Periodic MPO Representation

• Periodic MPO representation is a tensor network method that encodes the stroboscopic evolution of periodically driven, non-Markovian, open quantum systems using a periodic set of site tensors.
• It utilizes discretized path integrals and SVD-based compression to incorporate environmental memory effects while controlling computational complexity and errors.
• The method scales polynomially with system parameters, allowing simulations of large Hilbert spaces and capturing both transient and steady-state quantum dynamics.

A periodic matrix product operator (MPO) representation is a tensor network formalism that efficiently encodes the evolution superoperator (propagator) over one period for quantum systems subject to periodic driving. This construct generalizes the concept of an MPO from time-independent or Markovian settings to periodically driven, non-Markovian, and open quantum systems. The periodic MPO encapsulates all environmental memory effects, system-dynamics, and periodicity in a numerically tractable format, enabling exact stroboscopic and transient simulations in regimes inaccessible to exact diagonalization or brute-force techniques.

1. Formal Definition and General Structure

Consider a quantum system coupled to an environment, governed by a time-periodic Hamiltonian $H(t) = H(t+T)$. The system's evolution over one period $T$ is described by the stroboscopic Floquet propagator $\mathcal{F}(T)$, a linear map acting on the system's density matrix. When integrating out the environment, the influence of non-Markovian bath correlations can be captured by the Feynman–Vernon influence functional, which, upon discretization of the time interval into $M$ steps ($T = M \delta t$), yields a high-order tensor:

$\mathcal{I}^{(\mu_M \mu_M'), ... , (\mu_1 \mu_1')}$

This tensor can be factorized into a periodic sequence of MPO "site tensors" $\{ Q_n \}_{n=1}^M$, each of fixed bond dimension $\chi$, such that

$\mathcal{F}(T) = Q_M Q_{M-1} \cdots Q_1.$

The periodic boundary condition $Q_{n+M} = Q_n$ enforces $T$-periodicity at the MPO level (Mickiewicz et al., 11 Nov 2025).

2. MPO Construction and System-Bath Path Integral

The construction of the periodic MPO follows from the discretized system-bath path-integral. For Gaussian (e.g., bosonic) baths, the full influence functional over $N$ steps factorizes via boundary vectors $v_L$, $v_R$, and a set of MPO tensors $q^{(\mu \mu')}$,

$$\mathcal{I}_{\text{undriven}}^{ (\mu_N \mu_N'),...,(\mu_1 \mu_1')} = v_L \cdot q^{(\mu_N \mu_N')} \cdots q^{(\mu_1 \mu_1')} \cdot v_R.$$

With periodic system Hamiltonian $H_\mathrm{sys}(t)$, the physical MPO site tensors for each time step $Q_n$ take the form

$Q_n^{\mu \mu'}_{rr'} = \sum_{\alpha, \alpha'} U_\mathrm{sys}^{\mu \alpha}(t_n, t_n - \delta t/2) \, q^{(\alpha \alpha')}_{rr'} \, U_\mathrm{sys}^{\alpha' \mu'}(t_n - \delta t/2, t_n - \delta t),$

where $U_\mathrm{sys}$ is the system-only unitary evolution. The $Q_n$ inherit the period $M$ through $H_\mathrm{sys}(t+T) = H_\mathrm{sys}(t)$ (Mickiewicz et al., 11 Nov 2025).

Numerical efficiency is maintained by performing SVD-based compression on the temporal bond indices at each $Q_n$, truncating singular values below a selected threshold $\epsilon$. This allows for large environmental memory times and non-Markovian effects to be incorporated without exponential growth in computational cost.

3. Coarse-Graining and Floquet Propagator Extraction

The one-period Floquet MPO, denoted here as $Q_F$, is built as the ordered product of the periodic tensors:

$Q_F \equiv Q_M \cdots Q_1.$

To obtain the effective reduced system propagator, the external bath indices are contracted with the boundary vectors, yielding a $d^2 \times d^2$ stroboscopic superoperator $\mathcal{F}(T)$. Diagonalization of $Q_F$ yields its eigenvectors and eigenvalues, enabling explicit computation of stroboscopic evolution:

$\rho(nT) = [\mathcal{F}(T)]^n [\rho(0)],$

and—by inclusion of partial products of $Q_n$—detailed micro-motion within each period (Mickiewicz et al., 11 Nov 2025).

