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Fermionic cellular automata in one dimension

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Pith's one-line read In one dimension, fermionic cellular automata with the same index are connected by a finite-depth fermionic circuit, with no additional ancillary systems.

desk verdict Solid classification of nearest-neighbour fermionic automata, but the ancilla-free equivalence is built on a proof sketch. read the letter →

arxiv 2501.05349 v1 pith:QMHVEHRD submitted 2025-01-09 quant-ph cond-mat.stat-mechmath-phmath.MP

Lorenzo S. Trezzini , Matteo Lugli , Paolo Meda , Alessandro Bisio , Paolo Perinotti , Alessandro Tosini This is my paper · ORCID
classification quant-phcond-mat.stat-mechmath-phmath.MP
keywords fermioniccellularautomataquantumindextheoryfinite-depthcircuitsancillaremovalMargoluspartitionschemeforkingautomatonMajoranashifts
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper strengthens the classification of one-dimensional fermionic cellular automata. Earlier work showed that two automata with the same index—an invariant measuring the net left/right flow of information on the lattice—can be connected by a finite-depth circuit only if ancillary fermionic degrees of freedom are allowed. The paper shows that those ancillas can be removed: equal index now implies equivalence by a finite-depth fermionic circuit acting on the physical cells alone. It then completes the classification of nearest-neighbour automata with one fermionic mode per site, discovering a 'forking automaton' that is locally implementable but cannot be built from single-mode and controlled-phase gates composed with shifts, unlike every qubit cellular automaton. This matters because it shows fermionic locality sustains dynamics absent from qubit lattices while still admitting an index-based classification.

What carries the argument

The machinery is the fermionic index together with the support algebras of the cellular automaton. For a FCA $T$, the left and right support algebras $L_{2x}$ and $R_{2x+1}$ are the smallest graded subalgebras on which the evolved two-cell algebras are supported; the index is $\mathrm{ind}[T]=\sqrt{\dim[L_{2x}]/\dim[A_{2x}]}$, and it takes values $1$, $2^{\pm1}$, and $2^{\pm 1/2}$ for single-mode fermionic chains, with the half-integer powers of two coming from Majorana shifts. The paper's new step is an ancilla-removal procedure (Proposition 2): starting from the known fact that an index-one FCA is implemented by a Margolus partitioned scheme on the enlarged lattice with one ancilla per cell, it sweeps a 12-cell block across the lattice, using already-updated physical cells as temporary ancillas, to produce a finite-depth fermionic circuit on the physical system alone. The forking automaton is then isolated by a case analysis of the support algebras: if both supports are generated by odd anticommuting operators, the local rule must split the two odd generators between left and right neighbours. The classification is completed by combining these unit-index rules with shifts and Majorana shifts.

What would settle it

Simulate the sweeping construction on a finite chain of length $N$ with periodic boundary conditions and record the minimal circuit depth and the number of cells used as temporary ancillas per update; if either grows without bound as $N$ increases, the ancilla-free equivalence theorem fails.

Watch

Extended reading notes

Core claim

The central claim is that the stable-equivalence classification of one-dimensional fermionic cellular automata can be made ancilla-free. Two FCAs $T$ and $S$ with the same index are $F$-equivalent: there exists a finite-depth fermionic circuit $F$ such that $T=F\circ S$ (Corollary 2). For automata with one fermionic mode per site, every index-one FCA is $M$-implementable—realisable by a two-layer nearest-neighbour Margolus partition scheme—and the nearest-neighbour index-one FCAs are exhausted by the controlled-phase type of Proposition 3 and the forking automaton of Theorem 3. The forking automaton sends the two odd generators of a site to the two neighbouring sites, $T_0(\eta)=X\boxtimes I\boxtimes I$ and $T_0(\xi)=I\boxtimes I\boxtimes Y$, and cannot be expressed as single-mode and controlled-phase gates composed with shifts. This completes the classification: shifts and Majorana shifts cover all non-unit indices, and an index-one FCA is either a controlled-phase local unitary or a forking automaton.

Load-bearing premise

The result rests on assuming that the trick of borrowing already-updated cells as temporary ancillas, shown on a 12-cell block, keeps working on the infinite chain with a bounded number of borrowed cells and a bounded circuit depth, and that the ancilla-removal theorem from the ungraded qudit setting transfers unchanged to $\mathbb{Z}_2$-graded fermionic algebras.

