[Paper Review] Quantum adiabatic evolutions that can't be used to design efficient algorithms

Zhaohui Wei, Mingsheng Ying|ArXiv.org|Apr 11, 2006

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper demonstrates that quantum adiabatic algorithms with either a simple initial or final Hamiltonian suffer from an inverse exponential minimal energy gap, rendering them exponentially slow and thus ineffective for efficient computation. The authors prove via the quantum adiabatic theorem that such evolutions cannot achieve polynomial-time performance, even with arbitrary interpolation paths, thereby ruling out efficient quantum speedup for certain problems including search.

ABSTRACT

Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.

Motivation & Objective

• To investigate the performance limits of quantum adiabatic algorithms when the initial or final Hamiltonian is too simple.
• To determine whether quantum adiabatic evolutions can provide efficient solutions for minimizing classical functions.
• To establish that certain adiabatic paths lead to exponentially small energy gaps, invalidating efficient algorithm design.
• To generalize results from linear interpolation paths to arbitrary continuous interpolation paths.
• To provide a theoretical basis for identifying worst-case scenarios in quantum adiabatic optimization.

Proposed method

• The authors apply the quantum adiabatic theorem, using the condition that the minimal gap between the ground and first excited states must be large enough to ensure adiabatic evolution.
• They define the minimal gap $ g_{ ext{min}} = \min_t [E_1(t) - E_0(t)] $ and the rate of change of the Hamiltonian $ D_{\text{max}} = \max_t |\langle E_1,t | \frac{dH}{dt} | E_0,t \rangle| $.
• For a linear interpolation path $ H(t) = (1-t/T)H_0 + (t/T)H_1 $, they derive a lower bound on $ g_{\text{min}} $, showing it is less than $ \frac{2}{2^{n/2 - n/100}} $.
• They generalize the result to arbitrary continuous interpolation paths $ H(t) = f(t)H_0 + g(t)H_1 $, maintaining the same gap bound scaled by constants $ c_2 $.
• Using symmetry and unitary transformations, they show that the minimal gap remains bounded by the same exponential expression even when the final Hamiltonian is a simple projector.
• The proof relies on analyzing the characteristic polynomial of the time-dependent Hamiltonian and identifying roots in specific intervals to bound the energy gap.

Experimental results

Research questions

• RQ1 Can quantum adiabatic algorithms with a simple initial Hamiltonian achieve efficient performance for minimizing classical functions?
• RQ2 What is the minimal energy gap in adiabatic evolutions with a simple initial or final Hamiltonian, and how does it affect runtime?
• RQ3 Does the choice of interpolation path affect the minimal gap in such adiabatic evolutions?
• RQ4 Can the worst-case performance of quantum adiabatic algorithms be bounded using this framework?
• RQ5 Is it possible to achieve exponential speedup in quantum search using adiabatic evolution with a simple final Hamiltonian?

Key findings

• The minimal gap $ g_{\text{min}} $ in adiabatic evolutions with a simple initial Hamiltonian is bounded above by $ \frac{2}{2^{n/2 - n/100}} $, which is exponentially small in $ n $.
• As a result, the required evolution time $ T $ must be exponentially large in $ n $, making such algorithms inefficient.
• The same exponential lower bound on the minimal gap holds for arbitrary continuous interpolation paths, not just linear ones.
• The bound is generalized to include a scaling factor $ c_2 $, so $ g_{\text{min}} < \frac{2c_2}{2^{n/2 - n/100}} $ for any path satisfying $ c_1 < f(t) + g(t) < c_2 $.
• A similar result holds when the final Hamiltonian is simple (e.g., a projector), showing that such evolutions also suffer from exponentially small gaps.
• This implies that quantum adiabatic algorithms cannot provide exponential speedup for problems like unstructured search, even with optimal path choices.