Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm
X, The Moonshot Factory（X，登月工厂）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；University of Waterloo（滑铁卢大学）；Stanford University（斯坦福大学）；California Institute of Technology（加州理工学院）
Guillaume Verdon, Jacob Marks, Sasha Nanda, Stefan Leichenauer, Jack Hidary
We introduce a new class of generative quantum-neural-network-based models called Quantum Hamiltonian-Based Models (QHBMs). In doing so, we establish a paradigmatic approach for quantum-probabilistic hybrid variational learning, where we efficiently decompose the tasks of learning classical and quantum correlations in a way which maximizes the utility of both classical and quantum processors. In addition, we introduce the Variational Quantum Thermalizer (VQT) for generating the thermal state of a given Hamiltonian and target temperature, a task for which QHBMs are naturally well-suited. The VQT can be seen as a generalization of the Variational Quantum Eigensolver (VQE) to thermal states: we show that the VQT converges to the VQE in the zero temperature limit. We provide numerical results demonstrating the efficacy of these techniques in illustrative examples. We use QHBMs and the VQT on Heisenberg spin systems, we apply QHBMs to learn entanglement Hamiltonians and compression codes in simulated free Bosonic systems, and finally we use the VQT to prepare thermal Fermionic Gaussian states for quantum simulation.
We put forth a quantum simulation of a stroboscopic two-stroke thermal engine in the IBMQ processor. The system consists of a quantum spin chain connected to two baths at their boundaries, prepared at different temperatures using the variational quantum thermalizer algorithm. The dynamics alternates between heat and work strokes, which can be separately designed using independent quantum circuits. The results show good agreement with theoretical predictions, showcasing IBMQ as a powerful tool to study thermodynamics in the quantum regime, as well as the implementation of variational quantum al…
Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum (NISQ) computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We pro…
Effective low-energy theories represent powerful theoretical tools to reduce the complexity in modeling interacting quantum many-particle systems. However, common theoretical methods rely on perturbation theory, which limits their applicability to weak interactions. Here we introduce the Variational Adiabatic Gauge Transformation (VAGT), a non-perturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit that brings the Hamiltonian to either its block-diagonal or full-diagonal form. If a Hamiltonian can be diagonalized …
For the variational quantum eigensolver we propose to generate trial wavefunctions from a small amount of selected Pauli terms of the problem Hamiltonian. Two different approaches, one inspired by the quantum approximate optimization algorithm and the other by imaginary-time evolution, are proposed and studied in detail. Using numerical calculations, we study the efficiency of these trial wavefunctions for finding the ground-state energy of three molecules: H2, LiH and H2O. We find that only a small number of Pauli terms are needed to reach chemical accuracy, leading to short-depth quantum cir…
Electronic structure simulation is an anticipated application for quantum computers. Due to high-dimensional quantum entanglement in strongly correlated systems, the quantum resources required to perform such simulations are far beyond the capacity of current quantum devices. To reduce the quantum circuit complexity, it has been suggested to incorporate a part of the electronic correlation into an effective Hamiltonian, which is often obtained from a similarity transformation of the electronic Hamiltonian. In this work, we introduce a new transformation in the form of a product of a linear com…
Variational quantum eigensolver (VQE), which attracts attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called ansatz. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the ansatz, it has been difficult to analyze the performance of optimizers for the VQE systematically. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achi…
Variational Quantum Eigensolver (VQE) is a promising algorithm for near-term quantum machines. It can be used to estimate the ground state energy of a molecule by performing separate measurements of $O(N^4)$ terms. Several recent papers observed that this scaling may be reducible to $O(N^3)$ by partitioning the terms into linear-sized commuting families that can be measured simultaneously. We confirm these empirical observations by studying the MIN-COMMUTING-PARTITION problem at the level of the fermionic Hamiltonian and its encoding into qubits. Moreover, we provide a fast, pre-computable pro…
Dynamical quantum simulation may be one of the first applications to see quantum advantage. However, the circuit depth of standard Trotterization methods can rapidly exceed the coherence time of noisy quantum computers. This has led to recent proposals for variational approaches to dynamical simulation. In this work, we aim to make variational dynamical simulation even more practical and near-term. We propose a new algorithm called Variational Hamiltonian Diagonalization (VHD), which approximately transforms a given Hamiltonian into a diagonal form that can be easily exponentiated. VHD allows …
Calculating the energy spectrum of a quantum system is an important task, for example to analyse reaction rates in drug discovery and catalysis. There has been significant progress in developing algorithms to calculate the ground state energy of molecules on near-term quantum computers. However, calculating excited state energies has attracted comparatively less attention, and it is currently unclear what the optimal method is. We introduce a low depth, variational quantum algorithm to sequentially calculate the excited states of general Hamiltonians. Incorporating a recently proposed techniqu…
