[DO NOT MERGE] Experiment with fixing PurifiedMixedStateWithData API doc

api/qsharp/microsoft.quantum.preparation.purifiedmixedstatewithdata.md

@@ -1,4 +1,4 @@
----
+---
 uid: Microsoft.Quantum.Preparation.PurifiedMixedStateWithData
 title: PurifiedMixedStateWithData function
 ms.date: 1/25/2022 12:00:00 AM
@@ -7,51 +7,104 @@ qsharp.kind: function
 qsharp.namespace: Microsoft.Quantum.Preparation
 qsharp.name: PurifiedMixedStateWithData
 qsharp.summary: "Returns an operation that prepares a a purification of a given mixed\rstate, entangled with a register representing a given collection of data.\rA \"purified mixed state with data\" refers to a state of the form Σᵢ √\U0001D45Dᵢ |\U0001D456⟩ |\U0001D465ᵢ⟩ |garbageᵢ⟩,\rwhere each \U0001D465ᵢ is a bitstring encoding additional data associated with the register |\U0001D456⟩.\r\rSee https://arxiv.org/pdf/1805.03662.pdf?page=15 for further discussion."
----
+---
 
 # PurifiedMixedStateWithData function
 
 Namespace: [Microsoft.Quantum.Preparation](xref:Microsoft.Quantum.Preparation)
-
-Package: [Microsoft.Quantum.Standard](https://nuget.org/packages/Microsoft.Quantum.Standard)
 
+Package: [Microsoft.Quantum.Standard](https://nuget.org/packages/Microsoft.Quantum.Standard)
 