4. Computational Complexity and Error Control

The cost of constructing each $Q_n$ scales as $O(d^4 \chi^2)$ for a local Hilbert space of size $d$, bond dimension $\chi$, and with $M$ time slices per period, the overall cost is $O(M d^4 \chi^2)$. Diagonalization of the system-bond Floquet MPO $Q_F$ scales as $O(d^6)$.

Control of numerical error rests entirely on two parameters:

• Time-step $\delta t \rightarrow 0$, which suppresses Trotter errors $O(\delta t^3)$;
• Bond dimension $\chi \rightarrow \infty$ (or compression threshold $\epsilon \rightarrow 0$), which suppresses tensor truncation errors.

In practice, convergence is achieved when local singular values at each step fall below $10^{-8}$ and observables stabilize for successive reductions of $\delta t$ to within $10^{-6}$ (Mickiewicz et al., 11 Nov 2025).

5. Applications to Dissipative and Periodically Driven Systems

The periodic MPO formalism permits simulation of a wide class of open, strongly dissipative, and periodically driven quantum systems, including:

• Spin-boson models in a structured or Ohmic environment, accessed via construction of $Q_F$ for a single qubit coupled via $\sigma_z$ to the bath;
• Multi-qubit systems subjected to local driving and common dissipation channels (e.g., $S = [\sigma_z^A + \sigma_z^B]/2$ for two qubits), where entanglement dynamics and concurrence can be directly extracted from the stroboscopic steady-state solved via $Q_F$ diagonalization (Mickiewicz et al., 11 Nov 2025).

The approach yields not only stationary asymptotic states but also transient and multi-time observables through explicit products of the underlying periodic sequence $\{Q_n\}$.

6. Connections to Broader Tensor Network and Floquet Methods

Periodic MPO representations are a natural evolution of traditional MPO and tensor-network quantum dynamics methods, developed to accommodate

• Periodic system Hamiltonians and bath couplings,
• Non-Markovianity arising from finite bath memory encoded in finite $\chi$,
• Necessity to describe stroboscopic and transient dynamics exactly, not limited to Markovian or weak-coupling limits.

They provide a direct dissipative analogue to well-established Floquet-Sambe constructions for isolated systems, where large sparse Floquet matrices are diagonalized via Lanczos or time-evolution operators are built via composite-coefficient expansions (Mickiewicz et al., 11 Nov 2025, Parra-Murillo et al., 2015, Laptyeva et al., 2015), but extend these methods to open-system scenarios previously considered intractable.

7. Practical Considerations and Limitations

Efficient implementation relies on adaptive selection of $\chi$ and $\delta t$ consistent with the system's memory time and dynamical bandwidth. For physical systems with large environmental correlations or extreme periodic driving, convergence may necessitate significant bond dimensions or smaller time steps, though always within a polynomial scaling regime in $d$ and $M$. Unlike exact diagonalization which is limited to $d \sim 10^4$ (Laptyeva et al., 2015), periodic MPOs maintain feasibility for substantially larger total Hilbert spaces by exploiting separability and periodicity.

A plausible implication is that further generalizations of periodic MPOs may allow for scalable simulations of higher-dimensional or interacting systems, contingent upon developments in tensor network theory.

Summary Table: Key Features of Periodic MPO Representation

Aspect	Periodic MPO Approach	Sparse Floquet/Lanczos (Parra-Murillo et al., 2015)
System Class	Open, non-Markovian, periodic	Closed, many-body, periodic
Structure	Chain of M-site periodic tensors	Block-tridiagonal matrix in Sambe basis
Complexity	$O(M d^4 \chi^2 + d^6)$	$O(N_F^2)$
Error Control	$\delta t$, $\chi$	Matrix size, Krylov depth, tolerance

This approach provides a numerically exact, non-perturbative, and scalable framework for long-time, stroboscopic, and multi-time dynamics in strongly damped, periodic quantum systems, with flexibility to resolve both transient and stationary properties to arbitrary precision set by the computational resources and chosen error thresholds (Mickiewicz et al., 11 Nov 2025).