Upload nonlinear amplitude transformation demo by drdren · Pull Request #1738 · PennyLaneAI/demos

drdren · 2026-04-08T16:52:04Z

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github-actions · 2026-04-08T16:52:14Z

👋 Hey, looks like you've updated some demos!

🐘 Don't forget to update the dateOfLastModification in the associated metadata files so your changes are reflected in Glass Onion (search and recommendations).

Please hide this comment once the field(s) are updated. Thanks!

drdren

Try to fix pip install

…demo.py

drdren

Remove pip install

…demo.py

drdren

Add pip install pyqsp

…demo.py

KetpuntoG · 2026-04-08T21:03:26Z

+def MultiControlledZ(wires, control_values=None):
+    if control_values is None:
+        control_values = [0] * (len(wires) - 1)
+    qml.ctrl(qml.Z(wires=wires[-1]),
+             control=wires[:-1],
+             control_values=control_values)
+
+# R_gate implements the reflection R used in the block construction.
+def R(wires):
+    assert len(wires) % 2 == 1
+    n = len(wires)//2
+    qml.PauliX(wires=wires[0])
+    MultiControlledZ(wires=wires[1:n+1]+[wires[0]])
+    qml.PauliX(wires=wires[0])


Take a look to qml.Reflection(wires)
That should replace all this code 😄

KetpuntoG · 2026-04-08T21:05:03Z

+def ProjCtrlPhaseShift(control_wires, target_wire, phi):
+    qml.MultiControlledX(wires=control_wires + target_wire,
+                         control_values=[0] * len(control_wires))
+    qml.RZ(phi = 2 * phi, wires=target_wire)
+    qml.MultiControlledX(wires=control_wires + target_wire,
+                         control_values=[0] * len(control_wires))


Is this qml.PCPhase?

KetpuntoG · 2026-04-08T21:08:44Z

+def generate_poly(deg, func, odd):
+    poly = PolyTaylorSeries().taylor_series(
+        func=func, degree=deg, max_scale=0.9,
+        chebyshev_basis=True, cheb_samples=2*deg)
+    pcoefs = poly.coef
+    if odd:
+        pcoefs[0::2] = 0
+    else:
+        pcoefs[1::2] = 0
+    return pcoefs


This is the way I get coefs given an arbitrary function
So no pyqsp needed here :)

from numpy.polynomial import Chebyshev as T cheb_poly = T.interpolate(target_func, degree)

drdren

Remove pip install pyqsp

…demo.py

github-actions · 2026-04-17T20:28:45Z

Your preview is ready 🎉!

You can view your changes here

Deployed at: 2026-05-15 15:15:56 UTC

drdren

Update figures

…demo.py

…metadata.json

drdren

Fix figure formatting

…demo.py

drdren

Better format

…demo.py

drdren

Fix title

drdren

Update formatting

…demo.py

drdren

Manifest a bullet point list

…demo.py

drdren

Fix bullet point list

…demo.py

drdren

Fix bullet points

…demo.py

drdren

Fix bullets

…demo.py

drdren

Here are some suggestions!

drdren · 2026-04-20T21:21:47Z

+# As also described in previous demos, :doc:`block
+# encoding <demos/tutorial_block_encoding>`
+# [`1 <https://arxiv.org/abs/1804.01973>`__] and :doc:`quantum singular value
+# transformation <demos/tutorial_intro_qsvt>`
+# [`2 <https://arxiv.org/abs/1806.01838>`__] have become the “gold standard” for implementing matrix
+# functions. However, these techniques primarily transform the singular values (or eigenvalues) of an


The transition from the first paragraph this sentence is harsh. It's clear why nonlinear transformations -> ... previous demos, ..., block encoding, ...

How about:

Suggested change

# As also described in previous demos, :doc:`block

# encoding <demos/tutorial_block_encoding>`

# [`1 <https://arxiv.org/abs/1804.01973>`__] and :doc:`quantum singular value

# transformation <demos/tutorial_intro_qsvt>`

# [`2 <https://arxiv.org/abs/1806.01838>`__] have become the “gold standard” for implementing matrix

# functions. However, these techniques primarily transform the singular values (or eigenvalues) of an

# The "gold standard" methods to implement matrix transformations are :doc:`block

# encoding <demos/tutorial_block_encoding>`

# [`1 <https://arxiv.org/abs/1804.01973>`__] and :doc:`quantum singular value

# transformation <demos/tutorial_intro_qsvt>`

# [`2 <https://arxiv.org/abs/1806.01838>`__]. However, these techniques primarily transform the singular values (or eigenvalues) of an

drdren · 2026-04-20T21:25:48Z

+# operator. In many quantum machine learning settings - especially amplitude encoding - the data isn’t
+# stored in an operator at all. Instead, it lives directly in the amplitudes of a quantum state.
+# 
+# To transform these amplitudes nonlinearly, we need a generalized approach. The Nonlinear Amplitude


I'm quite confused at this point. The goal is clear: come up with a method to perform nonlinear transformations when a quantum computer only does linear operations.

