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arXiv:2507.09416

Local unitary decomposition of tripartite arbitrary leveled qudit stabilizer states into $p$-level-qudit EPR and GHZ state

Yat Wong, Liang Jiang

Published Jul 12, 2025• 0 citations
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Abstract

We study the entanglement structure of tripartite stabilizer states on $N$ qudits of dimension $D$, distributed across parties $A$, $B$, and $C$, under arbitrary local unitaries. Prior work by Bravyi et al. and Looi et al. showed that qubit and squarefree qudit stabilizer states can be transformed via local Clifford unitaries into tensor products of GHZ states, EPR pairs, and unentangled qudits [arXiv:quant-ph/0504208, arXiv:1107.1761]. We generalize this to arbitrary integer $D$ by introducing local unitaries beyond the Clifford group, enabling decomposition of prime-power qudit stabilizer states into $p$-level GHZ states, EPR pairs, and unentangled qudits. Our algorithm leverages subsystem phase matrices to characterize entanglement and applies to quantum protocols requiring efficient entanglement distribution.

DOI
10.48550/arxiv.2507.09416

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Local unitary decomposition of tripartite arbitrary leveled qudit stabilizer states into ppp-level-qudit EPR and GHZ state
We study the entanglement structure of tripartite stabilizer states on NNN qudits of dimension DDD, distributed across parties AAA, BBB, and CCC, under arbitrary local unitaries. Prior work by Bravyi et al. and Looi et al. showed that qubit and squarefree qudit stabilizer states can be transformed via local Clifford unitaries into tensor products of GHZ states, EPR pairs, and unentangled qudits [arXiv:quant-ph/0504208, arXiv:1107.1761]. We generalize this to arbitrary integer DDD by introducing local unitaries beyond the Clifford group, enabling decomposition of prime-power qudit stabilizer states into ppp-level GHZ states, EPR pairs, and unentangled qudits. Our algorithm leverages subsystem phase matrices to characterize entanglement and applies to quantum protocols requiring efficient entanglement distribution.
Consider a stabilizer state on nnn qudits, each of dimension DDD with DDD being a prime or squarefree integer, divided into three mutually disjoint sets or parts. Generalizing a result of Bravyi et al. [J. Math. Phys. 47, 062106 (2006)] for qubits (D=2D=2D=2), we show that up to local unitaries, the three parts of the state can be written as tensor product of unentangled signle-qudit states, maximally entangled Einstein-Podolsky-Rosen (EPR) pairs, and tripartite Greenberger-Horne-Zeilinger (GHZ) states. We employ this result to obtain a complete characterization of the properties of a class of channels associated with stabilizer error-correcting codes, along with their complementary channels.
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47 , 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2 , which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.
We introduce the qudit noisy stabilizer formalism, a framework for efficiently describing the evolution of stabilizer states in prime-power dimensions subject to generalized Pauli-diagonal noise under Clifford operations and generalized Pauli measurements. For arbitrary dimensions, the formalism remains applicable, though restricted to a subset of stabilizer states and operations. The computational complexity scales linearly with the number of qudits in the initial state and exponentially with the number of qudits in the final state. This ensures that when noisy qudit stabilizer states evolve via generalized Pauli measurements and Clifford operations to generate multipartite entangled states of a few qudits, their description remains efficient. We demonstrate this by analyzing the generation of a generalized Bell pair from a noisy linear cluster state subject to two distinct noise sources acting on each of the qudits.
Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.
Bravyi and Gosset [S. Bravyi and D. Gosset, Phys. Rev. Lett. 116, 250501 (2016)] recently gave classical simulation algorithms for quantum circuits dominated by Clifford operations. These algorithms scale exponentially with the number of TTT gates in the circuit, but polynomially in the number of qubits and Clifford operations. Here we extend their algorithm to qudits of odd prime dimension. We generalize their approximate stabilizer rank method for weak simulation to qudits and obtain the scaling of the approximate stabilizer rank with the number of single-qudit magic states. We also relate the canonical form of qudit stabilizer states to Gauss sum evaluations and give an O(n3)O({n}^{3})O(n3) algorithm for calculating the inner product of two nnn-qudit stabilizer states.
The Clifford hierarchy of unitary operators is a foundational concept for universal quantum computation. It was introduced to show that universal quantum computation can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. [S. X. Cui et al., Phys. Rev. A 95, 012329 (2017)] recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal unitaries whose action on a computational basis qudit state is described by a 2kth{2}^{k}\mathrm{th}2kth root of unity raised to some polynomial function of the state, and they established the level of such unitaries in the hierarchy as a function of kkk and the degree of the polynomial. For qubit systems, we consider kthk\mathrm{th}kth-level diagonal unitaries that can be described just by quadratic forms of the state over the ring Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ of integers modulo 2k{2}^{k}2k. The quadratic forms involve symmetric matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ that can be used to efficiently describe all two-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal unitaries from lower levels. The result involves symplectic matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ and hence our perspective unifies a subgroup of diagonal gates in the Clifford hierarchy with the binary symplectic framework for gates in the Clifford group. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to expensive magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+T\mathrm{Clifford}+TClifford+T set of gates, perhaps leading to shorter depth circuits. Our results suggest that some nondiagonal gates in the hierarchy might also be understood by generalizing other binary symplectic matrices to integer rings.