Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning
Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven
Abstract
We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">⌈</mml:mo><mml:msub><mml:mi>log</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo fence="false" stretchy="false">⌉</mml:mo></mml:math>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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>log</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>qubits on average. We apply it to the problem of learning<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-fermion RDMs in parallel, to precision<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϵ</mml:mi></mml:math>, by repeating a single quantum circuit for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>≲</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>times. This result is based on a method we develop here that allows one to determine individual elements of all<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-qubit RDMs in parallel, to precision<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϵ</mml:mi></mml:math>, by repeating a single quantum circuit for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>≲</mml:mo><mml:msup><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>times, independent of the system size. This improves over existing schemes for determining qubit RDMs.