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

drdren · 2026-04-08T16:52:04Z

Before submitting

Please complete the following checklist when submitting a PR:

Ensure that your tutorial executes correctly, and conforms to the
guidelines specified in the README.
Remember to do a grammar check of the content you include.
All tutorials conform to
PEP8 standards.
To auto format files, simply pip install black, and then
run black -l 100 path/to/file.py.

When all the above are checked, delete everything above the dashed
line and fill in the pull request template.

Title:

Summary:

Relevant references:

Possible Drawbacks:

Related GitHub Issues:

If you are writing a demonstration, please answer these questions to facilitate the marketing process.

GOALS — Why are we working on this now?

Eg. Promote a new PL feature or show a PL implementation of a recent paper.
AUDIENCE — Who is this for?

Eg. Chemistry researchers, PL educators, beginners in quantum computing.
KEYWORDS — What words should be included in the marketing post?
Which of the following types of documentation is most similar to your file?
(more details here)

Tutorial
[ X] Demo
How-to

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?

I don't think they are equivalent. I tested it numerically and they both apply different phase patterns.

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)

Replaced the code.

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: 2026-06-25 19:37:55 UTC

drdren

Update figures

…demo.py

…metadata.json

drdren

Fix figure formatting

…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 · 2026-04-20T21:25:48Z

+# 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?

QSVT transforms the singular values / eigenvalues of an operator A: A \mapsto P(A). It needs the data sitting in the spectrum of something you can block-encode. However, the data here is not in a spectrum so block encoding and QSVT don't work directly. NLAT makes it work by constructing an auxiliary unitary $U_\Psi$ that block-encodes the diagonal operator. Once the amplitudes are the diagonal entries of an operator, they are also its eigenvalues, i.e. its spectrum and now QSVT applies.

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.

The opening sentence of this section mentions "QSVT applies a polynomial to the singular values or eigenvalues of an operator." This sets the premise of why QSVT can't be directly applied to the amplitudes of a quantum state to perform nonlinear transformation.

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?

So we take the data-encoding unitary U, which encodes the coefficients of |\psi>, and use it to build a larger unitary U_psi that block-encodes the diagonal operator \psi, on which we can then perform QSVT.

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 · 2026-04-24T16:08:19Z

+    ],
+    "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?

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

rishabh-gupta28 · 2026-06-12T20:59:29Z

+######################################################################
+# Diagonal block encoding of amplitudes
+# -------------------------------------
+# 


Suggested change

#

# Before diving in, reinforce your knowledge of the basics with our Intro to QSVT [and](https://pennylane.ai/demos/tutorial_intro_qsvt) QSVT in Practice [tutorials.](https://pennylane.ai/demos/tutorial_apply_qsvt)

rishabh-gupta28 · 2026-06-12T21:21:21Z

+# invocations of :math:`U` and :math:`U^\dagger`. For the purposes of this demo, we treat
+# :math:`U_\Psi` as a primitive and focus on what it enables. The construction idea is intuitionally
+# similar to building a quantum walk operator, and interested readers are encouraged to read original
+# papers for details.


The construction idea is intuitionally similar to building a quantum walk operator, and interested readers are encouraged to read original papers for details [qubitization].

rishabh-gupta28 · 2026-06-25T17:46:13Z

+# 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
+# the construction tangible. The code below spells out the walk-style ingredients used in Guo et


Suggested change

# the construction tangible. The code below spells out the walk-style ingredients used in Guo et

# the construction tangible. Here n is the number of qubits and :math:`N = 2^n` is the size of the Hilbert space. The code below spells out the walk-style ingredients used in Guo et

rishabh-gupta28 · 2026-06-25T17:59:22Z

+# al. (2024): a reflection :math:`R`, controlled applications of the state-preparation unitary and its
+# adjoint, and a pair of composite steps :math:`W` and :math:`G` that together produce the desired
+# block structure. A phase toggle :math:`p \in \{0,1\}` switches between encoding the real part
+# (:math:`p=0`) and the imaginary part (:math:`p=1`); here we focus on the real case.