-Returns an operation that prepares a a purification of a given mixedstate, entangled with a register representing a given collection of data.A "purified mixed state with data" refers to a state of the form Σᵢ √𝑝ᵢ |𝑖⟩ |𝑥ᵢ⟩ |garbageᵢ⟩,where each 𝑥ᵢ is a bitstring encoding additional data associated with the register |𝑖⟩.See https://arxiv.org/pdf/1805.03662.pdf?page=15 for further discussion.
+
+Returns an operation that prepares a a purification of a given mixed
+state, entangled with a register representing a given collection of data.
+A "purified mixed state with data" refers to a state of the form Σᵢ √𝑝ᵢ |𝑖⟩ |𝑥ᵢ⟩ |garbageᵢ⟩,
+where each 𝑥ᵢ is a bitstring encoding additional data associated with the register |𝑖⟩.
+
+See https://arxiv.org/pdf/1805.03662.pdf?page=15 for further discussion.
 
 ```qsharp
 function PurifiedMixedStateWithData (targetError : Double, coefficients : (Double, Bool[])[]) : Microsoft.Quantum.Preparation.MixedStatePreparation
 ```
-
-
-## Description
-
-Uses the Quantum ROM technique to represent a given density matrix,returning that representation as a state preparation operation.In particular, given a list of $N$ coefficients $\alpha_j$, and abitstring $\vec{x}_j$ associated with each coefficient, thisfunction returns an operation that uses the Quantum ROM technique toprepare an approximation$$\begin{align}\tilde\rho = \sum_{j = 0}^{N - 1} p_j \ket{j}\bra{j} \otimes \ket{\vec{x}_j}\bra{\vec{x}_j}\end{align}$$of the mixed state$$\begin{align}\rho = \sum_{j = 0}^{N-1}\ frac{|alpha_j|}{\sum_k |\alpha_k|} \ket{j}\bra{j} \otimes \ket{\vec{x}_j}\bra{\vec{x}_j},\end{align}$$where each $p_j$ is an approximation to the given coefficient $\alpha_j$such that$$\begin{align}\left| p_j - \frac{ |\alpha_j| }{ \sum_k |\alpha_k| } \right| \le \frac{\epsilon}{N}\end{align}$$for each $j$.When passed an index register and a register of garbage qubits,initially in the state $\ket{0} \ket{00\cdots 0}$, the returned operationprepares both registers into the purification of $\tilde \rho$,$$\begin{align}\sum_{j=0}^{N-1} \sqrt{p_j} \ket{j} \ket{\vec{x}_j} \ket{\text{garbage}_j},\end{align}$$such that resetting and deallocating the garbage register enacts thedesired preparation to within the target error $\epsilon$.
-
-## Input
-
-### targetError : [Double](xref:microsoft.quantum.qsharp.valueliterals#double-literals)
-
-The target error $\epsilon$.
-
-
-### coefficients : ([Double](xref:microsoft.quantum.qsharp.valueliterals#double-literals),[Bool](xref:microsoft.quantum.qsharp.valueliterals#bool-literals)[])[]
-
-Array of $N$ coefficients specifying the probability of basis states,along with the bitstring $\vec{x}_j$ associated with each coefficient.Negative numbers $-\alpha_j$ will be treated as positive $|\alpha_j|$.
-
-
-
-## Output : [MixedStatePreparation](xref:Microsoft.Quantum.Preparation.MixedStatePreparation)
-
-An operation that prepares $\tilde \rho$ as a purification onto a jointindex and garbage register.
-
-## Remarks
-
-The coefficients provided to this operation are normalized following the1-norm, such that the coefficients are always considered to describe avalid categorical probability distribution.
-
-## References
-
-- Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity  Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, Hartmut Neven  https://arxiv.org/abs/1805.03662
-
-## See Also
-
-- [Microsoft.Quantum.Preparation.PurifiedMixedState](xref:Microsoft.Quantum.Preparation.PurifiedMixedState)
+
+
+## Description
+
+Uses the Quantum ROM technique to represent a given density matrix,
+returning that representation as a state preparation operation.
+
+In particular, given a list of $N$ coefficients $\alpha_j$, and a
+bitstring $\vec{x}_j$ associated with each coefficient, this
+function returns an operation that uses the Quantum ROM technique to
+prepare an approximation
+
+$$
+\begin{align}
+\tilde\rho = \sum_{j = 0}^{N - 1} p_j \ket{j}\bra{j} \otimes \ket{\vec{x}_j}\bra{\vec{x}_j}
+\end{align}
+$$
+
+of the mixed state
+
+$$
+\begin{align}
+\rho = \sum_{j = 0}^{N-1} \frac{|\alpha_j|}{\sum_k |\alpha_k|} \ket{j}\bra{j} \otimes \ket{\vec{x}_j}\bra{\vec{x}_j},
+\end{align}
+$$
+
+where each $p_j$ is an approximation to the given coefficient $\alpha_j$
+such that
+
+$$
+\begin{align}
+\left| p_j - \frac{ |\alpha_j| }{ \sum_k |\alpha_k| } \right| \le \frac{\epsilon}{N}
+\end{align}
+$$
+for each $j$.
+
+When passed an index register and a register of garbage qubits,
+initially in the state $\ket{0} \ket{00\cdots 0}$, the returned operation
+prepares both registers into the purification of $\tilde \rho$,
+$$
+\begin{align}
+\sum_{j=0}^{N-1} \sqrt{p_j} \ket{j} \ket{\vec{x}_j} \ket{\text{garbage}_j},
+\end{align}
+$$
+such that resetting and deallocating the garbage register enacts the
+desired preparation to within the target error $\epsilon$.
+
+## Input
+
+### targetError : [Double](xref:microsoft.quantum.qsharp.valueliterals#double-literals)
+
+The target error $\epsilon$.
+
+
+### coefficients : ([Double](xref:microsoft.quantum.qsharp.valueliterals#double-literals),[Bool](xref:microsoft.quantum.qsharp.valueliterals#bool-literals)[])[]
+
+Array of $N$ coefficients specifying the probability of basis states,
+along with the bitstring $\vec{x}_j$ associated with each coefficient.
+Negative numbers $-\alpha_j$ will be treated as positive $|\alpha_j|$.
+
+
+
+## Output : [MixedStatePreparation](xref:Microsoft.Quantum.Preparation.MixedStatePreparation)
+
+An operation that prepares $\tilde \rho$ as a purification onto a joint
+index and garbage register.
+
+## Remarks
+
+The coefficients provided to this operation are normalized following the
+1-norm, such that the coefficients are always considered to describe a
+valid categorical probability distribution.
+
+## References
+
+- Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity
+  Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, Hartmut Neven
+  https://arxiv.org/abs/1805.03662
+
+## See Also
+
+- [Microsoft.Quantum.Preparation.PurifiedMixedState](xref:Microsoft.Quantum.Preparation.PurifiedMixedState)