Quantum Walks Can Find a Marked Element on Any Graph

Abstract

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set M consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time HT(P, M ) of any reversible random walk P on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity HT + (P , M ) which we call extended hitting time.Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk P and the absorbing walk P ′ , whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not statetransitive. We also provide algorithms in the cases when only approximations or bounds on parameters p M (the probability of picking a marked vertex from the stationary distribution) and HT + (P , M ) are known.

Quantum mixing of Markov chains for special distributions

“…Having such a quantum output is desirable if our protocol is to be embedded in a larger algorithm where the preparation is just an initial step. Examples where this is assumed include hitting algorithms [9,10], and algorithms which aim at sampling from a (renormalized) part of the distribution [10,17]. We point out that this property is not a necessary feature of all quantum algorithms for mixing-there are promising approaches which utilize decoherence to speed up mixing [5], which may preclude a coherent output.…”

“…Moreover, the quantum algorithm we have provided realizes a coherent encoding of the stationary distribution, which can be used as a fully quantum subroutine, for instance in the preparation of initial states in e.g. hitting algorithms [9,10].…”

“…Measuring the first register of such a state will result in an element in M with a constant probability, which means that by iterating this process k times ensures an element in M is found with an exponentially increasing probability in k. However, since the state ψ | 〉 is also in span{ ,˜}, π π | 〉 | 〉 it is easy to see that the measured state, conditional on being in M, will also be distributed according toπ. This observation was used in [17], and also in [10] to produce an element sampled from the truncated stationary distribution˜,…”

“…) Ω on the state space size N. Nonetheless, Szegedy walk operators have been successfully employed in other contexts, mostly relying on decreasing so-called hitting times of random walks [8][9][10][11]. For a recent review on quantum walks see e.g.…”

Models of optical quantum computing

“…There are several other problems for which quantum systems provide a more modest advantage than an exponential speed-up. These problems include collision finding, which is applicable in secure hashing, solving differential equations using the finite element method [9], and search on graphs for marked vertices [10]. All these examples provide evidence that this model of computing is potentially more powerful than classical computing.…”