We consider a novel quantum population transfer protocol to solve binary optimization problems that exploits quantum many-body dynamics in the delocalized regime. Hard optimization problems are characterized by energy landscape with a large number of local minima separated by large Hamming distances which scale with the problem size. This landscape gives rise to an interesting computational primitive: given an initial bit-string, we are to produce other bit-strings within certain narrow range of energies around the initial state. We consider a specific model we call "impurity band": a system of n qubits in a transverse field, where a number of bitstrings $M<<2^n$ selected at random are assigned random energies distributed in a narrow window of width $W<<1$ around the mean energy $-n$. We demonstrate the existence of the many-body delocalized regime in this model when the spectrum of the model splits into many-body minibands, and a typical eigenstate wave function is a superposition of peaks centered at a large number of local minima. The typical width of the minibands in energy determines the efficiency of the population transfer protocol. We demonstrate theoretically that the population transfer protocol achieves Grover type speedup in the unstructured impurity band model.
We consider a novel quantum population transfer protocol to solve binary optimization problems that exploits quantum many-body dynamics in the delocalized regime. Hard optimization problems are characterized by energy landscape with a large number of local minima separated by large Hamming distances which scale with the problem size. This landscape gives rise to an interesting computational primitive: given an initial bit-string, we are to produce other bit-strings within certain narrow range of energies around the initial state. We consider a specific model we call "impurity band": a system of n qubits in a transverse field, where a number of bitstrings M<<2n selected at random are assigned random energies distributed in a narrow window of width W<<1 around the mean energy −n. We demonstrate the existence of the many-body delocalized regime in this model when the spectrum of the model splits into many-body minibands, and a typical eigenstate wave function is a superposition of peaks centered at a large number of local minima. The typical width of the minibands in energy determines the efficiency of the population transfer protocol. We demonstrate theoretically that the population transfer protocol achieves Grover type speedup in the unstructured impurity band model.
Quantum tunneling has been proposed as a physical mechanism for solving binary optimization problems on a quantum computer because it provides an alternative to simulated annealing by directly connecting deep local minima of the energy landscape separated by large Hamming distances. However, classical simulations using Quantum Monte Carlo (QMC) were found to efficiently simulate tunneling transitions away from local minima if the tunneling is effectively dominated by a single path. We analyze a new computational role of coherent multi-qubit tunneling that gives rise to bands of non-ergodic extended (NEE) quantum states each formed by a superposition of a large number of deep local minima with similar energies. NEE provide a coherent pathway for population transfer (PT) between computational states with similar energies. In this regime, PT cannot be efficiently simulated by QMC. PT can serve as a new quantum subroutine for quantum search, quantum parallel tempering and reverse annealing optimization algorithms. We study PT resulting from quantum evolution under a transverse field of an n-spin system that encodes the energy function E(z) of an optimization problem over the set of bit configurations z. Transverse field is rapidly switched on in the beginning of algorithm, kept constant for sufficiently long time and switched off at the end. Given an energy function of a binary optimization problem and an initial bit-string with atypically low energy, PT protocol searches for other bitstrings at energies within a narrow window around the initial one. We provide an analytical solution for PT in a simple yet nontrivial model: M randomly chosen marked bit-strings are assigned energies E(z) within a narrow strip [-n -W/2, n + W/2], while the rest of the states are assigned energy 0. The PT starts at a marked state and ends up in a superposition of L marked states inside the narrow energy window whose width is smaller than W. The best known classical algorithm for finding another marked state is the exhaustive search. We find that the scaling of a typical PT runtime with n and L is the same as that in the multi-target Grover's quantum search algorithm, except for a factor that is equal to exp(n /(2B^2)) for finite transverse field B >>1. Unlike the Hamiltonians used in analog quantum unstructured search algorithms known so far, the model we consider is non-integrable and the transverse field delocalizes the marked states. As a result, our PT protocol is not exponentially sensitive in n to the weight of the driver Hamiltonian and may be initialized with a computational basis state. We develop the microscopic theory of PT by constructing a down-folded dense Hamiltonian acting in the space of marked states of dimension M. It belongs to the class of preferred basis Levy matrices (PBLM) with heavy-tailed distribution of the off-diagonal matrix elements. Under certain conditions, the band of the marked states splits into minibands of non-ergodic delocalized states. We obtain an explicit form of the heavy-tailed distribution of PT times by solving cavity equations for the ensemble of down-folded Hamiltonians. We study numerically the PT subroutine as a part of quantum parallel tempering algorithm for a number of examples of binary optimization problems on fully connected graphs.