Suggested change

# al. (2024): a reflection :math:`R`, controlled applications of the state-preparation unitary and its

# adjoint, and a pair of composite steps :math:`W` and :math:`G` that together produce the desired

# block structure. A phase toggle :math:`p \in \{0,1\}` switches between encoding the real part

# (:math:`p=0`) and the imaginary part (:math:`p=1`); here we focus on the real case.

# al. (2024):

# - a reflection :math:`R`

# - controlled applications of the state-preparation unitary and its adjoint

# - a pair of composite steps :math:`W` and :math:`G` that together produce the desired block structure.

# A phase toggle :math:`p \in \{0,1\}` switches between encoding the real part (:math:`p=0`) and the imaginary part (:math:`p=1`); here we focus on the real case.

rishabh-gupta28 · 2026-06-25T18:08:15Z

+# Below we create a simple block‑encoding for :math:`n=2` and inspect its matrix to confirm that its
+# diagonal corresponds to the input amplitudes.


Suggested change

# Below we create a simple block‑encoding for :math:`n=2` and inspect its matrix to confirm that its

# diagonal corresponds to the input amplitudes.

# Sanity check: after building the circuit, we inspect its matrix representation and look at the

# top-left :math:`N\times N` block. For a correct block-encoding, this block should behave like

# :math:`\Psi` (up to known normalization conventions), meaning its diagonal entries should match the

# input amplitudes :math:`\{\psi_k\}`. This is the smallest-scale verification that the circuit is

# implementing the intended “amplitudes :math:`\rightarrow` diagonal operator” transformation before

# we move on to applying QSVT polynomials.

# Below we create a simple block‑encoding for :math:`n=2` and inspect its matrix to confirm that its

# diagonal corresponds to the input amplitudes.

rishabh-gupta28 · 2026-06-25T19:19:35Z

+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


Replaced the code.

rishabh-gupta28 · 2026-06-25T19:20:09Z

+# Two ways to run the transformation
+# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+# 
+# We compare two initializations:


Suggested change

# We compare two initializations:

# We compare two reference state initializations:

rishabh-gupta28 · 2026-06-25T19:23:53Z

+# So far, the demo has focused on the primitive itself: using diagonal block encodings and QSVT to
+# implement an elementwise nonlinear map on amplitude-encoded data. Finally, we showcase the nonlinear
+# transformation of complex amplitude (NTCA) method as a genuine non-linear activation layer within a
+# trainable quantum model. We build a small quantum analogue of a two-layer MLP: two trainable linear


It's already defined in the following sentence: two trainable linear layers (implemented as parameterized unitaries) separated by a tanh activation

rishabh-gupta28 · 2026-06-25T19:29:11Z

+labels = (pnp.array(ds.test['4']['labels'][:200])+1)/2
+
+accuracy(best_weight, data, labels)
+


Suggested change

The goal of this section is not state-of-the-art accuracy. It is to show that the NLAT activation can be dropped into an end-to-end differentiable quantum model and trained. The modest accuracy is expected given the deliberately small model (2 data qubits, a degree-4 polynomial approximation of tanh, and only 100 optimization steps). Scaling any of these is the natural next step, but is outside the scope of this

minimal demonstration.

rishabh-gupta28 · 2026-06-25T19:29:17Z

+data = pnp.array(ds.test['4']['inputs'][:200])
+labels = (pnp.array(ds.test['4']['labels'][:200])+1)/2
+
+accuracy(best_weight, data, labels)


Upload nonlinear amplitude transformation demo

3997239

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

drdren commented Apr 8, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py Outdated

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

a7b0e4a

…demo.py

drdren commented Apr 8, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py Outdated

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

335df76

…demo.py

drdren commented Apr 8, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py Outdated

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

d501d82

…demo.py

KetpuntoG reviewed Apr 8, 2026

View reviewed changes

drdren commented Apr 16, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py Outdated

drdren added 3 commits April 16, 2026 15:55

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

2cd34b0

…demo.py

Merge branch 'master' into dren-nonlinear-QSVT

78adab8

Add requirements to install pyqsp

25ffba0

drdren commented Apr 17, 2026

View reviewed changes

drdren added 8 commits April 17, 2026 17:15

Upload thumbnails and figure

eb77784

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

825b4b2

…demo.py

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

2109524

…demo.py

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

311cb47

…demo.py

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

852df44

…demo.py

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

b868a35

…metadata.json

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

ede1c47

…metadata.json

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

3d17876

…metadata.json

drdren commented Apr 17, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py Outdated

drdren added 2 commits April 17, 2026 17:22

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

af2ac92

…demo.py

Fix figure name

c372945

Merge branch 'master' into dren-nonlinear-QSVT

e6639f9

drdren commented Apr 20, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

a73c75b

…demo.py

drdren commented Apr 20, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py Outdated

