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arXiv:2404.03644 · Quantum

Hamiltonian simulation for low-energy states with optimal time dependence

Alexander Zlokapa, Rolando D. Somma

Published Aug 27, 2024• 12 citations
arXivHTMLPDF

Abstract

We consider the task of simulating time evolution under a Hamiltonian H within its low-energy subspace. Assuming access to a block-encoding of H ′ := ( H − E ) / λ , for some λ > 0 and E ∈ R , the goal is to implement an ϵ -approximation to the evolution operator e − i t H when the initial state is confined to the subspace corresponding to eigenvalues [ − 1 , − 1 + Δ / λ ] of H ′ , for Δ ≤ λ . We present a quantum algorithm that requires O ( t λ Γ + λ / Γ log ⁡ ( 1 / ϵ ) ) queries to the block-encoding for any choice of Γ such that Δ ≤ Γ ≤ λ . When the parameters satisfy log ⁡ ( 1 / ϵ ) = o ( t λ ) and Δ / λ = o ( 1 ) , this result improves over generic methods with query complexity Ω ( t λ ) . Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.For a given H , the block-encoding of its H ′ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as g a p − a m p l i f i a b l e . We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of H

DOI
10.22331/q-2024-08-27-1449

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We consider the task of simulating time evolution under a Hamiltonian H within its low-energy subspace. Assuming access to a block-encoding of H ′ := ( H − E ) / λ , for some λ > 0 and E ∈ R , the goal is to implement an ϵ -approximation to the evolution operator e − i t H when the initial state is confined to the subspace corresponding to eigenvalues [ − 1 , − 1 + Δ / λ ] of H ′ , for Δ ≤ λ . We present a quantum algorithm that requires O ( t λ Γ + λ / Γ log ⁡ ( 1 / ϵ ) ) queries to the block-encoding for any choice of Γ such that Δ ≤ Γ ≤ λ . When the parameters satisfy log ⁡ ( 1 / ϵ ) = o ( t λ ) and Δ / λ = o ( 1 ) , this result improves over generic methods with query complexity Ω ( t λ ) . Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.For a given H , the block-encoding of its H ′ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as g a p − a m p l i f i a b l e . We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of H
We present the problem of approximating the time-evolution operator e−iH^t to error ϵ , where the Hamiltonian H^=(⟨G|⊗I^)U^(|G⟩⊗I^) is the projection of a unitary oracle U^ onto the state |G⟩ created by another unitary oracle. Our algorithm solves this with a query complexity O(t+log⁡(1/ϵ)) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d -sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H^ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any H^ in an invariant SU(2) subspace. A large class of operator functions of H^ can then be computed with optimal query complexity, of which e−iH^t is a special case.
We present the problem of approximating the time-evolution operator e-iHt to error ϵ, where the Hamiltonian H = ((G|⊗I)U (|Gi⊗I) is the projection of a unitary oracle U onto the state |Gi created by another unitary oracle. Our algorithm solves this with a query complexity O ( t + log(1/ϵ) ) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H is a density matrix. A key technical result is 'qubitization', which uses the controlled version of these oracles to embed any H in an invariant SU(2) subspace. A large class of operator functions of H can then be computed with optimal query complexity, of which e-iHt is a special case.
Ground-state preparation, energy estimation, and property estimation are fundamental tasks in quantum simulation. Existing ground-state preparation algorithms achieve near-optimal complexity using block encodings or controlled Hamiltonian evolution, raising a natural question: is controlled Hamiltonian evolution necessary, or can the same scaling be attained in the weaker access model that provides access to only uncontrolled Hamiltonian evolution? We resolve this question by giving two algorithms that access the input Hamiltonian HHH only through queries to UHU_HUH​ and its inverse UH†U_H^\daggerUH†​, where ∥H∥≤αH\|H\|\leα_H∥H∥≤αH​ and τ=Θ(1/αH)τ=Θ(1/α_H)τ=Θ(1/αH​). Given an input state with ground-state weight at least ηηη and a spectral gap of HHH of at least γγγ, our first algorithm uses fresh copies of the input state and requires O~(αH/(γη))\widetilde O(α_H/(γη))O(αH​/(γη)) queries to UHU_HUH​ and UH†U_H^\daggerUH†​, while our second algorithm uses coherent access to the input-state preparation unitary and its inverse to improve this complexity to O~(αH/(γη))\widetilde O(α_H/(γ\sqrtη))O(αH​/(γη​)). The latter bound matches the known lower bound up to logarithmic factors, showing that algorithms with additional access to controlled Hamiltonian evolution cannot improve the query complexity for ground-state preparation beyond logarithmic factors. Both algorithms are based on a simple mechanism that we call the spectral descent principle, in which the algorithm starts from the input state ∣ψ⟩|ψ\rangle∣ψ⟩ and progressively descends through the energy spectrum, moving closer to the ground state until the desired fidelity is reached. We also extend this framework to ground-state property and energy estimation and give explicit complexity guarantees for both tasks.
Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes O~(d 3/2/ε)\tilde{\mathcal{O}}(d^{\,3/2}/ε)O~(d3/2/ε) queries to the time-evolution operator of the system and O~(d 3/2)\tilde{\mathcal{O}}(d^{\,3/2})O~(d3/2) queries to a block-encoding of the perturbation, for ddd cooling steps and an εεε-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.
Hamiltonian simulations are key subroutines in adiabatic quantum computation and quantum many-body physics, where quantum dynamics often happen in the low-energy sector. Previous studies have shown that the low-energy assumption can reduce the resource requirements of standard time-independent Hamiltonian simulation algorithms. However, whether such advantages extend to time-dependent Hamiltonian simulation remains open. In this paper, we consider the adiabatic regime where the relevant low-energy subspace is spanned by a fixed number of low-energy eigenstates and separated from the rest of the spectrum by a gap. We show that, for simulating spin Hamiltonians by product formulas, the explicit system size dependence in the leading commutator-scaling term can be replaced by a low-energy scale up to logarithmic factors. Technically, we derive the low-energy simulation error with commutator scaling for product formulas by leveraging adiabatic perturbation theory to analyze the time-variant energy spectrum of the underlying Hamiltonian. We further conduct numerical experiments on adiabatic state preparation of an illustrative example system to support our theoretical findings. Finally, we prove a lower bound of query complexity for generic time-dependent Hamiltonian simulations.
Simulating Hamiltonian dynamics is one of the most fundamental and significant tasks for characterizing quantum materials. Recently, a series of quantum algorithms employing block encoding of Hamiltonians have succeeded in providing efficient simulation of time-evolution operators on quantum computers. While time-independent Hamiltonians can be simulated by the quantum eigenvalue transformation (QET) or quantum singular value transformation with the optimal query complexity in time ttt and desirable accuracy $\ensuremath{\varepsilon}$, generic time-dependent Hamiltonians face at larger query complexity and more complicated oracles due to the difficulty of handling time-dependency. In this paper, we establish a QET-based approach for simulating time-dependent Hamiltonians with multiple time periodicity. Such time-dependent Hamiltonians involve a variety of nonequilibrium systems such as time-periodic systems (Floquet systems) and time-quasiperiodic systems. Overcoming the difficulty of time dependency, our protocol can simulate the dynamics under multiperiodic time-dependent Hamiltonians with optimal and nearly optimal query complexity both in time ttt and desirable accuracy $\ensuremath{\varepsilon}$, and simple oracles as well as the optimal algorithm for time-independent cases.