Dynamic, symmetry-preserving, and hardware-adaptable circuits for quantum computing many-body states and correlators of the Anderson impurity model

We present a hardware-reconfigurable ansatz on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>N</a:mi><a:mi>q</a:mi></a:msub></a:math>-qubits for the variational preparation of many-body states of the Anderson impurity model (AIM) with <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>N</b:mi><b:mtext>imp</b:mtext></b:msub><b:mo>+</b:mo><b:msub><b:mi>N</b:mi><b:mtext>bath</b:mtext></b:msub><b:mo>=</b:mo><b:msub><b:mi>N</b:mi><b:mi>q</b:mi></b:msub><b:mo>/</b:mo><b:mn>2</b:mn></b:mrow></b:math> sites, which conserves total charge and spin <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>z</c:mi></c:mrow></c:math> component within each variational search subspace. The many-body ground state of the AIM is determined as the minimum over all minima of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>O</d:mi><d:mo>(</d:mo><d:msubsup><d:mi>N</d:mi><d:mi>q</d:mi><d:mn>2</d:mn></d:msubsup><d:mo>)</d:mo></d:mrow></d:math> distinct charge-spin sectors. Hamiltonian expectation values are shown to require <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>ω</e:mi><e:mrow><e:mo>(</e:mo><e:msub><e:mi>N</e:mi><e:mi>q</e:mi></e:msub><e:mo>)</e:mo></e:mrow><e:mo><</e:mo><e:msub><e:mi>N</e:mi><e:mtext>meas.</e:mtext></e:msub><e:mo>≤</e:mo><e:mi>O</e:mi><e:mrow><e:mo>(</e:mo><e:msub><e:mi>N</e:mi><e:mtext>imp</e:mtext></e:msub><e:msub><e:mi>N</e:mi><e:mtext>bath</e:mtext></e:msub><e:mo>)</e:mo></e:mrow></e:mrow></e:math> symmetry-preserving, parallelizable measurement circuits, each amenable to postselection. To obtain the one-particle impurity Green's function we show how initial Krylov vectors can be computed via midcircuit measurement and how Lanczos iterations can be computed using the symmetry-preserving ansatz. For a single-impurity Anderson model with a number of bath sites increasing from one to seven, we show using numerical emulation that the ease of variational ground-state preparation is suggestive of linear scaling in circuit depth and subquartic scaling in optimizer complexity. We therefore expect that, combined with time-dependent methods for Green's function computation, our ansatz provides a useful tool to account for electronic correlations on early fault-tolerant processors. Finally, with a view towards computing real materials properties of interest like magnetic susceptibilities and electron-hole propagators, we provide a straightforward method to compute many-body, time-dependent correlation functions using a combination of time evolution, midcircuit measurement-conditioned operations, and the Hadamard test.