I noticed that in page 427 in Nielsen & Chuang's book Quantum Computation and Quantum Information, quantum error correction is possible because errors can be discretized.

In other hand, the master equation in the Lindblad form in the following seems describes errors continuously occurring on qubits,

$\frac{d\rho}{dt}=-i[H,\rho]+\frac{1}{2}\sum_{j}\lambda_j(2L_j\rho L_j^\dagger-L_j^\dagger L_j\rho-\rho L_j^\dagger L_j)$.

I want to say that "errors can be discretized" is one of the basis that "quantum error correction is possible". But then I found errors described by the master equation can also be corrected and these errors seems like continuous errors to me.

My question is if the bold sentence is wrong or the errors described by the master equation are actually discrete errors?


The errors that are described by the Master equation are continuous errors. The action of error correction is to discretize those errors.

For example, dephasing noise can be described by the Master equation. The net effect is that an initial state $\rho$ is transformed into $$ \rho\mapsto (1-p)\rho+pZ\rho Z, $$ where $p$ is a function of time.

However, imagine that you are protecting against such errors by using a majority vote in the $X$ basis. We encode as: $$ |\psi\rangle=\alpha|+++\rangle+\beta|---\rangle $$ so $\rho=|\psi\rangle\langle \psi|$. Under dephasing noise, this continuously transforms. However, at the moment that you choose to do error correction, you introduce ancillas, perform the syndrome extraction and measurement. At the moment you get the measurement result, you know whether you have, for example, $\rho$ or $Z_1\rho Z_1$ etc. So, it has been discretized and you only have to think about Pauli errors.

  • $\begingroup$ It is also important to note that the master equation involves evolution of the density matrix, and as such measures the average effect of noise on the system. That is to say that the master equation is what the system looks like when you average together infinitely meany realizations of the state vector evolution under noise. $\endgroup$ Jan 10 '20 at 15:34
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    $\begingroup$ But aren't "quantum jumps" natural discretizations of the Lindblad equation, so even without quantum error correction, sometimes the Lindblad equation can be considered to have discrete errors. $\endgroup$
    – Peter Shor
    Jan 10 '20 at 19:29

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