LCU Select/Prepare Oracles#

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

SelectBlockEncoding#

LCU based block encoding using SELECT and PREPARE oracles.

Builds the block encoding via

\[ B[H] = \mathrm{SELECT} \]

\[ \mathrm{SELECT} |l\rangle_a|\psi\rangle_s = |l\rangle_a U_l |\psi\rangle_s. \]

The Hamiltonian can be extracted via

\[ \langle G | B[H] | G \rangle = H / \alpha \]

where

\[ |G\rangle = \mathrm{PREPARE} |0\rangle_a = \sum_l \sqrt{\frac{w_l}{\alpha}} |l\rangle_a, \]

The ancilla register is at least of size \(\log L\).

In our implementations we typically split the ancilla registers into selection registers (i.e. the \(l\) registers above) and junk registers which are extra qubits needed by state preparation but not controlled upon during SELECT.

Parameters#

  • select: The bloq implementing the SelectOracle interface.

  • prepare: The bloq implementing the PrepareOracle interface.

Registers#

  • selection: The combined selection register.

  • junk: Additional junk registers not prepared upon.

  • system: The combined system register.

References#

from qualtran.bloqs.block_encoding import SelectBlockEncoding

Example Instances#

from qualtran.bloqs.chemistry.hubbard_model.qubitization import PrepareHubbard, SelectHubbard

# 3x3 hubbard model U/t = 4
dim = 3
select = SelectHubbard(x_dim=dim, y_dim=dim)
U = 4
t = 1
prepare = PrepareHubbard(x_dim=dim, y_dim=dim, t=t, u=U)
select_block = SelectBlockEncoding(select=select, prepare=prepare)

from qualtran.bloqs.chemistry.hubbard_model.qubitization import PrepareHubbard, SelectHubbard
from qualtran.bloqs.multiplexers.black_box_select import BlackBoxSelect
from qualtran.bloqs.state_preparation.black_box_prepare import BlackBoxPrepare

# 3x3 hubbard model U/t = 4
dim = 3
select = SelectHubbard(x_dim=dim, y_dim=dim)
U = 4
t = 1
prepare = PrepareHubbard(x_dim=dim, y_dim=dim, t=t, u=U)
black_box_select_block = SelectBlockEncoding(
    select=BlackBoxSelect(select), prepare=BlackBoxPrepare(prepare)
)

Graphical Signature#

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

Call Graph#

from qualtran.resource_counting.generalizers import ignore_split_join
select_block_g, select_block_sigma = select_block.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(select_block_g)
show_counts_sigma(select_block_sigma)
../../_images/dbb4f7b90f57a41808d6c63d95b0dc0ca55b466ce92534d2e144b3b0cb1459f2.svg

Counts totals:

  • SelectHubbard(3, 3): 1

LCUBlockEncoding#

LCU based block encoding using SELECT and PREPARE oracles.

Builds the standard block encoding from an LCU as

\[ B[H] = \mathrm{PREPARE}^\dagger \cdot \mathrm{SELECT} \cdot \mathrm{PREPARE}, \]
where
\[ \mathrm{PREPARE} |0\rangle_a = \sum_l \sqrt{\frac{w_l}{\alpha}} |l\rangle_a, \]
and
\[ \mathrm{SELECT} |l\rangle_a|\psi\rangle_s = |l\rangle_a U_l |\psi\rangle_s. \]

The Hamiltonian can be extracted via

\[ \langle 0 | B[H] | 0 \rangle = H / \alpha, \]
This differs from the SelectBlockEncoding which uses Prepare for the signal state, while here it is the identity operator.

The ancilla register is at least of size \(\log L\).

In our implementations we typically split the ancilla registers into selection registers (i.e. the \(l\) registers above) and junk registers which are extra qubits needed by state preparation but not controlled upon during SELECT.

Parameters#

  • select: The bloq implementing the SelectOracle interface.

  • prepare: The bloq implementing the PrepareOracle interface.

Registers#

  • selection: The combined selection register.

  • junk: Additional junk registers not prepared upon.

  • system: The combined system register.

References#

from qualtran.bloqs.block_encoding import LCUBlockEncoding

Example Instances#

from qualtran.bloqs.chemistry.hubbard_model.qubitization import PrepareHubbard, SelectHubbard

# 3x3 hubbard model U/t = 4
dim = 3
select = SelectHubbard(x_dim=dim, y_dim=dim)
U = 4
t = 1
prepare = PrepareHubbard(x_dim=dim, y_dim=dim, t=t, u=U)
lcu_block = LCUBlockEncoding(select=select, prepare=prepare)

from qualtran.bloqs.chemistry.hubbard_model.qubitization import PrepareHubbard, SelectHubbard
from qualtran.bloqs.multiplexers.black_box_select import BlackBoxSelect
from qualtran.bloqs.state_preparation.black_box_prepare import BlackBoxPrepare

