Pauli–MPS Form: Efficient Quantum Tensor Networks

• Pauli–MPS form is a tensor-network representation that transforms quantum states into the Pauli basis, enabling scalable computation of Pauli string amplitudes.
• It facilitates efficient evaluation of quantum measures like stabilizer Rényi entropies and Bell magic through standard MPS contraction techniques and compression.
• Its applications span integrable models and quantum devices, providing insights into nonstabilizerness and many-body exclusion principles.

The Pauli–MPS form is a matrix product state (MPS) representation of many-body quantum states in which the local physical basis is transformed from the conventional computational basis to the local Pauli operator basis. This construction provides an efficient tensor-network framework for encoding and manipulating the full spectrum of Pauli string amplitudes, enabling scalable computation of key quantum information measures—including nonstabilizerness, stabilizer Rényi entropies, stabilizer nullity, and Bell magic. The Pauli–MPS form is directly relevant for analyzing nonstabilizerness in quantum devices and characterizing complex many-body states, and it has seen application in diverse contexts such as Ising and XXZ spin chains, Rydberg atom arrays, and the root zero mode spaces of fractional quantum Hall Hamiltonians (Tarabunga et al., 2024, Lami et al., 2023, Bandyopadhyay et al., 2018).

1. Definition and Construction of the Pauli–MPS Form

For an $N$-qubit pure state $|\psi\rangle$, a conventional MPS expression in the computational basis is

$$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$

where each $A_i^{s_i}$ is a $\chi \times \chi$ matrix, and $\chi$ is the MPS bond dimension.

Switching to the Pauli basis $\{I,X,Y,Z\}$, one considers the Pauli expansion of the density matrix:

$$\rho = |\psi\rangle\langle\psi| = \frac{1}{2^N} \sum_{\vec{\alpha}} \langle\psi|P_{\alpha_1}\otimes \cdots \otimes P_{\alpha_N}|\psi\rangle \, P_{\alpha_1}\otimes\cdots\otimes P_{\alpha_N},$$

where each $P_{\alpha_i}$ is a Pauli operator and $\vec{\alpha}$ indexes $|\psi\rangle$0 possible strings.

The Pauli–MPS representation re-expresses the Pauli coefficients as amplitudes of an MPS with local Hilbert space dimension 4 and defines local tensors $|\psi\rangle$1 as follows:

$|\psi\rangle$2

so that

$|\psi\rangle$3

This yields an explicit tensor-network form for the complete Pauli operator expansion of $|\psi\rangle$4.

2. Tensor Network Structure and Gauge Conventions

In the Pauli–MPS form, each site is represented by a local tensor $|\psi\rangle$5 with bond dimension $|\psi\rangle$6. The mapping from original MPS tensors in the computational basis $|\psi\rangle$7 to Pauli–MPS tensors is explicit:

• $|\psi\rangle$8 (I)
• $|\psi\rangle$9 (X)
• $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$0 (Z)
• $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$1 (Y)

The right-canonical normalization of the original MPS ($$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$2) translates to the property $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$3 for the Pauli–MPS, ensuring orthonormality and norm preservation of the “Pauli vector” $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$4 representation.

Schematic tensor diagrams can depict the construction: each site carries a double physical index (original and conjugate), contracted with a Pauli box, producing a chain of $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$5-tensors with virtual dimension $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$6, capable of generating all Pauli string amplitudes.

3. Computational Implications and Efficient Contractions

The Pauli–MPS form enables the calculation of quantities that require sums over all $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$7 Pauli strings to be recast as efficient tensor contractions. Central to this are nonstabilizerness measures, such as stabilizer Rényi entropies and Bell magic, which can be expressed as sums of the form $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$8.

