Ground State Preparation Research Articles

Efficient Quantum Imaginary Time Evolution by Drifting Real-Time Evolution: An Approach with Low Gate and Measurement Complexity.

Quantum imaginary time evolution (QITE) is one of the promising candidates for finding the eigenvalues and eigenstates of a Hamiltonian on a quantum computer. However, the original proposal suffers from large circuit depth and measurements due to the size of the Pauli operator pool and Trotterization. To alleviate the requirement for deep circuits, we propose a time-dependent drifting scheme inspired by the qDRIFT algorithm [Campbell, E. Phys. Rev. Lett. 2019, 123, 070503]. We show that this drifting scheme removes the depth dependency on the size of the operator pool and converges inversely with respect to the number of steps. We further propose a deterministic algorithm that selects the dominant Pauli term to reduce the fluctuation for the ground state preparation. We also introduce an efficient measurement reduction scheme across Trotter steps that removes its cost dependence on the number of iterations. We analyze the main source of error for our scheme both theoretically and numerically. We numerically test the validity of depth reduction, convergence performance of our algorithms, and the faithfulness of the approximation for our measurement reduction scheme on several benchmark molecules. In particular, the results on the LiH molecule give circuit depths comparable to that of the advanced adaptive variational quantum eigensolver (VQE) methods while requiring much fewer measurements.

Perturbation theory with quantum signal processing

Perturbation theory is an important technique for reducing computational cost and providing physical insights in simulating quantum systems with classical computers. Here, we provide a quantum algorithm to obtain perturbative energies on quantum computers. The benefit of using quantum computers is that we can start the perturbation from a Hamiltonian that is classically hard to solve. The proposed algorithm uses quantum signal processing (QSP) to achieve this goal. Along with the perturbation theory, we construct a technique for ground state preparation with detailed computational cost analysis, which can be of independent interest. We also estimate a rough computational cost of the algorithm for simple chemical systems such as water clusters and polyacene molecules. To the best of our knowledge, this is the first of such estimates for practical applications of QSP. Unfortunately, we find that the proposed algorithm, at least in its current form, does not exhibit practical numbers despite of the efficiency of QSP compared to conventional quantum algorithms. However, perturbation theory itself is an attractive direction to explore because of its physical interpretability; it provides us insights about what interaction gives an important contribution to the properties of systems. This is in sharp contrast to the conventional approaches based on the quantum phase estimation algorithm, where we can only obtain values of energy. From this aspect, this work is a first step towards “explainable'' quantum simulation on fault-tolerant quantum computers.

Quantum imaginary-time control for accelerating the ground-state preparation

Quantum imaginary-time control for accelerating the ground-state preparation

Non-Hermitian ground-state-searching algorithm enhanced by a variational toolbox

Ground-state preparation for a given Hamiltonian is a common quantum-computing task of great importance and has relevant applications in quantum chemistry, computational material modeling, and combinatorial optimization. We consider an approach to simulate dissipative non-Hermitian Hamiltonian quantum dynamics using Hamiltonian simulation techniques to efficiently recover the ground state of a target Hamiltonian. The proposed method facilitates the energy transfer by repeatedly projecting ancilla qubits to the desired state, rendering the effective non-Hermitian Hamiltonian evolution on the system qubits. To make the method more resource friendly in the noisy intermediate-scale quantum (NISQ) and early fault-tolerant era, we combine the non-Hermitian projection algorithm with multiple variational gadgets, including variational module enhancement and variational state recording, to reduce the required circuit depth and avoid the exponentially vanishing success probability for postselections. We compare our method, the non-Hermitian-variational algorithm, with a pure variational method, the quantum approximate optimization algorithm (QAOA), for solving the 3-SAT problem and preparing the ground state for the transverse field Ising model. As demonstrated by numerical evidence, the non-Hermitian-variational algorithm outperforms QAOA in convergence speed with improved quantum resource efficiency.

Surrogate-based optimization for variational quantum algorithms

Variational quantum algorithms are a class of techniques intended to be used on near-term quantum computers. The goal of these algorithms is to perform large quantum computations by breaking the problem down into a large number of shallow quantum circuits, complemented by classical optimization and feedback between each circuit execution. One path for improving the performance of these algorithms is to enhance the classical optimization technique. Given the relative ease and abundance of classical computing resources, there is ample opportunity to do so. In this work, we introduce the idea of learning surrogate models for variational circuits using few experimental measurements, and then performing parameter optimization using these models as opposed to the original data. We demonstrate this idea using a surrogate model based on kernel approximations, through which we reconstruct local patches of variational cost functions using batches of noisy quantum circuit results. Through application to the quantum approximate optimization algorithm and preparation of ground states for molecules, we demonstrate the superiority of surrogate-based optimization over commonly-used optimization techniques for variational algorithms.

