## 13.12 *Remarks and exercises*

### 13.12.1 Decoherence-free subspaces

Which of the following sets of isometries are correctable?

\{V_0,V_1\} , where\begin{aligned} V_0 &= |00\rangle\langle 0| + |11\rangle\langle 1| \\V_1 &= \frac{1}{\sqrt{2}}\Big[(|01\rangle+|10\rangle)\langle 0| + (|01\rangle-|10\rangle)\langle 1|\Big]. \end{aligned} \{V_0,V_1\} , where\begin{aligned} V_0 &= |00\rangle\langle 0| + |11\rangle\langle 1| \\V_1 &= \frac{1}{\sqrt{2}}\Big[(|01\rangle+|10\rangle)\langle 0| + (|00\rangle-|11\rangle)\langle 1|\Big]. \end{aligned} \{U^{\otimes4}V_0 \mid U\text{ unitary}\} , where\begin{aligned} V_0 &= \frac{1}{2}\Big[(|01\rangle-|10\rangle)(|01\rangle-|10\rangle)\Big]\langle 0|. \\&+ \frac{1}{\sqrt{12}}\Big[2|0011\rangle+2|1100\rangle-(|01\rangle+|10\rangle)(|01\rangle+|10\rangle)\Big]\langle 1|. \end{aligned}

### 13.12.2 Repetition encoding and majority voting failure

Consider encoding a single classical bit as

### 13.12.3 Correcting Pauli rotations with three qubits

We protect an unknown single-qubit state

Conclude that the standard error-correcting protocols that we have discussed will also correct for this type of error.

### 13.12.4 More on Shor [[9,1,3]]

Give the

**logical codewords**^{277}|0_L\rangle and|1_L\rangle for the Shor[[9,1,3]] code.What is the smallest number of single-qubit operations needed to convert

|0_L\rangle into|1_L\rangle ?Can you identify the stabilisers and the

**logical operators**X_L andZ_L for this code?^{278}Note that these may not be unique.Write a table of the syndromes for all single-qubit

X orZ errors on this code, where the columns are labelled by the single-qubit error, and the row by the corresponding stabiliser.How can we detect and correct a

Y error occurring on the first qubit?If an error of the form

\sqrt{1-p}\mathbf{1}+i\sqrt{p}Y occurs on the first qubit, what are the different possible outcomes of measurement?Assume that there is some environment, initially in state

|e\rangle . Decoherence occurs on the qubit, transforming it via\begin{aligned} |0\rangle|e\rangle &\longmapsto |0\rangle|e_{00}\rangle \\|1\rangle|e\rangle &\longmapsto |0\rangle|e_{11}\rangle. \end{aligned} Show that, if we use the Shor[[9,1,3]] code and this decoherence only affects the first qubit in transmission, then we can correct for the resulting error.

### 13.12.5 Distillation for Bell pairs

Alice wants to send

To send her

The process by which a set of **distillation**.
This occurs at a rate **one-way** distillation, where only Alice can send messages.

Assuming knowledge of the optimal code (i.e. one that is guaranteed to succeed and is as small as possible), Alice could transmit encoded halves of Bell pairs, which Bob could then decode. What is a bound on the rate at which Alice and Bob can distill Bell pairs through this channel?

Alternatively, Alice could send Bob unencoded halves of Bell pairs, which they then distill to create a smaller number of perfect Bell pairs which Alice can then use to teleport the desired information. Assuming knowledge of the optimal distillation procedure (i.e. one that maximises

D_1 ), how does this protocol bound the distillation rate?

### 13.12.6 Composing quantum codes

Consider two quantum codes:

- How many physical qubits are involved in the encoding of a single logical qubit of the new code?
- What is the distance of the new code?

That is, the states corresponding to the encoding of

|0\rangle and|1\rangle .↩︎That is, the operators

X_L andZ_L that behave on|0\rangle_L and|1\rangle_L exactly howX andZ behave on|0\rangle and|1\rangle .*Hint: start from the encoding circuit with the eight ancillas all prepared in state*↩︎|0\rangle ; what are their stabilisers? Recall that the encoding operation is a Clifford circuit.