Fast-forwarding quantum evolution

Abstract

We investigate the problem of fast-forwarding quantum evolution, whereby the dynamics of certain quantum systems can be simulated with gate complexity that is sublinear in the evolution time. We provide a definition of fast-forwarding that considers the model of quantum computation, the Hamiltonians that induce the evolution, and the properties of the initial states. Our definition accounts for any asymptotic complexity improvement of the general case and we use it to demonstrate fast-forwarding in several quantum systems. In particular, we show that some local spin systems whose Hamiltonians can be taken into block diagonal form using an efficient quantum circuit, such as those that are permutation-invariant, can be exponentially fast-forwarded. We also show that certain classes of positive semidefinite local spin systems, also known as frustration-free, can be polynomially fast-forwarded, provided the initial state is supported on a subspace of sufficiently low energies. Last, we show that all quadratic fermionic systems and number-conserving quadratic bosonic systems can be exponentially fast-forwarded in a model where quantum gates are exponentials of specific fermionic or bosonic operators, respectively. Our results extend the classes of physical Hamiltonians that were previously known to be fast-forwarded, while not necessarily requiring methods that diagonalize the Hamiltonians efficiently. We further develop a connection between fast-forwarding and precise energy measurements that also accounts for polynomial improvements.

Provably accurate simulation of gauge theories and bosonic systems

“…One factor in this product is the evolution arising from the terms in the truncated Hamiltonian H which have Λ-dependent norms, the terms involving the electric field at each link. This evolution can be "fastforwarded" [58,59] because the Hamiltonian is diagonal in a natural basis, and the evolution operator is just the tensor product of simple unitary operators, each acting on a single link. The other factor in the product is the interaction-picture evolution operator generated by the time-dependent interaction-picture Hamiltonian, in which each term has Λ-independent norm because the evolution induced by the electric field has been "rotated away."…”

Parallel Quantum Algorithm for Hamiltonian Simulation

“…Is it possible to reduce the dependence on other parameters like the simulation time t by parallelization? Atia and Aharonov [8] studied the fast-forwarding of Hamiltonians (which is further explored in a recent work [58])the ability to simulate a Hamiltonian by a quantum circuit with depth significantly less than the simulation time t (e.g. polylog(t)), which is essentially the possibility to reduce the complexity dependence on t in the parallel setting.…”

Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

“…The Cole-Hopf transform is a special case of a diagonalizable Koopman system. These systems are analogous to fast forwardable systems in the quantum literature [58], which have special time-energy uncertainty relations [59].…”

“…Clearly, the major bottleneck of this type of approach resides in the implementation of the nonlinear observables. This may be done either via direct computation [58,60], or quantum machine learning methods may be used to attempt to find a diagonalization using optimization methods [61][62][63]. However, we note that the nonlinear observables need only be implemented once to map the initial state to the transformed space, then once more to invert the transformation.…”