Somehow, that became block encoding and QSVT, but those don't work?

What is the true goal and problem that the nonlinear amplitude transformation addresses? Does NLAT keep the data in the form of amplitude encoding?

drdren · 2026-04-20T21:58:58Z

+# 
+# So the amplitude vector :math:`{\psi_i}` appears as the first column of :math:`U`. The obstacle is
+# that QSVT does not act on a column of a unitary; it acts on the spectrum of an operator accessed
+# through a block encoding.


A similar confusion has occurred here. I could understand the definition of a block encoding. I could also understand that |\psi> = U |0> implies that the elements of \psi are in fact the left-most column vector of U.

"The obstacle that QSVT does not act ..." comes out of nowhere. I think some previewing of what we're trying to argue at the end of "Once :math:A is available in this form, QSVT can implement polynomial transformations of the encoded operator, informally :math:A \mapsto P(A), by acting on its spectrum." would be helpful.

drdren · 2026-04-20T22:04:41Z

+# 
+# This is in sense equivalent to “encoding the first column into a diagonal,” but the key point is
+# subtler: :math:`U` is not modified. Instead, an auxiliary unitary :math:`U_\Psi` is engineered so
+# that the amplitudes :math:`\psi_i` appear as the diagonal entries of the encoded operator. In the


So creating a block encoding that has diagonal entries has the spectrum we want but U_A does not have?

drdren · 2026-04-20T22:09:43Z

+# spectrum (here, as the diagonal entries of :math:`\Psi`), a polynomial transform :math:`P(\Psi)`
+# corresponds to applying :math:`\psi_k \mapsto P(\psi_k)` in parallel (up to postselection).
+# 
+# Here, we build :math:`U_\Psi` explicitly for a small system (:math:`n=2`, so :math:`N=4`) to make


I know from context that n is the number of qubits and N = 2^n is the size of the Hilbert space but did you define those anywhere?

drdren · 2026-04-24T16:06:19Z

+# As a broader perspective, the same “linear mixing + elementwise nonlinearity” motif underpins more
+# advanced architectures. Recent work has explored the feasibility of quantum implementations of
+# transformer-style inference under various access models and resource assumptions
+# [`6 <https://arxiv.org/abs/2402.16714>`__]. The QMLP here should be viewed as a minimal instance of


Suggested change

# [`6 <https://arxiv.org/abs/2402.16714>`__]. The QMLP here should be viewed as a minimal instance of

# [#qtransformer]_. The QMLP here should be viewed as a minimal instance of

drdren · 2026-04-24T16:06:42Z

+# violating linearity.
+# 
+# In this demo, we have implemented the nonlinear amplitude transformation described in Guo et
+# al. (2024) and Rattew and Rebentrost (2024) [`3 <https://arxiv.org/abs/2107.10764>`__,


Suggested change

# al. (2024) and Rattew and Rebentrost (2024) [`3 <https://arxiv.org/abs/2107.10764>`__,

# al. (2024) and Rattew and Rebentrost (2024) [#ntca]_,

drdren · 2026-04-24T16:06:53Z

+# 
+# In this demo, we have implemented the nonlinear amplitude transformation described in Guo et
+# al. (2024) and Rattew and Rebentrost (2024) [`3 <https://arxiv.org/abs/2107.10764>`__,
+# `4 <https://arxiv.org/abs/2309.09839>`__]. We verified the diagonal amplitude block encoding on a


Suggested change

# `4 <https://arxiv.org/abs/2309.09839>`__]. We verified the diagonal amplitude block encoding on a

# [#importancesampling]_. We verified the diagonal amplitude block encoding on a

drdren · 2026-04-24T16:08:19Z

+    ],
+    "seoDescription": "Learn about the nonlinear amplitude transformation using QSVT",
+    "doi": "",
+    "references": [


If the bibliography in demo.py is correct, then this references section should be changed accordingly.

drdren · 2026-04-24T16:08:48Z

+#   :width: 95%
+#   :align: center
+# 
+#   Figure 1: *A schematic of the nonlinear transformation transformation with QSVT*


Now that we have a great graphic explaining the process, maybe you want to point back to it throughout the demo?

Upload nonlinear amplitude transformation demo

3997239

drdren requested review from a team as code owners April 8, 2026 16:52

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	# [`6 <https://arxiv.org/abs/2402.16714>`__]. The QMLP here should be viewed as a minimal instance of
	# [#qtransformer]_. The QMLP here should be viewed as a minimal instance of

	# al. (2024) and Rattew and Rebentrost (2024) [`3 <https://arxiv.org/abs/2107.10764>`__,
	# al. (2024) and Rattew and Rebentrost (2024) [#ntca]_,

	# `4 <https://arxiv.org/abs/2309.09839>`__]. We verified the diagonal amplitude block encoding on a
	# [#importancesampling]_. We verified the diagonal amplitude block encoding on a

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drdren commented Apr 8, 2026

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github-actions Bot commented Apr 17, 2026 •

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