Quantum annealing (QA) is one of the ways to search the ground state of the problem Hamiltonian. Here, we propose the QA scheme to search arbitrary excited states of the problem Hamiltonian. In our scheme, an $n$-th excited state of the trivial Hamiltonian is initially prepared and is adiabatically changed into an $n$-th excited state of the target Hamiltonian. Although our scheme is general such that we can search any excited states, we especially discuss the first excited state search in this paper. As a comparison, we consider a non-adiabatic scheme to find the first excited state with non-adiabatic transitions from the ground state. By solving the Lindblad master equation, we evaluate the performance of each scheme under the influence of decoherence. Our conclusion is that the adiabatic scheme show better performance than the non-adiabatic scheme as long as the coherence time of qubits is sufficiently long. These results are important for applications in the area of quantum chemistry, quantum simulation, and post-quantum cryptography.
The shortest vector problem (SVP) is one of the lattice problems and is mathematical basis for the lattice-based cryptography, which is expected to be post-quantum cryptography. The SVP can be mapped onto the Ising problem, which in principle can be solved by quantum annealing (QA). However, one issue in solving the SVP using QA is that the solution of the SVP corresponds to the first excited state of the problem Hamiltonian. Therefore, QA, which searches for ground states, cannot provide a solution with high probability. In this paper, we propose to adopt an excited-state search of the QA to …
Quantum chemistry is one of the important applications of quantum information technology. Especially, an estimation of the energy gap between a ground state and excited state of a target Hamiltonian corresponding to a molecule is essential. In the previous approach, an energy of the ground state and that of the excited state are estimated separately, and the energy gap can be calculated from the subtraction between them. Here, we propose a direct estimation of the energy gap between the ground state and excited state of the target Hamiltonian with quantum annealing. The key idea is to combine …
Algorithms for searching and sorting data sets on quantum annealing systems are presented. Search algorithms for unordered data sets are developed. A sorting algorithm for data sets is provided, with a consideration of sort stability. Scalability of the algorithms, considering both the number of qubits required and the qubit connectivity, is characterized as a function of problem size.
We have proposed the energy landscape transformation of Ising problems (ELTIP), which changes the combination of the state and eigenvalue without changing all the original eigenvalues [arXiv:2202.05927]. We study how the ELTIP affects the anti-crossing between two levels of the ground and first excited states during quantum annealing. We use a 5-spin maximum-weighted independent set for the problem to numerically investigate the anticrossing. For comparison, we introduce a non-stoquastic Hamiltonian that adds antiferromagnetic interaction to the normal transverse magnetic field. Annealing with…
We study a quantum annealer where bosons mediate the Ising-type interactions between qubits. We compare the efficiency of ground state preparation for direct and mediated couplings, for which Ising and spin-boson Hamiltonian are employed respectively. This comparison is done numerically for a small frustrated antiferromagnet, with a careful choice of the optimal adiabatic passage that reveals the features of the boson-mediated interactions. Those features can be explained by taking into account what we called excited solutions: states with the same spin correlations as the ground-state but wit…
We provide a reusability report of the method presented by Chen et al. in "Optimizing quantum annealing schedules with Monte Carlo tree search enhanced with neural networks" and add further benchmarks on Max-Cut problems.
