Product#
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
Product#
Product of a sequence of block encodings.
Builds the block encoding \(B[U_1 * U_2 * \cdots * U_n]\) given block encodings \(B[U_1], \ldots, B[U_n]\).
When each \(B[U_i]\) is a \((\alpha_i, a_i, \epsilon_i)\)-block encoding of \(U_i\), we have that \(B[U_1 * \cdots * U_n]\) is a block encoding of \(U_1 * \cdots * U_n\) with normalization constant \(\prod_i \alpha_i\), ancilla bitsize \(n - 1 + \max_i a_i\), and precision \(\sum_i \alpha_i \epsilon_i\).
Following Fig. 2 in Dalzell et al. (2023), Ch. 10.2, the product is encoded by concatenating each constituent block encoding, using a shared ancilla register and a set of flag qubits to verify that the ancilla is left as zero after each use:
┌────────┐
|0> ─┤ ├─ |0> ───────────X──────X────
│ │ │
│ U_(AB) │ = ┌─────┐ │ ┌─────┐
|0> ─┤ ├─ |0> ─┤ ├──(0)──┤ ├─
│ │ │ U_B │ │ U_A │
|Psi> ─┤ ├─ |Psi> ─┤ ├───────┤ ├─
└────────┘ └─────┘ └─────┘
Parameters#
block_encodings: A sequence of block encodings.
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 Product
Example Instances#
from qualtran.bloqs.basic_gates import Hadamard, TGate
from qualtran.bloqs.block_encoding.unitary import Unitary
product_block_encoding = Product((Unitary(TGate()), Unitary(Hadamard())))
from qualtran.bloqs.basic_gates import Hadamard, TGate
from qualtran.bloqs.block_encoding.unitary import Unitary
u1 = Unitary(TGate(), alpha=0.5, ancilla_bitsize=2, resource_bitsize=1, epsilon=0.01)
u2 = Unitary(Hadamard(), alpha=0.5, ancilla_bitsize=1, resource_bitsize=1, epsilon=0.1)
product_block_encoding_properties = Product((u1, u2))
import sympy
from qualtran.bloqs.basic_gates import Hadamard, TGate
from qualtran.bloqs.block_encoding.unitary import Unitary
alpha1 = sympy.Symbol('alpha1')
a1 = sympy.Symbol('a1')
eps1 = sympy.Symbol('eps1')
alpha2 = sympy.Symbol('alpha2')
a2 = sympy.Symbol('a2')
eps2 = sympy.Symbol('eps2')
product_block_encoding_symb = Product(
(
Unitary(TGate(), alpha=alpha1, ancilla_bitsize=a1, epsilon=eps1),
Unitary(Hadamard(), alpha=alpha2, ancilla_bitsize=a2, epsilon=eps2),
)
)
Graphical Signature#
from qualtran.drawing import show_bloqs
show_bloqs([product_block_encoding, product_block_encoding_properties, product_block_encoding_symb],
['`product_block_encoding`', '`product_block_encoding_properties`', '`product_block_encoding_symb`'])
Call Graph#
from qualtran.resource_counting.generalizers import ignore_split_join
product_block_encoding_g, product_block_encoding_sigma = product_block_encoding.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(product_block_encoding_g)
show_counts_sigma(product_block_encoding_sigma)
Counts totals:
B[H]: 1B[T]: 1