Unitary causal decompositions: a combinatorial characterisation via lattice theory

• URL: http://arxiv.org/abs/2508.11762v1
• Date: Fri, 15 Aug 2025 18:22:29 GMT
• Title: Unitary causal decompositions: a combinatorial characterisation via lattice theory
• Authors: Tein van der Lugt, Robin Lorenz,
• Abstract summary: Causal decompositions are circuit decompositions in which multiple no-influence conditions are made apparent simultaneously.<n>Our methods are based on finite-dimensional operator algebra as well as the concept lattice construction.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: If a unitary transformation has a decomposition into a quantum circuit with no directed path from input $a$ to output $b$, then $a$ does not influence $b$ through the overall unitary. Conversely, it is known that if $a$ does not influence $b$, one may always find a circuit decomposition lacking a path between these systems, thus making the no-influence condition directly apparent in the connectivity of the circuit. Causal decompositions are circuit decompositions in which, more generally, multiple such no-influence conditions are made apparent simultaneously. They bridge two fundamental concepts in quantum causality: causal structure, as expressed by influences through unitary transformations (and related to signalling through quantum channels); and compositional structure, expressed in terms of the shape of quantum circuits or networks. The general existence of causal decompositions remains unknown. This work focusses on unitary causal decompositions, i.e. decompositions in terms of unitary circuits in the traditional quantum circuit formalism that do not require the generalisation to `extended' or `routed' quantum circuits prompted by earlier research on this topic. We identify a combinatorial condition that characterises precisely those sets of causal no-influence constraints $G$ for which any unitary transformation satisfying $G$ has a unitary causal decomposition compositionally representing those constraints. Our methods are based on finite-dimensional operator algebra as well as the concept lattice construction, which was recently shown to provide a canonical shape $L_G$ for causal decompositions. The combinatorial condition we identify can be formulated in terms of $G$ as the absence of a forbidden substructure $C_3$ and in terms of $L_G$ as the existence of no more than one path between each input and output.

Related papers

• Causal Decompositions of 1D Quantum Cellular Automata [0.0]
  We present advances in the research program of causal decompositions.<n>Our results concern one-dimensional Quantum Cellular Automata (1D QCAs)<n>We show that this decomposition can be taken to be translation-invariant QCAs.
  arXiv  Detail & Related papers  (2025-06-27T13:36:50Z)
• Entanglement dynamics and Page curves in random permutation circuits [0.0]
  We study the ensembles generated by quantum circuits that randomly permute the computational basis.<n>Our results highlight the implications of classical features on entanglement generation in many-body systems.
  arXiv  Detail & Related papers  (2025-05-09T16:09:48Z)
• Entanglement transition and suppression of critical phase of thermofield double state in monitored quantum circuit with unitary $R$ matrix gates [4.889561507168047]
  We study quantum circuits with gates composed randomly of identity operators, projectors, or a kind of $R$ matrices.<n>This enables us to translate the quantum circuit into a topological object with distinguished overcrossings and undercrossings.
  arXiv  Detail & Related papers  (2025-03-01T08:17:11Z)
• On the Constant Depth Implementation of Pauli Exponentials [49.48516314472825]
  We decompose $Zotimes n$ exponentials of arbitrary length into circuits of constant depth using $mathcalO(n)$ ancillae and two-body XX and ZZ interactions.<n>We prove the correctness of our approach, after introducing novel rewrite rules for circuits which benefit from qubit recycling.
  arXiv  Detail & Related papers  (2024-08-15T17:09:08Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• A Complete Equational Theory for Quantum Circuits [58.720142291102135]
  We introduce the first complete equational theory for quantum circuits. Two circuits represent the same unitary map if and only if they can be transformed one into the other using the equations.
  arXiv  Detail & Related papers  (2022-06-21T17:56:31Z)
• Adaptive constant-depth circuits for manipulating non-abelian anyons [65.62256987706128]
  Kitaev's quantum double model based on a finite group $G$. We describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement.
  arXiv  Detail & Related papers  (2022-05-04T08:10:36Z)
• Conditions for realizing one-point interactions from a multi-layer structure model [77.34726150561087]
  A heterostructure composed of $N$ parallel homogeneous layers is studied in the limit as their widths shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schr"odinger equation is given.
  arXiv  Detail & Related papers  (2021-12-15T22:30:39Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• Sub-bosonic (deformed) ladder operators [62.997667081978825]
  We present a class of deformed creation and annihilation operators that originates from a rigorous notion of fuzziness. This leads to deformed, sub-bosonic commutation relations inducing a simple algebraic structure with modified eigenenergies and Fock states. In addition, we investigate possible consequences of the introduced formalism in quantum field theories, as for instance, deviations from linearity in the dispersion relation for free quasibosons.
  arXiv  Detail & Related papers  (2020-09-10T20:53:58Z)
• Causal and compositional structure of unitary transformations [0.0]
  We study whether causal structure can be understood in terms of compositional structure of a unitary. We derive causally faithful extended circuit decompositions for a large class of unitaries. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.
  arXiv  Detail & Related papers  (2020-01-21T21:06:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.