Quantum Information Conic Solver

QICS is a primal-dual interior point solver fully implemented in Python, and is specialized towards problems arising in quantum information theory. QICS solves conic programs of the form

\[\min_{x \in \mathbb{R}^n} \quad c^\top x \quad \text{s.t.} \quad b - Ax = 0, \ h - Gx \in \mathcal{K},\]

where \(c\in\mathbb{R}^n\), \(b\in\mathbb{R}^p\), \(h\in\mathbb{R}^q\), \(A\in\mathbb{R}^{p\times n}\), \(G\in\mathbb{R}^{q\times n}\), and \(\mathcal{K}\subset\mathbb{R}^{q}\) is a Cartesian product of convex cones. Some notable cones that QICS supports include:

Cone	QICS class	Description
Positive semidefinite	qics.cones.PosSemidefinite	\(\{ X \in \mathbb{H}^n : X \succeq 0 \}\)
Quantum entropy	qics.cones.QuantEntr	\(\text{cl}\{ (t, u, X) \in \mathbb{R} \times \mathbb{R}_{++} \times \mathbb{H}^n_{++} : t \geq -u S(u^{-1} X) \}\)
Quantum relative entropy	qics.cones.QuantRelEntr	\(\text{cl}\{ (t, X, Y) \in \mathbb{R} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq S(X \| Y) \}\)
Quantum conditional entropy	qics.cones.QuantCondEntr	\(\text{cl}\{ (t, X) \in \mathbb{R}\times\mathbb{H}^{\Pi_in_i}_{++}: t \geq -S(X) + S(\text{tr}_i(X)) \}\)
Quantum key distribution	qics.cones.QuantKeyDist	\(\text{cl}\{ (t, X) \in \mathbb{R} \times \mathbb{H}^n_{++} : t \geq -S(\mathcal{G}(X)) + S(\mathcal{Z}(\mathcal{G}(X))) \}\)
Operator perspective epigraph	qics.cones.OpPerspecEpi	\(\text{cl}\{ (T, X, Y) \in \mathbb{H}^n \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : T \succeq P_g(X, Y) \}\)
\(\alpha\)-Renyi entropy, for \(\alpha\in[0, 1)\)	qics.cones.RenyiEntr	\(\text{cl} \{ (t, u, X, Y) \in \mathbb{R} \times \mathbb{R}_{++} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq u D_\alpha(u^{-1}X \| u^{-1}Y) \}\)
Sandwiched \(\alpha\)-Renyi entropy, for \(\alpha\in[1/2, 1)\)	qics.cones.SandRenyiEntr	\(\text{cl} \{ (t, u, X, Y) \in \mathbb{R} \times \mathbb{R}_{++} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq u \hat{D}_\alpha(u^{-1}X \| u^{-1}Y) \}\)
\(\alpha\)-Quasi-relative entropy, for \(\alpha\in[-1, 2]\)	qics.cones.QuasiEntr	\(\text{cl} \{ (t, X, Y) \in \mathbb{R} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq \pm \text{tr}[ X^\alpha Y^{1-\alpha} ] \}\)
Sandwiched \(\alpha\)-quasi-relative entropy, for \(\alpha\in[1/2, 2]\)	qics.cones.SandQuasiEntr	\(\text{cl} \{ (t, X, Y) \in \mathbb{R} \times \mathbb{H}^n_{++} \times \mathbb{H}^n_{++} : t \geq \pm \text{tr}[ ( Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}} )^\alpha ] \}\)

where we define the following functions

• Quantum entropy: \(S(X)=-\text{tr}[X\log(X)]\)

• Quantum relative entropy: \(S(X \| Y)=\text{tr}[X\log(X) - X\log(Y)]\)

• Noncommutative perspective: \(P_g(X, Y)=X^{1/2} g(X^{-1/2} Y X^{-1/2}) X^{1/2}\)

• \(\alpha\)-Renyi entropy: \(D_\alpha(X\|Y)=\frac{1}{1-\alpha} \log(\text{tr}[X^\alpha Y^{1-\alpha}])\)

• Sandwiched \(\alpha\)-Renyi entropy: \(\hat{D}_\alpha(X \| Y) = \frac{1}{1-\alpha} \log(\text{tr}[ (Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}})^\alpha ])\)

The full list of supported cones can be found here.

