Exponential improvement in precision for simulating sparse Hamiltonians

TL;DR

This work addresses the challenge of simulating sparse Hamiltonians with high precision. By linking Hamiltonian simulation to continuous- and fractional-query models and introducing oblivious amplitude amplification, the authors achieve near-optimal, ε-dependent resource bounds, specifically a query complexity of $Oig( au rac{\log( au/ )}{ ext{log log}( au/ )}ig)$ and gate complexity of $Oig( au rac{ ^2ig( au/ ig)^2}{ ext{log log}( au/ )}ig)n$ for a $d$-sparse $H$ with $ au = d^2 rm_{ ext{max}} t$. The approach avoids $n$-dependence in queries, handles time-dependent Hamiltonians with improved $h'$-scaling, and provides ε-dependent lower bounds showing near-optimality. The paper also offers a simplified, more direct analysis of the continuous-to-discrete conversion and clarifies the role of fractional-query simulation in achieving these gains. Overall, the results deliver tighter, scalable resource estimates for quantum simulation in sparse settings and illuminate fundamental limits of quantum query-based simulation models.

Abstract

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $ε$ using $O\big(τ\frac{\log(τ/ε)}{\log\log(τ/ε)}\big)$ queries and $O\big(τ\frac{\log^2(τ/ε)}{\log\log(τ/ε)}n\big)$ additional 2-qubit gates, where $τ= d^2 \|{H}\|_{\max} t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

Figures (3)

• Figure 1: The fractional-query gadget. After performing the controlled-$Q$ operation on the target state $|\psi\rangle$, the operation $Q^\alpha$ is performed with amplitude depending on $\alpha$.
• Figure 2: A segment to implement the fractional-query algorithm. The segment consists of many concatneated applications of the fractional-query gadget, interspersed with $x$-independent unitaries $U_i$. The state preparation is indicated in the dotted box, and the main operation is performed by the circuit in the dashed box. The additional ancilla at the top is introduced to reduce the amplitude for performing the correct operation to exactly $1/2$.
• Figure 3: The gadget for querying $x_i$. If $x_i=0$, no edges are present. If $x_i=1$, the solid edges have weight $1/2$ and the dashed edges have weight $-1/2$.

Theorems & Definitions (41)

• Theorem 1.1: Sparse Hamiltonian simulation
• Theorem 1.2: $\epsilon$-dependent lower bound for Hamiltonian simulation
• Definition 1: Fractional-query model
• Definition 2: Continuous-query model
• Theorem 1.3: Continuous-query simulation
• Theorem 3.1: Equivalence of continuous- and fractional-query models
• Lemma 3.1
• Lemma 3.1: Gadget Lemma CGM+09
• Lemma 3.1: Segment Lemma
• Lemma 3.1: Approximate Segment Lemma