Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter

Abstract

Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.

arXiv comment21 pages. v3: Minor revisions, accepted in Quantum. v2: Significant revision including new introduction with new figures, new Appendix B, new Section V D and expanded Discussion regarding role of different QCA in MBQC, and various smaller changes throughout

References in corpus (91)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Unpaired Majorana fermions in quantum wires
• Elementary gates for quantum computation
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Generalized Global Symmetries
• Measurement-based quantum computation with cluster states
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Multi-party entanglement in graph states
• Realizing Majorana zero modes in superconductor-semiconductor heterostructures
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Zoo of quantum-topological phases of matter
• Classifying quantum phases using Matrix Product States and PEPS
• Matrix product states represent ground states faithfully
• Local stabilizer codes in three dimensions without string logical operators
• Topological fault-tolerance in cluster state quantum computation
• Symmetry Protected Topological phases of Quantum Matter
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Entropy scaling and simulability by Matrix Product States
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• A fault-tolerant one-way quantum computer
• A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations
• Detection of Symmetry Protected Topological Phases in 1D
• Universality in Quantum Computation
• A symmetry principle for Topological Quantum Order
• A practical scheme for quantum computation with any two-qubit entangling gate
• PEPS as ground states: degeneracy and topology
• Exotic topological order in fractal spin liquids
• String order and symmetries in quantum spin lattices
• Novel schemes for measurement-based quantum computation
• Measurement-based quantum computation beyond the one-way model
• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Topological phases protected by point group symmetry
• Subsystem symmetry protected topological order
• Symmetry and Topological Order
• Building crystalline topological phases from lower-dimensional states
• Gauging spatial symmetries and the classification of topological crystalline phases
• Symmetry-protected phases for measurement-based quantum computation
• Fractal symmetries: Ungauging the cubic code
• Foliated fracton order from gauging subsystem symmetries
• Anyons and matrix product operator algebras
• Topological phases with generalized global symmetries
• Index theory of one dimensional quantum walks and cellular automata
• Quantum computation on the edge of a symmetry-protected topological order
• Matrix Product Unitaries: Structure, Symmetries, and Topological Invariants
• Identifying phases of quantum many-body systems that are universal for quantum computation
• Globally controlled quantum wires for perfect qubit transport, mirroring and computing
• Majorana Fermion Surface Code for Universal Quantum Computation
• The spectral gap for some spin chains with discrete symmetry breaking
• Fractal Symmetric Phases of Matter
• Architectures for quantum simulation showing a quantum speedup
• Classification of subsystem symmetry-protected topological phases
• A computationally universal phase of quantum matter
• Twisted Fracton Models in Three Dimensions
• Hierarchy of universal entanglement in 2D measurement-based quantum computation
• Computational Power of Symmetry-Protected Topological Phases
• Universal quantum computation using fractal symmetry-protected cluster phases
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Resource quality of a symmetry-protected topologically ordered phase for quantum computation
• Symmetry protected topological order at nonzero temperature
• Chiral topological spin liquids with projected entangled pair states
• Gapped boundaries, group cohomology and fault-tolerant logical gates
• Quantum computational renormalization in the Haldane phase
• Entanglement phases as holographic duals of anyon condensates
• Topologically protected braiding in a single wire using Floquet Majorana modes
• Anticoncentration theorems for schemes showing a quantum speedup
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Quantum computation via translation-invariant operations on a chain of qubits
• Symmetry-protected topological phases with uniform computational power in one dimension
• Symmetry protection of measurement-based quantum computation in ground states
• Fermionic projected entangled-pair states and topological phases
• Time Asymptotics and Entanglement Generation of Clifford Quantum Cellular Automata
• On the structure of Clifford quantum cellular automata
• A generalization of the injectivity condition for Projected Entangled Pair States
• Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states
• Quantum Information Processing with Delocalized Qubits under Global Control
• Qudit quantum computation on matrix product states with global symmetry
• Symmetry-protected topologically ordered states for universal quantum computation
• Normal projected entangled pair states generating the same state
• Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry
• Symmetry-Protected Topological Entanglement
• Universal measurement-based quantum computation in two-dimensional SPT phases
• Flow Ambiguity: A Path Towards Classically Driven Blind Quantum Computation
• Universal quantum computing using $({\mathbb{Z}}_d{)}^3$ symmetry-protected topologically ordered states
• Graph states as ground states of two-body frustration-free Hamiltonians
• Universal quantum computing with parafermions assisted by a half fluxon
• Composite symmetry protected topological order and effective models
• Projected Entangled Pair States with continuous virtual symmetries

