It's Quick to be Square: Fast Quadratisation for Quantum Toolchains

TL;DR

This paper considers a specific class of higher-level representations, that is, PUBOs, and devise novel automatic transformation mechanisms into widely used QUBOs that substantially improve efficiency and versatility over the state of the art.

Abstract

Many of the envisioned use-cases for quantum computers involve optimisation processes. While there are many algorithmic primitives to perform the required calculations, all eventually lead to quantum gates operating on quantum bits, with an order as determined by the structure of the objective function and the properties of target hardware. When the structure of the problem representation is not aligned with structure and boundary conditions of the executing hardware, various overheads degrading the computation may arise, possibly negating any possible quantum advantage. Therefore, automatic transformations of problem representations play an important role in quantum computing when descriptions (semi-)targeted at humans must be cast into forms that can be ``executed'' on quantum computers. Mathematically equivalent formulations are known to result in substantially different non-functional properties depending on hardware, algorithm and detail properties of the problem. Given the current state of noisy intermediate-scale quantum (NISQ) hardware, these effects are considerably more pronounced than in classical computing. Likewise, efficiency of the transformation itself is relevant because possible quantum advantage may easily be eradicated by the overhead of transforming between representations. In this paper, we consider a specific class of higher-level representations, that is, PUBOs, and devise novel automatic transformation mechanisms into widely used QUBOs that substantially improve efficiency and versatility over the state of the art. In addition, we conduct a comprehensive investigation of industry-relevant problem formulations and their conversion into a quantum-specific representation, identifying significant obstacles in scaling behaviour and demonstrating how these can be circumvented.

Figures (24)

• Figure 1: Multi-graph representation of $f(x_1, \ldots, x_6) = \pi x_1x_2x_3 -13 \textcolor{lfdblue}{x_2x_4x_5x_6} + 7 \textcolor{lfdyellow}{x_1x_3}$, where edge labels correspond to monomial indices according to \ref{['tab:QuarkPolyDict']}.
• Figure 2: Known reduction (black), introduced construction (blue) and graph evolution (yellow).
• Figure 3: Shows the inductive graph evolution for changing edges from $G_{f_t}$ to $G_{f_{t+1}}$. The penalty term induced edges $\{(i,h,z_k), (j,h,z_{k+1}), (i,j,z_{k+2})\}$ for a fitting $k\in \mathbb{N}$ can be excluded from $G_{f_{t+1}}$ (see \ref{['thm:PenaltyInvariant']}) and are not shown.
• Figure 4: Time in seconds (y-axis) for the quadratisation of a $\operatorname{deg}(f)= 4$ function $f$ vs problem size (x-axis: Number of variables). New lsr [lsr]lsr lsr algorithm (top row) compared to existing monomial-based (bottom row). Vertical facets: Different base polynomial densities (i.e., $d_1(f) = d_2(f) = d_3(f) = d_4(f) \in \{0.2, 0.4, \ldots , 1.0\}$). Selection type $1.0$ ( lsr [lsr]lsr lsr algorithm) comparable to Dense / better (monomial-based). The Dense / better and Medium type are cut off at $39$ variables prior to reduction, which limits the runtime.
• Figure 5: Time in seconds (y-axis) for the quadratisation of a $\operatorname{deg}(f)= 4$ function $f$ vs problem size (x-axis: Number of Terms). New lsr [lsr]lsr lsr algorithm (top row) compared to existing monomial-based (bottom row). Vertical facets: Different base polynomial densities (i.e., $d_1(f) = d_2(f) = d_3(f) = d_4(f) \in \{0.2, 0.4, \ldots , 1.0\}$). Selection type $1.0$ ( lsr [lsr]lsr lsr algorithm) comparable to Dense / better (monomial-based). The Dense / better and Medium type are cut off at $39$ variables prior to reduction, which limits the runtime.