We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the third-order formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest- and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases.

VL - 4 UR - https://arxiv.org/abs/2111.12177 U5 - https://doi.org/10.1103/PhysRevResearch.4.013191 ER - TY - JOUR T1 - Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning JF - Quantum Y1 - 2020 A1 - Zhang Jiang A1 - Amir Kalev A1 - Wojciech Mruczkiewicz A1 - Hartmut Neven AB -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 ⌈log3(2n+1)⌉ 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 log3(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 using the ternary-tree mapping one can determine the elements of all k-fermion RDMs, to precision ϵ, by repeating a single quantum circuit for ≲(2n+1)kϵ−2 times. This result is based on a method we develop here that allows one to determine the elements of all k-qubit RDMs, to precision ϵ, by repeating a single quantum circuit for ≲3kϵ−2 times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

VL - 4 UR - https://arxiv.org/abs/1910.10746 CP - 276 U5 - https://doi.org/10.22331/q-2020-06-04-276 ER -