workbench_algorithms.utils.dyson_utils

Utility functions for Dyson Series Qubrick and Dyson Series unit tests.

DysonSeriesUnaryIndex dataclass

DysonSeriesUnaryIndex(
    trunc_reg: Qubits,
    discret_reg: list[Qubits],
    be_anc: Qubits,
    system_reg: Qubits,
    t_disc_param: int,
    name: str,
)

Bases: CompositeRegister

Dataclass for the unary Dyson Series Composite Register.

initialize classmethod

initialize(
    qc,
    name,
    trunc_param: int,
    t_disc_param: int,
    syst: Union[int, Qubits],
    n_be_anc: Union[int, list],
)

The unary Dyson Series Expansion composite index register.

Parameters:

Name	Type	Description	Default
qc	QPU	a QPU instance.	required
name	str	name of composite register.	required
trunc_param	int	truncation parameter.	required
t_disc_param	int	time-discretization parameter.	required
syst	Union[int, Qubits]	the system register an operator is applied to.	required
n_be_anc	Union[int, list]	a register for the operator block encoding.	required

DysonSeriesBinaryIndex dataclass

DysonSeriesBinaryIndex(
    trunc_reg: Qubits,
    discret_reg: list[Qubits],
    be_anc: Qubits,
    system_reg: Qubits,
    name: str,
)

Bases: CompositeRegister

Dataclass for the binary Dyson Series Composite Register.

initialize classmethod

initialize(
    qc,
    name,
    trunc_param: int,
    t_disc_param: int,
    syst: Union[int, Qubits],
    n_be_anc: Union[int, list],
)

The binary Dyson Series Expansion composite index register.

Parameters:

Name	Type	Description	Default
qc	QPU	a QPU instance.	required
name	str	name of composite register.	required
trunc_param	int	truncation parameter.	required
t_disc_param	int	time-discretization parameter.	required
syst	Union[int, Qubits]	the system register an operator is applied to.	required
n_be_anc	Union[int, list]	a register for the operator block encoding.	required

DysonSeriesData dataclass

DysonSeriesData(
    discretization: int,
    truncation: int,
    collision: bool,
    hamiltonian: PauliSum,
    t0: float,
)

Dataclass for the DysonSeriesLCU Qubrick.

Parameters:

Name	Type	Description	Default
discretization	int	the time discretization value.	required
truncation	int	the truncation bound value.	required
collision	bool	whether to flag a (time) collision in the time register.	required
hamiltonian	PauliSum	a time independent part of the Hamiltinian to simulate.	required
t0	float	a time interval to simulate a Hamiltinian.	required

probabilities

probabilities()

Return the normalized probabilities for the state preparation routine.

DysonSelectStrategyNaive

DysonSelectStrategyNaive(
    truncation: int,
    fast_forward_exp: Qubrick,
    hamiltonian: PauliSum,
    block_encoding: Qubrick,
)

A helper class to apply an operator of the form \exp(-iH_0 t)V \exp(iH_0 t).

Parameters:

Name	Type	Description	Default
truncation	int	a value of the truncation parameter.	required
fast_forward_exp	Qubrick	a fast-forwardable part \exp(-iH_0 t) of a Hamiltonian to simulate.	required
hamiltonian	PauliSum	a block-encoded part BE(V) of a Hamiltonian to simulate.	required
block_encoding	Qubrick	a block-encoding/LCU subroutine.	required

truncation property

truncation

Getter for the truncation parameter.

hamiltonian property

hamiltonian

Getter for the block-encoded part of a Hamiltonian to simulate.

fast_forward_exp property

fast_forward_exp

Getter for the fast-forwardable part of the Hamiltonian.

block_encoding property

block_encoding

Getter for the block encoding subroutine.

apply_compression_gadget_block_encoding

apply_compression_gadget_block_encoding(
    register: DysonSeriesBinaryIndex,
    idx: int,
    ctrl: Union[Qubits, int] = 0,
)

Apply the Hamiltonian operator to the system register.

Parameters:

Name	Type	Description	Default
register	CompositeRegister	a composite register for Dyson Series.	required
idx	int	an index of the time iteration for the time discretization register.	required
ctrl	(Qubits, int)	a register to control this operation on. Default to 0.	0

num_qubits_DysonSeriesExpansionIndex

num_qubits_DysonSeriesExpansionIndex(
    trunc_param,
    t_disc_param,
    syst,
    n_be_anc,
    reg_type="unary",
) -> int

Calculate the total size of the Dyson register.

Parameters:

Name	Type	Description	Default
trunc_param	int	truncation parameter.	required
t_disc_param	int	time-discretization parameter	required
n_be_anc	int	size of the register for block-encoding ancilla qubits.	required
syst	Union[int, Qubits]	the system register an operator is applied to.	required
reg_type	str	type of the truncation register of the Dyson Series (unary or binary).	'unary'

Return

size (int): the total number of qubits required for the Dyson Series expansion.

norm_beta

norm_beta(t0, cutoff)

Compute normalization beta (Dyson series).

Parameters:

Name	Type	Description	Default
t0	float	time step	required
cutoff	int	Cutoff of the Dyson series	required

dyson_target_prob

dyson_target_prob(t0, cutoff)

Compute the normalized probabilities for the state preparation routine (Dyson Series).

Parameters:

Name	Type	Description	Default
t0	float	time step.	required
cutoff	int	Cutoff of the Dyson series.	required

dyson_series_numpy

dyson_series_numpy(
    trunc_param, t_disc_param, t, H0, V, flag_collision
)

Compute the matrix representation of the (K, M)-Dyson series.

Computes the matrix representation of the K-truncated & M-discretized dyson series. \(\sum_{k = 0}^{K} (\frac{- i t}{k! M})^k (k! B_k + C_k)\)

where

\(B_k = \sum_{0 <= m_1 < m_2 < ... < m_k < M} \prod_{j=1}^{k} \exp{(i H_0 t m_i/ M)} V \exp{(- iH_0 t m_i/ M)}\)

and \(C_k = 0\) if one desire to exclude the collisions, to keep the collisions, meaning that it captures terms simuilar to B_k but where at least one pair of indices \(m_j = m_d\) collide for \(j != d\). See eq.16 in https://arxiv.org/pdf/1805.00675.

Parameters:

Name	Type	Description	Default
trunc_param	int	Truncation.	required
t_disc_param	int	Discretization.	required
t	float	Time.	required
H0	ndarray	Hermitian operator (diaginal operator, time dependent).	required
V	ndarray	Hermitian operator (offdiaginal operator, time independent).	required
flag_collision	bool	If flag_collision == True, the collisions will be flagged and the flag being in the 0 states excludes the collisions (a.k.a. \(C_k = 0\)). If flag_collision == False, the collisions will not be flagged and thus, will be included (\(C_k \neq 0\)).	required

Return

dyson (ndarray): the matrix representation of the (trunc_param, t_disc_param)-Dyson series.

evaluate_ham

evaluate_ham(t, H0, V)

Evaluate the operator of the form \(V(t)=\exp(iH_0t)V\exp(-iH_0t)\).

Parameters:

Name	Type	Description	Default
t	float	Evolution time.	required
H0	array	a matrix representation of the fast-forwardable part of the Hamiltonian.	required
V	array	a matrix representation of the perturbation part of the Hamiltonian.	required

Return

result: the result of evaluating the operator \(V(t)\).