On the adiabatic condition and the quantum hitting time of Markov chains

Abstract

We present an adiabatic quantum algorithm for the abstract problem of searching marked vertices in a graph, or spatial search. Given a random walk (or Markov chain) $P$ on a graph with a set of unknown marked vertices, one can define a related absorbing walk $P^{\prime }$ where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian $H(s)$ from the interpolated Markov chain $P(s)=(1-s)P+s{P}^{\prime }$ and use it in an adiabatic quantum algorithm to drive an initial superposition over all vertices to a superposition over marked vertices. The adiabatic condition implies that for any reversible Markov chain and any set of marked vertices, the running time of the adiabatic algorithm is given by the square root of the classical hitting time. This algorithm therefore demonstrates a novel connection between the adiabatic condition and the classical notion of hitting time of a random walk. It also significantly extends the scope of previous quantum algorithms for this problem, which could only obtain a full quadratic speed-up for state-transitive reversible Markov chains with a unique marked vertex.

arXiv comment22 pages

References in corpus (10)

• Tight bounds on quantum searching
• Spatial search by quantum walk
• Quantum Search by Local Adiabatic Evolution
• Bounds for the adiabatic approximation with applications to quantum computation
• Inconsistency in the application of the adiabatic theorem
• Faster quantum walk algorithm for the two dimensional spatial search
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• Spatial search and the Dirac equation
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• Quadratic speedup for spatial search by continuous-time quantum walk
• Finding a marked node on any graph by continuous-time quantum walk
• Analog quantum algorithms for the mixing of Markov chains
• Irreconcilable Difference Between Quantum Walks and Adiabatic Quantum Computing
• Efficient quantum walk on the grid with multiple marked elements
• Faster quantum mixing of Markov chains in non-regular graph with fewer qubits
• Unstructured Adiabatic Quantum Optimization: Optimality with Limitations
• On Hitting Times for General Quantum Markov Processes
• Quantum Hitting Time according to a given distribution
• Stable Adiabatic Times For A Continuous Evolution Of Markov Chains