entanglement-assisted stabilizer formalism

We review the construction of an entanglement-assisted code (Brun et al. 2006). Suppose that

there is a nonabelian subgroup $\backslash mathcal\{S\}\backslash subset\backslash Pi^\{n\}$ of size $n-k=2c+s$.

Application of the fundamental theorem of symplectic geometry (Lemma 1 in the first external reference)

states that there exists a minimal set of independent generators

$\backslash left\backslash \{$

\bar{Z}_{1},\ldots,\bar{Z}_{s+c},\bar{X}_{s+1},\ldots,\bar{X}_{s+c}\right\}

for $\backslash mathcal\{S\}$ with the following commutation relations:

:$$

\left[ \bar{Z}_{i},\bar{Z}_{j}\right] = 0\ \ \ \ \ \forall

i,j,

:$$

\left[ \bar{X}_{i},\bar{X}_{j}\right] = 0\ \ \ \ \ \forall

i,j,

:$$

\left[ \bar{X}_{i},\bar{Z}_{j}\right] = 0\ \ \ \ \ \forall i\neq

j,

:$\backslash left\backslash \{\; \backslash bar\{X\}\_\{i\},\backslash bar\{Z\}\_\{i\}\backslash right\backslash \}\; =\; 0\backslash \; \backslash \; \backslash \; \backslash \; \backslash \; \backslash forall\; i.$

The decomposition of $\backslash mathcal\{S\}$ into the above minimal generating set

determines that the code requires $s$ ancilla qubits and $c$ ebits. The code

requires an ebit for every anticommuting pair in the minimal generating set.

The simple reason for this requirement is that an ebit is a simultaneous

$+1$-eigenstate of the Pauli operators $\backslash left\backslash \{\; XX,ZZ\backslash right\backslash \}$. The second qubit

in the ebit transforms the anticommuting pair $\backslash left\backslash \{\; X,Z\backslash right\backslash \}$ into a

commuting pair $\backslash left\backslash \{\; XX,ZZ\backslash right\backslash \}$. The above decomposition also

minimizes the number of ebits required for the code---it is an optimal decomposition.

We can partition the nonabelian group $\backslash mathcal\{S\}$ into two subgroups: the

isotropic subgroup $\backslash mathcal\{S\}\_\{I\}$ and the entanglement subgroup

$\backslash mathcal\{S\}\_\{E\}$. The isotropic subgroup $\backslash mathcal\{S\}\_\{I\}$ is a commuting

subgroup of $\backslash mathcal\{S\}$ and thus corresponds to ancilla

qubits:

:$\backslash mathcal\{S\}\_\{I\}=\backslash left\backslash \{\; \backslash bar\{Z\}\_\{1\},\backslash ldots,\backslash bar\{Z\}\_\{s\}\backslash right\backslash \}$.

The elements of the entanglement subgroup $\backslash mathcal\{S\}\_\{E\}$ come in

anticommuting pairs and thus correspond to ebits:

:$\backslash mathcal\{S\}\_\{E\}=\backslash left\backslash \{$

\bar{Z}_{s+1},\ldots,\bar{Z}_{s+c},\bar{X}_{s+1},\ldots,\bar{X}_{s+c}\right\}

.

The two subgroups $\backslash mathcal\{S\}\_\{I\}$ and $\backslash mathcal\{S\}\_\{E\}$ play a role in the

error-correcting conditions for the entanglement-assisted stabilizer

formalism. An entanglement-assisted code corrects errors in a set

$\backslash mathcal\{E\}\backslash subset\backslash Pi^\{n\}$ if for all $E\_\{1\},E\_\{2\}\backslash in\backslash mathcal\{E\}$,

:$E\_\{1\}^\{\backslash dagger\}E\_\{2\}\backslash in\backslash mathcal\{S\}\_\{I\}\backslash cup\backslash left(\; \backslash Pi^\{n\}-\backslash mathcal\{Z\}\backslash left($

\left\langle \mathcal{S}_{I},\mathcal{S}_{E}\right\rangle \right) \right) .

