openfermion.circuits.low_rank_two_body_decomposition

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Convert two-body operator into sum of squared one-body operators.

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Main aliases

openfermion.circuits.low_rank.low_rank_two_body_decomposition, openfermion.low_rank_two_body_decomposition

openfermion.circuits.low_rank_two_body_decomposition(
    two_body_coefficients,
    truncation_threshold=1e-08,
    final_rank=None,
    spin_basis=True
)

As in arXiv:1808.02625, this function decomposes $\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s$ as $\sum_{l} \lambda_l (\sum_{pq} g_{lpq} a^\dagger_p a_q)^2$ l is truncated to take max value L so that $\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x$

Args
`two_body_coefficients`	`ndarray` an N x N x N x N numpy array giving the $h_{pqrs}$ tensor. This tensor must be 8-fold symmetric (real integrals).
`truncation_threshold`	`optional Float` the value of x, above.
`final_rank`	`optional int` if provided, this specifies the value of L at which to truncate. This overrides truncation_threshold.
`spin_basis`	`bool` True if the two-body terms are passed in spin orbital basis. False if already in spatial orbital basis.

Returns
`eigenvalues`	`ndarray of floats` length L array giving the $\lambda_l$ .
`one_body_squares`	`ndarray of floats` L x N x N array of floats corresponding to the value of $g_{pql}$ .
`one_body_correction`	`ndarray` One-body correction terms that result from reordering to chemist ordering, in spin-orbital basis.
`truncation_value`	`float` after truncation, this is the value $\sum_{l=0}^{L-1} (\sum_{pq} \|g_{lpq}\|)^2 \|\lambda_l\| < x$

Raises
`TypeError`	Invalid two-body coefficient tensor specification.

openfermion.circuits.low_rank_two_body_decomposition

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