Fermionic quantum cellular automata and generalized matrix product unitaries

II. QCA AND INDEX THEORY

• One-dimensional fermionic QCA have a richer index classification than qudit QCA, including a square-root-of-two factor and stable equivalence by appending inert fermionic modes.The Majorana-shift operator translates Majorana modes rather than physical fermions.Source: §I · p. 1, §II · p. 3

A. Summary of our results

• Standard fermionic MPUs are not equivalent to fermionic QCA: some lack locality preservation, while some QCA cannot be represented by standard fMPUs.The Majorana-shift QCA is an example that cannot be written in the expected fMPU form.Source: A. Summary of our results · p. 2
• Generalized fMPUs add an operator on the auxiliary space, and a tensor condition characterizes the family equivalent to fQCA up to inert ancillary fermionic degrees of freedom.The condition generalizes the qudit simpleness condition; antiperiodic-boundary tensors become simple after blocking, unlike all periodic-boundary tensors.Source: A. Summary of our results · p. 2
• For locality-preserving fMPUs, a tensor-based index coincides with the fermionic QCA index and classifies them under smooth unitary-preserving deformations.The index has additional signs relative to the qudit construction and obeys the expected stability properties.Source: A. Summary of our results · p. 2
• The graded canonical form for fermionic MPSs is proved to exist and be unique for antiperiodic boundary conditions.It is based on irreducible fermionic tensors and may also be useful in other problems.Source: A. Summary of our results · p. 2

A. MPUs in qudit systems

• For qudit systems, every finite-bond-dimension MPU is a QCA and vice versa, so local operators spread only within a finite region.The characterization follows from the simpleness condition for the MPU tensor.Source: A. MPUs in qudit systems · pp. 4–5
• The paper investigates how the qudit MPU picture generalizes to fermionic one-dimensional systems, where significant differences arise.The qudit MPU framework uses a normalized transfer matrix with one nonzero eigenvalue equal to one.Source: A. MPUs in qudit systems · pp. 4–5

A. Fermionic MPSs

• Fermionic MPS coefficients use a parity operator in the periodic-boundary trace, whereas the antiperiodic-boundary trace omits it.The local tensors must satisfy the parity condition that gives the state well-defined parity.Source: A. Fermionic MPSs · p. 6
• Fermionic tensor networks use parity-graded physical and auxiliary spaces, and local even tensors ensure states have well-defined parity.Odd local tensors are also possible; blocking twice makes an odd tensor even.Source: A. Fermionic MPSs · p. 6
• Fermionic MPUs are fermionic MPOs that are unitary for every chain length, with separate definitions for periodic and antiperiodic boundaries.Blocking and composition can introduce additional tensor signs.Source: A. Fermionic MPSs · p. 7

A. Non-locality-preserving fMPUs

• The example’s transfer matrix has eigenvalues 1 and eight copies of 1/4, unlike the single nonzero eigenvalue of a qudit MPU.The nontrivial spectrum is consistent with unitarity of the operator.Source: A. Non-locality-preserving fMPUs · pp. 7–8
• A periodic fMPU can be unitary without being locality-preserving: conjugating a site-local operator can affect a mode arbitrarily far away.The example maps orthonormal bases onto one another, yet the transformed operator is not confined to any finite neighborhood.Source: A. Non-locality-preserving fMPUs · p. 8

B. The Majorana-shift operator

• The Majorana shift translates Majorana modes and defines a fermionic QCA, but its periodic operator is odd, unlike ordinary fMPUs of the specified form, which are even.Under a Jordan–Wigner transformation, its open-boundary MPO bulk tensors do not generate a periodic qudit MPU.Source: B. The Majorana-shift operator · pp. 8–9
• An odd auxiliary operator X commuting with the local tensors makes the periodic Majorana-shift representation translation-invariant.The shift has tensor-network representations for both periodic and antiperiodic boundary conditions.Source: B. The Majorana-shift operator · pp. 8–9

VI. GENERALIZED FERMIONIC MPUS

• Generalized fMPUs add an auxiliary-space operator to the fMPO representation, with parity and boundary-condition constraints that ensure translation invariance.The Majorana-shift operators are examples of generalized fMPUs.Source: §VI · p. 9
• The graded canonical form is attainable after blocking for any even tensor while preserving the fMPS with antiperiodic boundary conditions.For tensors in this form generating the same fMPS for all system sizes, the matrices are related by an even gauge transformation and a block permutation.Source: §VI · p. 10