The operation of an entanglement-assisted code is as follows. The sender

performs an encoding unitary on her unprotected qubits, ancilla qubits, and

her half of the ebits. The unencoded state is a simultaneous +1-eigenstate of

the following Pauli operators:

:$$

\left\{

Z_{1},\ldots,Z_{s}, Z_{s+1}|Z_{1},\ldots,Z_{s+c}|Z_{c},

X_{s+1}|X_{1},\ldots,X_{s+c}|X_{c}

\right\} .

The Pauli operators to the right of the vertical bars indicate the receiver's half

of the shared ebits. The encoding unitary transforms the unencoded Pauli operators

to the following encoded Pauli operators:

:$$

\left\{

\bar{Z}_{1},\ldots,\bar{Z}_{s},

\bar{Z}_{s+1}|Z_{1},\ldots,\bar{Z}_{s+c}|Z_{c},

\bar{X}_{s+1}|X_{1},\ldots,\bar{X}_{s+c}|X_{c}

\right\} .

The sender transmits all of her qubits over the noisy quantum channel. The

receiver then possesses the transmitted qubits and his half of the ebits. He

measures the above encoded operators to diagnose the error. The last step is

to correct the error.

We can interpret the rate of an entanglement-assisted code

in three different ways (Wilde and Brun 2007b).

Suppose that an entanglement-assisted quantum code encodes $k$ information

qubits into $n$ physical qubits with the help of $c$ ebits.

• The entanglement-assisted rate assumes that entanglement shared between sender and receiver is free. Bennett et al. make this assumption when deriving the entanglement assisted capacity of a quantum channel for sending quantum information. The entanglement-assisted rate is $k/n$ for a code with the above parameters.
• The trade-off rate assumes that entanglement is not free and a rate pair determines performance. The first number in the pair is the number of noiseless qubits generated per channel use, and the second number in the pair is the number of ebits consumed per channel use. The rate pair is $\backslash left(\; k/n,c/n\backslash right)$ for a code with the above parameters. Quantum information theorists have computed asymptotic trade-off curves that bound the rate region in which achievable rate pairs lie. The construction for an entanglement-assisted quantum block code minimizes the number $c$ of ebits given a fixed number $k$ and $n$ of respective information qubits and physical qubits.
• The catalytic rate assumes that bits of entanglement are built up at the expense of transmitted qubits. A noiseless quantum channel or the encoded use of noisy quantum channel are two different ways to build up entanglement between a sender and receiver. The catalytic rate of an $\backslash left[\; n,k;c\backslash right]$ code is $\backslash left(\; k-c\backslash right)\; /n$.

Which interpretation is most reasonable depends on the context in which we use

the code. In any case, the parameters $n$, $k$, and $c$ ultimately govern

performance, regardless of which definition of the rate we use to interpret

that performance.

We present an example of an entanglement-assisted code

that corrects an arbitrary single-qubit error (Brun et al. 2006). Suppose

the sender wants to use the quantum error-correcting properties of the

following nonabelian subgroup of $\backslash Pi^\{4\}$:

:$$

\begin{array}

[c]{cccc}

Z & X & Z & I\\

Z & Z & I & Z\\

X & Y & X & I\\

X & X & I & X

\end{array}

The first two generators anticommute. We obtain a modified third generator by

multiplying the third generator by the second. We then multiply the last

generator by the first, second, and modified third generators. The

error-correcting properties of the generators are invariant under these

operations. The modified generators are as follows:

:$$

\begin{array}

[c]{cccccc}

g_{1} & = & Z & X & Z & I\\

g_{2} & = & Z & Z & I & Z\\

g_{3} & = & Y & X & X & Z\\

g_{4} & = & Z & Y & Y & X

\end{array}

The above set of generators have the commutation relations given by the

fundamental theorem of symplectic geometry:

:$$

\left\{ g_{1},g_{2}\right\} =\left[ g_{1},g_{3}\right] =\left[

g_{1},g_{4}\right] =\left[ g_{2},g_{3}\right] =\left[ g_{2},g_{4}\right] =\left[

g_{3},g_{4}\right] =0.

