Chapter 11

Exercises are from QUANTUM COMPUTING: A GENTLE INTRODUCTION, by Eleanor Rieffel and Wolfgang Polak, published by The MIT Press.

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 11.1}$

For the code $C: \mathbf{Z}_2^1 \to \mathbf{Z}_2^3$ defined by generator matrix

(1)
\begin{align} G=\left(\begin{array}{c}1\\1\\1\end{array}\right) \end{align}

give

  • the set of code words
  • two distinct parity check matrices

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$\mathbf{Exercise\ 11.2}$ Computing a parity check matrix for a code specified by a generator matrix.

a) Show that adding a column of a generator matrix $G$ for a code $C$ to another column produces an alternative generator matrix $G'$ for the same code $C$.

b) Show that for any $[n,k]$ code there is a generator matrix of the form $\left(\begin{array}{c}A\\\hline I\end{array}\right)$ where $A$ is a $(n-k)\times k$ matrix and $I$ is the $k\times k$ identity matrix.

c) Show that if $G = \left(\begin{array}{c}A\\\hline I\end{array}\right)$ then the $(n-k)\times k$ matrix $P =(I|A)$, where $I$ is the $(n-k)\times(n-k)$ identity matrix, is a parity check matrix for the code $C$.

d) Show that if a parity check matrix $P'$ has the form $(A|I)$, then $G' = \left(\begin{array}{c}I\\\hline A\end{array}\right)$ is a generator matrix for the code.

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$\mathbf{Exercise\ 11.3}$

Show that the code $C_{PF}$ of section 11.1.2 corrects all linear combinations of single-qubit phase-flip errors $\{I, Z_2, Z_1, Z_0\}$ on any superposition $a\ket{\tilde 0} + b\ket{\tilde 1}$.

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$\mathbf{Exercise\ 11.4}$

Show that the code $C_{PF}$ of section 11.1.2 does not correct bit-flip errors.

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$\mathbf{Exercise\ 11.5}$

Show that if an $[[n,k,d]]$-quantum code is able to correct all errors of weight $t$ or less, $d \geq 2t +1$.

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$\mathbf{Exercise\ 11.6}$

Show that all Hamming codes have distance $3$ so correct single bit-flip errors.

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$\mathbf{Exercise\ 11.7}$

Show that Shor's code is a degenerate code.

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$\mathbf{Exercise\ 11.8}$ Alternative Steane code constructions

a) Find a parity check matrix of the form $[I | A]$ for the Steane code.

b) Construct an alternative circuit, based on the parity check matrix found in part a), that can serve as a syndrome extraction operator for the Steane code.

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$\mathbf{Exercise\ 11.9}$

Show that the generalized set of Pauli elements forms a basis for the linear transformations on the vector space associated with an $n$-qubit system.

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$\mathbf{Exercise\ 11.10}$

a) Show that for all $i$ and $j$ and for all orthonormal $\ket{c_1}\ne\ket{c_2}\in C$,

(2)
\begin{align} \bra {c_1} Z_j^\dagger Y_i \ket {c_2} &=& 0 \\ \bra {c_1} I Y_i \ket {c_2} &=& 0, \end{align}

b) Show that for all $j \ne i$ and for all orthonormal $\ket{c_1}\ne\ket{c_2}\in C$,

(3)
\begin{align} \bra {c_1} Y_j^\dagger Y_i \ket {c_2} = 0. \end{align}

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$\mathbf{Exercise\ 11.11}$

Describe how the Shor code can be used to correct single-qubit errors without making any measurements.

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$\mathbf{Exercise\ 11.12}$

Show that if two blocks encoded according to code $C$ are subjected to an error $E$ that is a superposition of errors $E = E_a\otimes E_b + E_c\otimes E_d$, where $E_a$, $E_b$, $E_c$, and $E_d$ are all elements of a correctable set of errors $\cal E$ for $C$, then $E$ can be corrected.

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$\mathbf{Exercise\ 11.13}$

Suppose a single qubit $\ket\psi = a\ket 0 + b\ket 1$ has been encoded using the Steane code and that the error $E = \frac{1}{2}X_2 + \frac{\sqrt{3}}{2}Z_3$ has occurred. Write down

a) the encoded state,

b) the state after the error has occurred,

c) for each phase of the error correction, the syndrome and the resulting state, and

d) each error correcting transformation applied and the state after each of these applications.

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$\mathbf{Exercise\ 11.14}$

Show that for a $[[n,k]]$ quantum stabilizer code there is, for any $k$ bit string $b_1\dots b_k$, a unique element of the code $C$ that is a $(-1)^{b_i}$-eigenstate of $Z_i$ for all $i$.

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$\mathbf{Exercise\ 11.15}$

Show that the subspaces $V_e$ of Section 11.4.3 are of dimension $2^{n -r}$.

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$\mathbf{Exercise\ 11.16}$

Show that the operators $\tilde{X_i}$ as defined for stabilizer codes in Section 11.4.4 act as a logical analog of the gates $X_i$ for the logical states obtained from the encoding $c$.

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$\mathbf{Exercise\ 11.17}$

Show that the $[[9,1]]$ Shor code is a stabilizer code.

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$\mathbf{Exercise\ 11.18}$

Find alternative ways of implementing operations corresponding to $X$ and $Z$ on the logical qubits of the $5$-qubit stabilizer code.

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$\mathbf{Exercise\ 11.19}$

Let $[[n, k, d]]$ be any non-degenerate code. Such a code can correct $t = \lfloor\frac{d-1}{2}\rfloor$ errors. Show that tracing any codeword over any $n-t$ qubits results in the totally mixed state $\rho = \frac{1}{2^t} I$ on the remaining $t$ qubits. Thus all codewords are highly entangled states.

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