Variational Ansatz Research Articles

Accurate numerical simulations of open quantum systems using spectral tensor trains.

Decoherence between qubits is a major bottleneck in quantum computations. Decoherence results from intrinsic quantum and thermal fluctuations as well as noise in the external fields that perform the measurement and preparation processes. With prescribed colored noise spectra for intrinsic and extrinsic noise, we present a numerical method, Quantum Accelerated Stochastic Propagator Evaluation (Q-ASPEN), to solve the time-dependent noise-averaged reduced density matrix in the presence of intrinsic and extrinsic noise. Q-ASPEN is arbitrarily accurate and can be applied to provide estimates for the resources needed to error-correct quantum computations. We employ spectral tensor trains, which combine the advantages of tensor networks and pseudospectral methods, as a variational ansatz to the quantum relaxation problem and optimize the ansatz using methods typically used to train neural networks. The spectral tensor trains in Q-ASPEN make accurate calculations with tens of quantum levels feasible. We present benchmarks for Q-ASPEN on the spin-boson model in the presence of intrinsic noise and on a quantum chain of up to 32 sites in the presence of extrinsic noise. In our benchmark, the memory cost of Q-ASPEN scales as a low-order polynomial in the size of the system once the number of system states surpasses the number of basis functions used in the spectral expansion.

Studying evaporating black hole using quantum computation algorithms on IBM quantum processor

Analyzing complex quantum systems using quantum computational algorithms is one of the most promising applications of quantum computers. This study focuses on evaluating the performance of a custom variational ansatz in the Variational Quantum Eigensolver (VQE) algorithm compared to predefined ansatzes. To achieve this, we employ the evaporating black hole model as a test bed for our analysis. Using the VQE approach, which integrates quantum and classical computing techniques, we aim to minimize the energy expectation value of the Hamiltonian. By training the circuit parameters of a trial wave function as a parameterized quantum circuit, we determine the upper bound for the ground state energy and assess the optimal variational form. We define a custom ansatz for the VQE protocol and compare its performance with other predefined ansatzes. Additionally, we test the performance of three different classical optimizers to further understand their impact on the VQE algorithm’s efficiency and accuracy.

Exploring ground states of Fermi-Hubbard model on honeycomb lattices with counterdiabaticity

Exploring the ground state properties of many-body quantum systems conventionally involves adiabatic processes, alongside exact diagonalization, in the context of quantum annealing or adiabatic quantum computation. Shortcuts to adiabaticity by counter-diabatic driving serve to accelerate these processes by suppressing energy excitations. Motivated by this, we develop variational quantum algorithms incorporating the auxiliary counter-diabatic interactions, comparing them with digitized adiabatic algorithms. These algorithms are then implemented on gate-based quantum circuits to explore the ground states of the Fermi-Hubbard model on honeycomb lattices, utilizing systems with up to 26 qubits. The comparison reveals that the counter-diabatic inspired ansatz is superior to traditional Hamiltonian variational ansatz. Furthermore, the number and duration of Trotter steps are analyzed to understand and mitigate errors. Given the model’s relevance to materials in condensed matter, our study paves the way for using variational quantum algorithms with counterdiabaticity to explore quantum materials in the noisy intermediate-scale quantum era.