The above set of generators is unitarily equivalent to the following canonical

generators:

:$$

\begin{array}

[c]{cccc}

X & I & I & I\\

Z & I & I & I\\

I & Z & I & I\\

I & I & Z & I

\end{array}

We can add one ebit to resolve the anticommutativity of the first two

generators and obtain the canonical stabilizer:

:$$

\begin{array}

[c]{c}

X\\

Z\\

I\\

I

\end{array}

\left\vert

\begin{array}

[c]{cccc}

X & I & I & I\\

Z & I & I & I\\

I & Z & I & I\\

I & I & Z & I

\end{array}

\right.

The receiver Bob possesses the qubit on the left and the sender Alice

possesses the four qubits on the right. The following state is an eigenstate

of the above stabilizer

:$$

\left\vert \Phi^{+}\right\rangle ^{BA}\left\vert 00\right\rangle

^{A}\left\vert \psi\right\rangle ^{A}.

where $\backslash left\backslash vert\; \backslash psi\backslash right\backslash rangle\; ^\{A\}$ is a qubit that the sender wants to

encode. The encoding unitary then rotates the canonical stabilizer to the following set of globally commuting

generators:

:$$

\begin{array}

[c]{c}

X\\

Z\\

I\\

I

\end{array}

\left\vert

\begin{array}

[c]{cccc}

Z & X & Z & I\\

Z & Z & I & Z\\

Y & X & X & Z\\

Z & Y & Y & X

\end{array}

\right.

The receiver measures the above generators upon receipt of all qubits to

detect and correct errors.

We continue with the previous example. We

detail an algorithm for determining an encoding circuit and the optimal number

of ebits for the entanglement-assisted code---this algorithm first appeared in the appendix of (Wilde and Brun 2007a) and later in the appendix of (Shaw et al. 2008). The operators in

the above example have the following representation as a binary

matrix (See the stabilizer code article):

:$$

H=\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 1 & 0\\

1 & 1 & 0 & 1\\

0 & 1 & 0 & 0\\

0 & 0 & 0 & 0

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 1 & 0 & 0\\

0 & 0 & 0 & 0\\

1 & 1 & 1 & 0\\

1 & 1 & 0 & 1

\end{array}

\right] .

Call the matrix to the left of the vertical bar the "$Z$

matrix" and the matrix to the right of the vertical bar the

"$X$ matrix."

The algorithm consists of row and column operations on the above matrix. Row

operations do not affect the error-correcting properties of the code but are

crucial for arriving at the optimal decomposition from the fundamental theorem

of symplectic geometry. The operations available for manipulating columns of

the above matrix are Clifford operations. Clifford

operations preserve the Pauli group $\backslash Pi^\{n\}$ under conjugation. The

CNOT gate, the Hadamard gate, and the Phase gate generate the Clifford group.

A CNOT gate from qubit $i$ to qubit $j$ adds column $i$ to column $j$ in the

$X$ matrix and adds column $j$ to column $i$ in the $Z$ matrix. A Hadamard

gate on qubit $i$ swaps column $i$ in the $Z$ matrix with column $i$ in the

$X$ matrix and vice versa. A phase gate on qubit $i$ adds column $i$ in the

$X$ matrix to column $i$ in the $Z$ matrix. Three CNOT gates implement a

qubit swap operation. The effect of a swap on qubits

$i$ and $j$ is to swap columns $i$ and $j$ in both the $X$ and $Z$ matrix.

The algorithm begins by computing the symplectic product between the first row

and all other rows. We emphasize that the symplectic product here is the

standard symplectic product. Leave the matrix as it is if the first row is not

symplectically orthogonal to the second row or if the first row is

symplectically orthogonal to all other rows. Otherwise, swap the second row

with the first available row that is not symplectically orthogonal to the

first row. In our example, the first row is not symplectically orthogonal to

the second so we leave all rows as they are.

Arrange the first row so that the top left entry in the $X$ matrix is one. A

CNOT, swap, Hadamard, or combinations of these operations can achieve this

result. We can have this result in our example by swapping qubits one and two.

