Quantum Power Flows: From Theory to Practice

TL;DR

This paper suggests potential, exponential quantum speedup by the use of the Harrow-Hassidim-Lloyd (HHL) algorithms for solving sparse linear systems of equations in Newton’s method of power-flow problems.

Abstract

Climate change is becoming one of the greatest challenges to the sustainable development of modern society. Renewable energies with low density greatly complicate the online optimization and control processes, where modern advanced computational technologies, specifically quantum computing, have significant potential to help. In this paper, we discuss applications of quantum computing algorithms toward state-of-the-art smart grid problems. We suggest potential, exponential quantum speedup by the use of the Harrow-Hassidim-Lloyd (HHL) algorithms for sparse matrix inversions in power-flow problems. However, practical implementations of the algorithm are limited by the noise of quantum circuits, the hardness of realizations of quantum random access memories (QRAM), and the depth of the required quantum circuits. We benchmark the hardware and software requirements from the state-of-the-art power-flow algorithms, including QRAM requirements from hybrid phonon-transmon systems, and explicit gate counting used in HHL for explicit realizations. We also develop near-term algorithms of power flow by variational quantum circuits and implement real experiments for 6 qubits with a truncated version of power flows.

Figures (9)

• Figure 1: Sparsity of Power-Flow Jacobian Matrix
• Figure 2: Condition Number of Power-Flow Jacobian Matrix obtained from different steps of Newton-Raphson algorithm. The red line segment represents the medium value for each testing case. The bottom and top edges of the blue box indicate 25 and 75 percentiles, respectively. Red points are the outliers that are more than 50% larger than the top of the blue box shown by the black line. Typically, the condition number increases at first and then declines along the iteration sequence.
• Figure 3: Bounds on the error rate $\varepsilon$ from quantum power-flow problem. Here we compute the required error rate from the precision of the problem $1-F$ and the size of the data $N$ assuming the time efficiency $T = \log N$. The region inside the red box is the state-of-the-art requirement for modern power flows in the smart grid.
• Figure 4: Bounds on the precision $1-F$ of the quantum power-flow problem from physical decoherence rate $\kappa+\gamma$. Here we assume $g_d=1 \text{ kHz} \times 2\pi$, $\nu = 10 \text{ MHz} \times 2\pi$, and $c_d =4.5$ which is the average of the CZ and SWAP gates inside the QRAM circuit as an estimate. The red line emphasizes the typical matrix scales used in the modern power-flow problem in the smart grid study.
• Figure 5: An example of the statistics of the LCU coefficients in the power-flow problem. Here we have 102 different $64 \times 64$ Hermitian matrices from the real power-flow problem. In this case, we have $835\pm 8$ non-zero coefficients in the LCU decomposition. The Probability Distribution Function (PDF) of the nonzero coefficients is shown.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2: Informal
• Theorem 3