Features

• Efficient quantum relative entropy programming

  We support optimizing over the quantum relative entropy cone, as well as related cones including the quantum conditional entropy cone, as well as slices of the quantum relative entropy cone that arise when solving quantum key rates of quantum cryptographic protocols. Numerical results show that QICS solves problems much faster than existing quantum relative entropy programming solvers, such as Hypatia, DDS, and CVXQUAD.

• Efficient semidefinite programming

  We implement an efficient semidefinite programming solver which utilizes state-of-the-art techniques for symmetric cone programming, including using Nesterov-Todd scalings and exploiting sparsity in the problem structure. Numerical results show that QICS has comparable performance to state-of-the-art semidefinite programming software, such as MOSEK, SDPA, SDPT3 and SeDuMi.

• Complex-valued matrices

  Users can specify whether cones involving symmetric matrices, such as the positive semidefinite cone or quantum relative entropy cone, are real-valued or complex-valued (i.e., Hermitian). Support for Hermitian matrices is embedded directly in the definition of the cone, which can be more computationally efficient than lifting into the real-valued symmetric cone.

Installation

QICS is currently supported for Python 3.8 or later, and can be directly installed from pip by calling

pip install qics

Note that the performance of QICS is highly dependent on the version of BLAS and LAPACK that Numpy and SciPy are linked to.

PICOS interface

The easiest way to use QICS is through the Python optimization modelling interface PICOS, which can be installed using

pip install picos

Below, we show how a simple nearest correlation matrix problem can be solved.

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)

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

Some additional details about how to use QICS with PICOS can be found here.

Native interface

Alternatively, advanced users can use the QICS’ native interface, which provides additional flexibilty in how the problem is parsed to the solver. Below, we show how the same nearest correlation matrix problem can be solved using QICS’ native interface.

import numpy
import qics

# Define the conic program
c = numpy.array([[1., 0., 0., 0., 0., 0., 0., 0., 0.]]).T
A = numpy.array([
   [0., 1., 0., 0., 0., 0., 0., 0., 0.],
   [0., 0., 1., 1., 0., 0., 0., 0., 0.],
   [0., 0., 0., 0., 1., 0., 0., 0., 0.],
   [0., 0., 0., 0., 0., 1., 0., 0., 0.],
   [0., 0., 0., 0., 0., 0., 0., 0., 1.]
])
b = numpy.array([[2., 2., 2., 1., 1.]]).T
cones = [qics.cones.QuantRelEntr(2)]
model = qics.Model(c=c, A=A, b=b, cones=cones)

# Solve the conic program
solver = qics.Solver(model)
info = solver.solve()

Additional details explaining this example can be found here.

Citing QICS

If you find our work useful, please cite our paper using:

@misc{he2024qics,
  title={{QICS}: {Q}uantum Information Conic Solver},
  author={Kerry He and James Saunderson and Hamza Fawzi},
  year={2024},
  eprint={2410.17803},
  archivePrefix={arXiv},
  primaryClass={math.OC},
  url={https://arxiv.org/abs/2410.17803},
}

If you found our sandwiched Renyi and quasi-relative entropy cones useful, please cite out paper using:

@misc{he2025operator,
  title={Operator convexity along lines, self-concordance, and sandwiched {R}\'enyi entropies},
  author={Kerry He and James Saunderson and Hamza Fawzi},
  year={2025},
  eprint={2502.05627},
  archivePrefix={arXiv},
  primaryClass={math.OC},
  url={https://arxiv.org/abs/2502.05627},
}