- as you would expect (the sum of the probabilities for all possible states is equal to 1)
Learn more at the Microsoft Quantum website: https://www.microsoft.com/en-us/quantum/
Download the Quantum Development Kit: https://www.microsoft.com/en-us/quantum/development-kit
Stay up to date with the Microsoft Quantum newsletter: https://info.microsoft.com/Quantum-Computing-Newsletter-Signup.html
Read about the latest developments on the Microsoft Quantum blog: https://cloudblogs.microsoft.com/quantum/