Editorial extensions

If this is right

  • Two fermionic cellular automata with the same index are equivalent by a finite-depth fermionic circuit without ancillas, matching the equivalence notion used for qubit cellular automata.
  • Every index-one FCA with one fermionic mode per site is implementable by a Margolus partitioned scheme, so index one is exactly the locally implementable class in this setting.
  • The nearest-neighbour index-one FCAs over $\mathrm{Mat}(\mathbb{C}^{1|1})$ are precisely the controlled-phase type and the forking automaton; composing with shifts and Majorana shifts exhausts all nearest-neighbour FCAs.
  • The forking automaton is a genuinely fermionic object: it is locally implementable but cannot be written as single-mode and controlled-phase gates composed with shifts, unlike every qubit cellular automaton.
  • Irrational index values $2^{\pm 1/2}$ remain a fermionic phenomenon, associated with Majorana shifts that move odd fermionic degrees of freedom by half a cell per step.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The forking automaton could serve as a primitive for fermionic quantum information processing that has no qubit analogue; one testable extension is whether its two independent Majorana legs can be used for fermionic state transfer on a translation-invariant chain.
  • The ancilla-removal argument is presented for one dimension and one ancilla per cell; the same strengthening of stable equivalence might hold for lattices with boundaries or for higher-dimensional graded algebras, but the authors do not prove that here.
  • With more than one fermionic mode per site, the support algebras can be richer, so new index-one local rules beyond the controlled-phase and forking forms may appear.
  • The forking automaton could be probed numerically on finite chains: although it is locally implementable, its correlation or entanglement structure after one step may differ from controlled-phase automata, giving an observable fermionic signature.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper studies one-dimensional fermionic cellular automata (FCAs) over the Z2-graded CAR algebra. The authors first review the index theory of FCAs, in which two automata with equal index are known to be stably equivalent up to finite-depth fermionic circuits and the addition of inert ancillas. Their first result claims to remove the ancillas: equal-index FCAs are F-equivalent, i.e., connected by a finite-depth fermionic circuit acting only on physical cells (Corollary 2), via an ancilla-removal procedure (Proposition 2). The second part classifies all nearest-neighbour FCAs with one fermionic mode per site: any index-one FCA is either a local unitary, a controlled-phase type automaton (Proposition 3), or a new “forking automaton” (Theorem 3) that is explicitly implemented by a Margolus scheme (Corollary 5); non-unit index automata reduce to shifts or Majorana shifts composed with index-one FCAs. The classification demonstrates a fermionic circuit that cannot be written as qubit-style single-mode and controlled-phase gates.

Significance. The claimed strengthening of the index classification is conceptually important: it would remove the caveat that the fermionic classification requires ancillas, aligning the FCA theory with the ungraded QCA case. The nearest-neighbour classification is a concrete and useful result, and the forking automaton is a genuinely new object with an explicit and checkable Margolus implementation (Corollary 5). The paper is clearly structured and uses standard algebraic tools. However, the ancilla-removal result, which underpins the paper's headline claim, is not proved with the same rigor as the classification part; the latter is supported by explicit constructions and appears sound.

major comments (2)
  1. [IV, Proposition 2] The proof of Proposition 2 is an informal finite-block construction (Figures 3-6) rather than a proof for the infinite lattice. It does not specify how the 12-cell block is tiled to cover all of Z, does not show that the resulting circuit has depth bounded by a constant independent of the block position, and does not prove that gates applied to cells that have already been “fully updated” (e.g., cell 4 in Step 4 and cell 3 in Step 3) leave those cells in their correct final state. Since Corollary 2 and the paper's first main result rest on this proposition, the claim that equal-index FCAs are F-equivalent without ancillas is not rigorously established as written. The text also invokes Ref. [41] for bounded ancilla removal, but that theorem is proved for ungraded qudit QCAs; the adaptation to the Z2-graded CAR algebra, where support-algebra lemmas involve graded commutators, is asserted rather than proved.
  2. [Corollary 2] The final sentence of Corollary 2 claims that “upon suitable regrouping of cells one can recast F in an MS.” This does not follow from the cited Lemma 5, which concerns stable M-equivalence with ancillary copies (Eq. (25): T ⊖ I = M2∘M1∘(S⊖I)) rather than an ancilla-free MS relation K = M∘J. A separate argument that the ancilla-free FDFC from Proposition 2 has bounded depth and can be recast as a Margolus scheme after blocking is needed; as written, the “Moreover” clause is unsupported.
minor comments (6)
  1. [VII] The section heading “AKNOWLEDGMENTS” is misspelled; it should be “ACKNOWLEDGMENTS.”
  2. [Definition 10] In Eq. (22), the second layer is defined with unitaries M^{(2)}_{2x+1} acting over {y,y+1}, but the variable y is not defined in that expression; please clarify the action site.
  3. [Lemma 10] In the proof of Lemma 10, point 2, “Theorem 8” should refer to “Lemma 8.”
  4. [Appendix B 1] The sentence “Repeating the computation for I ⊖ Σ_{2i+1} we get X_{2i+i} ↦→ and Y_{2i+1} ↦→Y_{2i+1} while X_{2i+1} ↦→ −Y_{2i}” contains an incomplete image for X_{2i+i} (with a typo “2i+i”) and should be corrected.
  5. [Eq. (42)] In Eq. (42), the identity factors I3 and I1 are undefined; also in Eq. (44) the tensor factors are not labeled by lattice sites, which makes the controlled-phase expression unnecessarily hard to parse.
  6. [Section V] The displayed index values “1, 2±1/2, 2±1” should read 1, 2^{±1/2}, 2^{±1}; please ensure the exponents are typeset correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the index-one classification is derived from local graded commutation and explicit circuit constructions, while the imported index and ancilla-removal results are external and do not assume the paper's conclusions.