We analyze a new computational role of coherent multi-qubit quantum tunneling that gives rise to bands of non-ergodic extended (NEE) quantum states each formed by a superposition of a large number of computational states (deep local minima of the energy landscape) with similar energies. NEE provide a mechanism for population transfer (PT) between computational states and therefore can serve as a new quantum subroutine for quantum search, quantum parallel tempering and reverse annealing optimization algorithms. We study PT in a quantum n-spin system subject to a transverse field where the energy function E(z) encodes a classical optimization problem over the set of spin configurations z. Given an initial spin configuration with low energy, PT protocol searches for other bitstrings at energies within a narrow window around the initial one. We provide an analytical solution for PT in a simple yet nontrivial model: M randomly chosen marked bit-strings are assigned energies E(z) within a narrow strip [−n−W/2,n+W/2], while the rest of the states are assigned energy 0. We find that the scaling of a typical PT runtime with n and L is the same as that in the multi-target Grover's quantum search algorithm, except for a factor that is equal to exp(n/(2B2)) for finite transverse field B≫1. Unlike the Hamiltonians used in analog quantum unstructured search algorithms known so far, the model we consider is non-integrable and population transfer is not exponentially sensitive in n to the weight of the driver Hamiltonian. We study numerically the PT subroutine as a part of quantum parallel tempering algorithm for a number of examples of binary optimization problems on fully connected graphs.
We consider the ability of local quantum dynamics to solve the ``energy-matching'' problem: given an instance of a classical optimization problem and a low-energy state, find another macroscopically distinct low-energy state. Energy matching is difficult in rugged optimization landscapes, as the given state provides little information about the distant topography. Here, we show that the introduction of quantum dynamics can provide a speedup over classical algorithms in a large class of hard optimization problems. Tunneling allows the system to explore the optimization landscape while approximately conserving the classical energy, even in the presence of large barriers. Specifically, we study energy matching in the random p-spin model of spin-glass theory. Using perturbation theory and exact diagonalization, we show that introducing a transverse field leads to three sharp dynamical phases, only one of which solves the matching problem: (1) a small-field ``trapped'' phase, in which tunneling is too weak for the system to escape the vicinity of the initial state; (2) a large-field ``excited'' phase, in which the field excites the system into high-energy states, effectively forgetting the initial energy; and (3) the intermediate ``tunneling'' phase, in which the system succeeds at energy matching. The rate at which distant states are found in the tunneling phase, although exponentially slow in system size, is exponentially faster than classical search algorithms.