The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

0 & 1 & 1 & 0\\

1 & 1 & 0 & 1\\

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0

\end{array}

\right\vert

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

1 & 1 & 1 & 0\\

1 & 1 & 0 & 1

\end{array}

\right] .

Perform CNOTs to clear the entries in the $X$ matrix in the top row to the

right of the leftmost entry. These entries are already zero in this example so

we need not do anything. Proceed to the clear the entries in the first row of

the $Z$ matrix. Perform a phase gate to clear the leftmost entry in the first

row of the $Z$ matrix if it is equal to one. It is equal to zero in this case

so we need not do anything. We then use Hadamards and CNOTs to clear the other

entries in the first row of the $Z$ matrix.

We perform the above operations for our example. Perform a Hadamard on qubits

two and three. The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 1\\

1 & 1 & 1 & 0\\

0 & 1 & 0 & 0

\end{array}

\right\vert

\begin{array}

[c]{cccc}

1 & 1 & 1 & 0\\

0 & 1 & 0 & 0\\

1 & 0 & 0 & 0\\

1 & 0 & 0 & 1

\end{array}

\right] .

Perform a CNOT from qubit one to qubit two and from qubit one to qubit three.

The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 1\\

1 & 1 & 1 & 0\\

1 & 1 & 0 & 0

\end{array}

\right\vert

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

1 & 1 & 1 & 0\\

1 & 1 & 1 & 1

\end{array}

\right] .

The first row is complete. We now proceed to clear the entries in the second

row. Perform a Hadamard on qubits one and four. The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

1 & 1 & 1 & 0\\

1 & 1 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 1 & 0 & 1\\

1 & 1 & 1 & 0\\

1 & 1 & 1 & 0

\end{array}

\right] .

Perform a CNOT from qubit one to qubit two and from qubit one to qubit four.

The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 1 & 1 & 0\\

1 & 1 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

1 & 0 & 1 & 1\\

1 & 0 & 1 & 1

\end{array}

\right] .

The first two rows are now complete. They need one ebit to compensate for

their anticommutativity or their nonorthogonality with respect to the

symplectic product.

Now we perform a "Gram-Schmidt

orthogonalization" with respect to the symplectic product.

Add row one to any other row that has one as the leftmost entry in its $Z$

matrix. Add row two to any other row that has one as the leftmost entry in its

$X$ matrix. For our example, we add row one to row four and we add row two to

rows three and four. The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 1 & 1 & 0\\

0 & 1 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 0 & 1 & 1\\

0 & 0 & 1 & 1

\end{array}

\right] .

The first two rows are now symplectically orthogonal to all other rows per the

fundamental theorem of symplectic geometry.

We proceed with the same algorithm on the next two rows. The next two rows are

symplectically orthogonal to each other so we can deal with them individually.

Perform a Hadamard on qubit two. The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 1 & 1 & 1\\

0 & 1 & 1 & 1

\end{array}

\right] .

Perform a CNOT from qubit two to qubit three and from qubit two to qubit

four. The matrix becomes

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 1 & 1 & 0\\

0 & 1 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 1 & 0 & 0

\end{array}

\right] .

Perform a phase gate on qubit two:

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 1 & 0 & 0

\end{array}

\right] .

Perform a Hadamard on qubit three followed by a CNOT from qubit two to qubit

three:

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 1 & 1 & 0

\end{array}

\right] .

Add row three to row four and perform a Hadamard on qubit two:

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 0 & 0 & 1

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 1 & 0

\end{array}

\right] .

Perform a Hadamard on qubit four followed by a CNOT from qubit three to qubit

four. End by performing a Hadamard on qubit three:

:$$

\left[ \left.

\begin{array}

[c]{cccc}

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 0 & 1 & 0

\end{array}

\right\vert

\begin{array}

[c]{cccc}

0 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0

\end{array}

\right] .

The above matrix now corresponds to the canonical Pauli operators. Adding one half of an ebit to the receiver's side

gives the canonical stabilizer whose

simultaneous +1-eigenstate is the above state.

The above operations in reverse order

take the canonical stabilizer to the encoded

stabilizer.

