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Some background knowledge

No - Cloning Theorem

You cannot copy an arbitrary quantum state.
So, if you have a state |ψ⟩ = α|0⟩ + β|1⟩, you cannot ever copy this exactly into let’s say |φ⟩
Classical example - cloning biased coin

Unitaries - any linear transformation that preserves length

This is basically a reversible linear operation on a qubit
For every unitary U, U^†U = I. This means all the unitary operations are reversible.
Examples - H, X, Y, Z

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What are Quantum errors?

In any information system, the transfer of the correct information is always affected by errors in the system.

Any information system, classical of quantum, does the following-

Transmission through space
Transmission through time
Redistribution into readable forms

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Why are there errors in information transmission

Noise in the channel (unwanted signals due to environment disturbances)
Issues with hardware / hardware parts
Lost bits due to the kind of transmission

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Classical Errors

Basic Example - transmission of a single bit

It either passes through the channel or gets flipped

Any idea how such encoding and decoding can work in the classical medium?

Error mitigation -

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Classical Single Bit Error correction Example

Suppose you are sending some classical information, and the bits get flipped with some probability 1 - p < ½.

How can you detect and mitigate errors?

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Repetition Codes

0 —(encode) → 000……0 (n times)

1 —(encode) → 111……1 (n times)

Decode function:

Output the majority bits

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Problems with Quantum Error Correction

There can be errors other than bit flip - rotation, phase/angle change
Cannot encode |ψ⟩⊗|ψ⟩⊗|ψ⟩⊗…….⊗|ψ⟩ - violates no cloning theorem
Cannot measure the coded message as it would destroy the information / render it useless

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How do we solve this?

Consider the simple bit flip example.

How can we detect this error in the quantum case?

|0⟩ → |000⟩

|1⟩ → |111⟩

α|0⟩ + β|1⟩ → α|000⟩ + β|111⟩

Is this allowed? Does this violate no-cloning?

How do you implement this unitary?

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Implementation of encoding and decoding

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How would we correct and decode this?

Will measuring help?

Compare consecutive bit pairs as -

|ψ’⟩

Example - |ψ’⟩ = α|010⟩ + β|100⟩

For α,

a = ?

b = ?

For β,

a = ?

b = ?

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The whole process

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Using this, how can we solve phase flip errors?

Note : Phase flip errors: Z

Z|0⟩ → |0⟩

Z|1⟩ → - |1⟩

Z|ψ⟩ = α|0⟩ - β|1⟩

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Detecting phase flip errors

What happens when you apply Z to |+⟩ and |-⟩?

Z|+⟩ = ?

Z|-⟩ = ?

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Detecting phase flip errors

What happens when you apply Z to |+⟩ and |-⟩?

Z|+⟩ = |-⟩

Z|-⟩ = |+⟩

So, phase flip error in the hadamard basis is just like the bit flip error.

How can we use the bit flip protocol?

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Encoding process

|0⟩ → |000⟩ → H|000⟩ → |+++⟩

|1⟩ → |111⟩ → H|111⟩ → |---⟩

α|0⟩ + β|1⟩ → α|+++⟩ + β|---⟩

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Phase Flip Correction - same as bit flip with encoding and decoding Hadamards

Keep this as reference, but don't just dump this on people without showing why it's useful (e.g. talk about traces along with density matrices)

%\ket{\psi} = \alpha \ket{0} + \beta \ket{1}

%\vec{v} = \ket{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}

%\begin{align*} \left(\vec{v}^{\,T}\right)^* &= \left[ v_1^*, v_2^*, \dots, v_n^* \right]^T \\ &= \vec{v}^{\,\dagger} \\ &= \bra{v} \end{align*}

%AB = \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} \begin{bmatrix} b_1 & b_2 \\ b_3 & b_4 \end{bmatrix} = \begin{bmatrix} a_1 b_2 + a_2 b_3 & a_2 b_4 + a_1 b_2 \\ a_3 b_1 + a_4 b_3 & a_4 b_4 + a_3 b_2 \end{bmatrix}

\ket{0} \otimes \ket{1} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \otimes \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \begin{bmatrix} 0 \\ 1 \end{bmatrix} \\ 0 \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \ket{01}

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