In this paper, we present a proof-of-concept quantum algorithm for simulating time-dependent Hamiltonian evolution by reducing the problem to simulating a time-independent Hamiltonian in a larger space using a discrete clock Hamiltonian construction. A similar construction was first explored for this simulation problem by Watkins, Wiebe, Roggero, and Lee [PRX Quantum, 2024]. Our algorithm improves upon their work in terms of the dependence on evolution time and precision. In addition, the complexity matches the state-of-the-art simulation algorithms using other approaches. To achieve this improvement, we use Duhamel's principle to treat the clock and system Hamiltonians separately and exploit properties of Gaussian quadrature to reduce the simulation cost. Our approach demonstrates that time-dependent Hamiltonian simulation can be as efficient in a simpler framework and hence provides a new angle to model and simulate time-dependent systems.
In this paper, we present a proof-of-concept quantum algorithm for simulating time-dependent Hamiltonian evolution by reducing the problem to simulating a time-independent Hamiltonian in a larger space using a discrete clock Hamiltonian construction. A similar construction was first explored for this simulation problem by Watkins, Wiebe, Roggero, and Lee [PRX Quantum, 2024]. Our algorithm improves upon their work in terms of the dependence on evolution time and precision. In addition, the complexity matches the state-of-the-art simulation algorithms using other approaches. To achieve this improvement, we use Duhamel's principle to treat the clock and system Hamiltonians separately and exploit properties of Gaussian quadrature to reduce the simulation cost. Our approach demonstrates that time-dependent Hamiltonian simulation can be as efficient in a simpler framework and hence provides a new angle to model and simulate time-dependent systems.
Compared with time-independent Hamiltonians, the dynamics of generic quantum Hamiltonians H ( t ) are complicated by the presence of time ordering in the evolution operator. In the context of digital quantum simulation, this difficulty prevents a direct adaptation of many time-independent simulation algorithms for time-dependent simulation. However, there exists a framework within the theory of dynamical systems that eliminates time ordering by adding a “clock” degree of freedom. In this work, we provide a computational framework, based on this reduction, for encoding time-dependent dynamics as time-independent systems. As a result, we make two advances in digital Hamiltonian simulation. First, we create a time-dependent simulation algorithm based on performing qubitization on the augmented clock system and, in doing so, provide the first qubitization-based approach to time-dependent Hamiltonians that goes beyond Trotterization of the ordered exponential. Second, we define a natural generalization of multiproduct formulas for time-ordered exponentials and then propose and analyze an algorithm based on these formulas. Unlike other algorithms of similar accuracy, the multiproduct approach achieves commutator scaling, meaning that this method outperforms existing methods for physically local time-dependent Hamiltonians with sufficient smoothness. Our work reduces the disparity between time-dependent and time-independent simulation and indicates a step toward optimal quantum simulation of time-dependent Hamiltonians. Published by the American Physical Society 2024
In this thesis, we present three innovations that can be used to simulate quantum dynamics on both quantum and classical computers. Our primary innovation is an efficient quantum algorithm for simulating time-dependent Hamiltonian evolution of general input states on a quantum computer. Given conditions on the smoothness of the Hamiltonian, the complexity of the algorithm is close to linear in the evolution time, and therefore is comparable to algorithms for time-independent Hamiltonians. In addition, we show how the complexity can be reduced by optimizing the time steps. The complexity of the algorithm is quantified by calls to an oracle, which yields information about the Hamiltonian. In contrast to previous work, which allowed an oracle query to yield an arbitrary number of bits or qubits, we assign a cost for each bit or qubit accessed, revealing hitherto unnoticed simulation costs. We also account for discretization errors in the time and the representation of the Hamiltonian. This work is enabled by our development of a decomposition scheme based on Lie-Trotter- Suzuki product formulae for ordered operator exponentials. We show, by counterexample, that Lie-Trotter-Suzuki approximations may be lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. We address this problem by presenting a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds for the approximation error. We also provide a new analysis of the evolution of quantum adiabatic systems in this thesis. Our work contains the tightest upper bounds that are currently known, and the first tight lower bounds, for the error in the adiabatic approximation applied to fixed Hamiltonians in the limit of slow evolution. We use our results to address the Marzlin- Sanders counterexample Hamiltonian. Specifically, we provide sufficiency conditions for the validity of the adiabatic approximation and demonstrate that the Marzlin-Sanders counterexample fails to satisfy them. We also find a new class of time-dependent Hamiltonians that are related to the Marzlin0Sanders counterexample Hamiltonian, but nonetheless obey the adiabatic approximation for sufficiently slow evolutions.
