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arXiv:2309.15903 · Physical review. B./Physical review. B

Pumping chirality in three dimensions

Lukasz Fidkowski, Matthew B. Hastings

Published Jun 20, 2024• 11 citations
arXivHTMLPDF

Abstract

Using bosonization, which maps fermions coupled to a ${\mathbb{Z}}_{2}$ gauge field to a qubit system, we give a simple form for the three-fermion nontrivial quantum cellular automaton (QCA) as realizing a phase depending on the framing of flux loops. We relate this framing-dependent phase to a pump of eight copies of a $p+ip$ state through the system. We give a resolution of an apparent paradox, namely that the pump is a shallow depth circuit (albeit with tails), while the QCA is nontrivial. We discuss also the pump of fewer copies of a $p+ip$ state and describe its action on topologically degenerate ground states. One consequence of our results is that a pump of $n p+ip$ states generated by a free Fermi evolution is a free fermion unitary characterized by a nontrivial winding number $n$ as a map from the third homotopy group of the Brilliouin Zone 3-torus to that of $SU({N}_{b})$, where ${N}_{b}$ is the number of bands. Using our simplified form of the QCA, we give higher-dimensional generalizations that we conjecture are also nontrivial QCAs, and we discuss the relation to Chern-Simons theory.

DOI
10.1103/physrevb.109.235142

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We give the definitive proof that the Dirac Quantum Cellular Automaton (QCA) used for both quantum simulation and algorithmic foundations of Quantum Field Theory (QFT), and especially of Quantum Electrodynamics (QED), as put forward in References https://doi.org/10.1007/s11128-019-2555-4 and https://doi.org/10.22331/q-2023-11-08-1179, does exhibit Fermion Doubling (FD), albeit thrice as less severe as discrete-time standard Lattice Gauge Theories (LGTs) [arXiv:2505.0790], which are naive regarding the spacetime discretization of differential operators acting on fermionic fields. The proof is done for the (1 + 1)D Dirac-QCA model. We show that the (one-time-step) two-point correlation function, also called Green's function (GF), of the Dirac QCA, is of astonishing simplicity, which is in contrast with the GF of the Dirac equation. We also compare, both qualitatively and quantitatively, this Dirac QCA to the continuous-time-LGT spatial discretization of Dirac fermions regarding how well these two lattice models approximate their naive continuum limit$\unicode{x2014}$which is nothing but the Dirac equation$\unicode{x2014}$even when far away from that limit, a situation which must be considered because of experimental limitations in quantum simulation: the Dirac QCA is better for ultrarelativistic regimes, whereas continuous-time LGT is better for non-relativistic regimes. Then, we compute the GF of the FD-fixed model put forward in the last cited reference, called Flavored Dirac QCA (FDQCA)$\unicode{x2014}$which staggers an extra, artificial flavor $only$, on a diamond spacetime lattice, and does not stagger chirality as staggered fermions in usual LGT. The structure of this FDQCA two-point correlation function is of extreme simplicity, and can be expressed in a very simple manner in terms of the four chiral components of the FD-suffering, original-model GF.