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 sy…
Hamiltonian diagonalization is at the heart of understanding physical properties and practical applications of quantum systems. It is highly desired to design quantum algorithms that can speedup Hamiltonian diagonalization, especially those can be implemented on near-term quantum devices. In this work, we propose a variational algorithm for Hamiltonians diagonalization (VQHD) of quantum systems, which explores the important physical properties, such as temperature, locality and correlation, of the system. The key idea is that the thermal states of the system encode the information of eigenvalu…
Variational quantum algorithms (VQAs) are a modern family of quantum algorithms designed to solve optimization problems using a quantum computer. Typically VQAs rely on a feedback loop between the quantum device and a classical optimization algorithm. The appeal of VQAs lies in their versatility, resistance to noise, and ability to demonstrate some results even with circuits of small depth. We primarily focus on the algorithm called variational quantum eigensolver (VQE), which takes a qubit Hamiltonian and returns its approximate ground state. We first present our numerical findings regarding …
Exploring critical states of the quantum Rabi model via Hamiltonian variational ansätze
South China Normal University（华南师范大学）；Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education)（原子与亚原子结构及量子控制教育部重点实验室）；Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter（广东省物质结构与基本相互作用卓越基础研究中心）；School of Physics, South China Normal University（华南师范大学物理学院）；Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials（广东省量子工程与量子材料重点实验室）；Guangdong-Hong Kong Joint Laboratory of Quantum Matter（广东-香港量子物质联合实验室）；Frontier Research Institute for Physics, South China Normal University（华南师范大学前沿物理研究院）
The numerical emulation of quantum physics and quantum chemistry often involves an intractable number of degrees of freedom and admits no known approximation in general form. In practice, representing quantum-mechanical states using available numerical methods becomes exponentially more challenging with increasing system size. Recently quantum algorithms implemented as variational models, have been proposed to accelerate such simulations. Here we study the effect of noise on the quantum phase transition in the Schwinger model, within a variational framework. The experiments are built using a f…
We present the meta-VQE, an algorithm capable to learn the ground state energy profile of a parametrized Hamiltonian. By training the meta-VQE with a few data points, it delivers an initial circuit parametrization that can be used to compute the ground state energy of any parametrization of the Hamiltonian within a certain trust region. We test this algorithm with a XXZ spin chain, an electronic H$_{4}$ Hamiltonian and a single-transmon quantum simulation. In all cases, the meta-VQE is able to learn the shape of the energy functional and, in some cases, resulted in improved accuracy in compari…
The accelerated progress in manufacturing noisy intermediate-scale quantum (NISQ) computing hardware has opened the possibility of exploring its application in transforming approaches to solving computationally challenging problems. The important limitations common among all NISQ computing technologies are the absence of error correction and the short coherence time, which limit the computational power of these systems. Shortening the required time of a single run of a quantum algorithm is essential for reducing environment-induced errors and for the efficiency of the computation. We have inve…
In this paper we discuss the utilization of Variational Quantum Solver (VQE) and recently introduced Generalized Unitary Coupled Cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N$_2$, H$_2$O, and C$_2$H$_4$, and benchmark systems to illustrate the perfo…
Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver
School of Applied and Engineering Physics, Cornell University（康奈尔大学应用工程物理学院）；Laboratory of Atomic and Solid State Physics, Cornell University（康奈尔大学原子与固态物理实验室）；Kavli Institute at Cornell for Nanoscale Science, Cornell University（康奈尔大学卡弗里纳米科学研究所）
Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum many-body states in analog quantum simulation. We describe a protocol where spatial deformations of the many-body Hamiltonian, physically realized on the quantum device, serve as an efficient variational ansatz for a local EH. Optimal variational parameters are determined in a feedback loop, involving quench dynamics with the deformed Hamiltonian as a quantum processing step, and classical optimization. We simulate the protocol for the ground state of Fermi-Hubbard models in quasi-1D geometries, fi…
Diffusion-Enhanced Optimization of Variational Quantum Eigensolver for General Hamiltonians
School of Physics, Beihang University（北京航空航天大学物理学院）；School of Physics, Beijing Institute of Technology（北京理工大学物理学院）；School of Instrument Science and Opto-Electronics Engineering, Beijing Information Science and Technology University（北京信息科技大学仪器科学与光电工程学院）
Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We approach this problem from the perspective of contextuality, and use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which addit…
Variational quantum algorithms involve training parameterized quantum circuits using a classical co-processor. An important variational algorithm, designed for combinatorial optimization, is the quantum approximate optimization algorithm. Realization of this algorithm on any modern quantum processor requires either embedding a problem instance into a Hamiltonian or emulating the corresponding propagator by a gate sequence. For a vast range of problem instances this is impossible due to current circuit depth and hardware limitations. Hence we adapt the variational approach -- using ion native H…
