QwaveMPS: MPS Simulations in Waveguide-QED

• QwaveMPS is an open-source Python package that simulates one-dimensional waveguide QED using matrix product states to capture complex photon dynamics.
• It employs time-bin discretization and SVD-based compression to efficiently model non-Markovian feedback and long delay interactions.
• The framework also inspires broader MPS workflows in quantum chemistry and quantum machine learning, reducing tractability issues in high-dimensional systems.

Searching arXiv for papers on "QwaveMPS" and closely related usages. QwaveMPS most specifically denotes an open-source Python package for simulating one-dimensional waveguide quantum electrodynamics using matrix product states, with an emphasis on non-Markovian delay lines, time-delayed feedback, and strongly nonlinear few- and many-photon dynamics (Regidor et al., 17 Feb 2026). In the recent literature, the same label also appears in broader or informal senses for several matrix-product-state-centered workflows in quantum chemistry, quantum classification, continuum quantum fields, continuous-variable circuit simulation, and wave simulation. The common thread is the use of MPS compression to make otherwise prohibitive Hilbert spaces or discretizations tractable, but the term does not designate a single universally standardized formalism across those domains.

1. Scope and terminological usage

The most concrete and explicit usage is the software package introduced in “QwaveMPS: An efficient open-source Python package for simulating non-Markovian waveguide-QED using matrix product states” (Regidor et al., 17 Feb 2026). That package targets one-dimensional waveguide QED with localized emitters, propagating quantum fields, Markovian and non-Markovian regimes, and observables such as populations, fluxes, spectra, correlations, and entanglement.

The supplied literature also uses “QwaveMPS” in a broader, domain-specific sense for MPS-based workflows rather than the waveguide-QED package alone. In quantum chemistry, it is used to describe a workflow combining an MPS-inspired quantum circuit with variational quantum imaginary-time evolution and, optionally, transcorrelated Hamiltonians (Li et al., 2024). In quantum machine learning, it denotes an MPS classifier built from sequential $R_y$ and CNOT blocks with an ancilla (Bhatia et al., 2019). In quantum computing for many-body systems, it is used synonymously with qMPS, a quantum realization of MPS prepared in canonical forms by QSVD and reshape operations (Liu et al., 10 Jun 2025).

Usage in the literature	Representative source	Core object
Open-source waveguide-QED package	(Regidor et al., 17 Feb 2026)	Time-bin MPS simulation of Markovian and non-Markovian waveguide systems
Quantum-chemistry workflow	(Li et al., 2024)	QCMPS + VarQITE + transcorrelated Hamiltonians
Quantum or classical wave / continuum simulation	(Kaplan et al., 22 Jun 2026, Gómez-Lozada et al., 17 Aug 2025, Lively et al., 15 Apr 2025, Michelsen et al., 8 Apr 2025)	MPS compression of continuum fields, PDEs, or CV circuits

This multiplicity of usage suggests that “QwaveMPS” functions as a family label for qubit-frugal or memory-frugal MPS workflows, while the 2026 waveguide-QED package is the primary named artifact.

2. Waveguide-QED formulation

In the package-centered meaning, QwaveMPS models a total Hamiltonian decomposed as

$$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$

Here $H_{\mathrm{sys}}$ acts on localized emitters, $H_{\mathrm{wg}}$ describes the waveguide continuum, and $H_{\mathrm{int}}$ couples emitters to waveguide modes (Regidor et al., 17 Feb 2026). In frequency space, the waveguide Hamiltonian is written as

$$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$

while a real-space form with linear dispersion is

$$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$

The package adopts a time-bin discretization. Quantum noise operators are defined by

$$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$

with commutation relations

$$\big[\Delta B_\alpha(t_k), \Delta B_{\alpha'}^\dagger(t_{k'})\big] = \Delta t\, \delta_{k,k'}\, \delta_{\alpha,\alpha'}.$$

Time-bin Fock states are then built with a local cutoff $n_{\max}$, and the short-time propagator is approximated as

$$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$0

A semi-infinite waveguide with a mirror at distance $$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$1 produces a round-trip delay

$$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$2

and more generally a feedback path length $$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$3 corresponds to

$$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$4

time bins in memory (Regidor et al., 17 Feb 2026). Delayed interactions are represented by including a feedback bin at time $$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$5, and the corresponding time evolution operator acts on three sites: the system bin, the present time bin, and the delayed bin.