Beyond Ground States: Physics-Inspired Optimization of Excited States of Classical Hamiltonians
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（光子科学研究所，巴塞罗那科技研究所）；ICREA-Institucio Catalana de Recerca i Estudis Avançats（加泰罗尼亚研究高级学院）
Diabatic quantum annealing (DQA) is an alternative algorithm to adiabatic quantum annealing (AQA) that can be used to circumvent the exponential slowdown caused by small minima in the annealing energy spectrum. We present the locally suppressed transverse-field (LSTF) protocol, a heuristic method for making stoquastic optimization problems compatible with DQA. We show that, provided an optimization problem intrinsically has magnetic frustration due to inhomogeneous local fields, a target qubit in the problem can always be manipulated to create a double minimum in the energy gap between the gro…
We confirmed the annealing time of Grover's search which is required to obtain desired success probability for quantum annealing by the imaginary-time and the real-time Schrödinger equation with two kinds of schedulings; one linearly decreases the quantum fluctuation and the other tunes the evolution rate of the Hamiltonian based on the adiabatic condition. With linear scheduling, the required annealing time for quantum annealing by the imaginary-time Schrödinger equation is of order $\log N$, which is very different from $O(N)$ required for the quantum annealing by the real-time Schrödinger e…
Quantum annealing is a practical approach to approximately implement the adiabatic quantum computational model under a real-world setting. The goal of an adiabatic algorithm is to prepare the ground state of a problem-encoded Hamiltonian at the end of an annealing path. This is typically achieved by driving the dynamical evolution of a quantum system slowly to enforce adiabaticity. Properly optimized annealing schedules often significantly accelerate the computational process. Inspired by the recent success of deep reinforcement learning such as DeepMind's AlphaZero, we propose a Monte Carlo T…
I describe how real quantum annealers may be used to perform local (in state space) searches around specified states, rather than the global searches traditionally implemented in the quantum annealing algorithm. The quantum annealing algorithm is an analogue of simulated annealing, a classical numerical technique which is now obsolete. Hence, I explore strategies to use an annealer in a way which takes advantage of modern classical optimization algorithms, and additionally should be less sensitive to problem mis-specification then the traditional quantum annealing algorithm.
I describe how real quantum annealers may be used to perform local (in state space) searches around specified states, rather than the global searches traditionally implemented in the quantum annealing algorithm. Such protocols will have numerous advantages over simple quantum annealing. By using such searches the effect of problem mis-specification can be reduced, as only energy differences between the searched states will be relevant. The quantum annealing algorithm is an analogue of simulated annealing, a classical numerical technique which has now been superseded. Hence, I explore two strat…
In this paper we present a novel strategy to solve optimization problems within a hybrid quantum-classical scheme based on quantum annealing, with a particular focus on QUBO problems. The proposed algorithm is based on an iterative structure where the representation of an objective function into the annealer architecture is learned and already visited solutions are penalized by a tabu-inspired search. The result is a heuristic search equipped with a learning mechanism to improve the encoding of the problem into the quantum architecture. We prove the convergence of the algorithm to a global opt…
In the computational model of quantum annealing, the size of the minimum gap between the ground state and the first excited state of the system is of particular importance, since it is inversely proportional to the running time of the algorithm. Thus, it is desirable to keep the gap as large as possible during the annealing process, since it allows the computation to remain under the protection of the adiabatic theorem while staying efficient. We propose steered quantum annealing as a new method to enlarge the gap throughout the process, in the case of diagonal final Hamiltonians, based on the…
It was recently shown that quantum annealing can be used as an effective, fast subroutine in certain types of matrix factorization algorithms. The quantum annealing algorithm performed best for quick, approximate answers, but performance rapidly plateaued. In this paper, we utilize reverse annealing instead of forward annealing in the quantum annealing subroutine for nonnegative/binary matrix factorization problems. After an initial global search with forward annealing, reverse annealing performs a series of local searches that refine existing solutions. The combination of forward and reverse …
Quantum annealing has great promise in leveraging quantum mechanics to solve combinatorial optimisation problems. However, to realize this promise to it's fullest extent we must appropriately leverage the underlying physics. In this spirit, I examine how the well known tendency of quantum annealers to seek solutions where more quantum fluctuations are allowed can be used to trade off optimality of the solution to a synthetic problem for the ability to have a more flexible solution, where some variables can be changed at little or no cost. I demonstrate this tradeoff experimentally using the re…
