1Google Research, Venice, CA 90291
2Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742-2420, USA
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Abstract
We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an $n$-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on $lceil log_3(2n+1)rceil$ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than $log_3(2n)$ qubits on average. We apply it to the problem of learning $k$-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all $k$-fermion RDMs in parallel, to precision $epsilon$, by repeating a single quantum circuit for $lesssim (2n+1)^k epsilon^{-2}$ times. This result is based on a method we develop here that allows one to determine individual elements of all $k$-qubit RDMs in parallel, to precision $epsilon$, by repeating a single quantum circuit for $lesssim 3^k epsilon^{-2}$ times, independent of the system size. This improves over existing schemes for determining qubit RDMs.
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Cited by
[1] Alexander Yu. Vlasov, “Clifford algebras, Spin groups and qubit trees”, arXiv:1904.09912.
[2] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, “Quantum computational chemistry”, Reviews of Modern Physics 92 1, 015003 (2020).
[3] Hsin-Yuan Huang, Richard Kueng, and John Preskill, “Predicting Many Properties of a Quantum System from Very Few Measurements”, arXiv:2002.08953.
[4] Adrian Chapman and Steven T. Flammia, “Characterization of solvable spin models via graph invariants”, arXiv:2003.05465.
[5] Ikko Hamamura and Takashi Imamichi, “Efficient evaluation of quantum observables using entangled measurements”, arXiv:1909.09119.
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