dyson_utils

dyson_utils

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: 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

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.

trunc_reg instance-attribute

trunc_reg: Qubits

discret_reg instance-attribute

discret_reg: list[Qubits]

be_anc instance-attribute

be_anc: Qubits

system_reg instance-attribute

system_reg: Qubits

name instance-attribute

name: str

initialize classmethod

initialize(qc, name, trunc_param: int, t_disc_param: int, syst: int | Qubits, n_be_anc: int | list)

The binary Dyson Series Expansion composite index register.

Parameters:

Name	Type	Description	Default
qc	QPU	A QPU instance.	required
name	str	The name of composite register.	required
trunc_param	int	Truncation parameter.	required
t_disc_param	int	Time-discretization parameter.	required
syst	int | Qubits	The system register an operator is applied to.	required
n_be_anc	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

discretization instance-attribute

discretization: int

truncation instance-attribute

truncation: int

collision instance-attribute

collision: bool

hamiltonian instance-attribute

hamiltonian: PauliSum

t0 instance-attribute

t0: float

probabilities

probabilities()

Return the normalized probabilities for the state preparation routine.

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.

trunc_reg instance-attribute

trunc_reg: Qubits

discret_reg instance-attribute

discret_reg: list[Qubits]

be_anc instance-attribute

be_anc: Qubits

system_reg instance-attribute

system_reg: Qubits

t_disc_param instance-attribute

t_disc_param: int

name instance-attribute

name: str

initialize classmethod

initialize(qc, name, trunc_param: int, t_disc_param: int, syst: int | Qubits, n_be_anc: int | list)

The unary Dyson Series Expansion composite index register.

Parameters:

Name	Type	Description	Default
qc	QPU	A QPU instance.	required
name	str	The name of composite register.	required
trunc_param	int	Truncation parameter.	required
t_disc_param	int	Time-discretization parameter.	required
syst	int | Qubits	The system register an operator is applied to.	required
n_be_anc	int | list	A register for the operator block encoding.	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:

\[\sum_{k = 0}^{K} (\frac{- i t}{k! M})^k (k! B_k + C_k)\]

Here

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

and \(C_k = 0\) if one desires to exclude the collisions. To keep the collisions, \(C_k\) captures terms simuilar to \(B_k\) but where at least one pair of indices \(m_j = m_d\) collide for \(j \neq d\). See eq.16 in "Hamiltonian Simulation in the Interaction Picture" (arXiv: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 (diagonal operator, time dependent).	required
V	ndarray	Hermitian operator (offdiagonal 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 (\(C_k = 0\)). If flag_collision == False, the collisions will not be flagged and thus, will be included (\(C_k \neq 0\)).	required

Returns:

Type	Description
ndarray	the matrix representation of the (trunc_param, t_disc_param)-Dyson series.

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

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	Matrix representation of the fast-forwardable part of the Hamiltonian.	required
V	array	Matrix representation of the perturbation part of the Hamiltonian.	required

Returns:

Type	Description
array	The result of evaluating the operator \(V(t)\).

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

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	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'

Returns:

Type	Description
int	The total number of qubits required for the Dyson Series expansion.