Qudit Visualizer – Discrete Wigner Function

This project visualizes the discrete Wigner function of a single qudit of arbitrary dimension $d \ge 2$.
It provides an interactive way to explore arbitrary states and dynamics in discrete phase space.

Online demo

A live demo is available here:

👉 https://lordrlo.github.io/qudit-visualizer/

Performance note: the online demo talks to a small remote backend and can be very slow for larger $d$ or long evolutions. For serious use (or just a smooth experience), you should run it locally.

You can:

• Define arbitrary initial states of any finite dimension
• Evolve them with arbitrary gates and Hamiltonians
• Inspect the resulting Wigner function $W(q,p,t)$

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Quick start

Install from PyPI with

pip install qudit-visualizer
qudit-visualizer

and enjoy! The website will open at http://localhost:8000.

Features

Qudit and Wigner function

• Single qudit with dimension $d \ge 2$ (odd or even).

• Discrete phase space with coordinates $(q,p)$ in $\mathbb{Z}_d \times \mathbb{Z}_d$.

• Wigner function computed as

  $$W(q,p) = \frac{1}{d}\mathrm{Tr}\big[\rho A_{q,p}\big]$$

  where $\rho = \ket{\psi}\bra{\psi}$ and $A_{q,p}$ are phase–point operators (see the theory section for details).

Initial states

Initial states are configured on the frontend and sent to the backend as a normalized vector $\lvert\psi\rangle$. Supported presets:

• Computational basis state $\lvert i\rangle$
  with a slider for the index $i$.

• Equal superposition

  $$\lvert\psi\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} \lvert k\rangle$$

• “Coherent”-like state $\lvert q,p\rangle$
  constructed via a discrete displacement $D_{q,p}\lvert 0\rangle$, with tunable $(q,p)$.

• Custom state

  • Specify amplitudes $\psi_i$ as mathjs expressions, e.g. exp(i*pi/3)/sqrt(2), cos(2*pi) + 3*i.
  • The backend normalizes the state when you press Generate.

Evolution modes

Two complementary evolution modes are available.

1. Gate mode

Apply a single unitary gate to the current state:

• Built-in gates:
  • $X$ – cyclic shift in the computational basis,
  • $Z$ – phase gate $Z\lvert q\rangle = \omega^q \lvert q\rangle$,
  • $F$ – discrete Fourier transform,
  • $T$ – simple quadratic phase gate.
• Custom gate:
  • Choose any $d \times d$ unitary matrix that will be used as a gate.

After applying the gate, the backend:

1. Normalizes the resulting state,
2. Computes the new Wigner function $W(q,p)$,
3. Returns the updated state and Wigner function.

2. Continuous evolution mode

Solve the time-dependent Schrödinger equation

$$ i \frac{d}{dt}\lvert\psi(t)\rangle = H \lvert\psi(t)\rangle $$

using Dynamiqs on top of JAX.

• Time parameters:
  • Final time $t_{\max}$,
• Hamiltonian choices:
  • Preset: diagonal quadratic spectrum $H_{i,i} = i^2 / d$,
  • Custom: full $d \times d$ Hermitian matrix specified in the UI.

For each saved time point, the backend:

1. Extracts $\lvert\psi(t_n)\rangle$,
2. Builds $\rho(t_n) = \lvert\psi(t_n)\rangle\langle\psi(t_n)\rvert$,
3. Computes $W(q,p;t_n)$,
4. Returns the time series ${t_n}$, states, and Wigner functions.

Visualization & controls

• Wigner function displayed as a heatmap on a canvas:
  • blue for negative values,
  • white near zero,
  • red for positive values.
• Hover tooltips show $W(q,p)$ at the cursor.
• For continuous evolution:
  • Playback controls (play/pause),
  • Frame slider,
  • Speed control.

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Project structure

Repository layout (top level):

.
├── qudit_visualizer/
│   ├── __init__.py          # Package init
│   ├── app.py               # FastAPI app: endpoints, dynamics, Wigner + static frontend
│   ├── models.py            # Pydantic models for requests/responses
│   ├── wigner.py            # Phase–point operators A(q,p) and Wigner map
│   └── static/              # Prebuilt frontend bundle for packaged use
├── frontend/
│   ├── .env.development     # VITE_API_BASE for local dev
│   ├── .env.production      # VITE_API_BASE for GitHub Pages demo
│   ├── .env.localapp        # VITE_API_BASE for packaged app (usually unset)
│   ├── index.html
│   ├── vite.config.ts
│   └── src/
│       ├── App.tsx              # Main React app: UI, controls, state management
│       ├── api.ts               # Typed API client for /simulate, /apply_gate
│       ├── config.ts            # API_BASE configuration
│       ├── components/
│       │   └── WignerHeatmap.tsx  # Canvas-based Wigner heatmap component
│       ├── App.css, index.css   # Styling
│       └── main.tsx             # React entry point
├── benchmark.py             # Local benchmark script for backend performance
├── pyproject.toml           # Packaging config for PyPI
└── README.md                # (This file)

