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The observation of 'double quantum' transitions which made it possible to determine the charge state of Ti as 2+ is reported. The EPR spectrum observed at 1.2 K is presented in a graph. The first derivative of the absorption is shown vs the magnetic field. The hyperfine patterns for the Ti-47 and Ti-49 isotopes are identified. Spin-Hamiltonian parameters for Ti(2+) in various cubic hosts are listed.
With the rapid digitization of the world, an increasing number of real-world applications are turning to nonEuclidean data, modeled as graphs. Due to their intrinsic high complexity and irregularity, learning from graph data demands tremendous computational power. Recently, CMOS-compatible Ising machines, i.e., dynamic systems composed of CMOS components, have emerged as a new approach that harnesses the inherent power of natural annealing within dynamic systems to efficiently resolve binary optimization problems and have been adopted for traditional graph computation, such as max-cut. However, when performing complex Graph Learning (GL) tasks, Ising machines face significant hurdles: (i) they are inherently binary and thus ill-suited for real-valued problems; (ii) their expensive all-to-all coupling network that guarantees effective natural annealing poses daunting scalability concerns. To address these challenges, this paper proposes a nature-powered graph learning framework dubbed DS-GL, which is the first effort to transform the process of solving graph learning problems into the natural annealing process within a parameterized dynamic system embodied as a CMOS chip. To tackle the two major hurdles, DS-GL first augments the Ising machine architecture to modify the self-reaction term of its Hamiltonian function from linear to quadratic, effectively serving as an energy regulator. This adjustment maintains the system’s original physical interpretation while enabling it to process continuous, real-valued data. Second, to address the scaling issue, DS-GL further upgrades the real-valued dense Ising machine by decomposing it into a mesh-based multi-PE dynamic system that supports efficient distributed spatial-temporal co-annealing across different PEs through sparse interconnects. By exploiting the inherent sparsity and component structures in real-world graphs, DS-GL is able to map complex graph learning tasks onto the scalable dynamic system while maintaining high accuracy. Evaluations with three diverse GL applications across six real-world datasets, including traffic flow and COVID-19 prediction, show that DS-GL can deliver from 102× to 106× speedups and 500× energy reduction over Graph Neural Networks on GPUs, with 5% - 20% accuracy enhancement.
Here, we analyze a steady-state thermoelectric engine, whose working substance consists of two capacitively coupled quantum dots. One dot is tunnel-coupled to a hot reservoir serving as a heat source, the other one to two electrically biased reservoirs at a colder temperature, such that work is extracted under the form of a steady-state current against the bias. In single realizations of the dynamics of this steady-state engine autonomous, four-stroke cycles can be identified. The cycles are purely stochastic, in contrast to mechanical autonomous engines which exhibit self-oscillations. In particular, these cycles fluctuate in direction and duration and occur in competition with other spurious cycles. Using a stochastic thermodynamic approach, we quantify the cycle fluctuations and relate them to the entropy produced during individual cycles. We identify the cycle mainly responsible for the engine performance and quantify its statistics with tools from graph theory. We show that such stochastic cycles are made possible because the work extraction mechanism is itself stochastic instead of the periodic time dependence in the working-substance Hamiltonian which can be found in conventional mechanical engines. Our investigation brings new perspectives about the connection between cyclic and steady-state engines.
Hamiltonian encoding is a methodology for revealing the mechanism behind the dynamics governing controlled quantum systems. In this paper, following Mitra and Rabitz \cite{abhra_1}, we define mechanism via pathways of eigenstates that describe the evolution of the system, where each pathway is associated with a complex-valued amplitude corresponding to a term in the Dyson series. The evolution of the system is determined by the constructive and destructive interference of these pathway amplitudes. Pathways with similar attributes can be grouped together into pathway classes. The amplitudes of pathway classes are computed by modulating the Hamiltonian matrix elements and decoding the subsequent evolution of the system rather than by direct computation of the individual terms in the Dyson series. The original implementation of Hamiltonian encoding was computationally intensive and became prohibitively expensive in large quantum systems. This paper presents two new encoding algorithms that calculate the amplitudes of pathway classes by using techniques from graph theory and algebraic topology to exploit patterns in the set of allowed transitions, greatly reducing the number of matrix elements that need to be modulated. These new algorithms provide an exponential decrease in both computation time and memory utilization with respect to the Hilbert space dimension of the system. To demonstrate the use of these techniques, they are applied to two illustrative state-to-state transition problems.
