Week 8 - my notes
Table of Contents
Week 8 (qubit by qubit)
Recap
Review:
- The Quantum Circuit Model is broken down in three parts
- The initial state
- The computation
- Measurements
- The final part of every circuit is the measurement.
- We can represent a quantum state as a vector with two elements
- We can represent a quantum gate as a matrix
- We learned how to mathematically apply a gate (Matrix times Vector = New Vector)
Measurement
What we know about quantum measurements
- By obersivign (measuring) a quantum state, we can change it
- When we measure something, we are basically asking if it is in a \(|0\rangle\) or \(|1\rangle\) state
- Those are the only two choices. After a measurement, its state needs to be in one state only.
- The outcome of a quantum measurement is sometimes random
- Quantum measurements operate in terms of probabilities.
- Our goal is to predict the result of our measurements
Today we will learn how we can predict quantum measurements
Statevector math
Born Rule
Here we got an equal contribuition of \(|0\rangle\) and \(|1\rangle\):
\[\frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix}\] For example, the contribuition of \(|0\rangle\) and its probability:
- Contribuition: \[\frac{1}{\sqrt{2}}\]
- Probability: [\frac{1}{2}]
- To get the probability above, we can simply square a state’s contribuition:
\[(\frac{1}{\sqrt{2}})^2=\frac{1}{2}\]
This rule works across the board!
- The probability of measuring a qubit in the \(|0\rangle\) state is equal to the square of the contribuition of \(|0\rangle\) to the qubit’s state.
- Similarly, the probability of measuring a qubit in the \(|1\rangle\) state is equal to the square of the contribuition of \(|1\rangle\) to the qubit’s state.
- The formal term for the contribuitions of \(|0\rangle\) and \(|1\rangle\) is the amplitude
- This is one of the reasons we learn vectors. The contain the probabilities hidden…
Some questions…
- What is the probability of measuring \(|0\rangle\) for this state? What about the probability of \(|1\rangle\)? \[\frac{1}{\sqrt{5}}\begin{bmatrix}1\\2\end{bmatrix}\]
- Contribuition of \(|0\rangle\): \[\frac{1}{\sqrt{5}}\]
- Probability of \(|0\rangle\): \[(\frac{1}{\sqrt{5}}^2)=\frac{1}{5}\]
- Contribuition of \(|1\rangle\): \[\frac{2}{\sqrt{5}}\]
- Probability of \(|1\rangle\): \[(\frac{2}{\sqrt{5}}^2)=\frac{4}{5}\]
That is why we normalize
- When we represent vectors in quantum state, the must be normalized, which means they must have a lenght of 1.
- Why 1?
- The numbers in the vectors show us probabilities of measuring a certain value. These probabilities must add up to 1.
- If we use 1 as the normalized state, it means that 100% of the time we can get that state.
Why does the Born rule work?
- No idea.
- In quantum mechanics we have rules for what happens, but sometimes we don’t know why they happen.
Measurement Bases
- Are 0 and 1 the only possible outcomes?
- Short answer: No! We can get different outcomes if we measure in different ways.
Rules for measurement
- The output can be one of two options. You only have these two options for a potential outcome.
- If your initial state is a superposition the measurement will default to one of the two options randomly.
- However, your set of two choices will be decided by which measurement basis you use.
Revisiting the Bloch Sphere
- The Bloch Sphere sits along 3 axes: X, Y and Z.
- The \(|0\rangle\) and \(|1\rangle\) states lie along the Z axis.
- The \(|+\rangle\) and \(|-\rangle\) states lie along the X axis.
- We won’t discuss the Y axis in this course (because it requires complex numbers…)
Measuring in the Z basis
- If our measurement only have either \(|0\rangle\) or \(|1\rangle\) results, we are measuring along the Z axis. This is also known as measuring along the Z basis.
- But we have other basis, remember!?
- When we make a measurement, we need to specify which axis we are measuring along, or which measurement basis we are using.
- basis = axis = direction
- The basis we choose will affect our measurement outcomes
- In circuits, we represent the measurement symbol with the Z inside it, but if you don’t, it also means it is in the Z basis.
Measuring in the X basis
- When measuring in this basis, we can only get either \(|+\rangle\) or \(|-\rangle\)
- In circuits, we represent the measurement symbol with the X inside it
- When you measure the \(|+\rangle\) in the X basis, you get a \(|+\rangle\) state
- This same state, measured along the Z basis, you randomly get \(|0\rangle\) or \(|1\rangle\)
- A \(|0\rangle\) state measured in the X basis is basically this: randomly \(|+\rangle\) or \(|-\rangle\) states, \(|0\rangle\) and \(|1\rangle\) are basically superpositions in the X basis.
- Every state is a superposition in some basis!
The Stern Gerlach Experiment
- Step 1: Measurement changes the state of a qubit
- Step 2: The results of quantum measurement are often random
- Step 3: How you measure determines the outcomes you get