Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang; Amir Kalev; Wojciech Mruczkiewicz; Hartmut Neven

doi:10.22331/q-2020-06-04-276

Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang¹, Amir Kalev², Wojciech Mruczkiewicz¹, and Hartmut Neven¹

¹Google Research, Venice, CA 90291
²Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742-2420, USA

Published:	2020-06-04, volume 4, page 276
Eprint:	arXiv:1910.10746v2
Doi:	https://doi.org/10.22331/q-2020-06-04-276
Citation:	Quantum 4, 276 (2020).

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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.

► BibTeX data

@article{Jiang2020optimalfermionto,
  doi = {10.22331/q-2020-06-04-276},
  url = {https://doi.org/10.22331/q-2020-06-04-276},
  title = {Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning},
  author = {Jiang, Zhang and Kalev, Amir and Mruczkiewicz, Wojciech and Neven, Hartmut},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {276},
  month = jun,
  year = {2020}
}

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