Solving Conic Programs over Sparse Graphs using a Variational Quantum Approach: The Case of the Optimal Power Flow

TL;DR

This work proposes a variational quantum paradigm for solving conic programs, including quadratically constrained quadratic programs (QCQPs) and semidefinite programs (SDPs), and proposes permuting the primal variables so that related observables are expressed in a banded form, enabling efficient measurement.

Abstract

Conic programs arise broadly in physics, quantum information, machine learning, and engineering, many of which are defined over sparse graphs. Although such problems can be solved in polynomial time using classical interior-point solvers, the computational complexity scales unfavorably with graph size. In this context, this work proposes a variational quantum paradigm for solving conic programs, including quadratically constrained quadratic programs (QCQPs) and semidefinite programs (SDPs). We encode primal variables via the state of a parameterized quantum circuit (PQC), and dual variables via the probability mass function of a second PQC. The Lagrangian function can thus be expressed as scaled expectations of quantum observables. A primal-dual solution can be found by minimizing/maximizing the Lagrangian over the parameters of the first/second PQC. We pursue saddle points of the Lagrangian in a hybrid fashion. Gradients of the Lagrangian are estimated using the two PQCs, while PQC parameters are updated classically using a primal-dual method. We propose permuting the primal variables so that related observables are expressed in a banded form, enabling efficient measurement. The proposed framework is applied to the OPF problem, a large-scale optimization problem central to the operation of electric power systems. Numerical tests on the IEEE 57-node power system using Pennylane's simulator corroborate that the proposed doubly variational quantum framework can find high-quality OPF solutions. Although showcased for the OPF, this framework features a broader scope, including conic programs with numerous variables and constraints, problems defined over sparse graphs, and training quantum machine learning models to satisfy constraints.

Figures (7)

• Figure 1: Workflow of approximately solving \ref{['eq:qc']} using a doubly variational approach. The primal PQC (bottom) parameterizes the primal variables. The dual PQC (top) parameterizes the dual variables.
• Figure 2: The XBM method of Kondo2022 groups the entries of an $N\times N$ Hermitian matrix $\mathbf{M}_m$ into $N$ groups or colors. This XBM grouping or entry coloring is shown here for an $8\times 8$ matrix associated with $\log 8=3$ qubits. Each color $c$ is identified by the binary form of its index $c$. Color $c$ includes all matrix entries with index pairs $(i,j)$ for which $i\oplus j =c$. For example, the grey color $c=7=\ket{111}$ includes all entries lying on the main anti-diagonal of $\mathbf{M}_m$. Note that the color arrangement is symmetric.
• Figure 3: The bandwidth of the original and permuted measurement matrices as a function of the logarithmic network size $(\log N)$ for the power system graphs in pglib. Nodes were permuted by the RCM algorithm. Permuted matrices feature patently smaller bandwidths. Upon data fitting and 5-fold cross-validation over different polynomial and exponential functions, the reduced bandwidth was numerically found to scale as $(\log N)^3$. The run time of the RCM algorithm (implemented by function scipy.sparse.csgraph.reverse_cuthill_mckee in Python 3.11.9) on the largest power system graph of $N=78,784$ is a few seconds using a MacBook laptop equipped with an M3 Pro processor and 36 GB of RAM.
• Figure 4: The number of colors in the original and permuted measurement matrices as a function of the logarithmic network size $(\log N)$ for the power system graphs in pglib. Permuting nodes using the RCM algorithm was numerically found to significantly reduce the number of colors in use. Data fitting and 5-fold cross validation showed that the number of colors scales as $(\log N)^3$.
• Figure 5: Implementation of rotation unitaries. Top: A unitary $\mathbf{U}_c$ is recompiled $2C-1$ times to consider for $2C-1$ rotation unitaries running in sequence. Bottom:$2C-1$ rotation unitaries running in parallel.

Theorems & Definitions (9)

• Lemma 1
• Lemma 2: Kondo2022
• Lemma 3
• Theorem 1: minimax
• Lemma 4
• Proposition 1
• Lemma 5
• proof
• proof : Proof of Lemma \ref{['le:var']}