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.

Constant-depth circuits for dynamic simulations of materials on quantum computers

“…As all H CD Hamiltonians of Table 1 are quadratic, it is possible to fast-forward simulations under their time-evolution (Atia and Aharonov 2017;Gu et al 2021). In Demonstration of constant-depth circuits section, we demonstrate simulations with our constant-depth circuits for two important models in H CD : (i) the XY model, where J z = 0 and h β = 0, and (ii) the TFIM, where J y = J z = 0.…”

“…According to the "no-fast-forwarding theorem", simulating the dynamics of a system under a generic Hamiltonian H for a time t requires (t) gates (Berry et al 2007;Childs and Kothari 2010), implying that circuit depths grow at least linearly with the number of time-steps. It has been shown, however, that quadratic Hamiltonians can be fast-forwarded, meaning the evolution of the systems under such Hamiltonians can be simulated with circuits whose depths do not grow significantly with the simulation time (Atia and Aharonov 2017;Gu et al 2021). A recent work took advantage of this to variationally compile approximate circuits with a hybrid classical-quantum algorithm for fast-forwarded simulations (Cîrstoiu et al 2020).…”

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 "fast-forwarded" [3,26] 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."…”

Thermal Multi-scale Entanglement Renormalization Ansatz for Variational Gibbs State Preparation

“…By the construction of the Hamiltonian, it is equivalent to a product state Hamiltonian conjugated by a logarithmic depth circuit. This mean time evolution by this Hamiltonian can be efficiently fast-forwarded [50]. This is true even in the case where the circuit is not a matchgate circuit and therefore is not equivalent to some quadratic fermion model.…”