Specialising neural-network quantum states for the Bose Hubbard model

Abstract Projected variational wavefunctions such as the Gutzwiller, many-body correlator and Jastrow ansatzes have provided crucial insight into the nature of superfluid-Mott insulator transition in the Bose Hubbard model (BHM) in two or more spatial dimensions. However, these ansatzes have no obvious tractable and systematic way of being improved. A promising alternative is to use Neural-network quantum states (NQS) based on Restricted Boltzmann Machines (RBMs). With binary visible and hidden units NQS have proven to be a highly effective at describing quantum states of interacting spin- 1 2 lattice systems. The application of NQS to bosonic systems has so far been based on one-hot encoding from machine learning where the multi-valued site occupation is distributed across several binary-valued visible units of an RBM. Compared to spin- 1 2 systems one-hot encoding greatly increases the number of variational parameters whilst also making their physical interpretation opaque. Here we revisit the construction of NQS for bosonic systems by reformulating a one-hot encoded RBM into a correlation operator applied to a reference state, analogous to the structure of the projected variational ansatzes. In this form we then propose a number of specialisations of the RBM motivated by the physics of the BHM and the ability to capture exactly the projected variational ansatzes. We analyse in detail the variational performance of these new RBM variants for a 10 × 10 BHM, using both a standard Bose condensate state and a pre-optimised Jastrow + many-body correlator state as the reference state of the calculation. Several of our new ansatzes give robust results as nearly good as one-hot encoding across the regimes of the BHM, but at a substantially reduced cost. Such specialised NQS are thus primed tackle bosonic lattice problems beyond the accuracy of classic variational wavefunctions.

Gauged Gaussian projected entangled pair states: A high dimensional tensor network formulation for lattice gauge theories

Gauge theories form the basis of our understanding of modern physics—ranging from the description of quarks and gluons to effective models in condensed matter physics. In the nonperturbative regime, gauge theories are conventionally treated discretely as lattice gauge theories. The resulting systems are evaluated with path-integral based Monte Carlo methods. These methods, however, can suffer from the sign problem and do not allow for a direct evaluation of real-time dynamics. In this work, we present a unified and comprehensive framework for gauged Gaussian projected entangled pair states (PEPS), a variational ansatz based on tensor networks. We review the construction of Hamiltonian lattice gauge theories, explain their similarities to PEPS, and detail the construction of the ansatz state. The estimation of ground states is based on a variational Monte Carlo procedure with the PEPS as an ansatz state. This sign-problem-free ansatz can be efficiently evaluated in any dimension with arbitrary gauge groups, and can include dynamical fermionic matter, suggesting new options for the simulation of nonperturbative regimes of gauge theories, including quantum chromodynamics. Published by the American Physical Society 2024

Quantum skyrmion dynamics studied by neural network quantum states

We study the dynamics of quantum skyrmions under a magnetic field gradient using neural network quantum states. First, we obtain a quantum skyrmion lattice ground state using variational Monte Carlo with a restricted Boltzmann machine as the variational ansatz for a quantum Heisenberg model with a Dzyaloshinskii-Moriya interaction. Then, using the time-dependent variational principle, we study the real-time evolution of quantum skyrmions after a Hamiltonian quench with an inhomogeneous external magnetic field. We show that field gradients are an effective way of manipulating and moving quantum skyrmions. Furthermore, we demonstrate that quantum skyrmions can decay when interacting with each other. This work shows that neural network quantum states offer a promising way of studying the real-time evolution of quantum magnetic systems that are outside the realm of exact diagonalization. Published by the American Physical Society 2024

Noise-Aware Variational Eigensolvers: A Dissipative Route for Lattice Gauge Theories

We propose a novel variational ansatz for the ground-state preparation of the Z2 lattice gauge theory (LGT) in quantum simulators. It combines dissipative and unitary operations in a completely deterministic scheme with a circuit depth that does not scale with the size of the considered lattice. We find that, with very few variational parameters, the ansatz can achieve >99% precision in energy in both the confined and deconfined phase of the Z2 LGT. We benchmark our proposal against the unitary Hamiltonian variational ansatz showing a reduction in the required number of variational layers to achieve a target precision. After performing a finite-size scaling analysis, we show that our dissipative variational ansatz can predict accurate critical exponents without requiring a number of layers that scales with the system size, which is the standard situation for unitary ansätze. Furthermore, we investigate the performance of this variational eigensolver subject to circuit-level noise, determining variational error thresholds that fix the error rate below which it would be beneficial to increase the number of layers. In light of these quantities and for typical gate errors p in current quantum processors, we provide a detailed assessment of the prospects of our scheme to explore the Z2 LGT on near-term devices. Published by the American Physical Society 2024

