Quantum Computation by Adiabatic Evolution

TL;DR

The paper introduces a quantum algorithm for satisfiability based on adiabatic evolution, where a time-dependent Hamiltonian smoothly interpolates from an easy initial ground state to a problem-specific final ground state encoding a satisfying assignment. The computational runtime hinges on the minimum spectral gap; while general guarantees are absent, the authors analyze several structured, scalable instances in which the gap scales polynomially with the number of bits, yielding polynomial-time evolution. They also demonstrate that naively Grover-like cases yield exponentially small gaps and no speedup, underscoring the role of problem structure. Additionally, the work connects the adiabatic approach to the conventional quantum circuit model via Trotterization, and discusses broader implications and outlook for adiabatic quantum computation in combinatorial search problems.

Abstract

We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.

Figures (12)

• Figure 1: The two eigenvalues of $\widetilde{H}(s)$ for a one-qubit example.
• Figure 2: The two eigenvalues of $\widetilde{H}(s)$ for a one-qubit example where $H_{\mathrm{B}}$ and $H_{\mathrm{P}}$ are diagonal in the same basis. The levels cross so $g_{\min}=0$.
• Figure 3: A small perturbation is added to the Hamiltonian associated with Fig.\ref{['fig:2']} and we see that the levels no longer cross.
• Figure 4: The four eigenvalues of $\widetilde{H}(s)$ associated with "2-bit disagree". The same levels are associated with "2-bit agree".
• Figure 5: The four eigenvalues of $\widetilde{H}(s)$ associated with the 2-bit imply clause.