Hamiltonian Learning via Shadow Tomography of Pseudo-Choi States

We introduce a new approach to learn Hamiltonians through a resource that we call the pseudo-Choi state, which encodes the Hamiltonian in a state using a procedure that is analogous to the Choi-Jamiolkowski isomorphism. We provide an efficient method for generating these pseudo-Choi states by querying a time evolution unitary of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>e</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&amp;#x2212;</mml:mo><mml:mi>i</mml:mi><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math> and its inverse, and show that for a Hamiltonian with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi></mml:math> terms the Hamiltonian coefficients can be estimated via classical shadow tomography within error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>-norm using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>O</mml:mi><mml:mo>&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>M</mml:mi><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math> queries to the state preparation protocol, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi><mml:mo>&amp;#x2264;</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mo symmetric="true">&amp;#x2016;</mml:mo><mml:mi>H</mml:mi><mml:mo symmetric="true">&amp;#x2016;</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:math>. We further show an alternative approach that eschews classical shadow tomography in favor of quantum mean estimation that reduces this cost (at the price of many more qubits) to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>O</mml:mi><mml:mo>&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>M</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mi>&amp;#x03F5;</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math>. Additionally, we show that in the case where one does not have access to the state preparation protocol, the Hamiltonian can be learned using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>O</mml:mi><mml:mo>&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>&amp;#x03B1;</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mi>M</mml:mi></mml:mrow><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math> copies of the pseudo-Choi state. The constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03B1;</mml:mi></mml:math> depends on the norm of the Hamiltonian, and the scaling in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03B1;</mml:mi></mml:math> can be improved quadratically if using pseudo-Choi states of the normalized Hamiltonian. Finally, we show that our learning process is robust to errors in the resource states and to errors in the Hamiltonian class. Specifically, we show that if the true Hamiltonian contains more terms than we believe are present in the reconstruction, then our methods give an indication that there are Hamiltonian terms that have not been identified and will still accurately estimate the known terms in the Hamiltonian.