PICOS for Quantum Information Science¶
PICOS was among the first convex optimization interfaces to natively support Hermitian semidefinite programming and subsystem manipulation operations such as the partial trace and partial transpose, which were implemented with feedback from the QIS community. This note outlines the features most relevant for the field and links to examples.
Cheat sheet¶
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trace norm |
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( and
denote the partial
trace and transpose of the
-th
subsystem, counted
from zero)
Hermitian semidefinite programming¶
PICOS makes use of the following identity to allow standard solvers to deal with hermitian LMIs:
Hermitian variables are vectorized such that is
passed to solvers via
real scalar variables. Alternatively, the
QICS solver is able
to directly handle hermitian variables.
Quantum relative entropy programming¶
As of version 2.5.0, PICOS supports solving quantum relative entropy programs with the solver QICS. A list of new expressions supported by PICOS and QICS is summarized below.
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Some examples for how to solve quantum relative entropy programs using PICOS can
be found here. Note that these functions are supported for both
real symmetric and complex hermitian matrices and
.
Examples and exercises¶
Quantum channel discrimination (exercise on Binder)
Course material¶
Jupyter notebooks for a hands-on workshop on practical semidefinite programming aimed at quantum information students are available on GitLab. The fourth notebook is based on [2], which also comes with Python/PICOS notebooks.
Recent articles¶
The following are peer-reviewed articles relating to quantum information that cite PICOS and were published within the last four years (last update: October 2024).
Vikesh Siddhu and John Smolin.
Maximum expectation of observables with restricted purity states.
Aby Philip, Soorya Rethinasamy,Vincent Russo, and M. Wilde.
Schrödinger as a quantum programmer: estimating entanglement via steering.
Piotr Mironowicz.
Semi-definite programming and quantum information.
Journal of Physics A: Mathematical and Theoretical 57, 2024. [pdf] [doi] [arXiv]
Yu Shi and Edo Waks.
Error metric for non-trace-preserving quantum operations.
Vincent Russo and Jamie Sikora.
Inner products of pure states and their antidistinguishability.
Armin Tavakoli, Alejandro Pozas-Kerstjens, Ming-Xing Luo, and Marc-Olivier Renou.
Bell nonlocality in networks.
Reports on Progress in Physics 85, 2022. [pdf] [doi] [arXiv]
Feng-Jui Chan et al.
Maxwell’s two-demon engine under pure dephasing noise.
Viktor Nordgren et al.
Certifying emergent genuine multipartite entanglement with a partially blind witness.
Vikesh Siddhu and Sridhar Tayur.
Five starter pieces: quantum information science via semidefinite programs.
Ulysse Chabaud, Pierre-Emmanuel Emeriau, and Frédéric Grosshans.
Witnessing Wigner negativity.
Ncpol2sdpa¶
Ncpol2sdpa [1] exposes SDP relaxations of (non-commutative) polynomial optimization problems as PICOS problem instances, see here.
References¶
Peter Wittek. Algorithm 950: Ncpol2sdpa—sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting Variables. ACM Transactions on Mathematical Software, 41(3), 21, 2015. DOI: 10.1145/2699464. arXiv: 1308.6029.
Vikesh Siddhu and Sridhar Tayur. Five starter pieces: quantum information science via semi-definite programs. Tutorials in Operations Research, 2022. DOI: 10.1287/educ.2022.0243. arXiv: 2112.08276.