[Paper Review] Local adiabatic quantum search with different paths

Recep Eryiğit, Y. Gündüç+1 more|ArXiv.org|Sep 28, 2003

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper proposes a generalized local adiabatic quantum search algorithm that uses a 'sure success' Hamiltonian as a control term to stabilize the minimum spectral gap, enabling the search time to approach a constant independent of database size N. By modifying the evolution path via a tailored driving Hamiltonian, the algorithm achieves asymptotic constant-time complexity, effectively outperforming Grover's O(√N) scaling without oracle access.

ABSTRACT

We report on a detailed analysis of generalization of the local adiabatic search algorithm. Instead of evolving directly from an initial ground state Hamiltonian to a solution Hamiltonian a different evolution path is introduced and is shown that the time required to find an item in a database of size $N$ can be made to be independent of the size of the database by modifying the Hamiltonian used to evolve the system.

Motivation & Objective

• To generalize the local adiabatic quantum search algorithm by introducing an alternative evolution path via a control Hamiltonian.
• To reduce the time complexity of adiabatic quantum search below the standard O(√N) scaling by manipulating the spectral gap.
• To analyze whether the minimum energy gap can be kept bounded away from zero during evolution, enabling faster adiabatic evolution.
• To investigate the trade-off between algorithmic speedup and the complexity of constructing the required Hamiltonian.

Proposed method

• Introduce a time-dependent Hamiltonian H̃(s) = (1−s)H₀ + sHₘ + s(1−s)H_D, where H_D is a control Hamiltonian derived from the 'sure success' Hamiltonian of Bae and Kwon.
• Use the local adiabatic approximation to dynamically adjust the evolution rate ds/dt based on the instantaneous spectral gap g(s) and matrix element |⟨dH̃/ds⟩₁₀|.
• Derive the evolution rate as ds/dt = ε·g²(s), ensuring the adiabatic condition is satisfied locally at each time step.
• Solve the resulting differential equation for s(t), which involves inverse hyperbolic tangent functions of roots of the gap function.
• Analyze the asymptotic behavior of the total evolution time T by evaluating s(t) at s=1, showing T → (1 + π/4) as N → ∞.
• Compare the resulting evolution path and time complexity with the standard local adiabatic algorithm (Roland and Cerf) and show faster convergence.

Experimental results

Research questions

• RQ1 Can a modified evolution path via a control Hamiltonian stabilize the minimum spectral gap and reduce the adiabatic evolution time?
• RQ2 Does the introduction of a 'sure success' Hamiltonian term allow the algorithm’s runtime to approach a constant as N increases?
• RQ3 How does the time complexity of this generalized adiabatic search compare to Grover’s O(√N) algorithm?
• RQ4 What is the asymptotic behavior of the total evolution time T in the large-N limit?
• RQ5 How does the choice of the driving Hamiltonian H_D affect the spectral gap and evolution speed?

Key findings

• The minimum spectral gap g_min is kept bounded away from zero due to the added H_D term, preventing the typical inverse-square-root scaling of the gap.
• The total evolution time T asymptotically approaches the constant value (1 + π/4) ≈ 1.785 as N → ∞, independent of system size.
• The algorithm achieves a runtime that is almost independent of N, demonstrating a significant improvement over the O(√N) scaling of standard adiabatic and Grover’s algorithms.
• The evolution path becomes significantly faster than the standard local adiabatic algorithm, as shown by s(t) curves in Fig. 2 for N=64.
• The time-complexity is effectively shifted from the search process to the construction of the Hamiltonian, which must include knowledge of the marked state |m⟩.
• The result is analogous to Das et al.’s approach but achieves a lower asymptotic time of 1 when using a different H_D parametrization with a(s)=b(s)=√(s(s+1)).