Calculation to walk through in class

Readings:

For introduction of decoherence and how it makes quantum errors, watch IQIS lecture 7.2-7.3 (16 minutes). https://www.youtube.com/watch?v=ZPfdKNzjWbo

As a recap:

Coherence is when the qubits can make superposition in itself and make entanglement with each other.

Each qubit is expressed as superposition $|\psi\rangle=a|0\rangle + b|1\rangle, |a|^2+|b|^2=1, a,b \in \mathbb{C}$

N entangled bits is $|\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle, a_i \in\mathbb{C}$

What is decoherence?

Qubits can interact with the environment by entanglement. The desirable interaction is measurement. Previously we assumed that the system is isolated so the only interaction is measurement. But in reality other interactions also occur outside the gates and measurements. Qubits with the environment more and more over time. There are several physical sources for decoherence (NS142 people can elaborate during class). This is because the interference in the qubit is weaker than some other interactions such as EM. For example, the pure environment state |e> can learn a quantum basis state to become mixed state |e_0> through the entanglement $|0\rangle|e\rangle → |0\rangle|e_0\rangle, |1\rangle|e\rangle → |1\rangle|e_1\rangle$.

How does decoherence absorb quantum states?

Each interaction loses some quantum information from the qubits. After learning the environmental bases are not guaranteed to be orthogonal though, $\langle e_0|e_1\rangle = v e^{i \alpha}$. Alpha is the phase angle between the environmental bases. V is visibility, or how much interference pattern remains. At the limit of v=0, the density matrix just becomes a diagonal matrix. You can get a simple derivation in the IQIS video. The state no longer contains superposition, is exactly like classical probability. We need superposition to store information, so we have to prevent errors caused by decoherence.

How does decoherence cause quantum errors?

The effect of the environment on the qubit state can be digitised into bit flip (X) and phase flip (Z) errors. The flips can be represented as operators (not necessarily unitary). They have a error probability of occuring. The matrices for bit flip, phase flip, both bit and phase flip are: $\sigma_x = [[0,1],[1,0]], \sigma_z=[[1,0],[0,-1]], \sigma_{xz} = [[0,-i],[i,0]]$. They each have physical sources such as depolarising noise for bit flip. In class we will use the Qiskit error model to test how the error correction codes deal with these noises.

The conceptual picture:

The quantum error correction problem:

1. there is an unknown quantum state.
   1. qubits can be material like atoms. but we can also use photon microwave to store qubit.
2. if you measure it, it will collapse.
3. the quantum computer has imperfect parts. it introduces impurity (error, decoherence) into the qubit system. you want to let the quantum information live long before it deprecates. more qubits have more errors faster.
4. if there is an error, you need to fix it (you need to know when there is an error and where it is). which error occurred (bit flip, phase flip, both), which of the N flips have the error.
   1. requires many syndrome measurements
   2. makes things worse by having more bits.

• the more IT/linalg than QM focus: is decoherence a really bad big error or just the process that generates any error?

The quantum error correction strategy: