Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers

TL;DR

This paper presents a gate-based quantum circuit implementation of the Laplacian using the binary reflected Gray code (BRGC), demonstrating improved efficiency over binary position encoding for quantum simulations.

Contribution

The authors develop a novel BRGC quantum circuit for the Laplacian, reducing circuit complexity and depth compared to traditional binary encoding methods.

Findings

Circuit cost scales as O(t^2 n A D / ε) with auxiliary qubits.

BRGC implementation scales as O(n) in cost and O(n) in depth, better than QFT.

Trotter error remains independent of system size for fixed lattice spacing.

Abstract

We study the gate-based implementation of the binary reflected Gray code (BRGC) and binary code of the unitary time evolution operator due to the Laplacian discretized on a lattice with periodic boundary conditions. We find that the resulting Trotter error is independent of system size for a fixed lattice spacing through the Baker-Campbell-Hausdorff formula. We then present our algorithm for building the BRGC quantum circuit. For an adiabatic evolution time $t$ with this circuit, and spectral norm error $\epsilon$, we find the circuit cost (number of gates) and depth required are $\mc{O}(t^2 n A D /\epsilon)$ with $n-3$ auxiliary qubits for a system with $2^n$ lattice points per dimension $D$ and particle number $A$; an improvement over binary position encoding which requires an exponential number of $n$-local operators. Further, under the reasonable assumption that $[T,V]$ bounds $\Delta t$, with $T$ the kinetic energy and $V$ a non-trivial potential, the cost of QFT (Quantum Fourier Transform ) implementation of the Laplacian scales as $\mc{O}\left(n^2\right)$ with depth $\mc{O}\left(n\right)$ while BRGC scales as $\mc{O}\left(n\right)$, giving an advantage to the BRGC implementation.