This time-bin construction belongs to a broader non-Markovian waveguide-QED tradition. Earlier comparative work emphasized that the MPS approach scales better when modelling multi-photon dynamics and long delay times, while a space-discretized waveguide model more easily adds off-chip decay and pure dephasing (Regidor et al., 2020).

3. Algorithms and software organization

QwaveMPS uses MPO applications on an MPS state with per-time-step unitaries, compressing after each step via SVD (Regidor et al., 17 Feb 2026). This is TEBD-like Trotterization: local system–present-bin coupling is applied in the Markovian case, and in non-Markovian cases delayed-bin coupling is handled after swapping the delayed bin into position. After each application, the MPS is re-canonicalized and truncated by keeping the largest singular values up to the user-specified maximum bond dimension $$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$6, denoted bond_max.

The package structure is explicit. parameters.py defines the InputParams and Bins data classes. states.py provides initial state builders such as tls_excited, tls_ground, vacuum, and fock_pulse. hamiltonians.py includes MPO builders such as hamiltonian_1tls, hamiltonian_1tls_feedback, hamiltonian_2tls_mar, and hamiltonian_2tls_nmar. simulation.py provides the time-evolution engines t_evol_mar and t_evol_nmar. operators.py implements single-time observables and entanglement. correlation.py implements two-time and steady-state correlation functions and spectra (Regidor et al., 17 Feb 2026).

The typical workflow is correspondingly explicit. One chooses $$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$7, $$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$8, the system Hilbert-space dimensions, the time-bin dimension per channel, couplings $$H = H_{\mathrm{sys}} + H_{\mathrm{wg}} + H_{\mathrm{int}}.$$9, and bond_max; prepares emitter and field initial states; constructs a Hamiltonian MPO; evolves with t_evol_mar or t_evol_nmar; and then evaluates populations, fluxes, correlations, spectra, or entanglement. Photon-number fluxes and correlation functions are represented by

$H_{\mathrm{sys}}$0

$H_{\mathrm{sys}}$1

$H_{\mathrm{sys}}$2

Entanglement entropies are computed from Schmidt coefficients, including a system–waveguide cut and, in the feedback case, a “TLSs + connecting waveguide” versus “rest of waveguide” cut (Regidor et al., 17 Feb 2026).

Accuracy is controlled by bond_max, the time step $H_{\mathrm{sys}}$3, and the local cutoff $H_{\mathrm{sys}}$4. Cost per local MPO application scales as approximately $H_{\mathrm{sys}}$5, and swap sequences for a delay of $H_{\mathrm{sys}}$6 bins add approximately $H_{\mathrm{sys}}$7 overhead (Regidor et al., 17 Feb 2026).

4. Benchmarks and reported performance

The package paper reports a broad benchmark set spanning vacuum decay, feedback, strong pumping, pulsed drives, and quantum Fock pulses (Regidor et al., 17 Feb 2026). For spontaneous emission into a waveguide, a single excited TLS with symmetric or chiral coupling exhibits exponential decay in the Markov limit,

$H_{\mathrm{sys}}$8

and QwaveMPS confirms quanta conservation through

$H_{\mathrm{sys}}$9

In the single-TLS mirror geometry with $H_{\mathrm{wg}}$0, the phase $H_{\mathrm{wg}}$1 distinguishes destructive and constructive feedback. For $H_{\mathrm{wg}}$2, feedback accelerates decay after $H_{\mathrm{wg}}$3; for $H_{\mathrm{wg}}$4, feedback traps photons and produces a long-lived steady state in the TLS and loop (Regidor et al., 17 Feb 2026). For two TLSs in the linear regime, population dynamics are reported to match a SciPy delay-ODE solver. In the nonlinear vacuum case, both $H_{\mathrm{wg}}$5 and $H_{\mathrm{wg}}$6 can reach 1 bit in the non-Markovian regime, while Markov entanglement remains lower.

