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arXiv:2310.07054 · Physical Review Research

Simplifying the simulation of local Hamiltonian dynamics

Ayaka Usui, Anna Sanpera, María García Díaz

Published Jun 4, 2024• 3 citations
arXivHTMLPDF

Abstract

Local Hamiltonians H k describe nontrivial k -body interactions in quantum many-body systems. Here, we address the dynamical simulatability of a k -local Hamiltonian by a simpler one H k ′ with k ′ < k , under the realistic constraint that both Hamiltonians act on the same Hilbert space. When it comes to exact simulation, we build upon known methods to derive examples of H k and H k ′ that simulate the same physics. We also address the most realistic case of approximate simulation. There, we upper-bound the error up to which a Hamiltonian can simulate another one, regardless of their internal structure, and show an example suggesting that the accuracy of a ( k ′ = 2 ) -local Hamiltonian to simulate H k with k > 2 increases with k . Finally, we propose a method to search for the k ′ -local Hamiltonian that simulates, with the highest possible precision, the short time dynamics of a given H k Hamiltonian. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.023243

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Local Hamiltonians H k describe nontrivial k -body interactions in quantum many-body systems. Here, we address the dynamical simulatability of a k -local Hamiltonian by a simpler one H k ′ with k ′ < k , under the realistic constraint that both Hamiltonians act on the same Hilbert space. When it comes to exact simulation, we build upon known methods to derive examples of H k and H k ′ that simulate the same physics. We also address the most realistic case of approximate simulation. There, we upper-bound the error up to which a Hamiltonian can simulate another one, regardless of their internal structure, and show an example suggesting that the accuracy of a ( k ′ = 2 ) -local Hamiltonian to simulate H k with k > 2 increases with k . Finally, we propose a method to search for the k ′ -local Hamiltonian that simulates, with the highest possible precision, the short time dynamics of a given H k Hamiltonian. Published by the American Physical Society 2024
We present a novel method to simulate the Lindblad equation, drawing on the relationship between Lindblad dynamics, stochastic differential equations, and Hamiltonian simulations. We derive a sequence of unitary dynamics in an enlarged Hilbert space that can approximate the Lindblad dynamics up to an arbitrarily high order. This unitary representation can then be simulated using a quantum circuit that involves only Hamiltonian simulation and tracing out the ancilla qubits. There is no need for additional postselection in measurement outcomes, ensuring a success probability of one at each stage. Our method can be directly generalized to the time-dependent setting. We provide numerical examples that simulate both time-independent and time-dependent Lindbladian dynamics with accuracy up to the third order. Published by the American Physical Society 2024
Understanding the collective behaviour of many-body quantum systems is an important subject in many areas of physics. With advances in ultra-cold gas experiments, the dynamics of strongly-interacting systems can now be studied in the lab. However, there is a paucity of theoretical techniques available to simulate such systems. One technique is phase-space methods, often known as the Truncated Wigner Approximation; however, its applicability in its naive form is limited. In this work, we expound on techniques to expand the regimes in which it can be effective. This involves creating a novel phase-space that is tailored to the problem at hand, and associated classical equations of motion. We show techniques for lattice systems with local finite Hilbert spaces, for fermionic systems, and for many-body localized systems. In all cases, we benchmark the accuracy of the approximation against exact results.
Recent progress in the development of quantum technologies has enabled the direct investigation of the dynamics of increasingly complex quantum many-body systems.This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage.Here, we present classical algorithms for approximating the dynamics of local observables and nonlocal quantities such as the Loschmidt echo, where the evolution is governed by a local Hamiltonian.For short times, their computational cost scales polynomially with the system size and the inverse of the approximation error.In the case of local observables, the proposed algorithm has a better dependence on the approximation error than algorithms based on the Lieb-Robinson bound.Our results use cluster-expansion techniques adapted to the dynamical setting, for which we give a novel proof of their convergence.This has important physical consequences besides our efficient algorithms.In particular, we establish a novel quantum speed limit, a bound on dynamical phase transitions, and a concentration bound for product states evolved for short times.
