南京大学：《计算理论之美》课程教学资源（课件讲稿）An introduction to quantum error-correction

An introduction to quantum error-correction Nanjing University Zhaohui Wei(Tsinghua University) July 2021

An introduction to quantum error-correction Nanjing University Zhaohui Wei (Tsinghua University) July 2021

Outline Background and motivation Some relevant properties of quantum information The Shor code-the first real quantum error-correcting code Quantum error-correction conditions Stabilizer codes Fault-tolerant quantum computation

Background and motivation
Some relevant properties of quantum information
The Shor code – the first real quantum error-correcting code
Quantum error-correction conditions
Stabilizer codes
Fault-tolerant quantum computation

Remarkable progresses on quantum computation (QC) It is widely believed that quantum supremacy has been achieved The power of a baby quantum computer can surpass that of the most powerful supercomputer on a specific task ÷Google(0ct2019) The task:sample the output of a random circuit(53 qubits) Quantum 3 min vs.Supercomputer 10000 years ÷"九章”quantum computer(Dec2020) The task:Boson sampling(76 photos) Quantum 3 min vs.Supercomputer 600 million years The rise of quantum engineering and quantum industry Quantum computation in chemistry,drug research,machine learning... 。Quantum software

It is widely believed that quantum supremacy has been achieved ◦ The power of a baby quantum computer can surpass that of the most powerful supercomputer on a specific task
Google (Oct 2019) ◦ The task: sample the output of a random circuit (53 qubits) ◦ Quantum 3 min vs. Supercomputer 10000 years
“九章”quantum computer (Dec 2020) ◦ The task: Boson sampling (76 photos) ◦ Quantum 3 min vs. Supercomputer 600 million years
The rise of quantum engineering and quantum industry ◦ Quantum computation in chemistry, drug research, machine learning… ◦ Quantum software

A fundamental difficulty in QC The sizes of quantum computers and other quantum schemes are still relatively small Main reason:Quantum information is extremely fragile The No.1 mission of the quantum computing:Implementing reliable quantum schemes (including quantum computers)with unreliable devices As a result,high precision of quantum controls is needed,and quantum error corrections (QEC)have to be performed

The sizes of quantum computers and other quantum schemes are still relatively small ◦ Main reason: Quantum information is extremely fragile
The No. 1 mission of the quantum computing: Implementing reliable quantum schemes (including quantum computers) with unreliable devices
As a result, high precision of quantum controls is needed, and quantum error corrections (QEC) have to be performed

Classical error correction The key idea:redundancy (cloning unknown information) 0→000 1→111 Suppose the noise flips the bit with probability p>0 1-p 0 0 1 1-p Decode the information by majority voting,then The probability that two or more of the bits are flipped:3p2-2p This is smaller than the original,whenever p<1/2

The key idea: redundancy (cloning unknown information)
Suppose the noise flips the bit with probability
Decode the information by majority voting, then ◦ The probability that two or more of the bits are flipped: ◦ This is smaller than the original, whenever

Quantum error correction is challenging Surprising quantum properties No cloning Errors are continuous Measurement destroys quantum information The main purpose of this talk is convinecing you that quantum error correction is possible!

Surprising quantum properties ◦ No cloning ◦ Errors are continuous ◦ Measurement destroys quantum information
The main purpose of this talk is convinecing you that quantum error correction is possible!

No-cloning theorem for quantum information There is no procedure that will copy or"clone"an arbitrary quantum state,i.e. wo)→ww〉 We now prove that such an operation is not linear,thus is not permitted by quantum mechanics. We can copy all the elements of an orthogonal set of states,but when we extend this operation linearly,no other states will be correctly cloned.For example,we can map 100〉→100》 o〉→) However (o+o)→00)+|)≠(o+0)+以

No-cloning theorem for quantum information ψ 0  ψ ψ There is no procedure that will copy or “clone” an arbitrary quantum state, i.e. We now prove that such an operation is not linear, thus is not permitted by quantum mechanics. We can copy all the elements of an orthogonal set of states, but when we extend this operation linearly, no other states will be correctly cloned. For example, we can map 0 0  0 0 1 0  1 1 However   0  1 0  0 0  1 1    0  1  0  1 

On quantum measurement Generally,when measuring unknown quantum states the outcomes is intrinsically random -even if one have the full knowledge of the measurement device and the measurement is performed perfectly The measurement usually changes the quantum state It is not easy to detect whether noises affect quantum information However,later we will see that this is also extremely helpful! Sometimes it is possible that the outcome is deterministic,and if this is the case,the state stays unchanged after measurement The more information you reveal,the more you disturb the state! Two quantum states that are not orthogonal cannot be distinguished perfectly One cannot find a measurement giving deterministic outcomes on both

Generally, when measuring unknown quantum states the outcomes is intrinsically random – even if one have the full knowledge of the measurement device and the measurement is performed perfectly ◦ The measurement usually changes the quantum state ◦ It is not easy to detect whether noises affect quantum information ◦ However, later we will see that this is also extremely helpful!
Sometimes it is possible that the outcome is deterministic, and if this is the case, the state stays unchanged after measurement ◦ The more information you reveal, the more you disturb the state!
Two quantum states that are not orthogonal cannot be distinguished perfectly ◦ One cannot find a measurement giving deterministic outcomes on both

On quantum operations The operator-sum representation:The behavior of an open quantum system can be modeled as p)=∑ExpE, where E are operation elements,satisfying EE=I if the quantum oper- ation is trace-preserving

A simple example:bit flip code(1) Bit flip channel:leaves the qubits untouched with probability 1-p, and flips the qubits (X))with probability p e We can use the encoding 10)→10z〉≡1000) 1)→1z〉≡111〉 Then the state a0)+b1)is encoded as a000)+6111) The encoding circuit is 10)》 10)

Bit flip channel: leaves the qubits untouched with probability , and flips the qubits ( ) with probability
We can use the encoding Then the state is encoded as
The encoding circuit is