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Quantum search algorithm for similar subgraph identification under fixed edge removal

Ruben Kara, Sven Danz, Tobias Stollenwerk, Andrea Benigni

Abstract

We introduce a novel quantum algorithm for similar subgraph identification in form of an NP-hard cardinality-constrained binary quadratic optimization problem. Given a weighted reference graph with Laplacian $\boldsymbol{B}$, our algorithm determines the subgraph featuring Laplacian $\boldsymbol{B'}$ on the same vertex set, but $x$ out of $N$ inactive edges, minimizing the Frobenius distance $||\boldsymbol{B} - \boldsymbol{B'}||_\mathrm{F}^2$. We represent the $\binom{N}{x}$ graph topologies by an equal-weight superposition in form of a Dicke state, enabling controlled transformations applied to the quantum state associated with the vectorized Laplacian of the reference graph. Combined with amplitude estimation and a minimum finding approach, our algorithm provides a polynomial speed up $\mathcal{O}(\sqrt{N^{x}/x!}N\log\log N)$ compared to $\mathcal{O}(N^{x+1}/x!)$ of classical brute-force search algorithms. We demonstrate the application of our method on standard test cases, which represent electric power grids, by reconstructing $||\boldsymbol{B} -\boldsymbol{B'}||_\mathrm{F}^2$ from measurements and show how our approach can be additionally used to calculate energy functional like quadratic forms of the Laplacians with respect to a given vector.

Quantum search algorithm for similar subgraph identification under fixed edge removal

Abstract

We introduce a novel quantum algorithm for similar subgraph identification in form of an NP-hard cardinality-constrained binary quadratic optimization problem. Given a weighted reference graph with Laplacian , our algorithm determines the subgraph featuring Laplacian on the same vertex set, but out of inactive edges, minimizing the Frobenius distance . We represent the graph topologies by an equal-weight superposition in form of a Dicke state, enabling controlled transformations applied to the quantum state associated with the vectorized Laplacian of the reference graph. Combined with amplitude estimation and a minimum finding approach, our algorithm provides a polynomial speed up compared to of classical brute-force search algorithms. We demonstrate the application of our method on standard test cases, which represent electric power grids, by reconstructing from measurements and show how our approach can be additionally used to calculate energy functional like quadratic forms of the Laplacians with respect to a given vector.

Paper Structure

This paper contains 22 sections, 1 theorem, 68 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results
  3. Algorithm
  4. Complexity analysis
  5. Runtime analysis
  6. Memory requirements
  7. Comparison to a classical approach
  8. Numerical Proof-of-Concept
  9. Discussion
  10. Methods
  11. Encoding grid topologies into quantum states
  12. Single edge removal
  13. Multiple edge removal
  14. Vectorizing matrices to define quantum states
  15. Block encoding of the incidence matrix
  16. ...and 7 more sections

Key Result

Theorem 1

Let $G$ be a graph with $N$ weighted edges from which we form all subgraphs by deactivating a fixed number $x$ of edges. The configurations of activated edges and with it the subgraphs are denoted ${\boldsymbol{\tilde{d}}}$. There exists a quantum algorithm with internal precision $\log \varepsilon^ steps using at most qubits. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Double logarithmic plot showing runtime comparison between our , with runtime $t_\mathrm{min}=\mathcal{O}(\sqrt{N^x/x!}N\log\log N )$, and a state-of-the-art classical alternative based on the computation of the Frobenius distance for all $S$ Dicke states, with a runtime $t_\mathrm{cla}=\mathcal{O}(N^{x+1}/x!)$. Here, $N$ denotes the total number of edges in our graph and $x$ the number of inactive ones. For simplicity, we assume a constant precision $\log\varepsilon^{-1}$ for both the classical and quantum approach.
  • Figure 2: Distance calculated with our (quantum) approach compared to the classical approach for selected subgraph configurations of the IEEE-9 instance with two inactive edges ($x=2$) using $S_\mathrm{total}=1\mathrm{E}8$ shots in total. The reference graph is shown in black on the left side. For the subgraphs, green indicates the inactivity of the corresponding edge and blue the activity. The corresponding binaries $d_0 d_1 \dots d_8$ indicating the configuration i.e. topology of the subgraph are shown below.
  • Figure 3: Precision of the quantum sampling approach given via the absolute difference $\Delta_x$ (see \ref{['eq:diff-quantum-classical-measure']}) in the sum of calculated distances between our quantum method and the (exact) classical approach, against the number of samples from the quantum circuit simulation for the IEEE-4 instance with two inactive edges ($x=2$). Double logarithmic presentation of the axes are displayed. We show the mean of 10 different random seeds for the measurement simulation as well as different percentiles. The grey line indicates the expected behavior of $\mathcal{O}(1/\sqrt{S_\mathrm{total}})$ for $S_\mathrm{total}$ number of shots, as a guide to the eye.
  • Figure 4: Illustration of the topology controlled operation for $N=4$ via the multi controlled X-gate. Circuit corresponds to example presented in \ref{['eq:ctrl_state_a', 'eq:ctrl_state_b', 'eq:ctrl_state_c', 'eq:ctrl_state_d']} where $\ket{i}=\ket{i_1} \ket{i_2}$.

Theorems & Definitions (1)

  • Theorem 1