In the Pauli–MPS representation, such sums map to computing the squared norm of an MPS where diagonal MPOs are applied $$|\psi\rangle = \sum_{s_1,\dots,s_N=0,1} \mathrm{Tr}\left[A_1^{s_1} \cdots A_N^{s_N}\right] |s_1,\dots,s_N\rangle,$$9 times and are contracted with the original MPS, reducing to standard MPS contraction operations. The computational cost per contraction is $A_i^{s_i}$0, with the effective bond dimension $A_i^{s_i}$1 and often truncated to $A_i^{s_i}$2 in gapped phases. Standard MPS compression techniques (SVD or variational) further control the bond dimension scaling, yielding practical costs of $A_i^{s_i}$3 or better for a single 2D “folded” network evaluation (Tarabunga et al., 2024).

4. Perfect Pauli Sampling in MPS and SRE Estimation

Sampling Pauli strings from the distribution $A_i^{s_i}$4 is essential for estimating stabilizer Rényi entropies (SRE) and related metrics. The perfect sampling algorithm introduced by Lami & Collura (Lami et al., 2023) exploits the chain-rule decomposition and the MPS structure to exactly sample $A_i^{s_i}$5 with overall computational complexity $A_i^{s_i}$6 per sample.

The algorithm proceeds sequentially: at each site, conditional probabilities $A_i^{s_i}$7 are computed using the current MPS “left environment.” Each site update involves contractions scaling as $A_i^{s_i}$8. The method generalizes to reduced density matrices by appropriate environment initialization and supports local operator bases beyond Pauli matrices.

Sampling-based estimators for SREs and related observables converge rapidly, with the variance of the estimator remaining controlled independent of $A_i^{s_i}$9 for $\chi \times \chi$0 and requiring $\chi \times \chi$1 samples in the worst case for $\chi \times \chi$2. This suggests the method is viable for large $\chi \times \chi$3 in physically relevant states with moderate $\chi \times \chi$4.

5. Applications: Nonstabilizerness, Magic, and Quantum Many-Body States

The Pauli–MPS framework enables efficient computation of several quantum resource monotones:

• Nonstabilizerness (magic): Multiple monotones (e.g., stabilizer nullity, stabilizer Rényi entropies, Bell magic) become tractable, providing benchmarks for systems with up to hundreds of logical qubits (Tarabunga et al., 2024, Lami et al., 2023).
• Stabilizer group learning: The tensor-network structure allows for explicit reconstruction of the stabilizer group associated with an MPS.
• Many-body systems: Efficacy has been demonstrated in ground states of integrable models (Ising, XXZ chains) and out-of-equilibrium dynamics (e.g., Rydberg atom arrays), and for circuit evolution scenarios realized experimentally.

The Pauli–MPS also underlies constructions in fractional quantum Hall zero mode spaces, where it naturally encodes generalized exclusion (entangled Pauli) principles tied to AKLT-type MPS structure and emergent SU(2) symmetry (Bandyopadhyay et al., 2018).

6. Extensions, Assumptions, and Limitations

The Pauli–MPS representation presupposes that the many-body state admits an MPS with finite bond dimension $\chi \times \chi$5 in right-canonical gauge. The approach applies directly to pure states, with immediate extension to mixed or reduced states by modifying input environments or working with matrix product operators. The crucial requirement is an orthonormal, single-site local operator basis; generalization to other bases $\chi \times \chi$6 is straightforward provided $\chi \times \chi$7. However, states exhibiting large entanglement may require rapidly growing bond dimension, limiting efficiency. A plausible implication is that physical gapped phases and finite-correlated systems are best suited for practical applications.

7. Relation to Generalized and Entangled Pauli Principles

Beyond standard fermionic exclusions, the Pauli–MPS form provides a natural language for encoding entangled Pauli exclusion rules—constraints that act on multi-site occupancy patterns, as in the Jain–221 sequence for fractional quantum Hall zero modes (Bandyopadhyay et al., 2018). In these contexts, the local tensors map virtual spin-$\chi \times \chi$8 degrees of freedom to spin-1 multiplets, and AKLT-type fusion/projector relations impose nontrivial null-space conditions (e.g., forbidding fusion into spin 2), which are not realizable by single-site projectors and reflect the fundamentally entangled nature of the many-body exclusion. The Pauli–MPS thus offers a unifying, tensor-network machinery for both conventional and generalized exclusion principles and their physical realizations in quantum matter.