13.12 Remarks and exercises

13.12.1 Decoherence-free subspaces

Which of the following sets of isometries are correctable?

  1. \{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}

  2. \{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}

  3. \{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 2k+1 bits using a repetition code, and then decoding with majority voting. If during the transmission process between encoding and decoding each bit is flipped with independent probability p, what is the probability of an error on the logical bit after the encoding–decoding process?

13.12.3 Correcting Pauli rotations with three qubits

We protect an unknown single-qubit state \alpha|0\rangle+\beta|1\rangle against bit-flip errors by encoding it with the three-qubit repetition code: |\psi\rangle = \alpha|000\rangle + \beta|111\rangle. An error of the form (\cos\theta)\mathbf{1}+(i\sin\theta)X occurs on the first qubit during transmission. When we perform the error syndrome measurements, what are the possible outcomes, and what are the corresponding output states?

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]]

  1. Give the logical codewords277 |0_L\rangle and |1_L\rangle for the Shor [[9,1,3]] code.

  2. What is the smallest number of single-qubit operations needed to convert |0_L\rangle into |1_L\rangle?

  3. Can you identify the stabilisers and the logical operators X_L and Z_L for this code?278 Note that these may not be unique.

  4. Write a table of the syndromes for all single-qubit X or Z errors on this code, where the columns are labelled by the single-qubit error, and the row by the corresponding stabiliser.

  5. How can we detect and correct a Y error occurring on the first qubit?

  6. 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?

  7. 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 m qubits of information to Bob. She can send quantum states, but only through a transmission channel that induces errors, though she can send classical information perfectly. Bob cannot send messages (neither quantum nor classical) to Alice, but both of them can perfectly implement quantum logic gates.

To send her m qubits in spite of the noise, Alice might encode them in an n-qubit error correcting code.

The process by which a set of N noisy Bell pairs is converted into a small number M of perfect Bell pairs is known as distillation. This occurs at a rate D_1=M/N, which is often considered in the limit of large N. The subscript 1 denotes that this is one-way distillation, where only Alice can send messages.

  1. 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?

  2. 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: C_1 is an [[n_1,1,d_1]] code, and C_2 is an [[n_2,1,d_2]] code. We decide to encode a qubit |\psi\rangle by first encoding it into n_1 qubits using C_1, and then encoding each of those resulting qubits into n_2 qubits using C_2. The overall effect is an encoding into the composite code C_2C_1.

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

  1. That is, the states corresponding to the encoding of |0\rangle and |1\rangle.↩︎

  2. That is, the operators X_L and Z_L that behave on |0\rangle_L and |1\rangle_L exactly how X and Z 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.↩︎