Cited by in corpus (71)

• A review of Quantum Cellular Automata
• Identification of symmetry-protected topological states on noisy quantum computers
• Anomaly Inflow for Subsystem Symmetries
• Twisted foliated fracton phases
• Quantum Cellular Automata, Tensor Networks, and Area Laws
• Subsystem Non-Invertible Symmetry Operators and Defects
• Strong planar subsystem symmetry-protected topological phases and their dual fracton orders
• Symmetry-protected sign problem and magic in quantum phases of matter
• Gauging permutation symmetries as a route to non-Abelian fractons
• Bifurcating entanglement-renormalization group flows of fracton stabilizer models
• Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices
• Subsystem symmetry enriched topological order in three dimensions
• Searching for Fracton Orders via Symmetry Defect Condensation
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Classifying local fractal subsystem symmetry protected topological phases
• Crystalline Quantum Circuits
• Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual-unitary circuits
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Quantum computational advantage with string order parameters of 1D symmetry-protected topological order
• Detecting subsystem symmetry protected topological order via entanglement entropy
• Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power
• A converse to Lieb-Robinson bounds in one dimension using index theory
• Spontaneously broken supersymmetric fracton phases with fermionic subsystem symmetries
• Bifurcating subsystem symmetric entanglement renormalization in two dimensions
• Fractonic Higher-Order Topological Phases in Open Quantum Systems
• A comparative study of universal quantum computing models: towards a physical unification
• The role of cohomology in quantum computation with magic states
• Detecting Subsystem Symmetry Protected Topological Order Through Strange Correlators
• Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements
• Fractionalization of subsystem symmetries in two dimensions
• Efficient verification of Affleck-Kennedy-Lieb-Tasaki states
• Inaccessible entanglement in symmetry protected topological phases
• 3-Fermion topological quantum computation
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Beyond the Freshman's Dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models
• Fractalizing quantum codes
• Does causal dynamics imply local interactions?
• Some aspects of Affleck-Kennedy-Lieb-Tasaki models: tensor network, physical properties, spectral gap, deformation, and quantum computation
• Entanglement dynamics and phase transitions of the Floquet cluster spin chain
• Efficient Verification of Ground States of Frustration-Free Hamiltonians
• Fracton-like phases from subsystem symmetries
• Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi-Point Strange Correlators
• Fundamental charges for dual-unitary circuits
• Non-local finite-depth circuits for constructing SPT states and quantum cellular automata
• The Gauge Theory of Measurement-Based Quantum Computation
• Classification of measurement-based quantum wire in stabilizer PEPS
• Variational measurement-based quantum computation for generative modeling
• Nearly optimal time-independent reversal of a spin chain
• Triggering Boundary Phase Transitions through Bulk Measurements in 2D Cluster States
• Measurement-based quantum computation from Clifford quantum cellular automata
• Subsystem Symmetry Fractionalization and Foliated Field Theory
• Preparing matrix product states via fusion: constraints and extensions
• The spin-orbital Kitaev model: from kagome spin ice to classical fractons
• Phases of Matrix Product States with Symmetric Quantum Circuits and Symmetric Measurements with Feedforward
• Optimally generating $\mathfrak{su}(2^N)$ using Pauli strings
• Topological Dipole Insulator
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• Fractonic superfluids. III. Hybridizing higher moments
• Redundant string symmetry-based error correction: Demonstrations on quantum devices
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Counter-intuitive yet efficient regimes for measurement based quantum computation on symmetry protected spin chains
• Phase transitions in (2 + 1)D subsystem-symmetric monitored quantum circuits
• Infinite-component $BF$ field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Renormalisation of Quantum Cellular Automata
• Symmetry-Accelerated Classical Simulation of Clifford-Dominated Circuits
• Scaling of Computational Order Parameters in Rydberg Atom Graph States
• Sub-Riemannian geometry of measurement based quantum computation
• Entanglement dynamics and ergodicity breaking in a quantum cellular automaton
• Classification of qubit cellular automata on hypercubic lattices