Abstract Quantum chemistry is interested in calculating ground and excited states of molecular systems by solving the electronic Schrödinger equation. The exact numerical solution of this equation, frequently represented as an eigenvalue problem, remains unfeasible for most molecules and requires approximate methods. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix in Slater determinant basis without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants (as off-diagonal elements) is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. A gauge metric based on perturbation theory is used to determine the lowest energy cluster. This cluster or sub-matrix of Slater determinants is used to calculate approximate ground state and excited state energies within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and bond dissociation cases. These examples provide proof-of-principle results for approximate solution of the electronic structure problem using quantum computing. This approach is general and shows potential to reduce the computational complexity of post-Hartree–Fock methods as future advances in quantum hardware become available.
Trapped atomic ion crystals are a leading platform for quantum simulations of spin systems, with programmable and long-range spin-spin interactions mediated by excitations of phonons in the crystal. In this study, we describe a complementary approach for quantum simulations of bosonic systems using phonons in trapped-ion crystals, here mediated by excitations of the trapped-ion spins. The scheme enables a high degree of programability across a dense graph of bosonic couplings, utilizing long-lived collective phonon modes in a trapped-ion chain. As such, it is well suited for tackling hard problems such as boson sampling and simulations of long-range bosonic and spin-boson Hamiltonians.
Farhi et al. recently proposed a class of quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), for approximately solving combinatorial optimization problems. A level-p QAOA circuit consists of steps in which a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The 2p times for which these two Hamiltonians are applied are the parameters of the algorithm. As p increases, however, the parameter search space grows quickly. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here, we analytically and numerically study parameter setting for QAOA applied to MAXCUT. For level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MAXCUT, the Ring of Disagrees, or the 1D antiferromagnetic ring, we provide an analysis for arbitrarily high level. Using a Fermionic representation, the evolution of the system under QAOA translates into quantum optimal control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of QAOA for any p. It also greatly simplifies numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional sub-manifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.
Moiré structures, along with line-graph-based d -electron systems, represent a setting to realize flat bands. One form of the associated strong correlation physics is the Kondo effect. Here, we address the recently observed Kondo-driven heavy fermion state and its destruction in AB-stacked hetero-bilayer transition metal dichalcogenides, which can be controlled by the gate voltages. By studying an effective interacting Hamiltonian using the slave spin approach, we obtained a phase diagram with the total filling factor and the displacement field strength as the tunable parameters. In an extended range of the tunable displacement field, our numerical results show that the relative filling of the d orbital, which is associated with the highest moiré band from the MoTe 2 layer, is enforced to be ν d ≈ 1 by the interaction. This agrees with the experimental observation. We also argue that the observed high coherence temperature scale could be explained by the non-negligible bandwidth of the d orbital. Our results set the stage to address the amplified quantum fluctuations that the Kondo effect may produce in these structures and new regimes that the systems open up for Kondo-destruction quantum criticality. Published by the American Physical Society 2024
Qualitative and quantitative orbital properties such as bonding/antibonding character, localization, and orbital energies are critical to how chemists understand reactivity, catalysis, and excited-state behavior. Despite this, representations of orbitals in deep learning models have been very underdeveloped relative to representations of molecular geometries and Hamiltonians. Here, we apply state-of-the-art equivariant deep learning architectures to the task of assigning global labels to orbitals, namely energies characterizations, given the molecular coefficients from Hartree–Fock or density functional theory. The architecture we have developed, the Cartesian Equivariant Orbital Network (CEONET), shows how molecular orbital coefficients are readily featurized as equivariant node features common to all graph-based machine-learned potentials. We find that CEONET performs well at predicting difficult quantitative labels such as the orbital energy and orbital entropy. Furthermore, we find that the CEONET representation provides an intuitive latent space for differentiating orbital character for the qualitative assignment of e.g. bonding or antibonding character. In addition to providing a useful representation for further integrating deep learning with electronic structure theory, we expect CEONET to be useful for automatizing and interpreting the results of advanced electronic structure methods such as complete active space self-consistent field theory. In particular, the ability of CEONET to infer multireference character via the orbital entropy paves the way toward the machine-learned selection of active spaces.
