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¶

Complex expression manipulation¶
on paper	in picos
$A \in \mathbb{C}^{m \times n}$	`A = pc.ComplexVariable("A", (m, n))`
$A = B + iC$	`A = B + 1j*C`
$\overline{A}$	`A.conj`
$A^\dagger$	`A.H`
$\Re(A)$	`A.real`
$\Im(A)$	`A.imag`
$\frac{1}{2} \left( A + A^\dagger \right)$	`A.opreal` / `A.hermitianized`
$\frac{1}{2i} \left( A - A^\dagger \right)$	`A.opimag`
$\langle \phi \vert \psi \rangle$	`phi.H * psi` / `(phi \| psi)`
$\rvert \phi \rangle \langle \psi \lvert$	`phi * psi.H`

Hermitian semidefinite programming¶
on paper	in picos
$\rho \in \mathbb{S}^n$	`rho = pc.HermitianVariable("ρ", n)`
$\rho \succeq 0$	`rho >> 0`
$\rho \succeq I$	`rho >> 1` / `rho >> pc.I(n)`
$\operatorname{Tr}(\rho) = 1$	`rho.tr == 1`
$\begin{bmatrix}A & B \\ C & D\end{bmatrix} \succeq 0$	`pc.block([[A, B], [C, D]]) >> 0`

Schatten norms¶
on paper	in picos	note / aka
${\lVert A \rVert}_1 = \operatorname{Tr}\left( \sqrt{A^\dagger A} \right)$	`pc.NuclearNorm(A)`	trace norm
${\lVert A \rVert}_\infty = \sqrt{\lambda_{\max}(A^\dagger A)}$	`pc.SpectralNorm(A)`	$\lambda_{\max}(A)$ for $A \in \mathbb{H}^n$

Subsystem manipulation (partial trace, partial transpose, realignment)¶
on paper	in picos	note / docs
$A = B \otimes C$	`A = B @ C`
$A_1 \otimes \cdots \otimes \operatorname{Tr}(A_i) \otimes \cdots \otimes A_n$	`A.partial_trace([i-1], shapes)`	`partial_trace`
$A_1 \otimes \cdots \otimes A_i^T \otimes \cdots \otimes A_n$	`A.partial_tranpose([i-1], shapes)`	`partial_transpose`
$A_{ij\;\mapsto\;ji} = A^T$	`A.reshuffled("ji")`	`reshuffled`
$A_{ijkl\;\mapsto\;kjil} = \operatorname{T}_1(A)$	`A.reshuffled("kjil")`	`reshuffled`
$\operatorname{Tr}_1(A),\;\ldots{},\;\operatorname{Tr}_4(A),\;\operatorname{Tr}_\text{last}(A)$	`A.tr0`, …, `A.tr3`, `A.trl`	$A \in \mathbb{H}^2 \otimes \cdots \otimes \mathbb{H}^2$
$\operatorname{T}_1(A),\;\ldots{},\;\operatorname{T}_4(A),\;\operatorname{T}_\text{last}(A)$	`A.T0`, …, `A.T3`, `A.Tl`	$A \in \mathbb{H}^2 \otimes \cdots \otimes \mathbb{H}^2$

( $\operatorname{Tr}_i$ and $\operatorname{T}_i$ denote the partial trace and transpose of the $i$ -th $2 \times 2$ subsystem, counted from zero)

Hermitian semidefinite programming¶

PICOS makes use of the following identity to allow standard solvers to deal with hermitian LMIs:

$A \succeq 0 \qquad \Longleftrightarrow \qquad \begin{bmatrix} \Re(A) & \Im(A) \\ -\Im(A) & \Re(A) \end{bmatrix} \succeq 0$

Hermitian variables are vectorized such that $\rho \in \mathbb{S}^n$ is passed to solvers via $n^2$ 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.

Quantum entropies and non-commutative perspectives¶
on paper	in picos	docs
$S(X) = -\operatorname{Tr}(X\log(X))$	`pc.quantentr(X)`	`QuantumEntropy`
$S(X \\| Y) = \operatorname{Tr}(X\log(X) - X\log(Y))$	`pc.quantrelentr(X, Y)`	`NegativeQuantumEntropy`
$S(X) - S(\operatorname{Tr}_i(X))$	`pc.quantcondentr(X, [i-1], shapes)`	`QuantumConditionalEntropy`
$S(\mathcal{G}(X) \\| \mathcal{Z}(\mathcal{G}(X)))$	`pc.quantkeydist(X, [i-1], shapes, K_list)`	`QuantumKeyDistribution`
$P_{\log}(X, Y) = X^{1/2} \log(X^{1/2} Y^{-1} X^{1/2}) X^{1/2}$	`pc.oprelentr(X, Y)`	`OperatorRelativeEntropy`
$X\,\#_t\,Y = X^{1/2} (X^{1/2} Y^{-1} X^{1/2})^t X^{1/2}$	`pc.mtxgeomean(X, Y, t)`	`MatrixGeometricMean`
$\Psi_{\alpha}(X, Y) = \operatorname{Tr}[ X^\alpha Y^{1-\alpha} ]$	`pc.quasientr(X, Y)`	`QuasiEntropy`
$\hat{\Psi}_{\alpha}(X, Y) = \operatorname{Tr}[ (Y^\frac{1-\alpha}{2\alpha} X Y^\frac{1-\alpha}{2\alpha} )^\alpha ]$	`pc.sandquasientr(X, Y)`	`SandQuasiEntropy`
$D_\alpha(X \\| Y) = \frac{1}{\alpha - 1} \log(\Psi_\alpha(X, Y))$	`pc.renyientr(X, Y)`	`RenyiEntropy`
$\hat{D}_\alpha(X \\| Y) = \frac{1}{\alpha - 1} \log(\hat{\Psi}_\alpha(X, Y))$	`pc.sandrenyientr(X, Y)`	`SandRenyiEntropy`

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 $X$ and $Y$ .

Examples and exercises¶

Fidelity between operators

Quantum relative entropy programs

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.

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.

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