Grid-based methods for chemistry simulations on a quantum computer.

First-quantized, grid-based methods for chemistry modeling are a natural and elegant fit for quantum computers. However, it is infeasible to use today's quantum prototypes to explore the power of this approach because it requires a substantial number of near-perfect qubits. Here, we use exactly emulated quantum computers with up to 36 qubits to execute deep yet resource-frugal algorithms that model 2D and 3D atoms with single and paired particles. A range of tasks is explored, from ground state preparation and energy estimation to the dynamics of scattering and ionization; we evaluate various methods within the split-operator QFT (SO-QFT) Hamiltonian simulation paradigm, including protocols previously described in theoretical papers and our own techniques. While we identify certain restrictions and caveats, generally, the grid-based method is found to perform very well; our results are consistent with the view that first-quantized paradigms will be dominant from the early fault-tolerant quantum computing era onward.

Quantum approximate optimization algorithm applied to the binary perceptron

We apply digitized quantum annealing (QA) and quantum approximate optimization algorithm (QAOA) to a paradigmatic task of supervised learning in artificial neural networks: the optimization of synaptic weights for the binary perceptron. At variance with the usual QAOA applications to MaxCut, or to quantum spin-chains ground-state preparation, here the classical cost function is characterized by highly nonlocal multispin interactions. Yet, we provide evidence for the existence of optimal smooth solutions for the QAOA parameters, which are transferable among typical instances of the same problem, and we prove numerically an enhanced performance of QAOA over traditional QA. We also investigate on the role of the classical cost-function landscape geometry in this problem. By artificially breaking this geometrical structure, we show that the detrimental effect of a gap-closing transition, encountered in QA, is also negatively affecting the performance of our QAOA implementation.

Neural network enhanced measurement efficiency for molecular groundstates

It is believed that one of the first useful applications for a quantum computer will be the preparation of groundstates of molecular Hamiltonians. A crucial task involving state preparation and readout is obtaining physical observables of such states, which are typically estimated using projective measurements on the qubits. At present, measurement data is costly and time-consuming to obtain on any quantum computing architecture, which has significant consequences for the statistical errors of estimators. In this paper, we adapt common neural network models (restricted Boltzmann machines and recurrent neural networks) to learn complex groundstate wavefunctions for several prototypical molecular qubit Hamiltonians from typical measurement data. By relating the accuracy ɛ of the reconstructed groundstate energy to the number of measurements, we find that using a neural network model provides a robust improvement over using single-copy measurement outcomes alone to reconstruct observables. This enhancement yields an asymptotic scaling near ɛ −1 for the model-based approaches, as opposed to ɛ −2 in the case of classical shadow tomography.

Phase-dependent strategy to mimic quantum phase transitions

This study proposes a hybrid quantum system of an ensemble of collective spins coupled to a surface acoustic wave (SAW) cavity through a sideband design. Assisted by a dichromatic optical drive with a phase-dependent control, this spin ensemble can effectively mimic different types of long-range Lipkin–Meshkov–Glick (LMG) interactions and then undergo quantum phase transitions (QPTs) due to phase-induced spontaneous symmetry breaking (SSB). In addition, this phase-controlled scheme also ensures the dynamical preparation of the spin-squeezed state (SSS), which may be a useful application in quantum measurement. This study is a fresh attempt at quantum manipulation based on acoustic control and also provides a promising route toward useful applications in quantum information processing, especially the adiabatic preparation of multiparticle-entangled ground states via QPTs; i.e., the Greenberger–Horne–Zeilinger (GHZ) or W-type states.

Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models

Hybrid quantum-classical embedding methods for correlated materials simulations provide a path towards potential quantum advantage. However, the required quantum resources arising from the multi-band nature of d and f electron materials remain largely unexplored. Here we compare the performance of different variational quantum eigensolvers in ground state preparation for interacting multi-orbital embedding impurity models, which is the computationally most demanding step in quantum embedding theories. Focusing on adaptive algorithms and models with 8 spin-orbitals, we show that state preparation with fidelities better than 99.9% can be achieved using about 214 shots per measurement circuit. When including gate noise, we observe that parameter optimizations can still be performed if the two-qubit gate error lies below 10−3, which is slightly smaller than current hardware levels. Finally, we measure the ground state energy on IBM and Quantinuum hardware using a converged adaptive ansatz and obtain a relative error of 0.7%.

