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arXiv:2207.11647

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

Ermal Rrapaj, Kenneth McElvain, Chia Cheng Chang, Yantao Wu, André Walker-Loud

Published Jul 24, 2022• 0 citations
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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 $ε$, we find the circuit cost (number of gates) and depth required are $\mc{O}(t^2 n A D /ε)$ 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 $Δt$, 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.

DOI
10.48550/arxiv.2207.11647

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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 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 ttt with this circuit, and spectral norm error εεε, we find the circuit cost (number of gates) and depth required are \mcO(t2nAD/ε)\mc{O}(t^2 n A D /ε)\mcO(t2nAD/ε) with n−3n-3n−3 auxiliary qubits for a system with 2n2^n2n lattice points per dimension DDD and particle number AAA; an improvement over binary position encoding which requires an exponential number of nnn-local operators. Further, under the reasonable assumption that [T,V][T,V][T,V] bounds ΔtΔtΔt, with TTT the kinetic energy and VVV a non-trivial potential, the cost of QFT (Quantum Fourier Transform ) implementation of the Laplacian scales as \mcO(n2)\mc{O}\left(n^2\right)\mcO(n2) with depth \mcO(n)\mc{O}\left(n\right)\mcO(n) while BRGC scales as \mcO(n)\mc{O}\left(n\right)\mcO(n), giving an advantage to the BRGC implementation.
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We introduce a quantum algorithm to perform the Laplace transform on quantum computers. Already, the quantum Fourier transform (QFT) is the cornerstone of many quantum algorithms, but the Laplace transform or its discrete version has not seen any efficient implementation on quantum computers due to its dissipative nature and hence non-unitary dynamics. However, a recent work has shown an efficient implementation for certain cases on quantum computers using the Taylor series. Unlike previous work, our work provides a completely different algorithm for doing Laplace Transform using Quantum Eigenvalue Transformation and Lap-LCHS, very efficiently at points which form an arithmetic progression. Our algorithm can implement N×NN \times NN×N discrete Laplace transform in gate complexity that grows as O((log N)3)O((log\,N)^3)O((logN)3), ignoring the state preparation cost, where N=2nN=2^nN=2n and nnn is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity O(N⋅log N)O(N\cdot log\,N)O(N⋅logN) for the same cases. Also, the circuit width grows as O(log N)O(log\,N)O(logN). Quantum Laplace Transform (QLT) may enable new Quantum algorithms for cases like solving differential equations in the Laplace domain, developing an inverse Laplace transform algorithm on quantum computers, imaginary time evolution in the resolvent domain for calculating ground state energy, and spectral estimation of non-Hermitian matrices.
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