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Using bosonization, which maps fermions coupled to a Z2{\mathbb{Z}}_{2}Z2​ gauge field to a qubit system, we give a simple form for the three-fermion nontrivial quantum cellular automaton (QCA) as realizing a phase depending on the framing of flux loops. We relate this framing-dependent phase to a pump of eight copies of a p+ipp+ipp+ip state through the system. We give a resolution of an apparent paradox, namely that the pump is a shallow depth circuit (albeit with tails), while the QCA is nontrivial. We discuss also the pump of fewer copies of a p+ipp+ipp+ip state and describe its action on topologically degenerate ground states. One consequence of our results is that a pump of np+ipn p+ipnp+ip states generated by a free Fermi evolution is a free fermion unitary characterized by a nontrivial winding number nnn as a map from the third homotopy group of the Brilliouin Zone 3-torus to that of SU(Nb)SU({N}_{b})SU(Nb​), where Nb{N}_{b}Nb​ is the number of bands. Using our simplified form of the QCA, we give higher-dimensional generalizations that we conjecture are also nontrivial QCAs, and we discuss the relation to Chern-Simons theory.
We construct a novel three-dimensional quantum cellular automaton (QCA) based on a system with short-range entangled bulk and chiral semion boundary topological order.We argue that either the QCA is nontrivial, i.e., not a finite-depth circuit of local quantum gates, or there exists a two-dimensional commuting projector Hamiltonian realizing the chiral semion topological order [characterized by U(1) 2 Chern-Simons theory].Our QCA is obtained by first constructing the Walker-Wang Hamiltonian of a certain premodular tensor category of order four, then condensing the deconfined bulk boson at the level of lattice operators.We show that the resulting Hamiltonian hosts chiral semion surface topological order in the presence of a boundary and can be realized as a non-Pauli stabilizer code on qubits, from which the QCA is defined.The construction is then generalized to a class of QCAs defined by non-Pauli stabilizer codes on 2 n -dimensional qudits that feature surface anyons described by U(1) 2 n Chern-Simons theory.Our results support the conjecture that the group of nontrivial three-dimensional QCAs is isomorphic to the Witt group of nondegenerate braided fusion categories.
Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the Z4\mathbb{Z}_4Z4​ toric code, we condense the boson e2m2e^2m^2e2m2 and identify the semion and anti-semion algebras. We then decompose each Z4\mathbb{Z}_4Z4​ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the zzz direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a U(1)4U(1)_4U(1)4​ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions d=4k−2d=4k{-}2d=4k−2, with one qubit or Z4\mathbb{Z}_4Z4​ qudit on each (2k−1)(2k{-}1)(2k−1)-cell for the semion or U(1)4U(1)_4U(1)4​ ISA, respectively.
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the Z8×Z2\mathbb{Z}_8\times\mathbb{Z}_2Z8​×Z2​ subgroup of the Witt group, including the U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ QCAs. The same algebraic structure yields new infinite families of generalized U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ non-Clifford QCAs in spatial dimensions d=4k−1d=4k-1d=4k−1. We also reformulate the 4-dimensional w2w3w_2w_3w2​w3​ QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families: w2nw3mw_2^nw_3^mw2n​w3m​ QCAs in dimension d=2n+3m−1d=2n+3m-1d=2n+3m−1 and w2w4k−1w_2w_{4k-1}w2​w4k−1​ QCAs in dimension d=4kd=4kd=4k. For the 5-dimensional w32w_3^2w32​ and w23w_2^3w23​ QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.
In three dimensions, there is a nontrivial quantum cellular automaton (QCA) which disentangles the three-fermion Walker--Wang model, a model whose action depends on Stiefel--Whitney classes of the spacetime manifold. Here we present a conjectured generalization to higher dimensions. For an arbitrary symmetry protected topological phase of time reversal whose action depends on Stiefel--Whitney classes, we construct a corresponding QCA that we conjecture disentangles that phase. Some of our QCA are Clifford, and we relate these to a classification theorem of Clifford QCA. We identify Clifford QCA in 4m+14m+14m+1 dimensions, for which we find a low-depth circuit description using non-Clifford gates but not with Clifford gates.
We prove that the three-dimensional chiral Zp\Z_pZp​ Clifford quantum cellular automata of Haah and of Sun--Yang--Wang--Tantivasadakarn--Chen are, for every odd prime ppp, not finite-depth quantum circuits, even after tensoring with arbitrary ancillary degrees of freedom and composing with translations. Previously such statements were known only inside the Clifford category. As a consequence, the group of three-dimensional QCA modulo stabilized circuits and translations is not finitely generated. The key ingredient is a new invariant of invertible subalgebras, which we conjecture to be the square of the total quantum dimension of the hosted anyon theory, modulo rational squares.
We give the definitive proof that the Dirac Quantum Cellular Automaton (QCA) used for both quantum simulation and algorithmic foundations of Quantum Field Theory (QFT), and especially of Quantum Electrodynamics (QED), as put forward in References https://doi.org/10.1007/s11128-019-2555-4 and https://doi.org/10.22331/q-2023-11-08-1179, does exhibit Fermion Doubling (FD), albeit thrice as less severe as discrete-time standard Lattice Gauge Theories (LGTs) [arXiv:2505.0790], which are naive regarding the spacetime discretization of differential operators acting on fermionic fields. The proof is done for the (1 + 1)D Dirac-QCA model. We show that the (one-time-step) two-point correlation function, also called Green's function (GF), of the Dirac QCA, is of astonishing simplicity, which is in contrast with the GF of the Dirac equation. We also compare, both qualitatively and quantitatively, this Dirac QCA to the continuous-time-LGT spatial discretization of Dirac fermions regarding how well these two lattice models approximate their naive continuum limit$\unicode{x2014}$which is nothing but the Dirac equation$\unicode{x2014}$even when far away from that limit, a situation which must be considered because of experimental limitations in quantum simulation: the Dirac QCA is better for ultrarelativistic regimes, whereas continuous-time LGT is better for non-relativistic regimes. Then, we compute the GF of the FD-fixed model put forward in the last cited reference, called Flavored Dirac QCA (FDQCA)$\unicode{x2014}$which staggers an extra, artificial flavor onlyonlyonly, on a diamond spacetime lattice, and does not stagger chirality as staggered fermions in usual LGT. The structure of this FDQCA two-point correlation function is of extreme simplicity, and can be expressed in a very simple manner in terms of the four chiral components of the FD-suffering, original-model GF.