Characterizing barren plateaus in quantum ansätze with the adjoint representation

Variational quantum algorithms, a popular heuristic for near-term quantum computers, utilize parameterized quantum circuits which naturally express Lie groups. It has been postulated that many properties of variational quantum algorithms can be understood by studying their corresponding groups, chief among them the presence of vanishing gradients or barren plateaus, but a theoretical derivation has been lacking. Using tools from the representation theory of compact Lie groups, we formulate a theory of barren plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra, covering a large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks. Our theory provides, for the first time, the ability to compute the exact variance of the gradient of the cost function of the quantum compound ansatz, under mixing conditions that we prove are commonplace.

Exponential qubit reduction in optimization for financial transaction settlement

We extend the qubit-efficient encoding presented in (Tan et al. in Quantum 5:454, 2021) and apply it to instances of the financial transaction settlement problem constructed from data provided by a regulated financial exchange. Our methods are directly applicable to any QUBO problem with linear inequality constraints. Our extension of previously proposed methods consists of a simplification in varying the number of qubits used to encode correlations as well as a new class of variational circuits which incorporate symmetries thereby reducing sampling overhead, improving numerical stability and recovering the expression of the cost objective as a Hermitian observable. We also propose optimality-preserving methods to reduce variance in real-world data and substitute continuous slack variables. We benchmark our methods against standard QAOA for problems consisting of 16 transactions and obtain competitive results. Our newly proposed variational ansatz performs best overall. We demonstrate tackling problems with 128 transactions on real quantum hardware, exceeding previous results bounded by NISQ hardware by almost two orders of magnitude.

Deep neural networks as variational solutions for correlated open quantum systems

In this work we apply deep neural networks to find the non-equilibrium steady state solution to correlated open quantum many-body systems. Motivated by the ongoing search to find more powerful representations of (mixed) quantum states, we design a simple prototypical convolutional neural network and show that parametrizing the density matrix directly with more powerful models can yield better variational ansatz functions and improve upon results reached by neural density operator based on the restricted Boltzmann machine. Hereby we give up the explicit restriction to positive semi-definite density matrices. However, this is fulfilled again to good approximation by optimizing the parameters. The great advantage of this approach is that it opens up the possibility of exploring more complex network architectures that can be tailored to specific physical properties. We show how translation invariance can be enforced effortlessly and reach better results with fewer parameters. We present results for the dissipative one-dimensional transverse-field Ising model and a two-dimensional dissipative Heisenberg model compared to exact values.

Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric.

Generalization is the ability of machine learning models to make accurate predictions on new data by learning from training data. However, understanding generalization of quantum machine learning models has been a major challenge. Here, we introduce the data quantum Fisher information metric (DQFIM). It describes the capacity of variational quantum algorithms depending on variational ansatz, training data, and their symmetries. We apply the DQFIM to quantify circuit parameters and training data needed to successfully train and generalize. Using the dynamical Lie algebra, we explain how to generalize using a low number of training states. Counterintuitively, breaking symmetries of the training data can help to improve generalization. Finally, we find that out-of-distribution generalization, where training and testing data are drawn from different data distributions, can be better than using the same distribution. Our work provides a useful framework to explore the power of quantum machine learning models.

Applicability of measurement-based quantum computation towards physically-driven variational quantum eigensolver

Variational quantum algorithms are considered one of the most promising methods for obtaining near-term quantum advantages; however, most of these algorithms are only expressed in the conventional quantum circuit scheme. The roadblock to developing quantum algorithms with the measurement-based quantum computation (MBQC) scheme is resource cost. Recently, we discovered that the realization of multi-qubit rotation operations only requires a constant number of single-qubit measurements with the MBQC scheme, providing a potential advantage in terms of resource cost. The structure of the Hamiltonian variational ansatz aligns well with this characteristic. Thus, we propose an efficient measurement-based quantum algorithm for quantum many-body system simulation tasks, called measurement-based Hamiltonian variational ansatz (MBHVA). We then demonstrate its effectiveness, efficiency, and advantages with the two-dimensional Heisenberg model and the Fermi–Hubbard chain. Numerical experiments show that MBHVA can have similar performance as circuit-based ansatz, and is expected to reduce operation counts during execution compared to quantum circuits, bringing the advantage of running time. We conclude that the MBQC scheme is potentially feasible for achieving near-term quantum advantages in the noisy intermediate-scale quantum era, especially in the presence of large multi-qubit rotation operations.