full rationale

The paper's central derivation chain is not circular. The index formalism (Definition 8, Lemmas 3–6) is imported from Refs. [15] and [39], which are external to the present author list and do not assume the paper's conclusions. Proposition 1 and Corollary 1 use that formalism to show stable M-implementability with one ancilla per site; the argument is a support-algebra dimension and isomorphism argument, not a restatement of the conclusion. The classification in Section V is self-contained: Proposition 3 and Theorem 3 follow from local graded-commutation constraints (Eqs. (32)–(36), Lemma 8, Corollary 4, and the appendices), and Corollary 5 provides an explicit Margolus scheme for the forking automaton, so Theorem 4 does not borrow its own target. The main risk is a proof gap rather than circularity: Proposition 2's ancilla-removal proof is a 12-cell graphical construction that is not shown to tile the infinite lattice with uniform bounded depth, and the paper asserts rather than proves that the ancilla-removal theorem of Ref. [41] (proved for ungraded qudit QCAs) carries over to Z2-graded CAR algebras. That gap affects Corollary 2, but the conclusion is not equivalent to the input by definition. Minor self-citations (e.g., Refs. [42] and [43]) are contextual and not load-bearing.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper introduces no fitted parameters. It relies on established index theory and a diagrammatic ancilla-removal argument; the main new mathematical object is the forking automaton, which is explicitly constructed.

assumptions (4)
  • domain assumption The support-algebra index formalism for fermionic graded C*-algebras (Lemmas 2 and 3) is correct as imported from Refs. [15,39].
    Used to define L2x, R2x+1 and to prove index values in Section III; if the graded version fails, the classification collapses.
  • domain assumption Bounded ancilla removal for quantum cellular automata, proved for ungraded qudit systems in Ref. [41], can be applied to fermionic graded algebras.
    Section IV states the possibility was proved in [41]; the paper supplies a diagrammatic proof for one ancilla per cell, but does not give a formal graded version.
  • domain assumption Parity superselection: physical fermionic states and allowed operations have definite parity.
    Theorem 1; constrains local rules and screens out forbidden superpositions.
  • standard math Wedderburn classification of semisimple Z2-graded algebras with trivial graded center (Lemma 4).
    Used to enumerate possible index values via Eq. (20).
invented entities (1)
  • Forking automaton independent evidence
    purpose: Exhibits a nearest-neighbour index-one FCA over Mat(C1|1) that is M-implementable but not expressible as single-mode and controlled-phase gates composed with shifts.
    Explicit local rule Eq. (45) and explicit Margolus circuit in Corollary 5, so it is a constructed object with a falsifiable gate implementation, not a postulated entity.

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Pith. "Pith review of Fermionic cellular automata in one dimension." pith.science (2026). https://pith.science/paper/QMHVEHRD

@misc{pith2026250105349,
  author       = {Pith},
  title        = {Pith review of: Fermionic cellular automata in one dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QMHVEHRD}},
  note         = {Machine review of arXiv:2501.05349}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2501.05349 by the authors.

Figure 1
Figure 1. FIG. 1. Graphical representation of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The first layer of gates ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The second layer of transformations ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Step 3 and Step 4. Notice that the ancilla’s transform [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. After Step 5 the FDFC is complete. The picture corresp [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Reference graph

Works this paper leans on

80 extracted references · 78 canonical work pages · cited by 2 Pith papers

  1. [41]

    Unpaired majorana fermions in quantum wires

    A Y u Kitaev. Unpaired majorana fermions in quantum wires. Physics-Uspekhi, 44(10S):131, oct 2001

    work page 2001

  2. [15]

    Potter, and Ashvin Vishwanath

    Lukasz Fidkowski, Hoi Chun Po, Andrew C. Potter, and Ashvin Vishwanath. Interacting invariants for floquet phas es of fermions in two dimensions. Phys. Rev. B , 99:085115, Feb 2019

    work page 2019

  3. [39]

