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We study the gate-based implementation of the binary reflected Gray code(BRGC) and binary code of the unitary time evolution operator due to theLaplacian discretized on a lattice with periodic boundary conditions. We findthat the resulting Trotter error is independent of system size for a fixedlattice spacing through the Baker-Campbell-Hausdorff formula. We then presentour algorithm for building the BRGC quantum circuit. For an adiabatic evolutiontime $t$ with this circuit, and spectral norm error $\epsilon$, we find thecircuit 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 pointsper dimension $D$ and particle number $A$; an improvement over binary positionencoding which requires an exponential number of $n$-local operators. Further,under the reasonable assumption that $[T,V]$ bounds $\Delta t$, with $T$ thekinetic energy and $V$ a non-trivial potential, the cost of QFT (QuantumFourier 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.

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