quantum_phase_estimation_utils

quantum_phase_estimation_utils

calculate_holevo_variance

calculate_holevo_variance(phases, probabilities)

Calculate the Holevo variance given a set of phases and their associated probabilities.

Parameters:

Name	Type	Description	Default
phases	List[float]	List of phases obtained from QPE, each between [0, \(2\pi\)].	required
probabilities	List[float]	List of probabilities associated with each phase.	required

Note

The phases from running QPE calculation are in the range [0, 1] and should be multiplied by \(2\pi\) to be scaled to [0, \(2\pi\)] before being passed to this function.

clopper_pearson_lower

clopper_pearson_lower(num_successes, num_trials, fail_prob)

Evaulates the estimated lower bound to the Binomial confidence interval, via a Clopper-Pearson test.

Parameters:

Name	Type	Description	Default
num_successes	int	The number of successful trials.	required
num_trials	int	The total number of trials.	required
fail_prob	float	The tolerable probability of failure (i.e. that the true value lies outside the confidence interval).	required

Returns:

Type	Description
float	The Clopper-Pearson lower bound to the confidence interval.

clopper_pearson_max_error

clopper_pearson_max_error(num_successes, num_trials, fail_prob)

Calculates the maximum error using Clopper-Pearson.

Returns the value furthest from the point estimate that still lies within the Clopper-Pearson confidence interval.

Parameters:

Name	Type	Description	Default
num_successes	int	The number of successful trials.	required
num_trials	int	The total number of trials.	required
fail_prob	float	The tolerable probability of failure (i.e. that the true value lies outside the confidence interval).	required

Returns:

Type	Description
float	The value furthest from the point estimate that still lies in the confidence inteval.

clopper_pearson_upper

clopper_pearson_upper(num_successes, num_trials, fail_prob)

Evaluates the estimated upper bound to the Binomial confidence interval, via a Clopper-Pearson test.

Parameters:

Name	Type	Description	Default
num_successes	int	The number of successful trials.	required
num_trials	int	The total number of trials.	required
fail_prob	float	The tolerable probability of failure (i.e. that the true value lies outside the confidence interval).	required

Returns:

Type	Description
float	The Clopper-Pearson upper bound to the confidence interval.

energy_to_phase_qubitization

energy_to_phase_qubitization(energy, norm)

Convert an eigenvalue to a phase.

Parameters:

Name	Type	Description	Default
energy	float	Energy eigenvalue of Hamiltonian.	required
norm	float	Norm of Hamiltonian.	required

Returns:

Type	Description
float	Exact phase corresponding to the energy eigenvalue

get_most_probable_phase

get_most_probable_phase(phase_qubits)

Get the most probable phase from a qubit register typically output after running phase estimation.

Note

This function is only designed for a qubit register of type QUFixed.

Parameters:

Name	Type	Description	Default
phase_qubits	QUFixed	A register of qubits, typically representing phase qubits after phase estimation.	required

Returns:

Type	Description
float	The most probable phase to read.

hartree_to_ev

hartree_to_ev(energy)

Converts Hartree to electronVolts.

Parameters:

Name	Type	Description	Default
energy	float	The energy in Hartree.	required

Returns:

Type	Description
float	The energy in eV.

num_trials_chernoff

num_trials_chernoff(precision, fail_prob)

Calculate the number of trials using the Chernoff bound.

Returns the number of trials needed for an estimate to be returned within a desired precision with a given fail probability, from the vanilla Chernoff bound.

Note

• Deprecated in favour of the Clopper-Pearson tests, below.

optimal_holevo_variance

optimal_holevo_variance(n_precision_qubits)

Calculate the optimal Holevo variance for a given number of bits of precision m as per Eq. 10 in "Optimum phase estimation with two control qubits" (arXiv:2303.12503 ⧉).

Parameters:

Name	Type	Description	Default
n_precision_qubits	int	Number of bits of precision.	required

peek_ipea_phase

peek_ipea_phase(qc, ancilla, phase_sign_fixed=False)

Extracts the exact phase given to an eigenstate.

Parameters:

Name	Type	Description	Default
qc	QPU	The QPU instance.	required
ancilla	Qubits	The output ancilla for IPEA.	required
phase_sign_fixed	bool	Whether the sign of the kicked-back phase is known in advance, or whether it needs to be determined. The former is useful for qubitization, and the latter for Trotterization.	False

Returns:

Type	Description
phase (float	The exact phase given to an eigenstate.

phase_from_cs

phase_from_cs(sine, cosine)

Calculates a phase from the sine and cosine.

Extracts an estimate of the phase, \(\phi\), given an estimate of \(\cos(2 \pi \phi)\) and \(\sin(2 \pi \phi)\).

Parameters:

Name	Type	Description	Default
sine	float	An estimate of \(\sin(2 \pi \phi)\).	required
cosine	float	An estimate of \(\cos(2 \pi \phi)\).	required

Returns:

Type	Description
float	An estimate of the phase, \(\phi\).

Note

The output phase is in \([0,1]\) (not \([0, 2\pi]\), or \([-\pi, \pi]\)).

phase_to_energy_qubitization

phase_to_energy_qubitization(phase, norm)

Convert a phase to an eigenvalue.

Parameters:

Name	Type	Description	Default
phase	float	Phase found using IPEA.	required
norm	float	Norm of Hamiltonian.	required

Returns:

Type	Description
float	Energy eigenvalue of Hamiltonian.

phase_to_energy_time_evolution

phase_to_energy_time_evolution(output, bits_precision, offset=0, rescaling_factor=1)

Converts a measured phase to an eigenenergy, when estimating the phase of \(\exp(iHt)\).

Parameters:

Name	Type	Description	Default
output	int	The bits of output from phase estimation.	required
bits_precision	int	The number of bits of precision.	required
offset	float	A constant factor to add to the eigenenergy (for example, if the Hamiltonian had an identity term that has been skimmed off).	0
rescaling_factor	float	A factor to account for rescaling of the Hamiltonian before simulation.	1

Returns:

Type	Description
float	The estimated eigenphase.