    In this paper, we can reformulate that notion for FCA as follows

    for ungraded QCA. In this paper, we can reformulate that notion for FCA as follows. Definition 12 (M-Implementability). A FCA T : A(Z) → A(Z) is said to be M-implementable if there exists a Margo- lus Partition Scheme (22) such that T = M. Based on the above notions we can introduce equiva- lence classes of Fermionic Cellular Automata. The origi- nal defini...

    work page

  4. [1]

    may be identified with the space F = Span R { ∏ x∈L (φ† x)px |Ω ⟩, p x = 0, 1 ∀x ∈ L } , (4) where px denotes the occupation number at the x-th site, and the product Π x∈L is well defined provided that a total (imma- terial) ordering is introduced on the denumerable set L. Using the creation/annihilation operators φx, φ† x, one can define the Fermionic Pauli...

    work page

  5. [2]

    Graded sum: The sum is defined only between elements of A with the same parity, i.e., O1, O2 ∈ Ap, O 1 + O2 ∈ Ap

    work page

  6. [3]

    Graded product: F or every two operators we have O1 ∈ Ap, O2 ∈ Aq, O 1O2 ∈ Ap⊕q, where ⊕ denotes sum modulo 2

    work page

  7. [4]

    g(T ) = 0 if T ∈ A0 and g(T ) = 1 if T ∈ A1, we have (T ⊠ I)(I ⊠ S) = ( −1)g(S)g(T )(I ⊠ S)(T ⊠ I), where I denotes the identity operator with g(I) = 0

    Graded tensor product: F or every two Z2-graded alge- bras A and B, we can define their graded tensor product A ⊠ B as A ⊠ B = ((A ⊠ B)0, (A ⊠ B)1), (A ⊠ B)i = ⨁ p,q =0, 1 p⊕q=i Ap ⊠ Bq, and denoting by g(T ) the grade of T , i.e. g(T ) = 0 if T ∈ A0 and g(T ) = 1 if T ∈ A1, we have (T ⊠ I)(I ⊠ S) = ( −1)g(S)g(T )(I ⊠ S)(T ⊠ I), where I denotes the identit...

    work page

  8. [5]

    T (A(Λ)) ⊂ A(Λ + N ) ∀Λ ⊂ Z (locality),

    work page

Show all 80 references
  1. [6]

    The homomorphism T0 : A0 − →A(N ) given by the restriction of the FCA to the 0 cell is called the local transition rule

    T ◦ τx = τx ◦ T ∀ x ∈ Z (homogeneity), where Λ + N := {y + x|y ∈ Λ , x ∈ N }. The homomorphism T0 : A0 − →A(N ) given by the restriction of the FCA to the 0 cell is called the local transition rule . Defining a FCA T as a ∗-automorphism over A(Z) = (A0, A1) implies that each T ...

  2. [7]

    The global *-homomorphism T is uniquely determined by the local transition rule T0

  3. [8]

    Majorna-shifts

    A *-homomorphism T0 : A0 − →AN is the transition rule of a FCA if and only if, for all x ∈ Z such that N ∩ (N + x) ⁄= ∅, the algebras T0(A0) and τxT0τ −1 x (Ax) graded-commute element-wise. The relation between the local rule and the global evolution is given by T (⊠x∈Λ Ax) = ...

  4. [9]

    As complex vector space, it has dimension (p + q)2

    Mat(Cp|q), the graded algebra of matrices acting on the graded space Cp|q. As complex vector space, it has dimension (p + q)2

  5. [10]

    Specifically, an element of C ℓ1(p|q) is of the form M1 ⊠ I + M2 ⊠ γ where Mi ∈ Mat(Cp|q), while γ is a Majorana mode

    C ℓ1(p|q) the Z2-graded algebra of (p + q) × (p + q) block-diagonal or anti-diagonal matrices with en- tries in the one dimensional complex Clifford algebra Cℓ1(C). Specifically, an element of C ℓ1(p|q) is of the form M1 ⊠ I + M2 ⊠ γ where Mi ∈ Mat(Cp|q), while γ is a Majorana ...

  6. [11]

    A.Y u. Kitaev. Fault-tolerant quantum computation by anyon s. Annals of Physics, 303(1):2–30, 2003

    2003

  7. [12]

    The forking automaton T0 Eq

    In the following corollary, proved in Appendix B 1, we provide an explicit MS for the forking FCA: Corollary 5. The forking automaton T0 Eq. (45) can be implemented through the following Margolus Partitioned Scheme: T0 = M2 ◦ M1 (47) M2(·) = ∏ x (U ⊠ U )M2x(·) ∏ y M † 2y(U ⊠ U...

