Hamiltonian simulation in the low-energy subspace

Abstract

We study the problem of simulating the dynamics of spin systems when the initial state is supported on a subspace of low energy of a Hamiltonian H. This is a central problem in physics with vast applications in many-body systems and beyond, where the interesting physics takes place in the low-energy sector. We analyze error bounds induced by product formulas that approximate the evolution operator and show that these bounds depend on an effective low-energy norm of H. We find improvements over the best previous complexities of product formulas that apply to the general case, and these improvements are more significant for long evolution times that scale with the system size and/or small approximation errors. To obtain these improvements, we prove exponentially decaying upper bounds on the leakage to high-energy subspaces due to the product formula. Our results provide a path to a systematic study of Hamiltonian simulation at low energies, which will be required to push quantum simulation closer to reality.

Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions

“…To make implementations of more complex gauge theories possible, including non-Abelian and higherdimensional models, unifying physics insights, algorithm optimization, hardware implementation, and postprocessing is required, as demonstrated in this work. In this context, it would be interesting to investigate whether more resource-efficient encodings of such theories exist, if optimal Trotter decompositions and term ordering schemes can be found, to what degree these preserve local gauge symmetries, whether information regarding the initial state and the symmetries can be incorporated to further tighten the algorithmic error bounds [70,[91][92][93][94][95][96], how to balance these errors with experimental errors, and whether symmetry-protection schemes are advantageous in suppressing algorithmic and experimental errors. While progress along these lines is already being made [23,24,27,30,32,36,67,[97][98][99][100][101][102][103][104][105], further technological advances in quantum hardware are essential to enable advanced gauge-theory simulations in the upcoming years.…”

Fast-forwarding quantum evolution

“…While a complexity improvement for simulating these systems under the assumption that the initial state is supported in a low-energy subspace was shown in Ref. [22], that result does not demonstrate fast-forwarding as the quantum complexity is still superlinear in t. Fermionic and bosonic Hamiltonians are also important, and fast simulation methods for these systems will play an important role in the general problem of simulating quantum field theories [35]. Our methods exploit different properties of these systems and go beyond the fast-forwarding approach based on diagonalization.…”

“…We present a method for polynomially fast-forwarding a class of Hamiltonians denominated as frustration-free [27,28,34], when the initial state is supported in a certain low-energy subspace. This setting can be relevant for studying quantum phase transitions, the simulation of adiabatic quantum state preparation, and more, where spectral gaps can decrease with the system size [22]. For a spin system, a frustration-free Hamiltonian is H = X⊂Λ h X , where each h X has the additional property h X ≥ 0 and the lowest eigenvalue of H is zero.…”

“…We also consider the problem of fast-forwarding in subspaces, which concerns the case where quantum evolution occurs only in a certain subspace of the full Hilbert space. This notion of subspace fast-forwarding is useful for simulating physical systems because often the evolution occurs in certain subspaces of, for example, low energies or certain preserved symmetries [22][23][24]. Moreover, since certain quantum systems (e.g., bosonic systems) cannot be directly simulated on a digital quantum computer, we provide a definition of fast-forwarding that is relative to a specified set of observables.…”

Trotter product formulae for $*$-automorphisms of quantum lattice systems

“…We further wish to point out two related results. Firstly, a slightly different approach to a product formula was taken in [18], focussing on the 'quasi-adiabatic' properties of product formulae, namely the error when projected onto the ground state space. Secondly, a similar spatial product factorization with sharp error bounds was derived in [19]: it is not based on the Trotter strategy but it uses rather directly the Lieb-Robinson bound, see also [20].…”