Ground States Of Hamiltonians Research Articles

Optimizing Variational Quantum Neural Networks Based on Collective Intelligence

Quantum machine learning stands out as one of the most promising applications of quantum computing, widely believed to possess potential quantum advantages. In the era of noisy intermediate-scale quantum, the scale and quality of quantum computers are limited, and quantum algorithms based on fault-tolerant quantum computing paradigms cannot be experimentally verified in the short term. The variational quantum algorithm design paradigm can better adapt to the practical characteristics of noisy quantum hardware and is currently one of the most promising solutions. However, variational quantum algorithms, due to their highly entangled nature, encounter the phenomenon known as the “barren plateau” during the optimization and training processes, making effective optimization challenging. This paper addresses this challenging issue by researching a variational quantum neural network optimization method based on collective intelligence algorithms. The aim is to overcome optimization difficulties encountered by traditional methods such as gradient descent. We study two typical applications of using quantum neural networks: random 2D Hamiltonian ground state solving and quantum phase recognition. We find that the collective intelligence algorithm shows a better optimization compared to gradient descent. The solution accuracy of ground energy and phase classification is enhanced, and the optimization iterations are also reduced. We highlight that the collective intelligence algorithm has great potential in tackling the optimization of variational quantum algorithms.

DMRG and Monte Carlo studies of CrI3 magnetic phases and the phase transition

The monolayer of $\textrm{CrI}_3$ has been reported to exhibit the ferromagnetic order, with a Curie temperature of $45 K$ and off-plane easy axis, which has attracted much attention in the community of condensed matter physics. Using the Density Matrix Renormalization Group method, we investigate the role of correlation effects and using classical Monte Carlo simulations analyze the nature of phase transitions in the XXZ Hamiltonian on a honeycomb lattice, which can effectively model $\textrm{CrI}_3$. We show, that the magnetic ordering of the Hamiltonian's ground state can be well approximated by classical models in a wide range of an anisotropy parameter space. Using classical Monte Carlo simulations we estimate the Curie temperature of $49.7 K$ which is in good agreement with experimental result.

Optimizing Parameterized Quantum Circuits With Free-Axis Single-Qubit Gates

Variational quantum algorithms, which utilize Parametrized Quantum Circuits (PQCs), are promising tools to achieve quantum advantage for optimization problems on near-term quantum devices. Their PQCs have been conventionally constructed from parametrized rotational angles of single-qubit gates around predetermined set of axes, and two-qubit entangling gates, such as CNOT gates. We propose a method to construct a PQC by continuous parametrization of both the angles and the axes of its single-qubit rotation gates. The method is based on the observation that when rotational angles are fixed, optimal axes of rotations can be computed by solving a system of linear equations whose coefficients can be determined from the PQC with small computational overhead. The method can be further simplified to select axes freely from continuous parameters with rotational angles fixed to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pi$</tex-math></inline-formula> . We show the simplified free-axis selection method has better expressibility against other structural optimization methods when measured with Kullback-Leibler (KL) divergence. We also demonstrate PQCs with free-axis selection are more effective to search the ground states of Hamiltonians for condensed matter physics, quantum chemistry and combinatorial optimization. Because free-axis selection allows designing PQCs without specifying their single-qubit rotational axes, it may significantly improve the handiness of PQCs.

Quadratic Speedup for Spatial Search by Continuous-Time Quantum Walk.

Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.

Peratic phase transition by bulk-to-surface response

The study of dynamical phase transitions has been attracting considerable research efforts in the last decade. One theme of present interest is to search for exotic scenarios beyond the framework of equilibrium phase transitions. Here, we establish a duality between many-body dynamics and static Hamiltonian ground states for both classical and quantum systems. We construct frustration free Hamiltonians whose ground state phase transitions have rigorous duality to chaotic transitions in dynamical systems. By this duality, we show the corresponding ground state phase transitions are characterized by bulk-to-surface response, which are then dubbed "peratic" meaning defined by response to the boundary. For the classical system, we show how the time-like dimension emerges in the static ground states. For the quantum system, the ground state is a superposition of geometrical lines on a two dimensional array, which encode the dynamical Floquet evolution history of one dimensional disordered spin chains. Our prediction of peratic phase transition has direct consequences in quantum simulation platforms such as Rydberg atoms and superconducting qubits, as well as anisotropic spin glass materials. The discovery would shed light on the unification of dynamical phase transitions with equilibrium systems.