  8. [13]

    The possibility of performing ancilla removal with a bounded number of ancillae for every physical cell was proved in [41]

    We observe that the local implementation of a FCA based on a FDFC on the physical cells only can be more involved than the simple MS needed with ancillae. The possibility of performing ancilla removal with a bounded number of ancillae for every physical cell was proved in [41]...

  9. [14]

    Doherty and Stephen D

    Andrew C. Doherty and Stephen D. Bartlett. Identifying phas es of quantum many-body systems that are universal for quantum computation. Phys. Rev. Lett., 103:020506, Jul 2009

    2009

  10. [16]

    EL (or ER) is trivial

  11. [17]

    EL = ER = {cI + dZ | c, d ∈ C}

  12. [18]

    with a suitable choice of basis, EL = {aI + bX | a, b ∈ C, ab = 0} and ER = {cI + dY | c, d ∈ C, cd = 0}. Proof. This straightforwardly follows from the above Lemma and the first condition in Eq. ( 36). One needs to consider that the only subalgebra of Mat(C1|1) with fixed parit...

  13. [19]

    is also site-independent, and hence can take values ind[T ] = √m p + q d . (20) B. Local implementability and equivalence relations As discussed in Refs. [15, 39], the index of a (Fermionic) Quantum Cellular Automaton determines its circuital imple- mentability, namely whether...

  14. [20]

    The Majorana shift σ± in Eq

    as proved in the following lemma: Lemma 9. The Majorana shift σ± in Eq. (12) satisfies ind(σ±) = 2 ± 1 2 , (37) ind(σ−1 ± ) = 2 ∓ 1 2 . (38) Proof. Applying σ± twice, as defined in Eq. ( 13), one can eas- ily find that (σ±)2 = τ±. Hence, exploiting the multiplicative property of ...

  15. [21]

    The graded commutation relation in Eq

    EL ≃ Mat(C1|1). The graded commutation relation in Eq. ( 36) and Corollary 3 immediately imply ER ≃ I. Thus, the second relation in Eq. ( 33) reduces to {[EL, EC ]} = 0 , which implies EC ≃ CI. Therefore, the automaton is isomorphic to a left shift which has not unit index

  16. [22]

    In this case, the condition in Eq

    EL contains an odd operator, that we will call X. In this case, the condition in Eq. ( 36) and Theorem 8 im- ply that either ER is generated by an odd operator that anticommutes with X—namely Y —or it is the trivial algebra ER ≃ I. In the former case we have indeed ER ≃ E L, w...

  17. [23]

    Then, ER is itself non-trivial and even, or it is trivial

    EL is non-trivial and even. Then, ER is itself non-trivial and even, or it is trivial. By similar arguments as those at point 2, the former case is the only admissible so- lution for an index one FCA, and thus EL ≃ E R ≃ {cI + dZ | c, d ∈ C}

  18. [24]

    Firstly, we apply the gates of the circu it F over a portion R of the lattice

    with a single ancilla per cell to implement a new circuit F ′ explicitly which uses cells of the system to play the role of the ancillae. Firstly, we apply the gates of the circu it F over a portion R of the lattice. In the meantime, we choose a set of cells not involved in th...

  19. [25]

    (27) ■ IV

    we have: T ⊠ I = M2 ◦ M1 ◦ (I ⊠ I) = M. (27) ■ IV . STABLE EQUIV ALENCE AND ANCILLA REMOV AL When we consider Eqs. ( 24),(25), the whole circuit ˜F acts trivially over the ancillary systems, but the single gates (M1 at first and M2 later, in Figure 1) composing the circuit ˜F d...

  20. [26]

    All the admissible cases are listed as follows: ER ≃ Mat(C1|1), ER is {cI + dZ | c, d ∈ C}, ER contains an odd operator—say X—or ER is trivial as well

    EL ≃ CI. All the admissible cases are listed as follows: ER ≃ Mat(C1|1), ER is {cI + dZ | c, d ∈ C}, ER contains an odd operator—say X—or ER is trivial as well. The first three cases are ruled out by exchanging the roles of EL and ER in items 1, 2 and 3. Therefore, we are only ...

  21. [27]

    EL ≃ E R ≃ {cI + dZ | c, d ∈ C}

  22. [28]

    Let us start with the trivial case 1

    EL = {aI + bOL | a, b ∈ C, ab = 0 } ≃ E R = {aI + bOR | a, b ∈ C, ab = 0}, with {[OL, OR]} = 0. Let us start with the trivial case 1. Since the evolution of an FCA is an automorphism, the only possible choice for the sup- port algebra EC is given by EC ≃ Mat(C1|1), i.e. the fu...

  23. [29]

    Furthermore, it was proved that forking FCAs are M-implementable (see Corollary 5)

    or by a forking FCA (see Theorem 3). Furthermore, it was proved that forking FCAs are M-implementable (see Corollary 5). To conclude this analysis we state the following results about one-dimensional FCA over Mat(C1|1), which provides a stronger version of the statements conta...