Ansatz optimization of the variational quantum eigensolver tested on the atomic Anderson model

We present a detailed analysis and optimization of the variational quantum algorithms required to find the ground state of a correlated electron model, using several types of variational ansatz. Specifically, we apply our approach to the atomic limit of the Anderson model, which is widely studied in condensed matter physics since it can simulate fundamental physical phenomena, ranging from magnetism to superconductivity. The method is developed by presenting efficient state preparation circuits that exhibit total spin, spin projection, particle number and time-reversal symmetries. These states contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace allowing to avoid irrelevant sectors of Hilbert space. Then, we show how to construct quantum circuits, providing explicit decomposition and gate count in terms of standard gate sets. We test these quantum algorithms looking at ideal quantum computer simulations as well as implementing quantum noisy simulations. We finally perform an accurate comparative analysis among the approaches implemented, highlighting their merits and shortcomings.

Lattice polaron in a Bose–Einstein condensate of hard-core bosons

Lattice polarons, quasiparticles arising from the interaction between an impurity and its surrounding bosonic environment confined to a lattice system, have emerged as a platform for generating complex few-body states, probing many-body phenomena, and addressing long-standing problems in physics. In this study, we employ a variational ansatz to investigate the quasiparticle and spectral properties of an impurity coupled to a condensate gas of hard-core bosons in a two-dimensional optical lattice. Our findings demonstrate that the polaron features can be tuned by adjusting the filling factor of the bath, revealing intriguing polaron characteristics in the strongly interacting regime. These results offer valuable insights for lattice polaron experiments with ultracold gases and can serve as a guide for new experiments in emergent quantum devices, such as moiré materials, where optical excitations can be described in terms of hard-core bosons.

NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians

Motivated by the ground state structure of quantum models with all-to-all interactions such as mean-field quantum spin glass models and the Sachdev-Ye-Kitaev (SYK) model, we propose a tensor network architecture which can accomodate volume law entanglement and a large ground state degeneracy. We call this architecture the non-local renormalization ansatz (NoRA) because it can be viewed as a generalization of MERA, DMERA, and branching MERA networks with the constraints of spatial locality removed. We argue that the architecture is potentially expressive enough to capture the entanglement and complexity of the ground space of the SYK model, thus making it a suitable variational ansatz, but we leave a detailed study of SYK to future work. We further explore the architecture in the special case in which the tensors are random Clifford gates. Here the architecture can be viewed as the encoding map of a random stabilizer code. We introduce a family of codes inspired by the SYK model which can be chosen to have constant rate and linear distance at the cost of some high weight stabilizers. We also comment on potential similarities between this code family and the approximate code formed from the SYK ground space.

Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks

We study the response of critical resonating valence bond (RVB) spin liquids to doping with longer-range singlets, and more generally of U(1)-symmetric tensor networks to nonsymmetric perturbations. Using a field theory description, we find that in the RVB, doping constitutes a relevant perturbation that immediately opens up a gap, contrary to previous observations. Our analysis predicts a very large correlation length even at significant doping, which we verify using high-accuracy numerical simulations. This emphasizes the need for careful analysis, but it also justifies the use of such states as a variational ansatz for critical systems. Finally, we give an example of a projected entangled pair state where nonsymmetric perturbations do not open up a gap and the U(1) symmetry reemerges. Published by the American Physical Society 2024

Variational manifolds for ground states and scarred dynamics of blockade-constrained spin models on two- and three-dimensional lattices

