The accurate computation of Hamiltonian ground, excited and thermal states on quantum computers stands to impact many problems in the physical and computer sciences, from quantum simulation to ...machine learning. Given the challenges posed in constructing large-scale quantum computers, these tasks should be carried out in a resource-efficient way. In this regard, existing techniques based on phase estimation or variational algorithms display potential disadvantages; phase estimation requires deep circuits with ancillae, that are hard to execute reliably without error correction, while variational algorithms, while flexible with respect to circuit depth, entail additional high-dimensional classical optimization. Here, we introduce the quantum imaginary time evolution and quantum Lanczos algorithms, which are analogues of classical algorithms for finding ground and excited states. Compared with their classical counterparts, they require exponentially less space and time per iteration, and can be implemented without deep circuits and ancillae, or high-dimensional optimization. We furthermore discuss quantum imaginary time evolution as a subroutine to generate Gibbs averages through an analogue of minimally entangled typical thermal states. Finally, we demonstrate the potential of these algorithms via an implementation using exact classical emulation as well as through prototype circuits on the Rigetti quantum virtual machine and Aspen-1 quantum processing unit.The quantum imaginary time evolution and Lanczos algorithms offer a resource-efficient way to compute ground or excited states of target Hamiltonians on quantum computers. This offers promise for quantum simulation on near-term noisy devices.
Abstract
The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for
N
molecular orbitals, the
$${\mathcal{O}}({N}^{4})$$
O
(
N
4
)
gate complexity of ...performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with
$${\mathcal{O}}({N}^{3})$$
O
(
N
3
)
gate complexity in small simulations, which reduces to
$${\mathcal{O}}({N}^{2})$$
O
(
N
2
)
gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with
$${\mathcal{O}}({N}^{3})$$
O
(
N
3
)
gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have
$${\mathcal{O}}({N}^{2})$$
O
(
N
2
)
depth on a linearly connected array, an improvement over the
$${\mathcal{O}}({N}^{3})$$
O
(
N
3
)
scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 10
5
non-Clifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.
Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high-resolution requirements and the exponential scaling of computational ...cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. Here, in this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations for systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.
Three different computational physics problems are discussed. The first project is solving the semi-classical Boltzmann transport equation (BTE) to compute the thermal conductivity of 1-D ...superlattices. We consider various spectral scattering models at each interface. This computation requires the inversion of a matrix whose size scales with the number of points used in the discretization of the Brillouin zone. We use spatial symmetries to reduce the size of data points and make the computation manageable. The other two projects involve quantum systems. Simulating quantum systems can potentially require exponential resources because of the exponential scaling of Hilbert space with system size. However, it has been observed that many physical systems, which typically exhibit locality in space or time, require much fewer resources to accurately simulate within some small error tolerance. The second project in the thesis is a two-step factorization of the electronic structure Hamiltonian that allows for efficient implementation on a quantum computer and also systematic truncation of small contributions. By using truncations that only incur errors below chemical accuracy, one is able to reduce the number of terms in the Hamiltonian from O (N 4 ) to O (N 3), where 푁 is the number of molecular orbitals in the system. The third project is a tensor network algorithm based on the concept of influence functionals (IFs) to compute long-time dynamics of single-site observables. IFs are high-dimensional objects that describe the influence of the bath on the dynamics of the subsystem of interest over all times, and we are interested in their low-rank approximations. We study two numerical models, the spin-boson model and a model of interacting hard-core bosons in a 1D harmonic trap, and find that the IFs can be efficiently computed and represented using tensor network methods. Consistent with physical intuition, the correlations in the IFs appear to decrease with increased bath sizes, suggesting that the low-rank nature of the IF is due to nontrivial cancellations in the bath.
The Vlasov-Maxwell equations provide an \textit{ab-initio} description of collisionless plasmas, but solving them is often impractical because of the wide range of spatial and temporal scales that ...must be resolved and the high dimensionality of the problem. In this work, we present a quantum-inspired semi-implicit Vlasov-Maxwell solver that utilizes the quantized tensor network (QTN) framework. With this QTN solver, the cost of grid-based numerical simulation of size \(N\) is reduced from \(\mathcal{O}(N)\) to \(\mathcal{O}(\text{poly}(D))\), where \(D\) is the ``rank'' or ``bond dimension'' of the QTN and is typically set to be much smaller than \(N\). We find that for the five-dimensional test problems considered here, a modest \(D=64\) appears to be sufficient for capturing the expected physics despite the simulations using a total of \(N=2^{36}\) grid points, \edit{which would require \(D=2^{18}\) for full-rank calculations}. Additionally, we observe that a QTN time evolution scheme based on the Dirac-Frenkel variational principle allows one to use somewhat larger time steps than prescribed by the Courant-Friedrichs-Lewy (CFL) constraint. As such, this work demonstrates that the QTN format is a promising means of approximately solving the Vlasov-Maxwell equations with significantly reduced cost.
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