Continuous-time quantum walks (CTQWs) on dynamic graphs, referred to as dynamic CTQWs, are a recently introduced universal model of computation that offers a new paradigm in which to envision quantum algorithms. In this work, we develop an algorithm that converts single-edge and self-loop dynamic CTQWs to the gate model of computation. We use this mapping to introduce an efficient sparse quantum state preparation framework based on dynamic CTQWs. Our approach utilizes combinatorics techniques such as minimal hitting sets, minimum spanning trees, and shortest Hamiltonian paths to reduce the number of controlled gates required to prepare sparse states. We show that our framework encompasses the current state of the art ancilla-free sparse state preparation method by reformulating this method as a CTQW. This CTQW-based framework offers an alternative to the uniformly controlled rotation method used by Qiskit by requiring fewer CX gates when the target state has a polynomial number of non-zero amplitudes.
In this work, we propose a polynomial-time algorithm for preparing the Gibbs state of the two-dimensional toric code Hamiltonian at any temperature, starting from any initial state, significantly improving upon prior estimates that suggested exponential scaling with inverse temperature. We prove that fast mixing at low temperature for the two-dimensional toric code can be achieved by augmenting local jump operators with simple global jump operators, which enable efficient transitions between logical sectors. To establish tight lower bounds on the spectral gap, we introduce a new reduction method that eventually maps the problem to estimating the spectral gap of a perturbed graph Laplacian on a stair graph. Our proof also shows that the Lindblad dynamics with a digitally implemented low-temperature local Davies generator is able to efficiently drive the quantum state toward the ground state manifold.
Abstract Artificial spiking neural networks have found applications in areas where the temporal nature of activation offers an advantage, such as time series prediction and signal processing. To improve their efficiency, spiking architectures often run on custom-designed neuromorphic hardware, but, despite their attractive properties, these implementations have been limited to digital systems. We describe an artificial quantum spiking neuron that relies on the dynamical evolution of two easy to implement Hamiltonians and subsequent local measurements. The architecture allows exploiting complex amplitudes and back-action from measurements to influence the input. This approach to learning protocols is advantageous in the case where the input and output of the system are both quantum states. We demonstrate this through the classification of Bell pairs which can be seen as a certification protocol. Stacking the introduced elementary building blocks into larger networks combines the spatiotemporal features of a spiking neural network with the non-local quantum correlations across the graph.
