workbench_algorithms.utils.trotter_utils

Utility functions for Trotterization.

get_ppr_args_from_ham

get_ppr_args_from_ham(ham_term)

Convenience to set up PPR arguments from a term in a PauliSum.

Parameters:

Name	Type	Description	Default
`ham_term`	`tuple`	A single term in a PauliSum, where the tuple is (coefficient, PauliMask).	required

Returns:

Type	Description
`list`	The arguments we need to pass. In particular: `angle` (float), `x_mask` (int), `z_mask` (int).

cmnrs_analytic_bound

cmnrs_analytic_bound(
    hamiltonian, trotter_error, trotter_order=2
)

Analytic bound for the Trotter error.

Implementation of the "analytic" Trotter error bound in 1711.10980 (see Eqs. 61, 70 therein), with the modification that \(L*\Lambda\) is replaced with the sum of the coefficients of the Hamiltonian.

Parameters:

Name	Type	Description	Default
`hamiltonian`	`PauliSum`	The Hamiltonian for which the number of Trotter steps is to be estimated.	required
`trotter_error`	`float`	The allowable Trotter error.	required
`trotter_order`	`int`	The order of the Trotterization.	`2`

Returns:

Type	Description
`int`	An upper bound on the number of Trotter steps required to reach precision given by `trotter_error`.

cmnrs_minimized_bound

cmnrs_minimized_bound(
    hamiltonian,
    trotter_error,
    trotter_order=2,
    max_exponent=100,
)

Calculate the minimized Trotter error commutator bound.

Implementation of the "minimized" Trotter error bound in 1711.10980 (see Eq. 73, 74 therein), by performing binary search over the number of trotter_steps in _cmnrs_min_func_log.

Parameters:

Name	Type	Description	Default
`hamiltonian`	`PauliSum`	The Hamiltonian for which the number of Trotter steps is to be estimated.	required
`trotter_error`	`float`	The allowable Trotter error.	required
`trotter_order`	`int`	The order of the Trotterization. Defaults to `trotter_order = 2`.	`2`
`max_exponent`	`int`	The maximum exponent over which binary search is to be performed; i.e. the maximum number of Trotter steps that is returned is bounded by 2^max_exponent. Defaults to `max_exponent = 100`.	`100`

Returns:

Type	Description
`int`	An upper bound on the number of Trotter steps required to reach precision given by `trotter_error`.

cmnrs_commutator_1

cmnrs_commutator_1(
    hamiltonian,
    trotter_error,
    max_exponent=100,
    verbose=False,
    mc=False,
    num_trials=100,
)

Commutator bound for first order Trotterization.

Returns the first order commutator bound in 1711.10980, by performing binary search over the number of trotter_steps in _cmnrs_commutator_func.

Parameters:

Name	Type	Description	Default
`hamiltonian`	`PauliSum`	The Hamiltonian for which the number of Trotter steps is to be estimated.	required
`trotter_error`	`float`	The allowable Trotter error.	required
`max_exponent`	`int`	The maximum exponent over which binary search is to be performed; i.e. the maximum number of Trotter steps that is returned is bounded by 2^max_exponent. Defaults to `max_exponent = 100`.	`100`
`verbose`	`bool`	Whether to output possibly useful debugging and timing data.	`False`
`mc`	`bool`	If `False`, evaluates the bound explicitly; if `True`, attempts a Monte Carlo estimate. Defaults to `mc = False`.	`False`
`num_trials`	`int`	The number of trials to run if `mc = True`. Defaults to `num_trials = 100`.	`100`

Returns:

Type	Description
`int`	An upper bound on the number of Trotter steps required to reach precision given by `trotter_error`.

Note

This bound is only applicable to first order Trotterization.

bounded_num_trotter_steps

bounded_num_trotter_steps(
    hamiltonian, trotter_error, verbose=False
)

Calculates Google's second order commutator bound for Trotterization.

Calculates Eq. (6) from 1902.10673; i.e. returns an upper bound on the number of Trotter steps needed to reach an allowable error given by trotter_error.

Parameters:

Name	Type	Description	Default
`hamiltonian`	`PauliSum`	The Hamiltonian for which the number of Trotter steps is to be estimated.	required
`trotter_error`	`float`	The allowable Trotter error.	required
`verbose`	`bool`	Whether to output possibly useful debugging and timing data.	`False`

Returns:

Type	Description
`int`	An upper bound on the number of Trotter steps required to reach precision given by `trotter_error`.

Note

This bound is only applicable to second order Trotterization.