quantum mechanics - Practical example of stabilizer codes

Given the Steane code $$ \left|0\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|0000000\right\rangle + \left|1010101\right\rangle + \left|0110011\right\rangle + \left|1100110\right\rangle + \left|0001111\right\rangle + \left|1011010\right\rangle + \left|0111100\right\rangle + \left|1101001\right\rangle) $$ $$ \left|1\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|1111111\right\rangle + \left|0101010\right\rangle + \left|1001100\right\rangle + \left|0011001\right\rangle + \left|1110000\right\rangle + \left|0100101\right\rangle + \left|1000011\right\rangle + \left|0010110\right\rangle) $$

and its relative stabilizers: $$ K^1 = IIIXXXX $$ $$ K^2 = XIXIXIX $$ $$ K^3 = IXXIIXX $$ $$ K^4 = IIIZZZZ $$ $$ K^5 = ZIZIZIZ $$ $$ K^6 = IZZIIZZ $$

The stabilizer set establishes valid codewords for a state if the equation $$s\left|\psi\right\rangle=\left|\psi\right\rangle,\;\;\;\forall s \in S \;\;\;\;\; (1)$$ is satisfied. That means $\left|\psi\right\rangle$ is a +1 eigenstate of $s$.

We then consider a practical example of the usage of these stabilizers

The state of the system is represented by: $$\left|\psi\right\rangle_F={1\over 2}(\left|\psi\right\rangle_I+U\left|\psi\right\rangle_I)\left|0\right\rangle + {1\over 2}(\left|\psi\right\rangle_I-U\left|\psi\right\rangle_I)\left|1\right\rangle$$

where $U \in \left\lbrace K^1,K^2,K^3\right\rbrace$.

We apply $U$ to the input state and we measure the ancilla qubits (syndrome measurement) to verify the integrity of the input (if $\left|\psi\right\rangle_I$ is +1 eigenstate of $K^1,K^2,K^3$). If the equation $(1)$ is not satisfied, then the corrupted qubit is corrected with a $Z$ gate addressed by the syndrome measurement.

This is how does the system work?

Answer

It is correct. We may summarize all the operations :

1) Encoding one logical qubit as $n$ physical qbits (codeword) , $\alpha|0\rangle + \beta |1\rangle \to \alpha|0\rangle_L + \beta |1\rangle_L$, here $n$ = 7 for the Steane code.

2) Preparing $m$ ancilla qbits, here $m = 3$, in your schema allowing to detect phase-flip errors $Z_i$

3) During transmission of the codeword, ther is exposition to noisy environment, and the codeword may suffer errors (bit -flip $X_i$, phase-flip $_iY$, bit-phase flip $Z_i$). In your schema, we are only intereseted in phase-flip errors $Z_i$.

4) With the help of the generators, we compute the syndrome and store it in the ancilla qbits without altering the $n$-qbit world.

5) Use the information provided by the error syndrome to locate the error of any one of the $n$ qbits of the codeword.

In your case, the generators $K_1, K_2, K_3$ are the only needed to check any error $Z_i$, and each syndrome $xyz$ identifies precisely the bit $i$ corresponding to the error $Z_i$. Here the syndromes for the $Z_i$ are $010,001,011,100,110,101,111$

6) Correct the error

Ref : Marinescu/Marinescu, Classical and Quantum information, Elsevier, p $462$, p $509$