Quantum adiabatic optimization seeks to solve combinatorial problems using quantum dynamics, requiring the Hamiltonian of the system to align with the problem of interest. However, these Hamiltonians are often incompatible with the native constraints of quantum hardware, necessitating encoding strategies to map the original problem into a hardware-conformant form. While the classical overhead associated with such mappings is easily quantifiable and typically polynomial in problem size, it is much harder to quantify their overhead on the quantum algorithm, e.g., the transformation of the adiabatic timescale. In this work, we address this challenge on the concrete example of the encoding scheme proposed in [Nguyen , PRX Quantum , 010316 (2023)], which is designed to map optimization problems on arbitrarily connected graphs into Rydberg atom arrays. We consider the fundamental building blocks underlying this encoding scheme and determine the scaling of the minimum gap with system size along adiabatic protocols. Even when the original problem is trivially solvable, we find that the encoded problem can exhibit an exponentially closing minimum gap. We show that this originates from a quantum coherent effect, which gives rise to an unfavorable localization of the ground-state wave function. On the QuEra Aquila neutral atom machine, we observe such localization and its effect on the success probability of finding the correct solution to the encoded optimization problem. Finally, we propose quantum-aware modifications of the encoding scheme that avoid this quantum bottleneck and lead to an exponential improvement in the adiabatic performance. This highlights the crucial importance of accounting for quantum effects when designing strategies to encode classical problems onto quantum platforms. Published by the American Physical Society 2025
ABSTRACT We compare two quantum Hamiltonian algorithms that address the maximum independent set problem: one based on the emergent non-Abelian gauge matrix in adiabatic evolution of an energetically isolated manifold of states; the other based on designed application of single-qubit operations. We demonstrate that they are mathematically equivalent in the sense that one is the other’s interaction picture. Despite their mathematical equivalence, our numerical simulations show significant differences between them in performance, which is explained analytically. Intriguingly, this equivalence unveils that the PXP model, recently prominent in quantum dynamics research, can be viewed as quantum diffusion over the median graph of all independent sets governed by the non-Abelian gauge matrix.
The application of quantum algorithms to the study of many-particle quantum systems requires the ability to prepare wave functions that are relevant in the behavior of the system under study. Hamiltonian symmetries are important instruments used to classify relevant many-particle wave functions and to improve the efficiency of numerical simulations. In this work, quantum circuits for the exact and approximate preparation of total spin eigenfunctions on quantum computers are presented. Two different strategies are discussed and compared: exact recursive construction of total spin eigenfunctions based on the addition theorem of angular momentum, and heuristic approximation of total spin eigenfunctions based on the variational optimization of a suitable cost function. The construction of these quantum circuits is illustrated in detail, and the preparation of total spin eigenfunctions is demonstrated on IBM quantum devices, focusing on three- and five-spin systems on graphs with triangle connectivity.
We propose several techniques to enhance the parallel scalability of a matrix-free eigensolver designed for studying many-body localization (MBL) of quantum spin chain models with nearest-neighbor interactions and on-site disorder. This type of problem is computationally challenging because the dimension of the associated Hamiltonian matrix grows exponentially with respect to the number of spins L, and we need to average over different realizations of the random disorder to obtain relevant statistical behavior. For each disorder realization, we need to compute eigenvalues from different regions of the spectrum and their corresponding eigenvectors. In previous work, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large L’s. For example, we need thousands of compute nodes on modern high performance computing infrastructures to go beyond L = 24. The matrix-free approach does not suffer from this memory bottleneck, however, its scalability is limited by a computation and communication load imbalance. To reduce this imbalance and to significantly enhance the scalability of the matrix-free eigensolver, we reorder the matrix and leverage the consistent space runtime, CSPACER. We also show its efficiency in managing irregular communication patterns at scale compared to optimized MPI non-blocking two-sided and one-sided RMA implementation variants. This effort enables us to study MBL for spin chains with a larger number of spins. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates on a massively parallel many-core high performance computer.
The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters, which is usually done using a variational optimization loop. Parameter transfer, or parameter concentration, is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable non-variational learning method for QAOA angle finding. In this work, we systematically study QAOA parameter transferability from small problem sizes (16 and 27 decision variables) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher order terms using the Julia based QAOA simulation tool \texttt{JuliQAOA} to perform classical angle finding for up to $49$ QAOA layers ($p$). Parameter transfer of the fixed angles is validated using a combination of full statevector, Projected Entangled Pair States (PEPS), Matrix Product State (MPS), and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to other (unseen) problem instances does not in general provide monotonically improving performance as a function of $p$ - there are many cases where the performance temporarily decreases as a function of $p$ - but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the $100+$ qubit instances. We also sample the hardware-compatible Ising models using the ensemble of transfer-learned QAOA parameters on several superconducting qubit IBM Quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM NISQ processors up to $p=5$ on \texttt{ibm\_fez}, up to $p=9$ on \texttt{ibm\_torino}, and up to $p=10$ on \texttt{ibm\_pittsburgh}.