# Jacobian function doesn't derive as expected

**URL:** <https://discuss.pennylane.ai/t/jacobian-function-doesnt-derive-as-expected/4636>\
**Category:** PennyLane Help\
**Created:** [May 20, 2024, 11:09am UTC](https://discuss.pennylane.ai/t/jacobian-function-doesnt-derive-as-expected/4636 "2024-05-20T11:09:25Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![ZivChen](https://avatars.discourse-cdn.com/v4/letter/z/edb3f5/32.png) [@ZivChen](https://discuss.pennylane.ai/u/ZivChen)\
**Post date:** [May 20, 2024, 11:09am UTC](https://discuss.pennylane.ai/t/jacobian-function-doesnt-derive-as-expected/4636/1 "2024-05-20T11:09:26Z")

</div>

TL;DR:  
My problem is that I use `qml.jacobian` to calculate the spatial derivative of an amplitude-encoded circuit, but the output of the jacobian is returning the original vector shape and not the derivative.

Details of this example:  
In this example, I have a circuit (`circuit_state_prep`) where I take a normalized sine and encode into a quantum state using amplitude encoding, and sample the probabilities (should have sine shaped probabilities).  
Afterwards I take the `sqrt(probs)` to get the quantum state (`|psi>`).  
My goal is to get the spatial derivative of the input, sine in this case, so I apply the Jacobian to the circuit and the take only it’s diagonal to calculate the derivative in each point (each probability), which yields a sine shaped results instead of cosine shaped as expected, as can be seen in the figure below.

```auto
#pennylane imports
import pennylane as qml
import pennylane.numpy as pnp

# other imports
import matplotlib.pyplot as plt
import numpy as np

```

```auto
nqubits = 5
dev = qml.device("default.qubit", wires=nqubits)
range_qubits = [i for i in range(nqubits)]
x=pnp.linspace(0, np.pi, 2**nqubits, requires_grad=True)
sin=np.sin(x)
sinnorm=sin/np.linalg.norm(sin)
sinnorm = pnp.array(sinnorm, requires_grad=True)
cos=np.cos(x)
cosnorm=cos/np.linalg.norm(cos)

```

```auto
@qml.qnode(dev)
def circuit_state_prep(sine_shaped_input_state):
    qml.StatePrep(sine_shaped_input_state, wires=range_qubits)
    prob=qml.probs(wires=range_qubits)
    return prob

print(qml.draw_mpl(circuit_state_prep)(sinnorm))

```

```auto
prob = circuit_state_prep(sinnorm) # circuit is only state_prep for this check
psi = pnp.sqrt(prob)
d_psi_dx_vec_jac = qml.jacobian(circuit_state_prep)(sinnorm) # spatial derivative of psi
d_psi_dx_vec = pnp.array([d_psi_dx_vec_jac[i,i] for i in range(len(d_psi_dx_vec_jac))]) # vector of the diagonal elements of the jacobian
plt.figure(figsize=(9, 6))
plt.plot(d_psi_dx_vec) # expected to be same as cosnorm
plt.plot(cosnorm, '--') #f = cosnorm
plt.plot(psi)
plt.plot(sinnorm, 'r:')
plt.legend(["d|psi>/dx vec", "cosnorm (expected derivative)", "|psi>", "sinnorm"])
plt.show()

```

 ![image](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/6/698f09988b6a7d0b2487bb957c8ad7025cb491d0.png)

> **The circuit draw:**
>
> ![image](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/a/afad5c348481f04b86e212db645ead036b66bcc8.png)

> **\`qml.about()\`.**
>
> Name: PennyLane  
> Version: 0.36.0  
> Summary: PennyLane is a cross-platform Python library for quantum computing, quantum machine learning, and quantum chemistry. Train a quantum computer the same way as a neural network.  
> Home-page: [GitHub - PennyLaneAI/pennylane: PennyLane is a cross-platform Python library for quantum computing, quantum machine learning, and quantum chemistry. Train a quantum computer the same way as a neural network.](https://github.com/PennyLaneAI/pennylane)  
> Author:  
> Author-email:  
> License: Apache License 2.0  
> Location: /home/cudaq/.local/lib/python3.10/site-packages  
> Requires: appdirs, autograd, autoray, cachetools, networkx, numpy, pennylane-lightning, requests, rustworkx, scipy, semantic-version, toml, typing-extensions  
> Required-by: PennyLane-qiskit, PennyLane\_Lightning, PennyLane\_Lightning\_GPU, PennyLane\_Lightning\_Kokkos
> 
> Platform info: Linux-5.15.146.1-microsoft-standard-WSL2-x86\_64-with-glibc2.35  
> Python version: 3.10.12  
> Numpy version: 1.26.4  
> Scipy version: 1.13.0  
> Installed devices:
> 
> - lightning.qubit (PennyLane\_Lightning-0.36.0)
> - lightning.kokkos (PennyLane\_Lightning\_Kokkos-0.36.0)
> - lightning.gpu (PennyLane\_Lightning\_GPU-0.36.0)
> - default.clifford (PennyLane-0.36.0)
> - default.gaussian (PennyLane-0.36.0)
> - default.mixed (PennyLane-0.36.0)
> - default.qubit (PennyLane-0.36.0)
> - default.qubit.autograd (PennyLane-0.36.0)
> - default.qubit.jax (PennyLane-0.36.0)
> - default.qubit.legacy (PennyLane-0.36.0)
> - default.qubit.tf (PennyLane-0.36.0)
> - default.qubit.torch (PennyLane-0.36.0)
> - default.qutrit (PennyLane-0.36.0)
> - default.qutrit.mixed (PennyLane-0.36.0)
> - null.qubit (PennyLane-0.36.0)
> - qiskit.aer (PennyLane-qiskit-0.36.0)
> - qiskit.basicaer (PennyLane-qiskit-0.36.0)
> - qiskit.basicsim (PennyLane-qiskit-0.36.0)
> - qiskit.ibmq (PennyLane-qiskit-0.36.0)
> - qiskit.ibmq.circuit\_runner (PennyLane-qiskit-0.36.0)
> - qiskit.ibmq.sampler (PennyLane-qiskit-0.36.0)
> - qiskit.remote (PennyLane-qiskit-0.36.0)
> 
> ```auto
> 
> ```

Thanks!

---

<div class="post-metadata">

**Author:** ![ZivChen](https://avatars.discourse-cdn.com/v4/letter/z/edb3f5/32.png) [@ZivChen](https://discuss.pennylane.ai/u/ZivChen)\
**Post date:** [May 20, 2024, 1:04pm UTC](https://discuss.pennylane.ai/t/jacobian-function-doesnt-derive-as-expected/4636/2 "2024-05-20T13:04:34Z")

</div>

I think that my problem here i that the jacobian’s differentiation only applies to the parametrized gates and I can’t get the spatial derivatives using it.  
I will just use `Numpy's.gradient()` on the `prob` output.

if there is still a better way I’ll be glad to hear about it.

---

<div class="post-metadata">

**Author:** ![CatalinaAlbornoz](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/catalinaalbornoz/32/1196_2.png) [@CatalinaAlbornoz](https://discuss.pennylane.ai/u/CatalinaAlbornoz)\
**Post date:** [May 22, 2024, 10:42pm UTC](https://discuss.pennylane.ai/t/jacobian-function-doesnt-derive-as-expected/4636/3 "2024-05-22T22:42:00Z")

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Hi @ZivChen ,

You’re right that the Jacobian is only differentiating with respect to the trainable parameters. Would maybe some of the higher order derivatives or utility functions [here](https://docs.pennylane.ai/en/stable/code/qml_gradients.html) help you?