We construct an Ising Hamiltonian with an engineered energy landscape such that it has a local energy minimum which is near to the true global minimum solution, and further away from a false minimum. Using a technique established in previous experiments, we design our experiment such that (at least on timescales relevant to our study) the false minimum is reached preferentially in forward annealing due to high levels of quantum fluctuations. This allows us to demonstrate the key principle of reverse annealing, that the solution space can be searched locally, preferentially finding nearby solut…
The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an…
Closed-system quantum annealing is expected to sometimes fail spectacularly in solving simple problems for which the gap becomes exponentially small in the problem size. Much less is known about whether this gap scaling also impedes open-system quantum annealing. Here we study the performance of a quantum annealing processor in solving such a problem: a ferromagnetic chain with sectors of alternating coupling strength that is classically trivial but exhibits an exponentially decreasing gap in the sector size. The gap is several orders of magnitude smaller than the device temperature. Contrary …
Classical and quantum annealing are two heuristic optimization methods that search for an optimal solution by slowly decreasing thermal or quantum fluctuations. Optimizing annealing schedules is important both for performance and fair comparisons between classical annealing, quantum annealing, and other algorithms. Here we present a heuristic approach for the optimization of annealing schedules for quantum annealing and apply it to 3D Ising spin glass problems. We find that if both classical and quantum annealing schedules are similarly optimized, classical annealing outperforms quantum anneal…
We study the effect of the anneal path control per qubit, a new user control feature offered on the D-Wave 2000Q quantum annealer, on the performance of quantum annealing for solving optimization problems by numerically solving the time-dependent Schrödinger equation for the time-dependent Hamiltonian modeling the annealing problems. The anneal path control is thereby modeled as a modified linear annealing scheme, resulting in an advanced and retarded scheme. The considered optimization problems are 2-SAT problems with 12 Boolean variables, a known unique ground state and a highly degenerate f…
This paper presents the details and testing of two implementations (in C++ and Python) of the hybrid quantum-classical algorithm Quantum Annealing Learning Search (QALS) on a D-Wave quantum annealer. QALS was proposed in 2019 as a novel technique to solve general QUBO problems that cannot be directly represented into the hardware architecture of a D-Wave machine. Repeated calls to the quantum machine within a classical iterative structure and a related convergence proof originate a learning mechanism to find an encoding of a given problem into the quantum architecture. The present work conside…
We show, in the context of quantum combinatorial optimization, or quantum annealing, how the nonlinear Schrödinger-Langevin-Kostin equation can dynamically drive the system toward its ground state. We illustrate, moreover, how a frictional force of Kostin type can prevent the appearance of genuinely quantum problems such as Bloch oscillations and Anderson localization which would hinder an exhaustive search.
We present a comparison between the Quantum Approximate Optimization Algorithm (QAOA) and two widely studied competing methods, Quantum Annealing (QA) and Simulated Annealing (SA). To achieve this, we define a class of optimization problems with respect to their spectral properties which are exactly solvable with QAOA. In this class, we identify instances for which QA and SA have an exponentially small probability to find the solution. Consequently, our results define a first demarcation line between QAOA, Simulated Annealing and Quantum Annealing, and highlight the fundamental differences bet…
Quantum chemistry calculations are important applications of quantum annealing. For practical applications in quantum chemistry, it is essential to estimate a ground state energy of the Hamiltonian with chemical accuracy. However, there are no known methods to guarantee the accuracy of the estimation of the energy calculated by quantum annealing. Here, we propose a way to improve the accuracy of the estimate of the ground state energy by combining quantum annealing with classical computation. In our scheme, before running the QA, we need a pre-estimation of the energies of the ground state and…
We explain why quantum adiabatic evolution and simulated annealing perform similarly in certain examples of searching for the minimum of a cost function of n bits. In these examples each bit is treated symmetrically so the cost function depends only on the Hamming weight of the n bits. We also give two examples, closely related to these, where the similarity breaks down in that the quantum adiabatic algorithm succeeds in polynomial time whereas simulated annealing requires exponential time.