A. Type I and type II generalized fMPUs

• Degenerate normal tensors can give vanishing coefficients in the ordinary periodic fMPU trace, whereas the added auxiliary operator can close the trace to yield nonzero contributions.The Majorana-shift operator provides this case.Source: A. Type I and type II generalized fMPUs · p. 10
• Type I and type II generalized fMPUs form a minimal subset that accommodates both degenerate and non-degenerate normal tensors.For periodic boundary conditions, type I uses S = e^iαZ; the antiperiodic definitions use different auxiliary operators.Source: A. Type I and type II generalized fMPUs · p. 10

B. FMPUs and QCA: antiperiodic boundary conditions

• For antiperiodic boundaries, some blocked tensor power becomes simple, and simpleness ensures the generated fMPU is locality-preserving.For a tensor generating either type, Proposition VI.8 gives a simple power U^k for some k ≤ D^4.Source: B. FMPUs and QCA: antiperiodic… · p. 11
• Up to appending inert ancillary fermionic degrees of freedom, antiperiodic type I and type II fMPUs are fermionic QCA and vice versa.The tensor generating a type I fMPU is graded normal non-degenerate, while one generating type II is graded normal degenerate.Source: B. FMPUs and QCA: antiperiodic… · pp. 10–11

C. FMPUs and QCA: periodic boundary conditions

• Up to inert ancillary fermionic degrees of freedom, any fermionic QCA can be represented by a type I or type II fMPU with either boundary condition.The stated periodic-boundary tensor constraints identify which such fMPUs are locality-preserving.Source: C. FMPUs and QCA: periodic boundary… · pp. 11–12
• With periodic boundaries, type I fMPUs are fermionic QCA when their tensor is simple and its transfer matrix obeys the Z-symmetry and trace-one conditions.The stated conditions are (Z ⊗ Z)E_U = E_U and tr(E_U) = 1.Source: C. FMPUs and QCA: periodic boundary… · p. 12
• With periodic boundaries, type II fMPUs are fermionic QCA when their tensor is simple and its transfer matrix obeys the X-symmetry and trace-two conditions.The stated conditions are (X ⊗ X)E_U = E_U and tr(E_U) = 2.Source: C. FMPUs and QCA: periodic boundary… · p. 12

VII. INDEX THEORY FOR GENERALIZED FMPUS

• The fermionic index is defined from the ranks r and ℓ of two maps as (1/2)(log2(r) − log2(ℓ)); the qudit construction needs additional signs in the fermionic case.The paper notes that the qudit-style definition gives a nonzero index for a quantum circuit, which is the wrong result.Source: §VII · p. 12
• For locality-preserving fMPUs, the index is unchanged by blocking, additive under tensoring and composition, robust under continuous changes, and equal exactly for equivalent tensors.For type I fMPUs, the exponentiated index is rational.Source: §VII · p. 12
• The index analysis uses antiperiodic boundaries, where generalized fMPUs are guaranteed locality-preserving and the graded canonical form is unique.This choice does not restrict the periodic case because locality-preserving periodic tensors also generate antiperiodic fMPUs.Source: §VII · p. 13

A. Standard form for type I fMPUs

• The decompositions use singular value decompositions of parity-blocked maps, choosing even isometries and diagonal matrices to produce even graded tensors.For one map, the fermionic signs enter as the factor (−1)^|n||m|+|α|+|m| before the singular value decomposition.Source: A. Standard form for type I fMPUs · pp. 13–14
• For any tensor, the constructed graded operator u is an isometry; for a simple tensor, u and v are unitary.The standard form follows by blocking a simple tensor twice.Source: A. Standard form for type I fMPUs · p. 14
• The type I standard form is essentially unique up to single-site unitary invertible matrices acting on the legs that define u or v.The paper also gives a criterion for when two simple tensors generate the same type I fMPU.Source: A. Standard form for type I fMPUs · pp. 14–15

B. Standard form for type II fMPUs

• A type II fMPU is composed from a type I fMPU and a Majorana shift; after sufficient blocking, its standard form is unique up to single-site even invertible matrices.The composed tensor is not necessarily in graded canonical form.Source: B. Standard form for type II fMPUs · p. 15

