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QuantumToolbox.jl

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A. Mercurio and Y.-T. Huang.

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Introduction

QuantumToolbox.jl is a cutting-edge Julia package designed for quantum physics simulations, closely emulating the popular Python QuTiP package. It uniquely combines the simplicity and power of Julia with advanced features like GPU acceleration and distributed computing, making simulation of quantum systems more accessible and efficient.

With this package, moving from Python to Julia for quantum physics simulations has never been easier, due to the similar syntax and functionalities.

Features

QuantumToolbox.jl is equipped with a robust set of features:

• Quantum State and Operator Manipulation: Easily handle quantum states and operators with a rich set of tools, with the same functionalities as QuTiP.
• Dynamical Evolution: Advanced solvers for time evolution of quantum systems, thanks to the powerful DifferentialEquations.jl package.
• GPU Computing: Leverage GPU resources for high-performance computing. Simulate quantum dynamics directly on the GPU with the same syntax as the CPU case.
• Distributed Computing: Distribute the computation over multiple nodes (e.g., a cluster). For example, you can run hundreds of quantum trajectories in parallel on a cluster, with, again, the same syntax as the simple case. See here for more information.
• Differentiable Programming: Enable gradient-based optimization for quantum algorithms. Compute gradients of quantum dynamics with respect to their parameters using automatic differentiation. See here for more information.
• Easy Extension: Easily extend the package, taking advantage of the Julia language features, like multiple dispatch and metaprogramming.

Installation

[!NOTE] QuantumToolbox.jl requires Julia 1.10+.

To install QuantumToolbox.jl, run the following commands inside Julia's interactive session (also known as REPL):

using Pkg
Pkg.add("QuantumToolbox")

Alternatively, this can also be done in Julia's Pkg REPL by pressing the key ] in the REPL to use the package mode, and then type the following command:

(1.10) pkg> add QuantumToolbox

More information about Julia's package manager can be found at Pkg.jl.

To load the package and check the version information, use either QuantumToolbox.versioninfo() or QuantumToolbox.about(), namely

using QuantumToolbox
QuantumToolbox.versioninfo()
QuantumToolbox.about()

Brief Example

We now provide a brief example to demonstrate the similarity between QuantumToolbox.jl and QuTiP.

Let's consider a quantum harmonic oscillator with a Hamiltonian given by:

$$ \hat{H} = \omega \hat{a}^\dagger \hat{a} $$

where $\hat{a}$ and $\hat{a}^\dagger$ are the annihilation and creation operators, respectively. We can define the Hamiltonian as follows:

using QuantumToolbox

N = 20 # cutoff of the Hilbert space dimension
ω = 1.0 # frequency of the harmonic oscillator

a = destroy(N) # annihilation operator

H = ω * a' * a

We now introduce some losses in a thermal environment, described by the Lindblad master equation:

$$ \frac{d \hat{\rho}}{dt} = -i [\hat{H}, \hat{\rho}] + \gamma \mathcal{D}[\hat{a}] \hat{\rho} $$

where $\hat{\rho}$ is the density matrix, $\gamma$ is the damping rate, and $\mathcal{D}[\hat{a}]$ is the Lindblad dissipator, defined as:

$$ \mathcal{D}[\hat{a}]\hat{\rho} = \hat{a}\hat{\rho}\hat{a}^\dagger - \frac{1}{2}\hat{a}^\dagger\hat{a}\hat{\rho} - \frac{1}{2}\hat{\rho}\hat{a}^\dagger\hat{a} $$

We now compute the time evolution of the system using the mesolve function, starting from the initial state $\ket{\psi (0)} = \ket{3}$:

γ = 0.1 # damping rate

ψ0 = fock(N, 3) # initial state

tlist = range(0, 10, 100) # time list

c_ops = [sqrt(γ) * a]
e_ops = [a' * a]

sol = mesolve(H, ψ0, tlist, c_ops, e_ops = e_ops)

We can extract the expectation value of the number operator $\hat{a}^\dagger \hat{a}$ with the command sol.expect, and the states with the command sol.states.

Support for GPU calculation

We can easily pass the computation to the GPU, by simply passing all the Qobjs to the GPU:

using QuantumToolbox
using CUDA
CUDA.allowscalar(false) # Avoid unexpected scalar indexing

a_gpu = cu(destroy(N)) # The only difference in the code is the cu() function

H_gpu = ω * a_gpu' * a_gpu

ψ0_gpu = cu(fock(N, 3))

c_ops = [sqrt(γ) * a_gpu]
e_ops = [a_gpu' * a_gpu]

sol = mesolve(H_gpu, ψ0_gpu, tlist, c_ops, e_ops = e_ops)

Performance comparison with other packages

Here we provide a brief performance comparison between QuantumToolbox.jl and other popular quantum physics simulation packages, such as QuTiP (Python), dynamiqs (Python - JAX) and QuantumOptics.jl (Julia). We clearly show that QuantumToolbox.jl is the fastest package among the four. A detailed code is available [here](https://github.com/albertomercurio/QuantumToolbox.