Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end

Abstract

We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses ($p$-spin model), and $k$-local constraint satisfaction problems ($k$-CSP). We show that either (a) the algorithm finds the optimal solution in time $O^*(2^{(0.5-c)n})$ for an $n$-independent constant $c$, a $2^{cn}$ advantage over Grover's algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a $(1-η)$ approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of $η$. Additionally, we show that for a large fraction of random instances from the $k$-spin model and for any sufficiently close-to-regular, fully satisfiable (or slightly frustrated) $k$-CSP formula, statement (a) is the case. The algorithm and its analysis is largely inspired by Hastings' short-path algorithm [$\textit{Quantum}$ $\textbf{2}$ (2018) 78].

Figures (3)

• Figure 1: Plot of the lowest three eigenvalues of $H_b$ as a function of $b$, for an $n=20$ instance randomly chosen from the $3$-spin ensemble, with $\eta = 0.5$. Eigenvalues were computed numerically using exact diagonalization. The two key features are that the spectral gap remains large and the ground state energy barely shifts from $-1$ until a relatively large value of $b$, namely $b \approx 0.85$.
• Figure 2: Plot of the relevant overlap values for the same $n=20$, $\eta=0.5$ 3-spin instance from Fig. \ref{['fig:spectrum_n20']}. Overlaps are determined by exact diagonalization of $H_b$. The overlap $\lvert\left\langle\pmb{+} | \psi_b\right\rangle\rvert$ (blue) determines the runtime of step 2 of Algorithm \ref{['algo:main_simple']}, and the overlap $\lvert\left\langle z^* | \psi_b\right\rangle\rvert$ (red) determines the runtime of step 3.
• Figure 3: Plot of $\lvert \left\langle z^* | \psi_b\right\rangle \rvert ^{-1}$ for several randomly chosen instances from the $3$-spin model at $b=0.7$ and $\eta = 0.5$ at values of $n$ ranging from $n=17$ to $n=23$. For each value of $n$, the median value is also plotted. A fit of the medians to an exponential yields the fit $\lvert \left\langle z^* | \psi_b\right\rangle \rvert ^{-1} = 0.276 \cdot 2^{0.427n}$.

Theorems & Definitions (65)

• Proposition 1: jump from $K_1 \rightarrow K_2$, simplified
• Theorem 1: runtime
• proof
• Definition 1: $\alpha$-depolarizing
• Proposition 2
• proof
• Definition 2: $\alpha$-subdepolarizing
• Proposition 3
• Lemma 2
• proof