C. Index

• For type I fMPUs, the exponentiated index is rational and remains unchanged by blocking.After a tensor becomes simple, its blocked right and left ranks scale with the physical dimension as stated in the index proposition.Source: C. Index · p. 15
• The index is additive under tensoring and composition, invariant under continuous changes, and equal for two tensors exactly when they are equivalent.Equivalence allows transformations by blocking and attaching ancillas.Source: C. Index · p. 15
• At zero index and fixed local dimension, fermionic tensors may lack a parity-preserving path between their standard-form unitaries because their graded spaces need not be isomorphic.Appending a C1|1 ancillary space permits a continuous path of even unitaries.Source: C. Index · p. 16
• For the Majorana-shift tensor, the blocked right rank is twice its left rank, and tracking contraction signs is crucial to this difference.The stated ranks are r_k = 2d^k and ℓ_k = d^k.Source: C. Index · p. 16
• For type II fMPUs, the exponentiated index is twice the index of the type I factor in the standard-form decomposition.The relation applies after blocking at least 2k sites, where the type I factor becomes simple after k blockings.Source: C. Index · p. 16
• For type II fMPUs, the index is unchanged by blocking and is robust under continuous changes of the tensor.The index of a composition or tensor product of two type II fMPUs is obtained by summing their indices.Source: C. Index · p. 17
• The index classifies type II tensors up to the paper’s equivalence relation, which permits blocking and ancillary graded spaces.The equivalence criterion also uses continuous deformation; the ancillary even and odd dimensions may be arbitrary.Source: C. Index · pp. 15, 17

VIII. CONCLUSIONS

• Fermionic MPUs are not necessarily locality-preserving, and standard fMPUs do not represent all fermionic QCA.The authors define generalized fMPUs and identify a locality-preserving subset equivalent to fQCA.Source: §VIII · p. 17
• The paper extracts the fermionic QCA index from fMPU tensors and introduces a graded canonical form for fermionic MPSs.The canonical form is presented as potentially useful beyond this work.Source: §VIII · p. 17
• An explicit standard form for type II fMPUs and classification with additional symmetries remain open questions.The authors specifically ask whether type II tensors have a standard form analogous to the one for type I.Source: §VIII · p. 17

1. General definitions

• The fiducial-state formalism generalizes a construction for qudits by projecting neighboring auxiliary fermionic modes onto maximally entangled pairs.The appendix uses maximally entangled Majorana modes as a simpler formulation than auxiliary real fermions.Source: §1 · p. 18
• Local tensors act on a physical site and adjacent auxiliary Majorana modes, and are required to have well-defined fermionic parity.The construction assumes even local tensors, which the text says can be achieved by blocking without restriction.Source: §1 · p. 18
• Concatenating local tensors with maximally entangled auxiliary pairs produces translation-invariant fMPSs with periodic or antiperiodic boundary conditions.The boundary condition is selected by the choice of the boundary projector.Source: §1 · p. 18
• Taking the auxiliary-vacuum expectation value recovers the fMPS coefficient form, with auxiliary parity encoded by a diagonal parity operator.The derivation uses fermionic anticommutation relations and a basis reordering gauge transformation.Source: §1 · pp. 18–19
• Replacing the fMPS local tensors with operator-valued local tensors gives an analogous fiducial-state construction for fMPOs.The resulting fMPO coefficients have the same tensor form as the fMPS construction.Source: §1 · p. 19

2. Elementary operations with fMPOs

• Fermionic MPOs are closed under composition and adjoint, with local tensor formulas that include fermionic signs.The composition tensors can be read directly from the fiducial-state expression.Source: §2 · pp. 19–20
• Blocking neighboring sites produces another fMPO, while the graded-tensor formulation makes the required sign conventions more transparent.The appendix also extends graphical notation with the signs prescribed by the fermionic tensor equations.Source: §2 · p. 20

Appendix B: Graded tensor networks

• A graded Hilbert space splits into even and odd sectors defined by a parity operator, and its tensor products use an isomorphism to account for reordering.The contraction map includes parity-dependent signs when graded spaces are reordered.Source: Appendix B · pp. 20–21
• Fermionic MPSs are built by concatenating local tensors and contracting auxiliary spaces, with local tensor parity required to be well-defined.The construction assumes even tensors without loss of generality.Source: Appendix B · p. 21
• The graded-tensor and fiducial-state formalisms give the same fMPS coefficients up to an even gauge transformation, and yield formulas for fMPO adjoints and compositions.The operator |n⟩⊗g⟨m| corresponds to the fermionic operator introduced in the fiducial-state formalism.Source: Appendix B · p. 21

1. Non-locality-preserving fMPUs

• The appendix tabulates all 16 local matrices used in the fMPU constructed in Sec. V A.Source: §1 · p. 21
• The vacuum contraction has only one nonzero summand, with n_j=m_j=0 for every j.Source: §1 · p. 22
• The restricted contraction translates three effective fermionic modes per site, up to a sign that does not affect unitarity.The modes are a1=a2, a2=a1, and a3=a1a2.Source: §1 · p. 22