Toward an alternative approach to the quantum mechanic ground state search, we theoretically introduce a protocol in which energy of two identical systems are deterministically exchanged. The protocol utilizes a quantum interference between "forward" and "backward" time evolved states with respect to a given Hamiltonian. In addition, to make use the protocol for the ground state search, we construct a network with which we may be able to efficiently apply the protocol successively among multiple systems so that energy of one of them is gradually approaching the lowest one. Although rigorous an…
Mississippi State University（密西西比州立大学）；Air Force Research Laboratory（空军研究实验室）；HPC2 Center for Computational Sciences（HPC2计算科学中心）
Restricted Boltzmann Machines trained with different numbers of iterations were used to provide a diverse set of energy functions each containing many local valleys (LVs) with different energies, widths, escape barrier heights, etc. They were used to verify the previously reported possibility of using the D-Wave quantum annealer (QA) to find potentially important LVs in the energy functions of Ising spin glasses that may be missed by classical searches. For classical search, extensive simulated annealing (SA) was conducted to find as many LVs as possible regardless of the computational cost. S…
We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties to find the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions…
Adiabatic quantum annealing is a paradigm of analog quantum computation, where a given computational job is converted to the task of finding the global minimum of some classical potential energy function and the search for the global potential minimum is performed by employing external kinetic quantum fluctuations and subsequent slow reduction (annealing) of them. In this method, the entire potential energy landscape (PEL) may be accessed simultaneously through a delocalized wave-function, in contrast to a classical search, where the searcher has to visit different points in the landscape (i.e…
A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of $N$ qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by $O(\log N)$ bottlenecks inside the spin glass phase, …
Clustering is a powerful machine learning technique that groups "similar" data points based on their characteristics. Many clustering algorithms work by approximating the minimization of an objective function, namely the sum of within-the-cluster distances between points. The straightforward approach involves examining all the possible assignments of points to each of the clusters. This approach guarantees the solution will be a global minimum, however the number of possible assignments scales quickly with the number of data points and becomes computationally intractable even for very small da…
Ising-machine-like Driving Condition Restrictions in SSBM and Additional Pseudo-annealing- Driving Method for SSBM that Achieves Best Cut Value Search -
Quantum annealing is a generic algorithm using quantum-mechanical fluctuations to search for the solution of an optimization problem. The present paper first reviews the fundamentals of quantum annealing and then reports on preliminary results for an alternative method. The review part includes the relationship of quantum annealing with classical simulated annealing. We next propose a novel quantum algorithm which might be available for hard optimization problems by using a classical-quantum mapping as well as the Jarzynski equality introduced in nonequilibrium statistical physics.
Frustration-enhanced quantum annealing correction models with additional inter-replica interactions
Graduate School of Science and Technology, Keio University（庆应义塾大学理工学部）；Department of Applied Physics and Physico-Informatics, Keio University（庆应义塾大学应用物理与物理信息学系）；Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University（庆应义塾大学可持续量子人工智能中心）；Human Biology-Microbiome-Quantum Research Center (WPI-Bio2Q), Keio University（庆应义塾大学人类生物学-微生物组-量子研究中心）
Quantum algorithm design usually assumes access to a perfect quantum computer with ideal properties like full connectivity, noise-freedom and arbitrarily long coherence time. In Noisy Intermediate-Scale Quantum (NISQ) devices, however, the number of qubits is highly limited and quantum operation error and qubit coherence are not negligible. Besides, the connectivity of physical qubits in a quantum processing unit (QPU) is also strictly constrained. Thereby, additional operations like SWAP gates have to be inserted to satisfy this constraint while preserving the functionality of the original ci…
The performance of open-system quantum annealing is adversely affected by thermal excitations out of the ground state. While the presence of energy gaps between the ground and excited states suppresses such excitations, error correction techniques are required to ensure full scalability of quantum annealing. Quantum annealing correction (QAC) is a method that aims to improve the performance of quantum annealers when control over only the problem (final) Hamiltonian is possible, along with decoding. Building on our earlier work [S. Matsuura et al., Phys. Rev. Lett. 116, 220501 (2016)], we study…
We present a very general construction for quantum annealing protocols to solve Combinational Circuit Fault Diagnosis (CCFD) problems that restricts the evolution to the space of valid diagnoses. This is accomplished by using special local drivers that induce a transition graph on the space of feasible configurations that is regular and instance independent for each given circuit topology. Analysis of small instances shows that the energy gap has a generic form, and that the minimum gap occurs in the last third of the evolution. We used these features to construct an improved annealing schedul…
