(Sub)Exponential Quantum Speedup for Optimization

TL;DR

This work establishes provable quantum advantages for optimization by constructing oracle-based objective families in both discrete and continuous settings. It leverages the GHV subexponential oracle separation for adiabatic quantum computing and bridges to continuous optimization via Quantum Hamiltonian Descent (QHD), by mapping adiabatic paths to time-linear transverse-field diagonal forms and then to Schrödinger-operator evolutions. The core technical contribution is a sequence of perturbative gadget-based reductions that preserve spectral gaps while converting stoquastic sparse Hamiltonians into tractable, linear TFIs and eventually into the QHD framework, enabling polynomial-time quantum procedures to find global minimizers where classical methods require exponential queries. The continuous extension shows that the same hard-discrete instances yield exponential quantum advantage in the continuous domain through a Schrödinger-operator embedding, highlighting a unified approach to quantum optimization with potential practical implications for quantum speedups in real-world tasks.

Abstract

We demonstrate provable (sub)exponential quantum speedups in both discrete and continuous optimization, achieved through simple and natural quantum optimization algorithms, namely the quantum adiabatic algorithm for discrete optimization and quantum Hamiltonian descent for continuous optimization. Our result builds on the Gilyén--Hastings--Vazirani (sub)exponential oracle separation for adiabatic quantum computing. With a sequence of perturbative reductions, we compile their construction into two standalone objective functions, whose oracles can be directly leveraged by the plain adiabatic evolution and Schrödinger operator evolution for discrete and continuous optimization, respectively.

Figures (4)

• Figure 1: Overview diagram of the proof structure illustrating the logical dependencies among the main lemmas and theorems. The dashed lines separate major steps, which are covered in the corresponding subsections of \ref{['sec:overview']}.
• Figure 4: Spectrum of $A(t)$ for different $\lambda$ values. A small but nonzero $\lambda$ perturbs the spectrum, ensuring an $\Theta(\lambda)$ spectral gap throughout and making the instantaneous ground state transition of $A(t)$ smooth.
• Figure 5: Visualizations of $|\chi_0\rangle,|\chi_1\rangle$ regarding $\widehat{X}$ and their linear combination $|\widehat{0}\rangle,|\widehat{1}\rangle$ defined in \ref{['eq:hatzeroone-def-ov']}. The parameter for $\widehat{X}$ is chosen to be $\lambda = 30$. The potential field $f_{\rm dw}$ (amplified 10 times) is plotted in a dotted line for reference.
• Figure 6: The Construction of $\widehat{D}$ from a diagonal Hamiltonian $D$ with parameter $w = 1/4$ in \ref{['def:hatD']}. It is efficient in the sense that a query to $\widehat{D}(\cdot)$ can be implemented by one query to $D$.

Theorems & Definitions (55)

• Theorem 1.1: Informal version of \ref{['thm:main']}
• Theorem 1.2: Informal version of \ref{['thm:qhd']}
• Definition 3.2: Interaction graph and interaction constraint
• Definition 3.3: Hamiltonian simulation bravyi2017complexitycubitt2018universal
• Remark
• Lemma 3.4: Ground state simulation bravyi2017complexity
• Lemma 3.8: bravyi2017complexity
• Lemma 3.9: First-order reduction bravyi2017complexity
• Lemma 3.10: Second-order reduction bravyi2017complexity
• Theorem 4.1: Formal version of \ref{['thm:main_informal']}