Abstract
Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with O~(N3/2λ) T complexity, where N is number of orbitals and λ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.
Highlights
Quantum computers were originally proposed as special purpose tools for efficiently modeling physical quantum mechanical systems [1]
Ever since quantum simulation has been central to the study of quantum computing [2] while regarded as one of its most promising applications
In the remainder of this section we quantify the costs of the QROM needed for the state preparation, which is the main contributor to the complexity, give the total cost
Summary
Quantum computers were originally proposed as special purpose tools for efficiently modeling physical quantum mechanical systems [1]. The work of [11] suggests resolving this problem by simulating the plane wave Hamiltonian in first quantization to achieve O(N 1/3η8/3) gate complexity, where η is the number of electrons With such low scaling in N , one might be able to use an extremely large plane wave basis. Prior papers to compile a quantum chemistry algorithm to the level of Clifford + T gates and estimate the resources required within an error-correcting code are [8, 9, 29] These papers focus on minimizing T complexity or Toffoli complexity because these gates cannot be transversely implemented within practical codes [30, 37]. In Appendix E we give the details for minor contributions to the cost, and in Appendix H we give circuits and exact costings for arithmetic
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