# 3x3 hubbard model U/t = 4
dim = 3
select = SelectHubbard(x_dim=dim, y_dim=dim)
U = 4
t = 1
prepare = PrepareHubbard(x_dim=dim, y_dim=dim, t=t, u=U)
black_box_lcu_block = LCUBlockEncoding(
    select=BlackBoxSelect(select), prepare=BlackBoxPrepare(prepare)
)

Graphical Signature#

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

Call Graph#

from qualtran.resource_counting.generalizers import ignore_split_join
lcu_block_g, lcu_block_sigma = lcu_block.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(lcu_block_g)
show_counts_sigma(lcu_block_sigma)
../../_images/9ddfeeece1e383750297bdb8829af357d6fd2bde3c984b9ea1a5569330f57ea1.svg

Counts totals:

  • PrepareHubbard: 1

  • PrepareHubbard†: 1

  • SelectHubbard(3, 3): 1

SelectOracle#

Abstract base class that defines the interface for a SELECT Oracle.

The action of a SELECT oracle on a selection register \(|l\rangle\) and target register \(|\Psi\rangle\) can be defined as:

\[ \mathrm{SELECT} = \sum_{l}|l \rangle \langle l| \otimes U_l \]

In other words, the SELECT oracle applies \(l\)’th unitary \(U_l\) on the target register \(|\Psi\rangle\) when the selection register stores integer \(l\).

\[ \mathrm{SELECT}|l\rangle |\Psi\rangle = |l\rangle U_{l}|\Psi\rangle \]
from qualtran.bloqs.multiplexers.select_base import SelectOracle

PrepareOracle#

Abstract base class that defines the API for a PREPARE Oracle.

Given a set of coefficients \(\{c_0, c_1, ..., c_{N - 1}\}\), the PREPARE oracle is used to encode the coefficients as amplitudes of a state \(|\Psi\rangle = \sum_{l=0}^{N-1} \sqrt{\frac{c_l}{\lambda}} |l\rangle\) where \(\lambda = \sum_l |c_l|\), using a selection register \(|l\rangle\). In order to prepare such a state, the PREPARE circuit is also allowed to use a junk register that is entangled with selection register.

Thus, the action of a PREPARE circuit on an input state \(|0\rangle\) can be defined as:

\[ \mathrm{PREPARE} |0\rangle = \sum_{l=0}^{N-1} \sqrt{ \frac{c_l}{\lambda} } |l\rangle |\mathrm{junk}_l\rangle \]
from qualtran.bloqs.state_preparation.prepare_base import PrepareOracle

BlackBoxSelect#

A ‘black box’ Select bloq.

The SELECT operation applies the \(l\)’th unitary \(U_{l}\) on the system register when the selection register stores integer \(l\). When implementing specific SelectOracle bloqs, it is helpful to have multiple selection registers each with semantic meaning. For example: you could have spatial or spin coordinates on different, named registers. The SelectOracle interface encourages this. BlackBoxSelect uses the properties on the SelectOracle interface to provide a “black box” view of a select operation that just has a selection and system register. During decomposition, this bloq will use the Partition utility bloq to partition and route the parts of the unified selection register to the Select bloq.

Parameters#

  • select: The bloq implementing the SelectOracle interface.

Registers#

  • selection: The combined selection register

  • system: The combined system register

from qualtran.bloqs.multiplexers.black_box_select import BlackBoxSelect

Example Instances#

from qualtran.bloqs.chemistry.hubbard_model.qubitization import SelectHubbard

select = SelectHubbard(2, 2)
black_box_select = BlackBoxSelect(select=select)

Graphical Signature#

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

Call Graph#

from qualtran.resource_counting.generalizers import ignore_split_join
black_box_select_g, black_box_select_sigma = black_box_select.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(black_box_select_g)
show_counts_sigma(black_box_select_sigma)
../../_images/18d8cb1db5bcd4ea4cdbe360f658a70db45c90b8d838e7578be4024eaf652209.svg

Counts totals:

  • SelectHubbard(2, 2): 1

BlackBoxPrepare#

Provide a black-box interface to Prepare bloqs.

This wrapper uses Partition to combine descriptive selection registers into one register named “selection”.

Parameters#

  • prepare: The bloq following the Prepare interface to wrap.

Registers#

  • selection: selection register.

  • junk: Additional junk registers not prepared upon.

from qualtran.bloqs.state_preparation.black_box_prepare import BlackBoxPrepare

Example Instances#

from qualtran.bloqs.chemistry.hubbard_model.qubitization import PrepareHubbard

prepare = PrepareHubbard(2, 2, 1, 4)
black_box_prepare = BlackBoxPrepare(prepare=prepare)

Graphical Signature#

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

Call Graph#

from qualtran.resource_counting.generalizers import ignore_split_join
black_box_prepare_g, black_box_prepare_sigma = black_box_prepare.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(black_box_prepare_g)
show_counts_sigma(black_box_prepare_sigma)
../../_images/c0c7cd375c10fc2becf7feb0dd3f20924cb1beca753601b587c10d91f8f800ca.svg

Counts totals:

  • PrepareHubbard: 1