2. Majorana-shift operator

• The fMPO corresponding to (52) represents translation of Majorana modes.Source: §2 · p. 22
• The fMPO corresponding to (52) is unitary because its local matrices generate the identity operator.Source: §2 · p. 22
• The fMPO corresponding to coefficients (54) represents the Majorana shift with periodic boundary conditions.Source: §2 · p. 23

Appendix D: Graded canonical form for antiperiodic boundary conditions

• Every even tensor can be put in Graded Irreducible Form (GIF) without changing its antiperiodic-boundary fMPS.GIF tensors have block-decomposed matrices with spectral radius one for each block’s transfer matrix, and each block is graded irreducible.Source: Appendix D · p. 23
• For a graded-irreducible tensor with a non-graded invariant subspace, D=2D_e and an even gauge gives the stated parity-block form; in that form, graded irreducibility is equivalent to no invariant subspace for the even subalgebra.The structural result also requires a nonzero odd matrix block.Source: Appendix D · pp. 23–25
• A graded-normal degenerate tensor has, after an even gauge, nonzero matrices of both even and odd parity, and blocking preserves its degenerate or non-degenerate status.The even and odd nonzero matrices occur in the block form (D1).Source: Appendix D · p. 25
• For graded-normal tensors, the completely positive map has a unique strictly positive even fixed point when non-degenerate, and an even–odd fixed-point pair when degenerate; an even gauge puts normal tensors in GCFII.In the degenerate case, the fixed points can be chosen with parity even and odd, respectively.Source: Appendix D · pp. 25–26
• Normal non-degenerate tensors have one transfer-matrix eigenvalue of modulus one with positive-definite left and right eigenvectors, while normal degenerate tensors have exactly two eigenvectors at eigenvalue 1 with even and odd parity.In the degenerate case, there are no other eigenvalues of modulus one, and the even eigenvectors are positive definite.Source: Appendix D · p. 26
• After blocking, any even tensor can be replaced by an even tensor in GCF that generates the same fMPS with antiperiodic boundary conditions.The construction first puts the tensor in GIF; in the irreducible case, peripheral eigenvalues that are roots of unity are handled by blocking before continuing the procedure.Source: Appendix D · p. 27
• A nonvanishing limiting overlap between antiperiodic-boundary fMPSs generated by graded normal tensors requires both tensors to be non-degenerate or both degenerate, with a phase, invertible gauge, and permutation relating them.The gauge commutes with one tensor’s parity operator, and the permutation maps the parity operators to each other up to sign.Source: Appendix D · p. 28
• If two GCF tensors generate proportional fMPSs for every N, they have the same number of graded-normal-tensor basis components, matched by phase, invertible gauge, and permutation.The corresponding parity operators have the same block-diagonal structure and coincide within each block up to a global minus sign.Source: Appendix D · p. 29

1. GCF of fMPUs with ABC

• For antiperiodic type-I fMPUs, the GCF is graded normal non-degenerate; for type-II fMPUs, it is graded normal degenerate.Source: §1 · p. 29
• For either antiperiodic type-I or type-II fMPUs, some power U^k with k≤D^4 is simple.Source: §1 · p. 30
• For type-II fMPUs, the local physical dimension is even and its even and odd subspaces have equal dimension.Source: §1 · p. 31
• The type-I and type-II fMPU class is closed under composition: two type-II fMPUs can be represented as a type-I fMPU.A product of type I and type II remains type II.Source: §1 · p. 32
• Up to appending inert fermionic ancillas, antiperiodic type-I or type-II fMPUs are 1D fermionic QCAs and vice versa.The construction uses mode translations, Majorana shifts, and parity-even depth-two circuits.Source: §1 · p. 33

• For q ≥ 2k, the cited passage says Lemma VII.9 gives the exponentiated index of Uq, but the supplied text does not state its value.Source: §2 · p. 33
• For ˜q ≥ 4k, the index of U_˜q equals the index computed in its corresponding graded canonical form.Source: §2 · p. 33
• The even projector ˜P defines ˜W = ˜P W ˜P, which represents the same fMPU as W and is a degenerate graded-normal tensor on ˜P's range.Source: §2 · pp. 33–34
• Invertible auxiliary-space transformations relate ˜W to √Φ1W, establishing that the tensors have the same left and right ranks.Source: §2 · p. 34