Linear Combination

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

LinearCombination

Linear combination of a sequence of block encodings.

Builds the block encoding \(B[\lambda_1 U_1 + \lambda_2 U_2 + \cdots + \lambda_n U_n]\) given block encodings \(B[U_1], \ldots, B[U_n]\) and coefficients \(\lambda_i \in \mathbb{R}\).

When each \(B[U_i]\) is a \((\alpha_i, a_i, \epsilon_i)\)-block encoding of \(U_i\), we have that \(B[\lambda_1 U_1 + \cdots + \lambda_n U_n]\) is a \((\alpha, a, \epsilon)\)-block encoding of \(\lambda_1 U_1 + \cdots + \lambda_n U_n\) where the normalization constant \(\alpha = \sum_i \lvert\lambda_i\rvert\alpha_i\), number of ancillas \(a = \lceil \log_2 n \rceil + \max_i a_i\), and precision \(\epsilon = (\sum_i \lvert\lambda_i\rvert)\max_i \epsilon_i\).

Under the hood, this bloq uses LCU Prepare and Select oracles to build the block encoding. These oracles will be automatically instantiated if not specified by the user.

Parameters

• block_encodings: A sequence of block encodings.

• lambd: Corresponding coefficients.

• lambd_bits: Number of bits needed to represent coefficients precisely.

• prepare: If specified, oracle preparing \(\sum_i \sqrt{|\lambda_i|} |i\rangle\) (state should be normalized and can have junk).

• select: If specified, oracle taking \(|i\rangle|\psi\rangle \mapsto \text{sgn}(\lambda_i) |i\rangle U_i|\psi\rangle\).

Registers

• system: The system register.

• ancilla: The ancilla register (present only if bitsize > 0).

• resource: The resource register (present only if bitsize > 0).

References

• Quantum algorithms: A survey of applications and end-to-end complexities. Dalzell et al. (2023). Ch. 10.2.

from qualtran.bloqs.block_encoding import LinearCombination

Example Instances

from qualtran.bloqs.basic_gates import Hadamard, TGate, XGate, ZGate
from qualtran.bloqs.block_encoding.unitary import Unitary

linear_combination_block_encoding = LinearCombination(
    (Unitary(TGate()), Unitary(Hadamard()), Unitary(XGate()), Unitary(ZGate())),
    lambd=(0.25, -0.25, 0.25, -0.25),
    lambd_bits=1,
)

Graphical Signature

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

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
linear_combination_block_encoding_g, linear_combination_block_encoding_sigma = linear_combination_block_encoding.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(linear_combination_block_encoding_g)
show_counts_sigma(linear_combination_block_encoding_sigma)

Counts totals:

• Prep: 1

• Prep†: 1

• SELECT: 1