On the quantum computational complexity of classical linear dynamics with geometrically local interactions: Dequantization and universality

TL;DR

This work thoroughly characterize the computational complexity of quantum algorithms for simulating classical dynamics governed by partial differential equations, and suggests a super-polynomial time advantage when restricting the computation to polynomial-space, or an exponential space advantage otherwise.

Abstract

The simulation of large-scale classical systems in exponentially small space on quantum computers has gained attention. The prior work demonstrated that a quantum algorithm offers an exponential speedup over any classical algorithm in simulating classical dynamics with long-range interactions. However, many real-world classical systems, such as those arising from partial differential equations, exhibit only local interactions. The question remains whether quantum algorithms can still provide exponential speedup under this condition. In this work, we thoroughly characterize the computational complexity of quantum algorithms for simulating such geometrically local systems. First, we dequantize the quantum algorithm for simulating short-time (polynomial-time) dynamics of such systems. This implies that the problem of simulating this dynamics does not yield any exponential quantum advantage. Second, we show that quantum algorithms for short-time dynamics have the same computational complexity as polynomial-time probabilistic classical computation. Third, we show that the computational complexity of quantum algorithms for long-time (exponential-time) dynamics is captured by exponential-time and polynomial-space quantum computation. This suggests a super-polynomial time advantage when restricting the computation to polynomial-space, or an exponential space advantage otherwise. This work offers new insights into the complexity of classical dynamics governed by partial differential equations, providing a pathway for achieving quantum advantage in practical problems.

Figures (5)

• Figure 1: The overview of this paper. $N=2^n$ is the system size. Here we consider the $O(n)$-qubit quantum algorithm that simulates the dynamics efficiently, i.e., the runtime of the quantum algorithm is at most $\mathrm{poly}(t,n)$. Result $(\mathrm{i})$ shows that the short-time dynamics can be simulated by $\mathrm{polylog}(N)$-time probabilistic classical computation. In result $(\mathrm{ii})$, we establish the opposite direction that the short-time dynamics can simulate $\mathrm{polylog}(N)$-time probabilistic classical computation. In result $(\mathrm{iii})$, we indicate that the long-time dynamics can simulate $\mathrm{poly}(N)$-time and $O(n)$-qubit quantum computation. The opposite direction is by the assumption that the quantum algorithm runs in $\mathrm{poly}(t,n)$-time.
• Figure 2: The one-dimensional system behind a tridiagonal matrix $A$. Each vertex corresponds to some index of $A$ and each edge represents the matrix element $A_{i,j}$. Clearly, we can consider non-symmetric matrices by introducing one-dimensional directed graph.
• Figure 3: The underlying two-dimensional (inherently one-dimensional) lattice of the reduced Hamiltonian in Eq. \ref{['eq:FK-classical-reversible']}.
• Figure 4: An example of the underlying two-dimensional system of the reduced Hamiltonian in Eq. \ref{['eq:FK-quantum-long-dilated']}.
• Figure 5: Two-dimensional system with local interaction which has the same operation as the system with non-local interactions in Fig. \ref{['fig:2d_system_quantum_circuit_long']}.

Theorems & Definitions (29)

• Definition 1: Query-access to a vector
• Definition 2: Query-access to a sparse square matrix
• Definition 3: Sampling-access to a vector
• Definition 4: Sampling-and-query-access to a vector
• Definition 5: Geometrically local matrix
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3: gall2023robust