We introduce a variational manifold of simple tensor network states for the study of a family of constrained models that describe spin-12 systems as realized by Rydberg atom arrays. Our manifold permits analytical calculation via perturbative expansion of one- and two-point functions in arbitrary spatial dimensions and allows for efficient computation of the matrix elements required for variational energy minimization and variational time evolution in up to three dimensions. We apply this framework to the PXP model on the hypercubic lattice in one, two, and three dimensions and show that, in each case, it exhibits quantum phase transitions breaking the sublattice symmetry in equilibrium, and hosts quantum many-body scars out of equilibrium. We demonstrate that our variational ansatz qualitatively captures all these phenomena and predicts key quantities with an accuracy that increases with the dimensionality of the lattice, and conclude that our method can be interpreted as a generalization of mean-field theory to constrained spin models. Published by the American Physical Society 2024

Adaptive projected variational quantum dynamics

We propose an adaptive quantum algorithm to prepare accurate variational time evolved wave functions. The method is based on the projected variational quantum dynamics (pVQD) algorithm, that performs a global optimization with linear scaling in the number of variational parameters. Instead of fixing a variational ansatz at the beginning of the simulation, the circuit is grown systematically during the time evolution. Moreover, the adaptive step does not require auxiliary qubits and the gate search can be performed in parallel on different quantum devices. We apply the algorithm, named adaptive pVQD, to the simulation of driven spin models and fermionic systems, where it shows an advantage when compared to both Trotterized circuits and nonadaptive variational methods. Finally, we use the shallower circuits prepared using the adaptive pVQD algorithm to obtain more accurate measurements of physical properties of quantum systems on hardware. Published by the American Physical Society 2024

Fast Quantum State Preparation and Bath Dynamics Using Non-Gaussian Variational Ansatz and Quantum Optimal Control.

Fast preparation of quantum many-body states is essential for myriad quantum algorithms and metrological applications. Here, we develop a new pathway for fast, nonadiabatic preparation of quantum many-body states that combines quantum optimal control with a variational Ansatz based on non-Gaussian states. We demonstrate our approach on the spin-boson model, a single spin interacting with the bath. We use a multipolaron Ansatz to prepare near-critical ground states. For one mode, we achieve a reduction in infidelity of up to ≈60 (≈10) times compared to linear (optimized local adiabatic) ramps; for many modes, we achieve a reduction in infidelity of up to ≈5 times compared to nonadiabatic linear ramps. Further, we show that the typical control quantity, the leakage from the variational manifold, provides only a loose bound on the state's fidelity. Instead, in analogy to the bond dimension of matrix product states, we suggest a controlled convergence criterion based on the number of polarons. Finally, motivated by the possibility of realizations in trapped ions, we study the dynamics of a system with bath properties going beyond the paradigm of (sub- and/or super-) Ohmic couplings.

Variational quantum eigensolver with embedded entanglement using a tensor-network ansatz

In this paper, we introduce a tensor network (TN) scheme into the entanglement augmentation process of the synergistic optimization framework by Rudolph [] to build its process systematically for inhomogeneous systems. Our synergistic approach first embeds the variational optimal solution of the TN state with the entropic area law, which can be perfectly optimized in conventional (classical) computers, in a quantum variational circuit ansatz inspired by the TN state with the entropic volume law. Next, the framework performs a variational quantum eigensolver (VQE) process with embedded states as the initial state. We applied the synergistic to the ground-state analysis of the all-to-all coupled random transverse-field Ising, XYZ, Heisenberg model, employing the binary multiscale entanglement renormalization ansatz (MERA) state and branching MERA states as TN states with entropic area law and volume law, respectively. We then show that the synergistic accelerates VQE calculations in the three models without an initial parameter guess of the branching-MERA-inspired ansatz and can avoid a local solution trapped by a standard VQE with the ansatz in the Ising model. The improvement of optimizers for MERA in all-to-all coupled inhomogeneous systems, enhancement, and potential synergistic applications are also discussed. Published by the American Physical Society 2024