    2020

  24. [30]

    Cellular Automata Model- ing of Physical Systems

    Bastien Chopard and Michel Droz. Cellular Automata Model- ing of Physical Systems . Collection Alea-Saclay: Monographs and Texts in Statistical Physics. Cambridge University Pre ss, 1998

    1998

  25. [31]

    Vichniac

    G´ erard Y . Vichniac. Simulating physics with cellular automata. Physica D: Nonlinear Phenomena , 10(1):96–116, 1984

    1984

  26. [32]

    Theory of self-reproducing automata

    J von Neumann. Theory of self-reproducing automata. Edited by Arthur W . Burks, 1966

    1966

  27. [33]

    Reversible quantum cellular automata

    Benjamin Schumacher and Reinhard F Werner. Reversible quantum cellular automata. arXiv preprint quant-ph/0405174 , 2004

    2004 arXiv

  28. [34]

    D. J. Shepherd, T. Franz, and R. F. Werner. Universally programmable quantum cellular automaton. Phys. Rev. Lett. , 97:020502, Jul 2006

    2006

  29. [35]

    Quantum cellular automaton for univer sal quantum computation

    Robert Raussendorf. Quantum cellular automaton for univer sal quantum computation. Phys. Rev. A, 72:022301, Aug 2005

    2005

  30. [36]

    V . Murg F. V erstraete and J.I. Cirac. Matrix product states, projected entangled pair states, and variational renormal ization group methods for quantum spin systems. Advances in Physics, 57(2):143–224, 2008

    2008

  31. [37]

    Evenbly and G

    G. Evenbly and G. Vidal. Tensor network states and geometry. Journal of Statistical Physics , 145(4):891–918, 2011. 13

    2011

  32. [38]

    Matrix product unitaries: structure, sy mme- tries, and topological invariants

    J Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank V erstraete. Matrix product unitaries: structure, sy mme- tries, and topological invariants. Journal of Statistical Mechan- ics: Theory and Experiment , 2017(8):083105, aug 2017

    2017

  33. [40]

    Ignacio Cirac

    Lorenzo Piroli and J. Ignacio Cirac. Quantum cellular au- tomata, tensor networks, and area laws. Phys. Rev. Lett. , 125:190402, Nov 2020

    2020

  34. [42]

    Jason Alicea, Y uval Oreg, Gil Refael, Felix von Oppen, and Matthew P . A. Fisher. Non-abelian statistics and topologic al quantum information processing in 1d wire networks. Nature Physics, 7(5):412–417, 2011

    2011

  35. [43]

    Scalar fermionic ce llu- lar automata on finite cayley graphs

    Paolo Perinotti and Leopoldo Poggiali. Scalar fermionic ce llu- lar automata on finite cayley graphs. Phys. Rev. A, 98:052337, Nov 2018

    2018

  36. [44]

    Cellular automata in operational probabi listic theories

    Paolo Perinotti. Cellular automata in operational probabi listic theories. Quantum, 4:294, July 2020

    2020

  37. [45]

    are M-implementable according to Definition

  38. [46]

    A review of Quantum Cellular Automata

    Terry Farrelly. A review of Quantum Cellular Automata. Quan- tum, 4:368, November 2020

    2020

  39. [47]

    Fermionic quantum cellular automata and general - ized matrix-product unitaries

    Lorenzo Piroli, Alex Turzillo, Sujeet K Shukla, and J Igna- cio Cirac. Fermionic quantum cellular automata and general - ized matrix-product unitaries. Journal of Statistical Mechanics: Theory and Experiment, 2021(1):013107, jan 2021

    2021

  40. [48]

    Potter, and Ashvin Vishwanath

    Hoi Chun Po, Lukasz Fidkowski, Takahiro Morimoto, An- drew C. Potter, and Ashvin Vishwanath. Chiral floquet phases of many-body localized bosons. Phys. Rev. X , 6:041070, Dec 2016

    2016

  41. [49]

    Classification of interact ing floquet phases with u(1) symmetry in two dimensions

    Carolyn Zhang and Michael Levin. Classification of interact ing floquet phases with u(1) symmetry in two dimensions. Phys. Rev. B, 103:064302, Feb 2021

    2021

  42. [50]

    Ellison, Nathanan Tantivasadakarn, and Dominic J

    Wilbur Shirley, Y u-An Chen, Arpit Dua, Tyler D. Ellison, Nathanan Tantivasadakarn, and Dominic J. Williamson. Thre e- dimensional quantum cellular automata from chiral semion s ur- face topological order and beyond. PRX Quantum , 3:030326, Aug 2022

