"The dictionary meaning of 'Coherent' is 'Clear and easy to understand' ."
What is Coherent Error?
A simple definition for Coherent Errors is 'The errors caused due to the Unitary Operations or the Gates used in the circuit. It can be due to over or under-rotation of the state-vector on the Bloch Sphere.' This is the simplest type of coherent error.
Ideal Evolution on Bloch Sphere
To understand Bloch Sphere read this blog:
https://medium.com/@aanshsavla2453/what-is-bloch-sphere-of-a-qubit-2e59b058cd32
Time Evolution of X Gate
Relation between X and Rx Gate with Global Phase
Hence Rx(π) = X with a global phase of -i. However, the global phase does not change the quantum state. Hence the state remains unchanged.
Miscalibration of Gate
A miscalibrated gate provides some over or under-rotation to the Bloch Sphere.
- Ideal case of X Gate:
- Gate with miscalibration:
Decomposition of Gate with Ordering:
The order in which gates appear in a Quantum Circuit is the reverse of how we write in algebra. As time evolution of the Quantum Circuit happens from left to write. But while writing unitary operations in algebra, the operations happen from right to left.
- Quantum Circuit
- Equation in Algebraic form:
Hence the noisy gate can be decomposed as follows:
Noisy Gate in Quantum Circuit:
- Ideal
Now according to De-Moivre's theorem:
Analyzing the above circuit:
Using De-Moivre's Theorem:
The final state of the quantum circuit:
When d is odd:
When d is even:
Avoiding the global phase:
- Noisy Gate Case:
X gate and Noisy-X gate are some Unitary matrices.
If A and B are matrix then:
Now the equation for the above circuit:
Now the final quantum state:
Ideal v/s Noisy Observable
For a basic definition of the Expectation value of an observable read the 'Expectation value' section in this blog:
https://medium.com/@aanshsavla2453/even-odd-algorithm-in-quantum-computing-2c853bbb8bdc
- Ideal Observable:
Let's define the Bra-Ket version of the final quantum state:
Let's define the expectation value of the observable:
The plot for the above equation is:
(Note: Diagrams may not be accurate)
- Noisy Observable:
Let's define the Bra-Ket version of the final quantum state:
Expanding Rx gate for the error part:
Expanding the final quantum state:
Now
The final Bra and Ket versions of the quantum state are:
Let's define the expectation value of the observable:
The extra factor of 'd' times 'ε' shows that the noise is amplified 'd' times. The above equation depicts oscillation. The plot for the above equation for some constant 'ε' is :
(Note: Diagrams may not be accurate)
This type of sequence where we take a noisy gate and repeat it many times is a perfect way to amplify the noise, study noise and characterize it, understand it.
If we look at the real experiment plot:
(Note: Diagrams may not be accurate)
The behavior of the real experiment is similar as compared to the noisy gate behavior. However, the decaying is due to incoherent noise. Also, the initial state and fluctuations are due to other types of noise sources.
Coherent error is bad quadratically:
Coherent errors have in general, quadratic worst-case performance since this error can grow quadratically to the first order. The order of growth can be found in such ways:
Substituting the value in the above equation we get:
Final summary:
Coherent Errors can be described by unitary operations(Rx(dε)).
Coherent Errors are ubiquitous.
Since there is no decay, coherent errors do not lose quantum information.
These errors can create oscillations in the data.
These errors have a quadratic impact on algorithmic accuracy.
I have learned the above content from Qiskit Global Summer School 2023 by Zlatko Minev. Thank You, IBM!