Toward the first quantum simulation with quantum speedup

TL;DR

This paper investigates the first practical quantum simulation with genuine quantum advantage by focusing on a 1D Heisenberg spin chain with random z-field as a benchmark. It develops and implements three leading digital simulation algorithms—product formulas, Taylor series/LCU, and quantum signal processing—within a unified circuit framework (Quipper), and provides concrete resource estimates. Through refined analytical bounds (notably commutator-based PF bounds) and optimized subroutines (e.g., a binary-tree select(V)), the authors compare rigorous and empirical performance, finding that segmented QSP and high-order PF with commutator bounds offer the best rigorous guarantees, while empirical PF bounds can dramatically reduce circuit size. The results indicate spin-system Hamiltonian simulation requires far fewer resources than factoring or quantum chemistry and thus stands out as a practical early demonstration of quantum speedup. The work also outlines key open problems and future directions toward tighter bounds, efficient angle computation for QSP, and architecture-aware implementations.

Abstract

With quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, employing diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry.

Figures (13)

• Figure 1: Gate counts for optimized implementations of the PF algorithm (using the fourth-order formula with commutator bound and the better of the fourth- or sixth-order formula with empirical error bound), the TS algorithm, and the QSP algorithm (using the segmented version with analytic error bound and the non-segmented version with empirical Jacobi-Anger error bound) for system sizes between 10 and 100. Left: $\textsc{cnot}$ gates for Clifford+$R_z$ circuits. Right: $T$ gates for Clifford+$T$ circuits.
• Figure 2: Number of qubits used by the PF, TS, and QSP algorithms.
• Figure 3: Total gate counts in the Clifford+$R_z$ basis for product formula algorithms using the minimized (left), commutator (center), and empirical (right) bounds, for system sizes between 13 and 500.
• Figure 4: Comparison of the values of $r$ using the commutator and empirical bounds for formulas of order $1$, $2$, and $4$, and values of $r$ for the empirical bound for formulas of order $6$ and $8$. Straight lines show power-law fits to the data. The error bars for product formulas of order greater than $1$ are negligibly small, so we omit them from the plots.
• Figure 5: Binary tree encoding the circuit that implements the control generation for the $\mathop{\mathrm{select}}\nolimits(V)$ operation.

Theorems & Definitions (32)

• Lemma F.1
• proof
• Lemma F.2
• proof
• Proposition F.3
• proof
• Proposition F.4
• Definition 1
• Definition 2
• Theorem F.5: First-order commutator bound