Abstract
We analyze the role of coherent tunneling that gives rise to bands of delocalized quantum states providing a coherent pathway for population transfer (PT) between computational states with similar energies. Given an energy function ${\cal E}(z)$ of a binary optimization problem and a bit-string $z_i$ with atypically low energy, our goal is to find other bit-strings with energies within a narrow window around ${\cal E}(z_i)$. We study PT due to quantum evolution under a transverse field $B_\perp$ of an n-qubit system that encodes ${\cal E}(z)$. We focus on a simple yet nontrivial model: $M$ randomly chosen "marked" bit-strings ($2^n \gg M$) are assigned energies in the interval ${\cal E}(z)\in[-n -W/2, n + W/2]$ with $W << B_\perp$, while the rest of the states are assigned energy $0$. The PT starts at a marked state $z_i$ and ends up in a superposition of $\sim \Omega$ marked states inside the PT window. The scaling of a typical runtime for PT with $n$ and $\Omega$ is the same as in the multi-target Grover's algorithm, except for a factor that is equal to $\exp(n \,B_{\perp}^{-2}/2)$ for $n \gg B_{\perp}^{2} \gg 1$. Unlike the Hamiltonians used in analog quantum search algorithms, the model we consider is non-integrable, and the transverse field delocalizes the marked states. PT protocol is not sensitive to the value of B and may be initialized at a marked state. We develop microscopic theory of PT. Under certain conditions, the band of the system eigenstates splits into mini-bands of non-ergodic delocalized states, whose width obeys a heavy-tailed distribution directly related to that of PT runtimes. We find analytical form of this distribution by solving nonlinear cavity equations for the random matrix ensemble. We argue that our approach can be applied to study the PT protocol in other transverse field spin glass models, with a potential quantum advantage over classical algorithms.
Highlights
The idea to use quantum computers for the solution of search and discrete optimization problems has been actively pursued for decades, most notably in connection to Grover’s algorithm [1], quantum annealing [2,3,4,5,6,7,8,9,10], and, more recently, quantum approximate optimization [11].2160-3308=20=10(1)=011017(51)Published by the American Physical SocietyPHYS
We demonstrate that transport within the minibands can be used for an efficient quantum search in spin-glass problems and propose a population transfer (PT) protocol based on this theoretical insight
We developed the first well-controlled theoretical description of the eigenstate structure and quantum dynamics in the nonergodic extended (NEE) phase in a quantum spin glass
Summary
The idea to use quantum computers for the solution of search and discrete optimization problems has been actively pursued for decades, most notably in connection to Grover’s algorithm [1], quantum annealing [2,3,4,5,6,7,8,9,10], and, more recently, quantum approximate optimization [11]. A transverse field applied to a nonergodic classical model in Eq (1) gives rise to tunneling matrix elements between its deep local minima. In this regime, NEE eigenstates could be formed by coherent superpositions of local minima separated by large Hamming distances. We see no reason to expect a direct transition between the MBL and ergodic phases without an intermediate nonergodic phase similar to the case of ordinary Anderson localization in finitedimensional space This difference is due to the fact that the number of relevant bit strings at a given Hamming distance d from a given one increases for spin-glass models exponentially with d, or even quicker, whereas for finitedimensional models this increase is only polynomial.
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