dorsal/arxiv
View SchemaEigenvalue Estimation of Differential Operators
| Authors | Thomas Szkopek, Vwani Roychowdhury, Eli Yablonovitch, Daniel S. Abrams |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0408137 |
| URL | https://arxiv.org/abs/quant-ph/0408137 |
| DOI | 10.1103/PhysRevA.72.062318 |
| Journal | Phys. Rev. A 72, 062318 (2005) |
Abstract
We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions psi(x_1,x_2,...,x_D) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy Theta(1/N^2) is Theta((2(S+1)(1+1/nu)+D)log N) qubits and O(N^{2(S+1)(1+1/nu)} (D log N)^c) gate operations, where N is the number of points to which each argument is discretized, nu and c are implementation dependent constants of O(1). Optimal classical methods require Theta(N^D) bits and Omega(N^D) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D > 2(S+1)(1+1/nu). In the case of Schrodinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.
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"abstract": "We demonstrate how linear differential operators could be emulated by a\nquantum processor, should one ever be built, using the Abrams-Lloyd algorithm.\nGiven a linear differential operator of order 2S, acting on functions\npsi(x_1,x_2,...,x_D) with D arguments, the computational cost required to\nestimate a low order eigenvalue to accuracy Theta(1/N^2) is\nTheta((2(S+1)(1+1/nu)+D)log N) qubits and O(N^{2(S+1)(1+1/nu)} (D log N)^c)\ngate operations, where N is the number of points to which each argument is\ndiscretized, nu and c are implementation dependent constants of O(1). Optimal\nclassical methods require Theta(N^D) bits and Omega(N^D) gate operations to\nperform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby\nleads to exponential reduction in memory and polynomial reduction in gate\noperations, provided the domain has sufficiently large dimension D \u003e\n2(S+1)(1+1/nu). In the case of Schrodinger\u0027s equation, ground state energy\nestimation of two or more particles can in principle be performed with fewer\nquantum mechanical gates than classical gates.",
"arxiv_id": "quant-ph/0408137",
"authors": [
"Thomas Szkopek",
"Vwani Roychowdhury",
"Eli Yablonovitch",
"Daniel S. Abrams"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.72.062318",
"journal_ref": "Phys. Rev. A 72, 062318 (2005)",
"title": "Eigenvalue Estimation of Differential Operators",
"url": "https://arxiv.org/abs/quant-ph/0408137"
},
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