Non-Hermitian systems exhibiting topological properties are attracting growing interest. In this work, we propose an algorithm for solving the ground state of a non-Hermitian system in the matrix product state (MPS) formalism based on a companion Hermitian Hamiltonian. If the eigenvalues of the non-Hermitian system are known, the companion Hermitian Hamiltonian can be directly constructed and solved using Hermitian variational methods. When the eigenvalues are unknown, a gradient descent along with the companion Hermitian Hamiltonian yields both the ground state eigenenergy and the eigenstate.…
A large ongoing research effort focuses on Variational Quantum Algorithms (VQAs), representing leading candidates to achieve computational speed-ups on current quantum devices. The scalability of VQAs to a large number of qubits, beyond the simulation capabilities of classical computers, is still debated. Two major hurdles are the proliferation of low-quality variational local minima, and the exponential vanishing of gradients in the cost function landscape, a phenomenon referred to as barren plateaus. Here we show that by employing iterative search schemes one can effectively prepare the grou…
A Variational Framework for Time-Dependent Quantum Systems with Applications to Floquet Hamiltonians
Memorial University of Newfoundland and Labrador（纽芬兰与拉布拉多纪念大学）；National Research Council Canada（加拿大国家研究委员会）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）；Compute Everything Technologies Ltd.（Compute Everything科技有限公司）
We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a Symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A(~α) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ hamiltonian using rotation and reflection symmetry and H2 hamiltonian within S_z = 0 …
In this paper we reconsider the notion of an optimal effective Hamiltonian for the semiclassical propagation of the Wigner distribution in phase space. An explicit expression for the optimal effective Hamiltonian is obtained in the short time limit by minimizing the Hilbert-Schmidt distance between the semiclassical approximation and the real state of the system. The method is illustrated for the quartic oscillator.
Variational quantum eigensolver~(VQE) typically optimizes variational parameters in a quantum circuit to prepare eigenstates for a quantum system. Its applications to many problems may involve a group of Hamiltonians, e.g., Hamiltonian of a molecule is a function of nuclear configurations. In this paper, we incorporate derivatives of Hamiltonian into VQE and develop some hybrid quantum-classical algorithms, which explores both Hamiltonian and wavefunction spaces for optimization. Aiming for solving quantum chemistry problems more efficiently, we first propose mutual gradient descent algorithm …
The contextual subspace variational quantum eigensolver (CS-VQE) is a hybrid quantum-classical algorithm that approximates the ground-state energy of a given qubit Hamiltonian. It achieves this by separating the Hamiltonian into contextual and noncontextual parts. The ground-state energy is approximated by classically solving the noncontextual problem, followed by solving the contextual problem using VQE, constrained by the noncontextual solution. In general, computation of the contextual correction needs fewer qubits and measurements compared with solving the full Hamiltonian via traditional …
Calculating the ground state properties of a Hamiltonian can be mapped to the problem of finding the ground state of a smaller Hamiltonian through the use of embedding methods. These embedding techniques have the ability to drastically reduce the problem size, and hence the number of qubits required when running on a quantum computer. However, the embedding process can produce a relatively complicated Hamiltonian, leading to a more complex quantum algorithm. In this paper we carry out a detailed study into how density matrix embedding theory (DMET) could be implemented on a quantum computer to…
Integer factorization has been one of the cornerstone applications of the field of quantum computing since the discovery of an efficient algorithm for factoring by Peter Shor. Unfortunately, factoring via Shor's algorithm is well beyond the capabilities of today's noisy intermediate-scale quantum (NISQ) devices. In this work, we revisit the problem of factoring, developing an alternative to Shor's algorithm, which employs established techniques to map the factoring problem to the ground state of an Ising Hamiltonian. The proposed variational quantum factoring (VQF) algorithm starts by simplify…
Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves measurement of the Hamiltonian expectation value. Current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as $O(N^4)$ with the size of the system, $N$. To optimize VQE measurement one can try to group as many Hamiltonian terms as possib…
Variational quantum eigensolver (VQE) optimizes parameterized eigenstates of a Hamiltonian on a quantum processor by updating parameters with a classical computer. Such a hybrid quantum-classical optimization serves as a practical way to leverage up classical algorithms to exploit the power of near-term quantum computing. Here, we develop a hybrid algorithm for VQE, emphasizing the classical side, that can solve a group of related Hamiltonians simultaneously. The algorithm incorporates a snake algorithm into many VQE tasks to collectively optimize variational parameters of different Hamiltonia…
Measuring the expectation value of the molecular electronic Hamiltonian is one of the challenging parts of the variational quantum eigensolver. A widely used strategy is to express the Hamiltonian as a sum of measurable fragments using fermionic operator algebra. Such fragments have an advantage of conserving molecular symmetries that can be used for error mitigation. The number of measurements required to obtain the Hamiltonian expectation value is proportional to a sum of fragment variances. Here, we introduce a new method for lowering the fragments' variances by exploiting flexibility in th…