• {{cite journal | last1=Brun | first1=T. |author-link=Todd Brun| last2=Devetak | first2=I. | last3=Hsieh | first3=M.-H. | title=Correcting Quantum Errors with Entanglement | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=314 | issue=5798 | date=2006-10-20 | issn=0036-8075 | doi=10.1126/science.1131563 | pages=436–439| pmid=17008489 |arxiv=quant-ph/0610092| bibcode=2006Sci...314..436B | s2cid=18106089 }}
• Min-Hsiu Hsieh. Entanglement-assisted Coding Theory. Ph.D. Dissertation, University of Southern California, August 2008. Available at https://arxiv.org/abs/0807.2080
• Mark M. Wilde. Quantum Coding with Entanglement. Ph.D. Dissertation, University of Southern California, August 2008. Available at https://arxiv.org/abs/0806.4214
• {{cite journal | last1=Hsieh | first1=Min-Hsiu | last2=Devetak | first2=Igor | last3=Brun | first3=Todd | title=General entanglement-assisted quantum error-correcting codes | journal=Physical Review A | volume=76 | issue=6 | date=2007-12-19 | issn=1050-2947 | doi=10.1103/physreva.76.062313 | page=062313|arxiv=0708.2142| bibcode=2007PhRvA..76f2313H | s2cid=119155178 }}
• {{cite journal | last1=Kremsky | first1=Isaac | last2=Hsieh | first2=Min-Hsiu | last3=Brun | first3=Todd A. | title=Classical enhancement of quantum-error-correcting codes | journal=Physical Review A | volume=78 | issue=1 | date=2008-07-21 | issn=1050-2947 | doi=10.1103/physreva.78.012341 | page=012341|arxiv=0802.2414| bibcode=2008PhRvA..78a2341K | s2cid=119252610 }}
• {{cite journal | last1=Wilde | first1=Mark M. | last2=Brun | first2=Todd A. | title=Optimal entanglement formulas for entanglement-assisted quantum coding | journal=Physical Review A | volume=77 | issue=6 | date=2008-06-19 | issn=1050-2947 | doi=10.1103/physreva.77.064302 | page=064302|arxiv=0804.1404| bibcode=2008PhRvA..77f4302W | s2cid=118411793 }}
• {{cite conference | last1=Wilde | first1=Mark M. | last2=Krovi | first2=Hari | last3=Brun | first3=Todd A. | title=2010 IEEE International Symposium on Information Theory | chapter=Convolutional entanglement distillation | publisher=IEEE | year=2010 | pages=2657–2661 | isbn=978-1-4244-7892-7 | doi=10.1109/isit.2010.5513666 |arxiv=0708.3699}}
• {{cite journal | last1=Wilde | first1=Mark M. | last2=Brun | first2=Todd A. | title=Entanglement-assisted quantum convolutional coding | journal=Physical Review A | volume=81 | issue=4 | date=2010-04-30 | issn=1050-2947 | doi=10.1103/physreva.81.042333 | page=042333|arxiv=0712.2223| bibcode=2010PhRvA..81d2333W | s2cid=8410654 }}
• {{cite journal | last1=Wilde | first1=Mark M. | last2=Brun | first2=Todd A. | title=Quantum convolutional coding with shared entanglement: general structure | journal=Quantum Information Processing | publisher=Springer Science and Business Media LLC | volume=9 | issue=5 | date=2010-06-08 | issn=1570-0755 | doi=10.1007/s11128-010-0179-9 | pages=509–540|arxiv=0807.3803| s2cid=18185704 }}
• {{cite journal | last1=Shaw | first1=Bilal | last2=Wilde | first2=Mark M. | last3=Oreshkov | first3=Ognyan | last4=Kremsky | first4=Isaac | last5=Lidar | first5=Daniel A. |author-link5=Daniel Lidar| title=Encoding one logical qubit into six physical qubits | journal=Physical Review A | volume=78 | issue=1 | date=2008-07-18 | issn=1050-2947 | doi=10.1103/physreva.78.012337 | page=012337|arxiv=0803.1495| bibcode=2008PhRvA..78a2337S | s2cid=40040752 }}

{{Quantum computing}}

Category:Linear algebra