We address the long-standing problem of the structure of the low-energy eigenstates and long-time coherent dynamics in quantum spin-glass models.Below the spin-glass freezing transition, the energy landscape of the spin system is characterized by a proliferation of local minima where classical dynamics gets trapped.A theoretical description of quantum dynamics in this regime is challenging due to the complex nature of the distribution of the tunneling matrix elements between the local minima of the energy landscape.We study the transverse-field-induced quantum dynamics of the following "impurity band" (IB) spin model: zero energy of all spin configurations except for a small fraction of spin configurations ("marked states") that form a narrow band at a large negative energy.At a zero transverse field, the IB model demonstrates the freezing transition at inverse temperature β f ∼ 1 characterized by a nonzero value of the Edwards-Anderson order parameter.At a finite transverse field, the low-energy dynamics can be described by the effective downfolded Hamiltonian that acts in the Hilbert subspace involving only the marked states.We obtain in an explicit form the heavy-tailed probability distribution of the off-diagonal matrix elements of the down-folded Hamiltonian.This Hamiltonian is dense and belongs to the class of preferred basis Levy matrices.Analytically solving nonlinear cavity equations for the ensemble of down-folded Hamiltonians allows us to describe the statistical properties of the eigenstates.In a broad interval of transverse fields, they are nonergodic, albeit extended.It means that the band of marked states splits into a set of narrow minibands.Accordingly, the quantum evolution that starts from a particular marked state leads to a linear combination of the states belonging to a particular miniband.An analytical description of this qualitatively new type of quantum dynamics is a key result of our paper.Based on our analysis, we propose the population transfer (PT) algorithm: The quantum evolution under constant transverse field B ⊥ starts at a low-energy spin configuration and ends up in a superposition of Ω spin configurations inside a narrow energy window.This algorithm crucially relies on the nonergodic nature of delocalized low-energy eigenstates.In the considered model, the run-time of the best classical algorithm (exhaustive search) is t cl ¼ 2 n =Ω.For ffiffiffi n p ≫ B ⊥ ≫ 1, the typical runtime of the quantum PT algorithm ffiffiffiffi ffi t cl p e n=ð2B 2 ⊥ Þ scales with n and Ω as that of Grover's quantum search, except for the small correction to the exponent.Unlike the Hamiltonians proposed for analog quantum unstructured search algorithms, the model we consider is nonintegrable and the transverse field delocalizes the marked states.As a result, our PT protocol does not require fine-tuning of the transverse field and may be initialized in a computational basis state.We find that the run-times of the PT algorithm are distributed according to the alpha-stable Levy law with tail index 1.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.
Graph partitioning is a fundamental problem in computer science with applications in diverse fields like network design, compiler optimization, and data analysis. Traditional algorithms for graph partitioning often suffer from computational complexity, particularly when dealing with large graphs. This paper explores the potential of utilizing quantum computing to accelerate graph partitioning. Specifically, we investigate the application of Grover's algorithm, a quantum search algorithm, to the problem. Our core claim is that by leveraging the superposition and interference capabilities inherent in quantum mechanics, a quantum solution can achieve a provable speedup compared to classical approaches. We present a theoretical framework outlining the algorithm's operation and derive the expected speedup. The key mathematical formulation involves representing the graph partitioning problem as an unstructured search problem amenable to Grover's algorithm. We define the problem as minimizing the cut size, represented by the objective function: *f(S) = Σi∈S Σj∈Sc d(i, j)* where *S* is a subset of vertices, *Sc* is the complement of *S* in the graph, and *d(i, j)* is the shortest path distance between vertices *i* and *j*. The algorithm then constructs a quantum oracle that maps the problem state to a superposition of all possible partitions. Grover's algorithm is then applied to amplify the amplitude of the solution state, resulting in a high probability of finding an optimal or near-optimal partition. The speedup is directly proportional to the square root of the number of iterations performed by Grover's algorithm, which is influenced by the problem size and the quality of the oracle. We analyze the complexity of the quantum algorithm and compare it to the complexity of the best-known classical algorithms, demonstrating a potential for significant speedup. Further research will focus on optimizing the oracle construction and exploring other quantum algorithms suitable for graph partitioning.
Establishing quantum speedup for computationally hard problems of practical relevance, particularly combinatorial optimization problems, remains a central challenge in quantum computation. In this work, we identify a structurally defined family of classically hard maximum independent set (MIS) instances, and design and analyze a non-stoquastic adiabatic quantum optimization algorithm that exploits this structure. The algorithm runs in polynomial time and achieves an exponential speedup over both transverse-field quantum annealing and state-of-the-art classical solvers on these instances, under assumptions supported by analytical and numerical evidence. We identify the essential quantum mechanism enabling the speedup as the use of a non-stoquastic XX-driver to access a larger sign-structured admissible subspace beyond the stoquastic regime, which allows sign-generating quantum interference to create smooth evolution paths that bypass tunneling. This identifies a distinctive quantum mechanism underlying the speedup and explains why no efficient classical analogue is likely to exist. In addition, our analysis produces scalable small-scale models, derived from our structural reduction, that capture the essential dynamics of the algorithm. These models provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.