    2022

  43. [51]

    Stephen, Hendrik Poulsen Nautrup, Juani Bermejo- V ega, Jens Eisert, and Robert Raussendorf

    David T. Stephen, Hendrik Poulsen Nautrup, Juani Bermejo- V ega, Jens Eisert, and Robert Raussendorf. Subsystem symme- tries, quantum cellular automata, and computational phase s of quantum matter. Quantum, 3:142, May 2019

    2019

  44. [52]

    Derivation of the dirac equation from principles of information processi ng

    Giacomo Mauro D’Ariano and Paolo Perinotti. Derivation of the dirac equation from principles of information processi ng. Phys. Rev. A, 90:062106, Dec 2014

    2014

  45. [53]

    Thirring quantum cellular automato n

    Alessandro Bisio, Giacomo Mauro D’Ariano, Paolo Perinotti , and Alessandro Tosini. Thirring quantum cellular automato n. Phys. Rev. A, 97:032132, Mar 2018

    2018

  46. [54]

    A quantum cellular automaton for one-dimensional qed

    Pablo Arrighi, C´ edric B´ eny, and Terry Farrelly. A quantum cellular automaton for one-dimensional qed. Quantum Infor- mation Processing, 19(3):88, 2020

    2020

  47. [55]

    Quantum cellular automaton theory of light

    Alessandro Bisio, Giacomo Mauro D’Ariano, and Paolo Perinotti. Quantum cellular automaton theory of light. Annals of Physics, 368:177–190, 2016

    2016

  48. [56]

    Childs, Y uan Su, Minh C

    Andrew M. Childs, Y uan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. Theory of trotter error with commutator scalin g. Phys. Rev. X, 11:011020, Feb 2021

    2021

  49. [57]

    Childs and Y uan Su

    Andrew M. Childs and Y uan Su. Nearly optimal lattice simu- lation by product formulas. Phys. Rev. Lett., 123:050503, Aug 2019

    2019

  50. [58]

    Tobias J. Osborne. Efficient approximation of the dynamics of one-dimensional quantum spin systems. Phys. Rev. Lett. , 97:157202, Oct 2006

    2006

  51. [59]

    C´ edric B´ eny and Tobias J. Osborne. Information-geometric ap- proach to the renormalization group. Phys. Rev. A, 92:022330, Aug 2015

    2015

  52. [60]

    Scatt er- ing and perturbation theory for discrete-time dynamics

    Alessandro Bisio, Nicola Mosco, and Paolo Perinotti. Scatt er- ing and perturbation theory for discrete-time dynamics. Phys. Rev. Lett., 126:250503, Jun 2021

    2021

  53. [61]

    Quantum walks with a one-dimensional coin

    Alessandro Bisio, Giacomo Mauro D’Ariano, Marco Erba, Paolo Perinotti, and Alessandro Tosini. Quantum walks with a one-dimensional coin. Phys. Rev. A, 93:062334, Jun 2016

    2016

  54. [62]

    M., Perinotti, P ., and Tosini, A

    Bibeau-Delisle, A., Bisio, A., D’Ariano, G. M., Perinotti, P ., and Tosini, A. Doubly special relativity from quantum cellu lar automata. EPL, 109(5):50003, 2015

    2015

  55. [63]

    Special relativity in a discrete quantum univer se

    Alessandro Bisio, Giacomo Mauro D’Ariano, and Paolo Perinotti. Special relativity in a discrete quantum univer se. Phys. Rev. A, 94:042120, Oct 2016

    2016

  56. [64]

    Symmetries of the dirac quantum walk and emergence of the de sitter group

    Luca Apadula, Alessandro Bisio, Giacomo Mauro D’Ariano, and Paolo Perinotti. Symmetries of the dirac quantum walk and emergence of the de sitter group. Journal of Mathematical Physics, 61(8):082202, 08 2020

    2020

  57. [65]

    J. Watrous. On one-dimensional quantum cellular automata. In Proceedings of IEEE 36th Annual F oundations of Computer Science, pages 528–537, 1995

    1995

  58. [66]

    Unitari ty plus causality implies localizability

    Pablo Arrighi, Vincent Nesme, and Reinhard Werner. Unitari ty plus causality implies localizability. Journal of Computer and System Sciences , 77(2):372–378, 2011. Adaptivity in Hetero- geneous Environments

    2011

  59. [67]

    Partitioned quantum c el- lular automata are intrinsically universal

    Pablo Arrighi and Jonathan Grattage. Partitioned quantum c el- lular automata are intrinsically universal. Natural Computing, 11(1):13–22, 2012

    2012

  60. [68]

