Reflections

from qualtran import Bloq, CompositeBloq, BloqBuilder, Signature, Register
from qualtran import QBit, QInt, QUInt, QAny
from qualtran.drawing import show_bloq, show_call_graph, show_counts_sigma
from typing import *
import numpy as np
import sympy
import cirq

ReflectionUsingPrepare

Applies reflection around a state prepared by prepare_gate

Applies \(R_{s, g=1} = g (I - 2|s\rangle\langle s|)\) using \(R_{s} = P(I - 2|0\rangle\langle0|)P^{\dagger}\) s.t. \(P|0\rangle = |s\rangle\).

Here:

• \(|s\rangle\): The state along which we want to reflect.

• \(P\): Unitary that prepares that state \(|s\rangle \) from the zero state \(|0\rangle\)

• \(R_{s}\): Reflection operator that adds a -1 phase to all states in the subspace spanned by \(|s\rangle\).

• \(g\): The global phase to control the behavior of the reflection. For example: We often use \(g=-1\) in literature to denote the reflection operator as \(R_{s} = -1 (I - 2|s\rangle\langle s|) = 2|s\rangle\langle s| - I\)

The composite gate corresponds to implementing the following circuit:

|control> ------------------ Z -------------------
                             |
|L>       ---- PREPARE^† --- o --- PREPARE -------

Parameters

• prepare_gate: An implementation of PREPARE for state preparation.

• control_val: If 0/1, a controlled version of the reflection operator is constructed. Defaults to None, in which case the resulting reflection operator is not controlled.

• global_phase: The global phase to apply in front of the reflection operator. When building a controlled reflection operator, the global phase translates into a relative phase.

• eps: precision for implementation of rotation. Only relevant if global_phase is arbitrary angle and gate is not controlled.

References

• Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity. Babbush et al. 2018. Figure 1.

from qualtran.bloqs.reflections.reflection_using_prepare import ReflectionUsingPrepare

Example Instances

from qualtran.bloqs.state_preparation import StatePreparationAliasSampling

data = [1] * 5
eps = 1e-2
prepare_gate = StatePreparationAliasSampling.from_probabilities(data, precision=eps)

refl_using_prep = ReflectionUsingPrepare(prepare_gate)

refl_around_zero = ReflectionUsingPrepare.reflection_around_zero(
    bitsizes=(1, 2, 3), global_phase=-1, control_val=1
)

Graphical Signature

from qualtran.drawing import show_bloqs
show_bloqs([refl_using_prep, refl_around_zero],
           ['`refl_using_prep`', '`refl_around_zero`'])

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
refl_using_prep_g, refl_using_prep_sigma = refl_using_prep.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(refl_using_prep_g)
show_counts_sigma(refl_using_prep_sigma)

Counts totals:

• C^3ZGate: 1

• StatePreparationAliasSampling: 1

• StatePreparationAliasSampling†: 1

• XGate: 2

PrepareIdentity

An identity PrepareOracle.

This is helpful for creating a reflection about zero and as a signal state for block encodings.

Parameters

• selection_regs: The selection registers for state prepareation. These are the incilla the state will be prepared over.

Registers

• selection_registers: The selection registers.

from qualtran.bloqs.reflections.prepare_identity import PrepareIdentity

Example Instances

prepare_identity = PrepareIdentity.from_bitsizes((10, 4, 1))

Graphical Signature

from qualtran.drawing import show_bloqs
show_bloqs([prepare_identity],
           ['`prepare_identity`'])

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
prepare_identity_g, prepare_identity_sigma = prepare_identity.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(prepare_identity_g)
show_counts_sigma(prepare_identity_sigma)

Counts totals:

• I: 1

• I: 1

• I: 1