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We introduce an algorithm, the Orthogonal Operator Polynomial Expansion(OOPEX), to approximately compute expectation values in energy eigenstates atfinite energy density of non-integrable quantum many-body systems withpolynomial effort, whereas exact diagonalization (ED) of the Hamiltonian $H$ isexponentially hard. The OOPEX relies on the eigenstate thermalizationhypothesis, which conjectures that eigenstate expectation values of physicalobservables in such systems vary smoothly with the eigenstate energy (and othermacroscopic conserved quantities, if any), and computes them through a seriesgenerated by repeated multiplications, rather than diagonalization, of $H$ andwhose successive terms oscillate faster with the energy. The hypothesisguarantees that only the first few terms of this series contribute appreciably.We further show that the OOPEX, in a sense, is the most optimum algorithm basedon series expansions of $H$ as it avoids computing the many-body density ofstates which plagues other similar algorithms. Then, we argue non-rigorouslythat working in the Fock space of operators, rather than that of states as isusually done, yields convergent results with computational resources that scalepolynomially with $N$. We demonstrate the polynomial scaling by applying theOOPEX to the non-integrable Ising chain and comparing with ED andhigh-temperature expansion (HTX) results. The OOPEX provides access to muchlarger $N$ than ED and HTX do, which facilitates overcoming finite-size effectsthat plague the other methods to extract correlation lengths in chaoticeigenstates. In addition, access to large systems allows testing a recentconjecture that the Renyi entropy of chaotic eigenstates has positive curvatureif the Renyi index $>1$, and we find encouraging supporting evidence.

A polynomial time algorithm for studying physical observables in chaotic eigenstates