Noisy intermediate-scale quantum algorithm for semidefinite programming

Semidefinite programs (SDPs) are convex optimization programs with vast applications in control theory, quantum information, combinatorial optimization and operational research. Noisy intermediate-scale quantum (NISQ) algorithms aim to make an efficient use of the current generation of quantum hardware. However, optimizing variational quantum algorithms is a challenge as it is an NP-hard problem that in general requires an exponential time to solve and can contain many far from optimal local minima. Here, we present a current term NISQ algorithm for solving SDPs. The classical optimization program of our NISQ solver is another SDP over a lower dimensional ansatz space. We harness the SDP based formulation of the Hamiltonian ground state problem to design a NISQ eigensolver. Unlike variational quantum eigensolvers, the classical optimization program of our eigensolver is convex, can be solved in polynomial time with the number of ansatz parameters and every local minimum is a global minimum. We find numeric evidence that NISQ SDP can improve the estimation of ground state energies in a scalable manner. Further, we efficiently solve constrained problems to calculate the excited states of Hamiltonians, find the lowest energy of symmetry constrained Hamiltonians and determine the optimal measurements for quantum state discrimination. We demonstrate the potential of our approach by finding the largest eigenvalue of up to $2^{1000}$ dimensional matrices and solving graph problems related to quantum contextuality. We also discuss NISQ algorithms for rank-constrained SDPs. Our work extends the application of NISQ computers onto one of the most successful algorithmic frameworks of the past few decades.

NISQ Algorithm for Hamiltonian simulation via truncated Taylor series

Simulating the dynamics of many-body quantum systems is believed to be one of the first fields that quantum computers can show a quantum advantage over classical computers. Noisy intermediate-scale quantum (NISQ) algorithms aim at effectively using the currently available quantum hardware. For quantum simulation, various types of NISQ algorithms have been proposed with individual advantages as well as challenges. In this work, we propose a new algorithm, truncated Taylor quantum simulator (TQS), that shares the advantages of existing algorithms and alleviates some of the shortcomings. Our algorithm does not have any classical-quantum feedback loop and bypasses the barren plateau problem by construction. The classical part in our hybrid quantum-classical algorithm corresponds to a quadratically constrained quadratic program (QCQP) with a single quadratic equality constraint, which admits a semidefinite relaxation. The QCQP based classical optimization was recently introduced as the classical step in quantum assisted eigensolver (QAE), a NISQ algorithm for the Hamiltonian ground state problem. Thus, our work provides a conceptual unification between the NISQ algorithms for the Hamiltonian ground state problem and the Hamiltonian simulation. We recover differential equation-based NISQ algorithms for Hamiltonian simulation such as quantum assisted simulator (QAS) and variational quantum simulator (VQS) as particular cases of our algorithm. We test our algorithm on some toy examples on current cloud quantum computers. We also provide a systematic approach to improve the accuracy of our algorithm.

Iterative quantum-assisted eigensolver

The task of estimating the ground state of Hamiltonians is an important problem in physics with numerous applications ranging from solid-state physics to combinatorial optimization. We provide a hybrid quantum-classical algorithm for approximating the ground state of a Hamiltonian that builds on the powerful Krylov subspace method in a way that is suitable for current quantum computers. Our algorithm systematically constructs the Ansatz using any given choice of the initial state and the unitaries describing the Hamiltonian. The only task of the quantum computer is to measure overlaps and no feedback loops are required. The measurements can be performed efficiently on current quantum hardware without requiring any complicated measurements such as the Hadamard test. Finally, a classical computer solves a well characterized quadratically constrained optimization program. Our algorithm can reuse previous measurements to calculate the ground state of a wide range of Hamiltonians without requiring additional quantum resources. Further, we demonstrate our algorithm for solving problems consisting of thousands of qubits. The algorithm works for almost every random choice of the initial state and circumvents the barren plateau problem.

Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information

The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model withdparameters, however, is computationally expensive and generally requiresO(d2)function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.

Neural network agent playing spin Hamiltonian games on a quantum computer

Quantum computing is expected to provide new promising approaches for solving the most challenging problems in material science, communication, search, machine learning and other domains. However, due to the decoherence and gate imperfection errors modern quantum computer systems are characterized by a very complex, dynamical, uncertain and fluctuating computational environment. We develop an autonomous agent effectively interacting with such an environment to solve magnetism problems. By using reinforcement learning the agent is trained to find the best-possible approximation of a spin Hamiltonian ground state from self-play on quantum devices. We show that the agent can learn the entanglement to imitate the ground state of the quantum spin dimer. The experiments were conducted on quantum computers provided by IBM. To compensate the decoherence we use a local spin correction procedure derived from a general sum rule for spin–spin correlation functions of a quantum system with an even number of antiferromagnetically-coupled spins in the ground state. Our study paves a way to create a new family of neural network eigensolvers for quantum computers.

Validating a two-qubit nonstoquastic Hamiltonian in quantum annealing

We propose a two qubit experiment for validating tunable antiferromagnetic $XX$ interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be non-stoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian's ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic $XX$ interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state. This change is associated with a suppression of one of the Ising ground states, which can then be detected using solely computational basis measurements. We show that a semiclassical approximation of the annealing protocol fails to reproduce this feature, making it a candidate `quantum signature' of the evolution.

Many-Body Multifractality throughout Bosonic Superfluid and Mott Insulator Phases.

We demonstrate many-body multifractality of the Bose-Hubbard Hamiltonian's ground state in Fock space, for arbitrary values of the interparticle interaction. Generalized fractal dimensions unambiguously signal, even for small system sizes, the emergence of a Mott insulator that cannot, however, be naively identified with a localized phase in Fock space. We show that the scaling of the derivative of any generalized fractal dimension with respect to the interaction strength encodes the critical point of the superfluid to the Mott insulator transition, and provides an efficient way to accurately estimate its position. We further establish that the transition can be quantitatively characterized by one single wave function amplitude from the exponentially large Fock space.

Verification of Many-Qubit States

Verification is a task to check whether a given quantum state is close to an ideal state or not. In this paper, we show that a variety of many-qubit quantum states can be verified with only sequential single-qubit measurements of Pauli operators. First, we introduce a protocol for verifying ground states of Hamiltonians. We next explain how to verify quantum states generated by a certain class of quantum circuits. We finally propose an adaptive test of stabilizers that enables the verification of all polynomial-time-generated hypergraph states, which include output states of the Bremner-Montanaro-Shepherd-type instantaneous quantum polynomial time (IQP) circuits. Importantly, we do not make any assumption that the identically and independently distributed copies of the same states are given: Our protocols work even if some highly complicated entanglement is created among copies in any artificial way. As applications, we consider the verification of the quantum computational supremacy demonstration with IQP models, and verifiable blind quantum computing.

Computation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algorithm

Harnessing the full power of nascent quantum processors requires the efficient management of a limited number of quantum bits with finite lifetime. Hybrid algorithms leveraging classical resources have demonstrated promising initial results in the efficient calculation of Hamiltonian ground states--an important eigenvalue problem in the physical sciences that is often classically intractable. In these protocols, a Hamiltonian is parsed and evaluated term-wise with a shallow quantum circuit, and the resulting energy minimized using classical resources. This reduces the number of consecutive logical operations that must be performed on the quantum hardware before the onset of decoherence. We demonstrate a complete implementation of the Variational Quantum Eigensolver (VQE), augmented with a novel Quantum Subspace Expansion, to calculate the complete energy spectrum of the H2 molecule with near chemical accuracy. The QSE also enables the mitigation of incoherent errors, potentially allowing the implementation of larger-scale algorithms without complex quantum error correction techniques.

