PICOS interface¶

The easiest way to use QICS is by using the optimization modelling software PICOS, which provides a high-level interface to parse convex optimization problems to a solver. Notably, PICOS supports the following functions, which can be used to formulate conic problems to be solved using QICS.

**Quantum entropy and noncommutative perspectives supproted by PICOS**¶
Function	PICOS expression	Convexity	Description
Quantum entropy	`picos.quantentr()`	Concave	\(S(X) = -\text{tr}[X\log(X)]\)
Quantum relative entropy	`picos.quantrelentr()`	Convex	\(S(X \\| Y) = \text{tr}[X\log(X) - X\log(Y)]\)
Quantum conditional entropy	`picos.quantcondentr()`	Concave	\(S(X) - S(\text{tr}_i(X))\)
Quantum key distribution	`picos.quantkeydist()`	Convex	\(-S(\mathcal{G}(X)) + S(\mathcal{Z}(\mathcal{G}(X)))\)
Operator relative entropy	`picos.oprelentr()`	Operator convex	\(-P_{\log}(X, Y) = X^{1/2} \log(X^{1/2} Y^{-1} X^{1/2}) X^{1/2}\)
Matrix geometric mean	`picos.mtxgeomean()`	Operator convex if \(t\in[-1, 0]\cup[1, 2]\) Operator concave if \(t\in[0, 1]\)	\(X\,\#_t\,Y = X^{1/2} (X^{-1/2} Y^{-1} X^{-1/2})^t X^{1/2}\)
Renyi entropy	`picos.renyientr()`	Operator convex for \(\alpha\in[0, 1)\)	\(D_\alpha(X \\| Y) = \frac{1}{1-\alpha} \log(\text{tr}[X^\alpha Y^{1-\alpha}])\)
Sandwiched Renyi entropy	`picos.sandrenyientr()`	Operator convex for \(\alpha\in[1/2, 1)\)	\(\hat{D}_\alpha(X \\| Y) = \frac{1}{1-\alpha} \log(\text{tr}[ (Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}})^\alpha ])\)
Quasi-relative entropy	`picos.quasientr()`	Concave for \(\alpha\in[0, 1]\) Convex for \(\alpha\in[-1, 0]\cup[1, 2]\)	\(\text{tr}[ X^\alpha Y^{1-\alpha} ]\)
Sandwiched quasi-relative entropy	`picos.sandquasientr()`	Concave for \(\alpha\in[1/2, 1]\) Convex for \(\alpha\in[1, 2]\)	\(\text{tr}[ ( Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}} )^\alpha ]\)

Scalar functions (i.e., quantum entropy, quantum relative entropy, quantum conditional entropy, and quantum key distribution) can be used by either incorporating them in the objective function, e.g.,

P.set_objective("min", picos.quantrelentr(X, Y))

or as an inequality constraint, e.g.,

P.add_constraint(t > picos.quantrelentr(X, Y))

Matrix-valued functions (i.e., operator relative entropy and matrix geometric mean) can be used in a matrix inequality expression, e.g.,

P.add_constraint(T >> picos.oprelentr(X, Y))

or composed with a trace function to represent the corresponding scalar valued function

P.set_objective("min", picos.trace(picos.oprelentr(X, Y)))

Note that these expressions need to define a convex optimization problem. Once a PICOS problem has been defined, it can be solved using QICS by calling

P.solve(solver="qics")

Example¶

Below, we show an example of how we can solve the same problem nearest correlation matrix problem introduced in Getting started, i.e.,

\[\min_{Y \in \mathbb{S}^2} \quad S( X \| Y ) \quad \text{s.t.} \quad Y_{11} = Y_{22} = 1, \ Y \succeq 0,\]

where

\[\begin{split}X = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}.\end{split}\]

import picos

# Define the conic program
P = picos.Problem()
X = picos.Constant("X", [[2., 1.], [1., 2.]])
Y = picos.SymmetricVariable("Y", 2)

P.set_objective("min", picos.quantrelentr(X, Y))
P.add_constraint(picos.maindiag(Y) == 1)

print(P)

# Solve the conic program
P.solve(solver="qics")

print("\nOptimal matrix variable Y is:")
print(Y)

Quantum Relative Entropy Program
  minimize S(X‖Y)
  over
    2×2 symmetric variable Y
  subject to
    maindiag(Y) = [1]

Optimal matrix variable Y is:
[ 1.00e+00  5.00e-01]
[ 5.00e-01  1.00e+00]

Further examples for how PICOS can be used with QICS to solve problems arising in quantum information theory can be found in Quantum relative entropy programming.