The Hamiltonians of quantum many-body systems can display an extremely diverse range of behaviours.In this thesis we show that there exist simple systems which can simulate all other quantum many-body systems, including all of their physical properties.We call these universal Hamiltonians.We first define a notion of analogue simulation which captures what it means for the Hamiltonian of one system to be reproduced in the Hamiltonian of another physical system.Then we obtain a full classification of simulation power in the case of families of Hamiltonians built up as a linear combination of 2-local qubit interactions.The generalised Heisenberg model and generalised XY model are important special cases which are universal in this classification.We also prove universality for other restricted classes of Hamiltonians such as those for which all the interaction strengths have the same sign or for Hamiltonians with spatially restricted interaction graphs.And we prove universality for large classes of qudit Hamiltonians (where the local dimension d > 2).Universality of a family of Hamiltonians implies that the problem of estimating the ground state energy, known as the Local Hamiltonian problem, is QMA-complete for Hamiltonians from that family.Our results prove that this is the case for many restricted families of Hamiltonians, for which the complexity of the Local Hamiltonian problem was not previously known.We also show that the Local Hamiltonian problem is QMA EXPcomplete for a translationally invariant Hamiltonian of 4-local interactions between spins of dimension d = 4, arranged on a face-centred cubic lattice, by directly encoding a verifier circuit into the interaction terms.Finally we consider the problem of simulating a local measurement in the ground space of a local Hamiltonian, introduced by Ambainis as APX-SIM.We simplify some of the prior work on this problem and prove some previously unknown complexity class inequalities.We show that our notion of simulation preserves hardness of this problem and obtain a complexity classification of APX-SIM for Hamiltonians of 2-local qubit interactions.i Dedication and acknowledgements First and foremost, I must thank my supervisor, Ashley Montanaro, for all of his help over the last few years.He has always made himself available to discuss work, and I am grateful for all of his advice, his problem suggestions, and his excellent proof-reading skills!I must also thank all of my other co-
The advent of near-term digital quantum computers could offer us an exciting opportunity to investigate quantum many-body phenomena beyond that of classical computing. To make the best use of the hardware available, it is paramount that we have methods that accurately simulate Hamiltonian dynamics for limited circuit depths. In this paper, we propose a method to classically optimize unitary brickwall circuits to approximate quantum time evolution operators. Our method is scalable in system size through the use of tensor networks. We demonstrate that, for various three-body Hamiltonians, our approach produces quantum circuits that can outperform trotterization in both their accuracy and the quantum circuit depth needed to implement the dynamics, with the exact details being dependent on the Hamiltonian. We also explain how to choose an optimal time step that minimizes the combined errors of the quantum device and the brickwall circuit approximation. Published by the American Physical Society 2024
We study spectral properties of quantum many-body Hamiltonians through a subsystem-based framework. Given a Hamiltonian of the form H=∑X⊆ΛΦ(X)H = \sum_{X \subseteq Λ} Φ(X)H=∑X⊆Λ​Φ(X) acting on a tensor product Hilbert space, we associate to each subset S⊆ΛS \subseteq ΛS⊆Λ a subsystem Hamiltonian HSH_SHS​ and its spectrum S(S)=σ(HS)\mathcal{S}(S) = σ(H_S)S(S)=σ(HS​). This produces a family of spectra indexed by subsystems, allowing spectral data to be organized according to interaction structure. We show that subsystem Hamiltonians admit local approximations: HSH_SHS​ can be approximated by operators supported on finite neighborhoods with an error bounded by ∥HS−HS,r∥≤∣S∣e−μr∥Φ∥μ\|H_S - H_{S,r}\| \le |S| e^{-μr} \|Φ\|_μ∥HS​−HS,r​∥≤∣S∣e−μr∥Φ∥μ​. As a consequence, subsystem spectra are stable under truncation in the sense that dH(S(S),σ(HS,r))≤∣S∣e−μr∥Φ∥μ.d_H(\mathcal{S}(S), σ(H_{S,r})) \le |S| e^{-μr} \|Φ\|_μ.dH​(S(S),σ(HS,r​))≤∣S∣e−μr∥Φ∥μ​. We then prove that for disjoint subsets S1,S2⊆ΛS_1, S_2 \subseteq ΛS1​,S2​⊆Λ, the subsystem spectrum is approximately additive: dH(S(S1∪S2),S(S1)+S(S2))≤(∣S1∣+∣S2∣)e−μD∥Φ∥μ,d_H\big(\mathcal{S}(S_1 \cup S_2), \mathcal{S}(S_1) + \mathcal{S}(S_2)\big) \le (|S_1| + |S_2|) e^{-μD} \|Φ\|_μ,dH​(S(S1​∪S2​),S(S1​)+S(S2​))≤(∣S1​∣+∣S2​∣)e−μD∥Φ∥μ​, where D=d(S1,S2)D = d(S_1, S_2)D=d(S1​,S2​). In the finite-range case, this relation becomes exact. The results show that spectral properties reflect the locality of interactions not only at the level of operators, but also at the level of spectra. The framework provides a way to study many-body systems in which interaction geometry directly shapes spectral behavior.