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arXiv:2306.03171

Note on quantum cellular automata and strong equivalence

Carolyn Zhang

Published Jun 5, 2023• 0 citations
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Abstract

In this note, we present some results on the classification of quantum cellular automata (QCA) in 1D under strong equivalence rather than stable equivalence. Under strong equivalence, we only allow adding ancillas carrying the original on-site representation of the symmetry, while under stable equivalence, we allow adding ancillas carrying any representation of the symmetry. The former may be more realistic, because in physical systems especially in AMO/quantum computing contexts, we would not expect additional spins carrying arbitrary representations of the symmetry to be present. Ref.~\onlinecite{mpu} proposed two kinds of symmetry-protected indices (SPIs) for QCA with discrete symmetries under strong equivalence. In this note, we show that the more refined of these SPIs still only has a one-to-one correspondence to equivalence classes of $\mathbb{Z}_N$ symmetric QCA when $N$ is prime. We show a counter-example for $N=4$. We show that QCA with $\mathbb{Z}_2$ symmetry under strong equivalence, for a given on-site representation, are classified by $\mathbb{Z}^{pq}$ where $p$ is the number of prime factors of the on-site Hilbert space dimension and $q$ is the number of prime factors of the trace of the nontrivial on-site $\mathbb{Z}_2$ element. Finally, we show that the GNVW index has a formulation in terms of a $\mathbb{Z}_2$ SPI in a doubled system, and we provide a direct connection between the SPI formulation of the GNVW index and a second Renyi version of the mutual information formula for the GNVW index.

DOI
10.48550/arxiv.2306.03171

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In this note, we present some results on the classification of quantum cellular automata (QCA) in 1D under strong equivalence rather than stable equivalence. Under strong equivalence, we only allow adding ancillas carrying the original on-site representation of the symmetry, while under stable equivalence, we allow adding ancillas carrying any representation of the symmetry. The former may be more realistic, because in physical systems especially in AMO/quantum computing contexts, we would not expect additional spins carrying arbitrary representations of the symmetry to be present. Ref.~\onlinecite{mpu} proposed two kinds of symmetry-protected indices (SPIs) for QCA with discrete symmetries under strong equivalence. In this note, we show that the more refined of these SPIs still only has a one-to-one correspondence to equivalence classes of ZN\mathbb{Z}_NZN​ symmetric QCA when NNN is prime. We show a counter-example for N=4N=4N=4. We show that QCA with Z2\mathbb{Z}_2Z2​ symmetry under strong equivalence, for a given on-site representation, are classified by Zpq\mathbb{Z}^{pq}Zpq where ppp is the number of prime factors of the on-site Hilbert space dimension and qqq is the number of prime factors of the trace of the nontrivial on-site Z2\mathbb{Z}_2Z2​ element. Finally, we show that the GNVW index has a formulation in terms of a Z2\mathbb{Z}_2Z2​ SPI in a doubled system, and we provide a direct connection between the SPI formulation of the GNVW index and a second Renyi version of the mutual information formula for the GNVW index.
We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra – the symmetric subalgebra under a finite Abelian group symmetry G . For systems where each site carries a regular representation of G , we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a ( 2 + 1 ) d G gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of G -symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a Z 2 symmetric subalgebra, demonstrating that it maps to the e - m permutation in the two-dimensional toric code and has an irrational index of 2 . Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.
Over an arbitrary commutative ring RRR, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space Q(X)\mathbf{Q}(X)Q(X) of quantum cellular automata (QCA) on a given metric space XXX. In most cases of interest, π0Q(X)π_0 \mathbf{Q}(X)π0​Q(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences Q(∗)≃ΩnQ(Zn)\mathbf{Q}(*) \simeq Ω^n \mathbf{Q}(\mathbb{Z}^n)Q(∗)≃ΩnQ(Zn) for all nnn, which shows that the classification of QCA on Euclidean lattices is given by an ΩΩΩ-spectrum indexed by the dimension nnn. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya RRR-algebras, which may be of independent interest. We also include a section leading to the ΩΩΩ-spectrum for QCA over C∗C^*C∗-algebras with unitary circuits.
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.
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs. © 2020 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften. All rights reserved.
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have} been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs.
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs. © 2020 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften. All rights reserved.