QuasiTheory

QuasiTheory.CPTNDecompose(JF, dim_list)
\[\min\{p_1 + p_2:\mathcal{J}_F = p_1\mathcal{J}_1 - p_2\mathcal{J}_2\}\]
Parameters:

  • JF – The input matrix to be decomposed.

  • dim_list – List specifying the dimensions for the decomposition.

Returns:

J1: The choi matrix of the channel \(\mathcal{D}_1\).

J2: The choi matrix of the channel \(\mathcal{D}_2\).

p1: The sampling overhead of J1

p2: The sampling overhead of J2

Return type:

[matrix, matrix, numeric, numeric]

QuasiTheory.GammaPPT(JN, dim)
\[\gamma_{\operatorname{PPT}}(\mathcal{N}) = \min\{p_+ + p_-:\mathcal{N} = p_+\mathcal{M_+} - p_-\mathcal{M_-}\},\]
Parameters:

  • JN (numeric) – The Choi matrix of the bipartite channel.

  • dim (numeric) – The array storing input and output dimensions.

Returns:

The PPT-assisted sampling overhead of bipartite channel.

Return type:

numeric

Raises:

error – If either input/output dimension does not match, an error is raised.

Note

Jing, M., Zhu, C., & Wang, X. (2024). Circuit Knitting Faces Exponential Sampling Overhead Scaling Bounded by Entanglement Cost. arXiv preprint arXiv:2404.03619.

QuasiTheory.TCDecompose(JF, dim_list)
\[\min\{a:\mathcal{J}_F = a(\mathcal{J}_1 - \mathcal{J}_2)\}\]
Parameters:

  • JF – The input matrix to be decomposed.

  • dim_list – List specifying the dimensions for the decomposition.

Returns:

J1: The choi matrix of the channel \(\mathcal{D}_1\).

J2: The choi matrix of the channel \(\mathcal{D}_2\).

a: The sampling overhead

Return type:

[matrix, matrix, numeric]

QuasiTheory.VirtualRecovery(rho, dim)
\[R^v_{rec}(\rho_{ABC}) = \log\min\{c_1+c_2:(c_1\mathcal{N}_1 - c_2\mathcal{N}_2)(\rho_{AB}) = \rho_{ABC}, c_{1,2}\geq 0, \mathcal{N}_{1,2}\in\operatorname{CPTP}(B,B\otimes C)\}\]
Parameters:

  • JN (numeric) – The Choi matrix of the bipartite channel.

  • dim (numeric) – The array storing input and output dimensions.

Returns:

The virtual recovery of tripartite state.

Return type:

numeric

Raises:

error – If either input/output dimension does not match, an error is raised.

Note

Chen, Y. A., Zhu, C., He, K., Jing, M., & Wang, X. (2023). Virtual Quantum Markov Chains. arXiv preprint arXiv:2312.02031.