Variational quantum eigensolver for the Heisenberg antiferromagnet on the kagome lattice

Establishing the nature of the ground state of the Heisenberg antiferromagnet (HAFM) on the kagome lattice is well-known to be a prohibitively difficult problem for classical computers. Here, we give a detailed proposal for a variational quantum eigensolver (VQE) intending to solve this physical problem on a quantum computer. At the same time, this VQE constitutes an explicit experimental proposal for showing a useful quantum advantage on noisy intermediate-scale quantum devices because of its natural hardware compatibility. We classically emulate noiseless and noisy quantum computers with either 2D-grid or all-to-all connectivity and simulate patches of the kagome HAFM of up to 20 sites. In the noiseless case, the ground-state energy, as found by the VQE, approaches the true ground-state energy exponentially as a function of the circuit depth. Furthermore, VQEs for the HAFM on any graph can inherently perform their quantum computations in a decoherence-free subspace that protects against collective longitudinal and collective transversal noise, adding to the noise resilience of these algorithms. Nevertheless, the extent of the effects of other noise types suggests the need for error mitigation and performance targets alternative to high-fidelity ground-state preparation, even for essentially hardware-native VQEs.

Simulations of frustrated Ising Hamiltonians using quantum approximate optimization.

Novel magnetic materials are important for future technological advances. Theoretical and numerical calculations of ground-state properties are essential in understanding these materials, however, computational complexity limits conventional methods for studying these states. Here we investigate an alternative approach to preparing materials ground states using the quantum approximate optimization algorithm (QAOA) on near-term quantum computers. We study classical Ising spin models on unit cells of square, Shastry-Sutherland and triangular lattices, with varying field amplitudes and couplings in the material Hamiltonian. We find relationships between the theoretical QAOA success probability and the structure of the ground state, indicating that only a modest number of measurements ([Formula: see text]) are needed to find the ground state of our nine-spin Hamiltonians, even for parameters leading to frustrated magnetism. We further demonstrate the approach in calculations on a trapped-ion quantum computer and succeed in recovering each ground state of the Shastry-Sutherland unit cell with probabilities close to ideal theoretical values. The results demonstrate the viability of QAOA for materials ground state preparation in the frustrated Ising limit, giving important first steps towards larger sizes and more complex Hamiltonians where quantum computational advantage may prove essential in developing a systematic understanding of novel materials. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Variational counterdiabatic driving of the Hubbard model for ground-state preparation

Counterdiabatic (CD) protocols enable fast driving of quantum states by invoking an auxiliary adiabatic gauge potential (AGP) that suppresses transitions to excited states throughout the driving process. Usually, the full spectrum of the original unassisted Hamiltonian is a prerequisite for constructing the exact AGP, which implies that CD protocols are extremely difficult for many-body systems. Here, we apply a variational CD protocol recently proposed by P. W. Claeys et al. [Phys. Rev. Lett. 123, 090602 (2019)] to a two-component fermionic Hubbard model in one spatial dimension. This protocol engages an approximated AGP expressed as a series of nested commutators. We show that the optimal variational parameters in the approximated AGP satisfy a set of linear equations whose coefficients are given by the squared Frobenius norms of these commutators. We devise an exact algorithm that escapes the formidable iterative matrix-vector multiplications and evaluates the nested commutators and the CD Hamiltonian in analytic representations. We then examine the CD driving of the one-dimensional Hubbard model up to $L = 14$ sites with driving order $l \leqslant 3$. Our results demonstrate the usefulness of the variational CD protocol to the Hubbard model and permit a possible route towards fast ground-state preparation for many-body systems.

Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices

Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with Trotter based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground state preparation and energy estimating using a constant number of ancilla qubits (no more than 3).

State Preparation Boosters for Early Fault-Tolerant Quantum Computation

Quantum computing is believed to be particularly useful for the simulation of chemistry and materials, among the various applications. In recent years, there have been significant advancements in the development of near-term quantum algorithms for quantum simulation, including VQE and many of its variants. However, for such algorithms to be useful, they need to overcome several critical barriers including the inability to prepare high-quality approximations of the ground state. Current challenges to state preparation, including barren plateaus and the high-dimensionality of the optimization landscape, make state preparation through ansatz optimization unreliable. In this work, we introduce the method of ground state boosting, which uses a limited-depth quantum circuit to reliably increase the overlap with the ground state. This circuit, which we call a booster, can be used to augment an ansatz from VQE or be used as a stand-alone state preparation method. The booster converts circuit depth into ground state overlap in a controllable manner. We numerically demonstrate the capabilities of boosters by simulating the performance of a particular type of booster, namely the Gaussian booster, for preparing the ground state of N2 molecular system. Beyond ground state preparation as a direct objective, many quantum algorithms, such as quantum phase estimation, rely on high-quality state preparation as a subroutine. Therefore, we foresee ground state boosting and similar methods as becoming essential algorithmic components as the field transitions into using early fault-tolerant quantum computers.