The package also treats driven regimes. For a classical CW drive with $H_{\mathrm{wg}}$7, the non-Markovian spectrum shows suppression of the central Mollow-triplet peak and the emergence of new resonances in the constructive-feedback case. For a Gaussian $H_{\mathrm{wg}}$8-pulse with width $H_{\mathrm{wg}}$9 and center $H_{\mathrm{int}}$0, the non-Markovian geometry shows population trapping in the TLS and loop. For quantum Fock pulses, the package reproduces $H_{\mathrm{int}}$1 for a single-photon Gaussian pulse and the known “bird-shape” correlation pattern for a two-photon pulse (Regidor et al., 17 Feb 2026).

Representative numerical settings are also reported. The examples use $H_{\mathrm{int}}$2, $H_{\mathrm{int}}$3 for vacuum decay and approximately $H_{\mathrm{int}}$4 for CW steady state, with bond dimensions $H_{\mathrm{int}}$5 for linear vacuum, $H_{\mathrm{int}}$6 for two-quanta vacuum, and $H_{\mathrm{int}}$7 for CW drive (Regidor et al., 17 Feb 2026). Performance is modest: vacuum decay for a single TLS in the Markovian case is reported at $H_{\mathrm{int}}$8, the same system with delay at $H_{\mathrm{int}}$9, and CW drive with delay at $$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$0. Peak memory remained below approximately $$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$1 across all cases, and all examples ran on a laptop-class workstation.

5. Broader MPS-based workflows associated with the name

Outside waveguide QED, the supplied literature attaches “QwaveMPS” to distinct MPS-based methods. In quantum chemistry, it denotes a workflow that marries an MPS-inspired quantum circuit, QCMPS, with variational quantum imaginary-time evolution guided by McLachlan’s variational principle (Li et al., 2024). The state ansatz is

$$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$2

while the variational update uses

$$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$3

That workflow reports chemically accurate or better results for $$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$4, $$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$5, and active-space $$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$6, and, with transcorrelated Hamiltonians, Be and $$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$7 using only three qubits in the $$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$8 setting (Li et al., 2024).

In quantum machine learning, “QwaveMPS” denotes an MPS classifier in which angle-encoded inputs

$$H_{\mathrm{wg}} = \sum_{\alpha \in \{L,R\}} \int_{\mathcal{B}} d\omega \, \omega \, b_\alpha^\dagger(\omega) b_\alpha(\omega),$$9

are processed by a 1D chain of $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$0 rotations, CNOTs, an ancilla initialized to $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$1, and sequential compression to one output qubit (Bhatia et al., 2019). Training minimizes

$$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$2

The paper reports 4-qubit Iris experiments and 6-qubit evapotranspiration classification experiments on IBMQX4, with testing accuracies ranging from $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$3 to $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$4 on the Iris pairwise tasks and from $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$5 to $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$6 on the Agri tasks (Bhatia et al., 2019).

In quantum many-body simulation on quantum hardware, the label is used synonymously with qMPS, where each site is a small quantum register partitioned into left auxiliary, physical, and right auxiliary qubits, with local optimization on only $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$7 qubits (Liu et al., 10 Jun 2025). Canonical forms are enforced by QSVD and quantum reshape, the local effective Hamiltonian is assembled from measured $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$8 tensors, and benchmarked local problems of 3, 5, and 7 qubits are reported for $$H_{\mathrm{wg}} = \int dx\, \left[-i v_g\, a^\dagger(x) \, \partial_x a(x)\right].$$9 on the Heisenberg XXZ chain (Liu et al., 10 Jun 2025).

