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Fermionic cellular automata in one dimension

Lorenzo S. Trezzini, Matteo Lugli, Paolo Meda, Alessandro Bisio, Paolo Perinotti, Alessandro Tosini

TL;DR

This work develops a complete index-theoretic framework for one-dimensional Fermionic cellular automata (FCAs), removing the need for ancilla in the equivalence classification and clarifying how FCAs with nearest-neighbour locality behave. It shows that index-one FCAs are either the standard qubit-based 1D QCA or a novel forking FCA, with the latter realizable by a Margolus-partitioned circuit and not expressible solely via single-mode plus controlled-phase gates. The analysis reveals irrational indices (e.g., from Majorana shifts) and nontrivial FCA beyond the qubit paradigm, highlighting intrinsic differences between Fermionic and qubit QCAs. The results pave the way for extending FCA classifications beyond 1D and for exploring experimental implementations that exploit parity grading and Majorana-type dynamics.

Abstract

We consider quantum cellular automata for one-dimensional chains of Fermionic modes and study their implementability as finite depth quantum circuits. Fermionic automata have been classified in terms of an index modulo circuits and the addition of ancillary systems. We strengthen this result removing the ancilla degrees of freedom in defining the equivalence classes. A complete characterization of nearest-neighbours automata is given. A class of Fermionic automata is found which cannot be expressed in terms of single mode and controlled-phase gates composed with shifts, as is the case for qubit cellular automata.

Fermionic cellular automata in one dimension

TL;DR

This work develops a complete index-theoretic framework for one-dimensional Fermionic cellular automata (FCAs), removing the need for ancilla in the equivalence classification and clarifying how FCAs with nearest-neighbour locality behave. It shows that index-one FCAs are either the standard qubit-based 1D QCA or a novel forking FCA, with the latter realizable by a Margolus-partitioned circuit and not expressible solely via single-mode plus controlled-phase gates. The analysis reveals irrational indices (e.g., from Majorana shifts) and nontrivial FCA beyond the qubit paradigm, highlighting intrinsic differences between Fermionic and qubit QCAs. The results pave the way for extending FCA classifications beyond 1D and for exploring experimental implementations that exploit parity grading and Majorana-type dynamics.

Abstract

We consider quantum cellular automata for one-dimensional chains of Fermionic modes and study their implementability as finite depth quantum circuits. Fermionic automata have been classified in terms of an index modulo circuits and the addition of ancillary systems. We strengthen this result removing the ancilla degrees of freedom in defining the equivalence classes. A complete characterization of nearest-neighbours automata is given. A class of Fermionic automata is found which cannot be expressed in terms of single mode and controlled-phase gates composed with shifts, as is the case for qubit cellular automata.
Paper Structure (15 sections, 26 theorems, 123 equations, 2 figures)

This paper contains 15 sections, 26 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Algebra of a Fermionic chain
  3. CAR and $\mathbb{Z}_2$-graded quasi-local algebras
  4. One Dimensional Fermionic Cellular Automata
  5. Index for FCA
  6. Local implementability and equivalence relations
  7. Properties of the Fermionic index
  8. Stable Equivalence and Ancilla Removal
  9. Classification of One dimensional Nearest Neighbours FCA
  10. Unit Index FCA
  11. Outlook
  12. Aknowledgments
  13. Proof of Proposition \ref{['prop:Werner']}
  14. Proof of Theorem \ref{['thm:main']}
  15. Proof of Corollary \ref{['cor:Fork']}

Key Result

Theorem 1

Density matrices representing Fermionic states commute with the total parity operator $P$ given in Eq. eq:parity.

Figures (2)

  • Figure 1: Graphical representation of $\mathcal{M}$-Implementability of index one FCA. The white cells represent the cells of our register, the orange ones represent the ancillae, and the blue ones represent the cells evolved through $\mathcal{T}$ with $\operatorname{ind}[\mathcal{T}]=1$.
  • Figure 2: Graphical representation of the action of the gates $M_1$ in Eq. \ref{['eq:Mequiv']}. Here the purple and the green cells represent evolved sites and ancillary states respectively.

Theorems & Definitions (61)

  • Theorem 1: Parity Superselection
  • Definition 1: $\mathbb{Z}_2$-graded algebra
  • Definition 2: Graded commutator
  • Definition 3: Shift $\tau_x$
  • Definition 4
  • Theorem 2
  • Definition 5: Conjugation $\mathscr{U}_x$
  • Definition 6: Majorana-shift $\sigma_{\pm}$
  • Remark 1: Majorana modes
  • Definition 7: Support Algebra
  • ...and 51 more