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arXiv:2202.05442 · PRX Quantum

Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond

Wilbur E. Shirley, Yu-An Chen, Arpit Dua, Tyler D. Ellison, Nathanan Tantivasadakarn, Dominic J. Williamson

Published Aug 23, 2022• 44 citations
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Abstract

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.

DOI
10.1103/prxquantum.3.030326

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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 $4m+1$ dimensions, for which we find a low-depth circuit description using non-Clifford gates but not with Clifford gates.

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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.
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a Zp\mathbb{Z}_pZp​ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality SSS, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT TTT. In the continuum, they form a central extension of PSL(2,Z4)PSL(2,\mathbb{Z}_4)PSL(2,Z4​) for p=2p=2p=2, and of SL(2,Zp)SL(2,\mathbb{Z}_p)SL(2,Zp​) for odd primes ppp, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For p=2p=2p=2 the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a Zp\mathbb{Z}_pZp​ 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For p=2p=2p=2 they are generated by the non-trivial semion QCA, and for odd prime ppp they are generated by the non-trivial Zp\mathbb{Z}_pZp​ Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.
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.
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.
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.