Efficiently Controllable Graphs.

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that, when a controllable quantum network is described by such a graph and the gaps in eigenfrequencies and in transition frequencies are bounded exponentially in the number of vertices, the network is efficiently controllable, in the sense that universal quantum computation can be performed using a control sequence polynomial in the size of the network while controlling a vanishingly small fraction of subsystems. We show that networks corresponding to finite-dimensional lattices are efficiently controllable and explore generalizations to percolation clusters and random graphs. We show that the classical computational complexity of estimating the ground state of Hamiltonians described by controllable graphs is polynomial in the number of subsystems or qubits.

STIRAP schemes for atomic Bose-Einstein condensates

In coherent control of Bose-Einstein condensates (BEC) the major limitation to successful high fidelity transfer using optical Raman transitions is spontaneous emission of radiation and the decoherence that this causes. We present a scheme based on stimulated Raman adiabatic passage (STIRAP) designed specifically for Rubidium 87. STIRAP is a method of population transfer that relies on the adiabatic theorem of quantum mechanics. Instead of transferring population between Hamiltonian ground states, one adiabatically evolves the ground states into some desired superposition. We show that one may thereby implement arbitrary rotations of spinor BECs with extremely high fidelities. Our simulations show that given a BEC with N = 104 atoms, we may do an arbitrary unitary rotation with an infidelity of about 10-7 that takes approximately 100 ns.

Observing Geometric Frustration with Thousands of Coupled Lasers

Geometric frustration, the inability of an ordered system to find a unique ground state plays a key role in a wide range of systems. We present a new experimental approach to observe large-scale geometric frustration with 1500 negatively coupled lasers arranged in a kagome lattice. We show how dissipation drives the lasers into a phase-locked state that directly maps to the classical XY spin Hamiltonian ground state. In our system, frustration is manifested by the lack of long range phase ordering. Finally, we show how next-nearest-neighbor coupling removes frustration and restores order.

Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

GENUINE CORRELATIONS IN FINITE-SIZE SPIN SYSTEMS

Genuine multipartite correlations in finite-size XY chains are studied as a function of the applied external magnetic field. We find that, for low temperatures, multipartite correlations are sensitive to the parity change in the Hamiltonian ground state, given that they exhibit a minimum every time that the ground state becomes degenerate. This implies that they can be used to detect the factorizing point, that is, the value of the external field such that, in the thermodynamical limit, the ground state becomes the tensor product of single-spin states.

BCS ansatz for superconductivity in the light of the Bogoliubov approach and the Richardson–Gaudin exact wave function

The Bogoliubov approach to superconductivity provides a strong mathematical support to the wave function ansatz proposed by Bardeen, Cooper and Schrieffer (BCS). Indeed, this ansatz — with all pairs condensed into the same state — corresponds to the ground state of the Bogoliubov Hamiltonian. Yet, this Hamiltonian only is part of the BCS Hamiltonian. As a result, the BCS ansatz definitely differs from the BCS Hamiltonian ground state. This can be directly shown either through a perturbative approach starting from the Bogoliubov Hamiltonian, or better by analytically solving the BCS Schrödinger equation along Richardson–Gaudin exact procedure. Still, the BCS ansatz leads not only to the correct extensive part of the ground state energy for an arbitrary number of pairs in the energy layer where the potential acts — as recently obtained by solving Richardson–Gaudin equations analytically — but also to a few other physical quantities such as the electron distribution, as here shown. The present work also considers arbitrary filling of the potential layer and evidences the existence of a super dilute and a super dense regime of pairs, with a gap different from the usual gap. These regimes constitute the lower and upper limits of density-induced BEC–BCS cross-over in Cooper pair systems.