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Tech stack

Backend

• Python 3
• FastAPI – HTTP API server
• Dynamiqs – solvers for Schrödinger dynamics on top of JAX
• JAX / jax.numpy – linear algebra and fast array operations
• uvicorn – ASGI server for development and deployment

Main HTTP endpoints (see qudit_visualizer/app.py):

• POST /simulate
  Runs continuous time evolution with chosen Hamiltonian and initial state.
• POST /apply_gate
  Applies a single unitary gate and returns the updated state and Wigner function.

Frontend

• React + TypeScript
• Vite – dev server and bundler
• mathjs – parsing and evaluating complex-valued expressions in the UI
• Plain CSS for layout and styling; plotting uses a simple HTML <canvas>

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Theory overview

1. Generalized Pauli operators
   For a $d$-dimensional Hilbert space with computational basis ${\lvert q\rangle : q \in \mathbb{Z}_d}$, define

   • $X\lvert q\rangle = \lvert q + 1 \bmod d\rangle$,
   • $Z\lvert q\rangle = \omega^q \lvert q\rangle$, with $\omega = e^{2\pi i / d}$.

   These generate the discrete Weyl–Heisenberg group and underpin the phase–space structure.

2. Phase space and phase–point operators
   The discrete phase space is

   $$\Gamma_d = {(q,p)}= \mathbb{Z}_d \times \mathbb{Z}_d.$$

   To each phase–space point $(q,p)$ we associate a Hermitian phase–point operator $A_{q,p}$.
   The Wigner function is then

   $$W(q,p) = \frac{1}{d}\mathrm{Tr}\big[\rho A_{q,p}\big]$$

   and satisfies:

   • Real-valuedness,
   • Normalization $\sum_{q,p} W(q,p) = 1$,
   • Correct marginals (line sums reproduce measurement probabilities),
   • Possible negativity for nonclassical states.

3. Odd vs even dimension

   • For odd $d$ there is a simple closed-form expression for $A_{q,p}$ in terms of shifts in the computational basis, which is implemented directly in the code (see the phase_point_ops_odd path in wigner.py). One convenient expression is

     $$A_{q,p} = \sum_{s=0}^{d-1} \omega^{2ps} \lvert q + s\rangle\langle q - s\rvert$$

     with indices understood modulo $d$.

   • For even $d$, constructing a Wigner function on a $d \times d$ lattice with good axiomatic properties is subtler. The implementation in this repository follows the stencil-based framework of:

     Lucky K. Antonopoulos, Joseph F. Fitzsimons, Adam G. M. Lewis, and Antoine Tilloy,
     “Grand Unification of All Discrete Wigner Functions on $d \times d$ Phase Space”,
     PRA 112, 052219 (2025), https://doi.org/10.1103/s5wn-mysr

     In short:

     1. Build a “parent” Wigner kernel on a $2d \times 2d$ phase space using Weyl operators $X^m Z^n$ and a parity operator $P$.

     2. Construct parent phase–point operators $A^{(2d)}(m_1,m_2)$ on that larger lattice.

     3. Compress the result to a $d \times d$ phase space by averaging over a $2 \times 2$ stencil:

        $$A(q,p) = \frac{1}{2} \sum_{b_1,b_2 \in {0,1}}A^{(2d)}(2q + b_1, 2p + b_2).$$

     This guarantees a consistent Wigner function on the $d \times d$ lattice for even $d$.

4. Dynamics

   • Gate mode: $\lvert\psi'\rangle = U \lvert\psi\rangle$ for a chosen or custom unitary $U$.
   • Continuous mode: integrate $i, d\lvert\psi\rangle/dt = H\lvert\psi\rangle$ with Dynamiqs, sample $\lvert\psi(t_n)\rangle$, and compute $W(q,p;t_n)$ at each time.

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Optional: backend benchmark

To get a quick feeling for backend-only performance:

python benchmark.py --runs 5 --d 10 --steps 200 --tmax 10.0

This runs a few local simulations (without HTTP) and prints timing statistics for both simulate and apply_gate.

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Roadmap and ideas

Some natural extensions that this codebase is ready for:

• Multi-qudit phase spaces and Wigner functions,
• Open-system dynamics (Lindblad) with Dynamiqs,
• Saving/loading parameter presets and simulation results.

Contributions, bug reports, and feature requests are welcome!