ProcessTensors.jl

ProcessTensors.jl is a Julia package for MPS-based open quantum dynamics, Liouville-space simulation, and process tensors.

Why this project exists

Most open-quantum-softwares that exist today handles Markovian dynamics well, but say little about what happens when the environment remembers. As quantum devices push into regimes of strong coupling and structured environments, memory effects become the rule rather than the exception, and process tensors have emerged as the natural language for describing them.

The tools for working with process tensors, however, remain sparse for a field this active. ProcessTensors.jl exists to close that gap. This package primarily uses MPS/MPO infrastructure to define and manipulate process tensors in a memory-efficient way. It's built natively on ITensorMPS.jl, deliberately separating low-level tensor-network machinery from high-level physics workflows, so that the codes appear as natural and close to the theory. Alongside the code, the documentation is written to teach the basics of Liouville-space and process-tensor formalism, paired with runnable examples that mirror the underlying equations closely.

Today, ProcessTensors.jl covers the essentials: single-mode and small multimode process tensors, spin and bosonic baths, reduced dynamics, and multi-time correlations. The next concrete milestone is extending this to large, realistic non-Markovian environments via the ACE algorithm — a near-term, committed goal rather than a distant aspiration.

Beyond that, the longer-term vision is for this package to grow from a clean, accessible implementation into a genuine research platform: a place where process-tensor and open-system tensor-network algorithms — TEMPO, PT-TEMPO, TEDOPA, and other influence-functional-based methods — can be implemented side by side, benchmarked against each other, taught to newcomers, and reused by researchers who'd rather build on solid infrastructure than rebuild it from scratch.

Where to start

Most of what makes this package distinctive appears in two tutorials:

• Dissipative Dynamics — open-system evolution in Liouville space: Lindblad generators, liouvillian_mpo, TEBD/TDVP on density matrices.
• Single-Mode Process Tensor — process-tensor construction, bath memory, instruments, evolve, and evaluate_process.

Everything else in the documentation supports those two pages: ITensor syntax, MPS/MPO objects, Hilbert-versus-Liouville conventions, and closed-system dynamics as stepping stones.

Choose a path that suits you best:

Quick start

If you already know where you are headed, the snippet below builds a one-spin system coupled to one spin bath mode and constructs a process tensor. For context and checks along the way, use Single-Mode Process Tensor.

using ITensors
using ProcessTensors

dt = 0.1
nsteps = 24

# Physical Hilbert-space sites
sys = siteinds("S=1/2", 1)
bath = siteinds("S=1/2", 1)

# Liouville-space sites
sysL = liouv_sites(sys)
bathL = liouv_sites(bath)

# System Hamiltonian
Hsys = OpSum()
Hsys += 1.0, "Sx", 1
system = spin_system(sys, Hsys)

# Bath initial state
ψmps = MPS(bath, ["Up"])
ρmpo = to_dm(ψmps)
ρbath0 = to_liouville(ρmpo; sites=bathL)

# Bath Hamiltonian
Hbath = OpSum()
Hbath += 1.0, "Sx", 1

# System-bath coupling
Hcoupling = OpSum()
Hcoupling += 1.0, "Sz", 1, "Sz", 2

mode = spin_mode(bathL, Hbath, ρbath0; coupling = Hcoupling)
environment = spin_bath([mode])

pt = build_process_tensor(system, system.sites[1]; environment, dt, nsteps)

The process tensor can now be used to evolve a reduced system state.

ρsys0 = to_dm(MPS(sys, ["Up"]))

trajectory = evolve(pt, ρsys0)

Or it can be contracted with an explicit sequence of instruments.

obs = OpSum()
obs += 1.0, "Sz", 1

seq = default_schedule(pt)
add!(seq, 0, state_preparation(ρsys0))
add!(seq, nsteps, observable_measurement(obs))

expectation = evaluate_process(pt, seq)

Find a doc page by topic

Citing and contributing

ProcessTensors.jl is under active development. Contributions from expert developers, bug reports from users, new examples, and discussions about future directions are very welcome.

If you use the package in research, please cite the ProcessTensors.jl repository and any relevant process-tensor or tensor-network literature cited in the theory pages.