    Gross, V

    D. Gross, V . Nesme, H. V ogts, and R. F. Werner. Index theory of one dimensional quantum walks and cellular automata. Com- munications in Mathematical Physics , 310(2):419–454, 2012

    2012

  61. [69]

    Hastings

    Michael Freedman and Matthew B. Hastings. Classification of quantum cellular automata. Communications in Mathematical Physics, 376(2):1171–1222, 2020

    2020

  62. [70]

    Hastings

    Michael Freedman, Jeongwan Haah, and Matthew B. Hastings. The group structure of quantum cellular automata. Communi- cations in Mathematical Physics , 389(3):1277–1302, 2022

    2022

  63. [71]

    Classification of qubit cellular automata on hypercubic lat tices

    Andrea Pizzamiglio, Alessandro Bisio, and Paolo Perinotti . Classification of qubit cellular automata on hypercubic lat tices. arXiv preprint arXiv:2408.04493, 2024

    2024 arXiv

  64. [72]

    Bravyi and Alexei Y u

    Sergey B. Bravyi and Alexei Y u. Kitaev. Fermionic quantum computation. Annals of Physics, 298(1):210–226, 2002

    2002

  65. [73]

    M., Manessi, F., Perinotti, P ., and Tosini, A

    D’Ariano, G. M., Manessi, F., Perinotti, P ., and Tosini, A. Fermionic computation is non-local tomographic and violates monogamy of entanglement. Europhysics Letters , 107(2):20009, 2014

    2014

  66. [74]

    The feynman problem and fermionic entanglement: Fermionic theory versus qubit theory

    Giacomo Mauro D’Ariano, Franco Manessi, Paolo Perinotti, and Alessandro Tosini. The feynman problem and fermionic entanglement: Fermionic theory versus qubit theory. Interna- tional Journal of Modern Physics A , 29(17):1430025, 2014

    2014

  67. [75]

    Fermionic systems for quantum information people

    Szil´ ard Szalay, Zolt´ an Zimbor´ as, Mih´ aly M´ at´ e, Gergely Bar- cza, Christian Schilling, and ¨Ors Legeza. Fermionic systems for quantum information people. Journal of Physics A: Mathe- matical and Theoretical, 54(39):393001, oct 2021

    2021

  68. [76]

    Operator Algebras and Quantum Statistical Mechanics 1: C*- and W*-Algebras

    Ola Bratteli and Derek William Robinson. Operator Algebras and Quantum Statistical Mechanics 1: C*- and W*-Algebras. Symmetry Groups. Decomposition of States . Springer Science 14 & Business Media, 2012

    2012

  69. [77]

    Operator Alge- bras and Quantum Statistical Mechanics 2: Equilibrium Stat es Models in Quantum Statistical Mechanics

    Ola Bratteli and Derek William Robinson. Operator Alge- bras and Quantum Statistical Mechanics 2: Equilibrium Stat es Models in Quantum Statistical Mechanics . Springer Science & Business Media, 2013

    2013

  70. [78]

    Derezi´ nski.Introduction to Representations of the Canonical Commutation and Anticommutation Relations , pages 63–143

    J. Derezi´ nski.Introduction to Representations of the Canonical Commutation and Anticommutation Relations , pages 63–143. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006. Appendix A: Proof of Proposition 3 We want now to prove Proposition 3. For convenience of exposition...

    2006

  71. [79]

    This means that E decomposes as: E = (E00, E11), E00 = [Aξ ⊠ Y, X ⊠ Aξ], E11 = Aξ ⊠ [Y, Cξ] + [Bξ, X] ⊠ Aξ, 16 in which each component should vanish

    ⊕ (A1 0 ⊠ A1 1). This means that E decomposes as: E = (E00, E11), E00 = [Aξ ⊠ Y, X ⊠ Aξ], E11 = Aξ ⊠ [Y, Cξ] + [Bξ, X] ⊠ Aξ, 16 in which each component should vanish. We then have: [Aξ ⊠ Y, X ⊠ Aξ] = 0 . (B5) Writing Aξ = ⃗ p· ⃗ σwith ⃗ p· ˆz = 0, we get: px(⃗ p× ˆy)I ⊠ Z + py...

  72. [80]

    Consider the Forking automaton T0 that acts over X, Y , i.e.: I ⊠ T (X, Y ) ⊠ I = (X ⊠ I ⊠ I, I ⊠ I ⊠ Y )

    Proof of Corollary 5 Proof. Consider the Forking automaton T0 that acts over X, Y , i.e.: I ⊠ T (X, Y ) ⊠ I = (X ⊠ I ⊠ I, I ⊠ I ⊠ Y ). 17 This is the general form in Eq. ( 45) composed with a rotation that brings η, ξ in X, Y . In general, this acts over X, Y as: Xi → Xi−1, Y ...

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