Continuum and wave simulations provide a further extension of the name. “Wavelet Matrix Product States for Quantum Fields” introduces wMPS built on Daubechies scaling functions with $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$0, exact projected MPOs, and iterative multiresolution refinement; for the Lieb–Liniger model the exact MPO bond dimensions are reported as $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$1 for $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$2 and $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$3 for $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$4 (Kaplan et al., 22 Jun 2026). “Simulating Quantum Turbulence with Matrix Product States” applies an MPS solver to the Gross–Pitaevskii equation and reports memory reductions ranging from $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$5 to over $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$6, including a $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$7 vortex-reconnection case where a DNS snapshot would require approximately $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$8 while the MPS stores approximately $$\Delta B_{\alpha}^{(\dagger)}(t_k) = \int_{t_k}^{t_{k+1}} dt'\, b_{\alpha}^{(\dagger)}(t'),$$9 (Gómez-Lozada et al., 17 Aug 2025). “A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks” uses an MPS plus MPO-QFT construction for the isotropic wave equation and extrapolates simulation of approximately $$\big[\Delta B_\alpha(t_k), \Delta B_{\alpha'}^\dagger(t_{k'})\big] = \Delta t\, \delta_{k,k'}\, \delta_{\alpha,\alpha'}.$$0 grid points on a laptop (Lively et al., 15 Apr 2025). “Functional matrix product state simulation of continuous variable quantum circuits” develops an FMPS real-space simulator for shallow non-Gaussian CV circuits, including loss via purification, and reports superior performance to Fock-truncated approaches for cat and GKP inputs under the tested conditions (Michelsen et al., 8 Apr 2025).

These usages are distinct from the waveguide-QED package. A plausible implication is that “QwaveMPS” names a methodological style rather than a single fixed algorithm: explicit exploitation of low-entanglement structure, locality, or multiscale organization to compress otherwise intractable state spaces.

6. Limitations, assumptions, and outlook

In its package form, QwaveMPS is explicitly one-dimensional, uses two propagation directions $$\big[\Delta B_\alpha(t_k), \Delta B_{\alpha'}^\dagger(t_{k'})\big] = \Delta t\, \delta_{k,k'}\, \delta_{\alpha,\alpha'}.$$1 or a single channel for a symmetric feedback loop, and in the reported examples assumes the rotating-wave approximation and linear waveguide dispersion (Regidor et al., 17 Feb 2026). The local time-bin Hilbert space is finite, so $$\big[\Delta B_\alpha(t_k), \Delta B_{\alpha'}^\dagger(t_{k'})\big] = \Delta t\, \delta_{k,k'}\, \delta_{\alpha,\alpha'}.$$2 must be chosen large enough for the photon number per bin expected from the dynamics. Very long memories increase swap overhead and can require larger $$\big[\Delta B_\alpha(t_k), \Delta B_{\alpha'}^\dagger(t_{k'})\big] = \Delta t\, \delta_{k,k'}\, \delta_{\alpha,\alpha'}.$$3 because of accumulated entanglement. The current implementation supports a single feedback loop; multi-loop or multi-delay topologies would require extensions. Off-chip decay is described as implemented in ongoing work, and pure dephasing as planned (Regidor et al., 17 Feb 2026).

Those limitations mirror a broader feature of the other QwaveMPS-associated workflows. In quantum chemistry, the main cost is the measurement of $$\big[\Delta B_\alpha(t_k), \Delta B_{\alpha'}^\dagger(t_{k'})\big] = \Delta t\, \delta_{k,k'}\, \delta_{\alpha,\alpha'}.$$4 and $$\big[\Delta B_\alpha(t_k), \Delta B_{\alpha'}^\dagger(t_{k'})\big] = \Delta t\, \delta_{k,k'}\, \delta_{\alpha,\alpha'}.$$5, and the paper notes that approximate QFI or hardware-efficient variants are needed to reduce that overhead (Li et al., 2024). In qMPS on quantum hardware, accuracy improves with bond dimension until local optimization depth and iteration limits dominate (Liu et al., 10 Jun 2025). In wavelet MPS, larger Daubechies order increases smoothness but also sharply increases MPO bond dimension (Kaplan et al., 22 Jun 2026). In quantum turbulence and wave simulation, efficiency depends on keeping entanglement or bond growth modest enough that compression remains effective (Gómez-Lozada et al., 17 Aug 2025, Lively et al., 15 Apr 2025).

This suggests that the enduring significance of QwaveMPS lies not in a single package namespace, but in a recurring computational principle: exploit structured entanglement, finite-range operator representations, or mode factorizations so that quantum dynamics, continuum fields, and non-Gaussian states can be simulated within an MPS manifold at costs controlled by bond growth rather than by full Hilbert-space dimension.