Recently, there has been growing interest in simulating time-dependent Hamiltonians using quantum algorithms, driven by diverse applications, such as quantum adiabatic computing. While techniques for simulating time-independent Hamiltonian dynamics are well established, time-dependent Hamiltonian dynamics is less explored and it is unclear how to systematically organize existing methods and to find new methods. Sambe-Howland's continuous clock elegantly transforms time-dependent Hamiltonian dynamics into time-independent Hamiltonian dynamics, which means that by taking different discretizations, existing methods for time-independent Hamiltonian dynamics can be exploited for time-dependent dynamics. In this work, we systemically investigate how Sambe-Howland's clock can serve as a unifying framework for simulating time-dependent Hamiltonian dynamics. First, we demonstrate the versatility of this approach by showcasing its compatibility with analog quantum computing and digital quantum computing. Second, for digital quantum computers, we illustrate how this framework, combined with time-independent methods (e.g., product formulas, multiproduct formulas, qDrift, and linear combination of unitaries-Taylor), can facilitate the development of efficient algorithms for simulating time-dependent dynamics. This framework allows us to (1) resolve the problem of finding minimum-gate time-dependent product formulas; (2) establish a unified picture of both Suzuki's and Huyghebaert and De Raedt's approaches; (3) generalize Huyghebaert and De Raedt's first- and second-order formula to arbitrary orders; (4) answer an unsolved question in establishing time-dependent multiproduct formulas; (5) and recover continuous qDrift on the same footing as time-independent qDrift. Third, we demonstrate the efficacy of our higher-order Huyghebaert and De Raedt's algorithm through digital adiabatic simulation.
We present a low-space overhead simulation algorithm based on the truncated Dyson series for time-dependent quantum dynamics. This algorithm is applied to simulating time-independent Hamiltonians by transitioning to the interaction picture, where some portions are made time-dependent. This can provide a favorable complexity trade-off as the algorithm scales exponentially better with derivatives of the time-dependent component than the original Hamiltonian. We show that this leads to an exponential improvement in gate complexity for simulating some classes of diagonally dominant Hamiltonian. Additionally we show that this can reduce the gate-complexity scaling for simulating N-site Hubbard models for time t with arbitrary long-range interactions as well as reduce the cost of quantum chemistry simulations within a similar-sized plane-wave basis to O(N2t) from O(N11/3t). We also show a quadratic improvement in query complexity for simulating sparse time-dependent Hamiltonians, which may be of independent interest.
We provide a general method for efficiently simulating time-dependent Hamiltonian dynamics on a circuit-model-based quantum computer. Our approach is based on approximating the truncated Dyson series of the evolution operator, extending the earlier proposal by Berry et al. [Phys. Rev. Lett. 114, 090502 (2015)] to evolution generated by explicitly time-dependent Hamiltonians. Two alternative strategies are proposed to implement time ordering while exploiting the superposition principle for sampling the Hamiltonian at different times. The resource cost of our simulation algorithm retains the optimal logarithmic dependence on the inverse of the desired precision.
Abstract Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case, it corresponds to having a time-dependent Hamiltonian. Designing quantum algorithms for time-dependent Hamiltonian is generally much more complicated than time-independent Hamiltonian problems, due to the challenge in handling the time-ordering; moreover, almost all existing such algorithms are developed for qubit-based quantum devices. Here we propose an alternative formalism based on Sambe-Howland’s clock that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e. quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrödingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.