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py Outdated

drdren added 2 commits April 20, 2026 13:12

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

300a2b3

…demo.py

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

a782e96

…demo.py

drdren commented Apr 20, 2026

View reviewed changes

Comment thread demonstrations_v2/tutorial_nonlinear_amplitude_transformation/demo.py

drdren added 2 commits April 20, 2026 13:36

Update demonstrations_v2/tutorial_nonlinear_amplitude_transformation/…

9670073

…demo.py

Change qml to qp

1d756dc

drdren commented Apr 24, 2026

View reviewed changes

drdren requested a review from rishabh-gupta28 May 15, 2026 15:09

drdren assigned rishabh-gupta28 May 15, 2026

drdren and others added 17 commits May 15, 2026 11:09

Merge branch 'master' into dren-nonlinear-QSVT

038c4d7

Apply suggestion from @drdren

a6cb8b5

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

bd3b1f8

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

a87cfc6

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

abcbf97

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

feddad0

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

7fcf98a

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

2653f1f

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

94830fe

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

952ec09

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

df2d547

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

e9bb93b

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

0d7fd1d

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

f6a4a71

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

8bd9563

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

33c09fd

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

Apply suggestion from @drdren

59a5532

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>

rishabh-gupta28 reviewed Jun 25, 2026

View reviewed changes

	# [`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

	#
	# Before diving in, reinforce your knowledge of the basics with our Intro to QSVT [and](https://pennylane.ai/demos/tutorial_intro_qsvt) QSVT in Practice [tutorials.](https://pennylane.ai/demos/tutorial_apply_qsvt)

	# the construction tangible. The code below spells out the walk-style ingredients used in Guo et
	# the construction tangible. Here n is the number of qubits and :math:`N = 2^n` is the size of the Hilbert space. The code below spells out the walk-style ingredients used in Guo et

		# Below we create a simple block‑encoding for :math:`n=2` and inspect its matrix to confirm that its
		# diagonal corresponds to the input amplitudes.

-# Below we create a simple block‑encoding for :math:`n=2` and inspect its matrix to confirm that its
-# diagonal corresponds to the input amplitudes.
+# Sanity check: after building the circuit, we inspect its matrix representation and look at the
+# top-left :math:`N\times N` block. For a correct block-encoding, this block should behave like
+# :math:`\Psi` (up to known normalization conventions), meaning its diagonal entries should match the
+# input amplitudes :math:`\{\psi_k\}`. This is the smallest-scale verification that the circuit is
+# implementing the intended “amplitudes :math:`\rightarrow` diagonal operator” transformation before
+# we move on to applying QSVT polynomials.
+# Below we create a simple block‑encoding for :math:`n=2` and inspect its matrix to confirm that its
+# diagonal corresponds to the input amplitudes.

	# We compare two initializations:
	# We compare two reference state initializations:

		labels = (pnp.array(ds.test['4']['labels'][:200])+1)/2

		accuracy(best_weight, data, labels)


	The goal of this section is not state-of-the-art accuracy. It is to show that the NLAT activation can be dropped into an end-to-end differentiable quantum model and trained. The modest accuracy is expected given the deliberately small model (2 data qubits, a degree-4 polynomial approximation of tanh, and only 100 optimization steps). Scaling any of these is the natural next step, but is outside the scope of this
	minimal demonstration.

Uh oh!

Conversation

drdren commented Apr 8, 2026

Before submitting

Uh oh!

github-actions Bot commented Apr 8, 2026

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

github-actions Bot commented Apr 17, 2026 • edited Loading Uh oh! There was an error while loading. Please reload this page.

Uh oh!

Your preview is ready 🎉!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

drdren left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

Choose a reason for hiding this comment

Uh oh!

github-actions Bot commented Apr 17, 2026 •

edited

Loading