Effects of cosine tapering window on quantum phase estimation

We provide a modification to the quantum phase estimation algorithm (QPEA) inspired on classical windowing methods for spectral density estimation. From this modification we obtain an upper bound in the cost that implies a cubic improvement with respect to the algorithm's error rate. Numerical evaluation of the costs also demonstrates an improvement. Moreover, with similar techniques, we detail an iterative projective measurement method for ground state preparation that gives an exponential improvement over previous bounds using QPEA. Numerical tests that confirm the expected scaling behavior are also obtained. For these numerical tests we have used a Lattice Thirring model as testing ground. Using well-known perturbation theory results, we also show how to more appropriately estimate the cost scaling with respect to state error instead of evolution operator error.

Bayesian learning for optimal control of quantum many-body states in optical lattices

Strongly correlated quantum many-body states provide invaluable resources for state-of-the-art quantum information science, ranging from quantum simulation to quantum metrology. In this work, we propose a supervised machine learning algorithm for optimal control of quantum many-body atomic states in optical lattices by numerical simulations based on the Bayesian method of Gaussian process regression (GPR). We combine this method with the time evolving block decimation (TEBD) algorithm for the preparation of the Heisenberg antiferromagnetic state in a system of bosonic atoms confined with a one-dimensional optical lattice. The quantum many-body ground state of 80 atoms is efficiently optimized within a few hundred machine learning iterations, reaching a state fidelity above $96%$. With a multistep learning strategy, we find the machine-learning-based optimal control method is scalable to large systems owing to its transferability. Its robustness against noise is demonstrated by considering imperfections that are typically present in optical lattice experiments. In the application of the GPR method to the preparation of the two-dimensional (2D) antiferromagnetic state, a state fidelity of $94%$ is reached for a 2D array of $6\ifmmode\times\else\texttimes\fi{}6$ spins through the time dependent variational principle (TDVP) algorithm, confirming the generalizability of our method. We further optimize the Hamiltonian ramping sequence crossing a quantum phase transition of the quantum $XXZ$ spin chain with the exact diagonalization method, from which a control protocol for generating the long-sought atomic Greenberger-Horne-Zeilinger state is obtained. We believe the proposed optimal control method for quantum Hamiltonian ground-state preparation would benefit present ultracold-atom experiments in the study of strongly correlated many-body physics.

Two-photon Raman transition channels of NaCs predicted from ab initio calculations

Obtaining effective spontaneous radiative transition channels is a fundamental topic in the preparation of ultracold rovibrational ground states of molecules. Herein, the ranges of prospective transition channels in the NaCs dimer are predicted via exploring the complex energy level structure and transition properties. The potential energy curves of \ensuremath{\Lambda}-$S$ states and \ensuremath{\Omega} states, spectroscopy constants, absolute energies of bound states, and transition information (Frank-Condon factors, lifetimes, transition dipole moments) have been researched. With this purpose, the results of our experiment on binding energies of the ${b}^{3}{\mathrm{\ensuremath{\Pi}}}_{0}{}^{+}$ rovibrational states are analyzed; the transition from the higher (photoassociated) ${}^{1}\mathrm{\ensuremath{\Pi}}$ state to the ground ${X}^{1}{\mathrm{\ensuremath{\Sigma}}}^{+}$ state is discussed. The wavelengths of the two lasers are predicted as 622.8-656.7 nm and 896.3-969.8 nm with the ${B}^{1}{\mathrm{\ensuremath{\Pi}}}_{1}$ and ${c}^{3}{\mathrm{\ensuremath{\Sigma}}}_{1}^{+}$ states, respectively, for the stimulated raman adiabatic passage method.

Adiabatic Spectroscopy and a Variational Quantum Adiabatic Algorithm

A toolbox for variational quantum adiabatic algorithms is proposed that allows to combine the strengths of the adiabatic and the variational approaches for fast and high-fidelity ground state preparation.

Hybrid quantum-classical approach for coupled-cluster Green's function theory

The three key elements of a quantum simulation are state preparation, time evolution, and measurement. While the complexity scaling of time evolution and measurements are well known, many state preparation methods are strongly system-dependent and require prior knowledge of the system's eigenvalue spectrum. Here, we report on a quantum-classical implementation of the coupled-cluster Green's function (CCGF) method, which replaces explicit ground state preparation with the task of applying unitary operators to a simple product state. While our approach is broadly applicable to many models, we demonstrate it here for the Anderson impurity model (AIM). The method requires a number ofTgates that grows asO(N5)per time step to calculate the impurity Green's function in the time domain, whereNis the total number of energy levels in the AIM. Since the number ofTgates is analogous to the computational time complexity of a classical simulation, we achieve an order of magnitude improvement over a classical CCGF calculation of the same order, which requiresO(N6)computational resources per time step.