PurifiedMixedStateWithData function

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Warning

This documentation refers to the Classic QDK, which has been replaced by the Modern QDK.

Please see https://aka.ms/qdk.api for the API documentation for the Modern QDK.

Namespace: Microsoft.Quantum.Preparation

Package: Microsoft.Quantum.Standard

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.

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 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

The target error $\epsilon$.

coefficients : (Double,Bool[])[]

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

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