Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers
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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 (cid:15) , we find the circuit cost (number of gates) and depth required are O ( t 2 nAD/(cid:15) ) 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 ∆ 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 O (cid:0) n 2 (cid:1) with depth O ( n ) while BRGC scales as O ( n ) , giving an advantage to the BRGC implementation.