Estimating Green's functions with a quantum Arnoldi method
ORAL
Abstract
While Green's functions are a powerful tool for studying quantum many-body systems, there are several bottlenecks to accessing them through quantum simulation algorithms.
The most significant is that it rarely suffices to know a Green's function at a single value of its argument (a frequency or energy), and it is preferable to know it at $N \gg 1$ values.
In that setting, straightforward application of the quantum singular value transformation (QSVT) to estimate a degree-$d$ approximation to a Green's function would require $O(Nd)$ queries to a Hamiltonian block encoding with subnormalization factor $\lambda$.
Quantum Krylov methods remove the scaling with $N$ in the query complexity but will introduce a $O(\lambda^d)$ scaling if the subspace is generated directly from the Hamiltonian block encoding.
We describe a quantum Arnoldi method that avoids this exponential scaling with $d$, describe the relevant design space for a particular implementation, and focus on controlling various sources of error.
We will also consider prospective applications of our method in materials and nuclear physics.
The most significant is that it rarely suffices to know a Green's function at a single value of its argument (a frequency or energy), and it is preferable to know it at $N \gg 1$ values.
In that setting, straightforward application of the quantum singular value transformation (QSVT) to estimate a degree-$d$ approximation to a Green's function would require $O(Nd)$ queries to a Hamiltonian block encoding with subnormalization factor $\lambda$.
Quantum Krylov methods remove the scaling with $N$ in the query complexity but will introduce a $O(\lambda^d)$ scaling if the subspace is generated directly from the Hamiltonian block encoding.
We describe a quantum Arnoldi method that avoids this exponential scaling with $d$, describe the relevant design space for a particular implementation, and focus on controlling various sources of error.
We will also consider prospective applications of our method in materials and nuclear physics.
*SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525.
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Presenters
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Jacob Nelson
- University of New Mexico