Translationally Invariant Universal Classical Hamiltonians

Kohler, Tamara; Cubitt, Toby S.

doi:10.1007/s10955-019-02295-3

Cited by 9 publications

(9 citation statements)

References 44 publications

(155 reference statements)

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“…The original proof actually demonstrates that AME(2m, q) states are the purification of a threshold quantum secret sharing (QSS) scheme, however every pure QSS scheme is equivalent to a quantum MDS code [7] so the result follows immediately. 27 The proof of this theorem is a straightforward generalisation of [25,Theorem 2] which will give:…”

Section: B (Pseudo-)perfect Tensors and Quantum Error Correcting Codesmentioning

confidence: 99%

“…The existence of translationally invariant universal quantum Hamiltonians is an open question (in the classical case it has been shown that translationally invariant universal Hamiltonians do exist [27]). If translationally invariant universal quantum models were found it may be possible to construct a HQECC where the boundary Hamiltonian exhibits full translational invariance.…”

Section: G Translational Invariance In the Boundary Modelmentioning

confidence: 99%

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Toy models of holographic duality between local Hamiltonians

Kohler

Cubitt

2019

J. High Energ. Phys.

Self Cite

131

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Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between models, encompassing the relationship between bulk and boundary energy scales and time dynamics.It also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-tobulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically. ContentsF Perturbative simulations 55 G Translational invariance in the boundary model 59 2. The Hilbert space of the boundary consists of a triangulation of M by triangles of O(1) area, with a qubit at the centre of each triangle, and a total of O n(log n) 4 triangles/qubits. 3. Any local observable/measurement M in the bulk has a set of corresponding observables/measurements {M } on the boundary with the same outcome. A local bulk operator M can be reconstructed on a boundary region A if M acts within the greedy entanglement wedge of A, denoted E[A]. 3 4. H boundary consists of 2-local, nearest-neighbour interactions between the boundary qubits. Furthermore, H boundary can be chosen to have full local SU (2) symmetry; i.e. the local interactions can be chosen to all be Heisenberg interactions: H boundary5. H boundary is a (∆ L , , η)-simulation of H bulk in the rigorous sense of [9, Definition 23], with , η = 1/ poly(∆ L ), ∆ L = Ω H bulk 6 , and where the interaction strengths in H boundary scale as max ij |α ij | = O ∆ poly(n log(n)) L .This result allows us to extend toy models of holographic duality such as [5,21,34,49] to include a mapping between local Hamiltonians. In doing so we show that the expected relationship between bulk and boundary energy scales can be realised by local boundary models. In particular, in our construction toy models of static black holes (as originally proposed in [34]) correspond to high-energy states of the local boundary model, as would be expected in AdS/CFT. Moreover, in our toy model we can say something about how dynamics in the bulk correspond to dynamics on the boundary. Even without writing down a specific bulk Hamiltonian, we are able to demonstrate that the formation of a (toy model) static black hole in the bulk corresponds to the boundary unitarily evolving to a state outside of...

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Section: B (Pseudo-)perfect Tensors and Quantum Error Correcting Codesmentioning

confidence: 99%

Section: G Translational Invariance In the Boundary Modelmentioning

confidence: 99%

Toy models of holographic duality between local Hamiltonians

Kohler

Cubitt

2019

J. High Energ. Phys.

Self Cite

131

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“…This fact reflects the intrinsic computational power of near-neighbor interactions. In fact, Ising models have been shown to be universally complete [13,14], with a map between any given logic circuit to the ground states of some 2D Ising model Hamiltonian.…”

Section: Critical Phenomena and The 2d Ising Modelmentioning

confidence: 99%

The 2D Ising model, criticality and AIT

Ruffini¹,

Deco

2021

Preprint

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In this short note we study the 2D Ising model, a universal computational model which reflects phase transitions and critical phenomena, as a framework for establishing links between systems that exhibit criticality with the notions of complexity. This is motivated in the context of neuroscience applications stemming from algorithmic information theory (AIT). Starting with the original 2D Ising model, we show that — together with correlation length of the spin lattice, susceptibility to a uniform external field — the correlation time of the magnetization time series, the compression ratio of the spin lattice, the complexity of the magnetization time series — as derived from Lempel-Ziv-Welch compression—, and the rate of information transmission in the lattice, all reflect the effects of the phase transition, which results in spacetime pockets of uniform magnetization at all scales. We also show that in the Ising model the insertion of sparse long-range couplings has a direct effect on the critical temperature and other parameters. The addition of positive links extends the ordered regime to higher critical temperatures, while negative links have a stronger, disordering influence at the global scale. We discuss some implications for the study of long-range (e.g., ephaptic) interactions in the human brain and the effects of weak perturbations in neural dynamics.

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“…But this has sown the ground for many other results: An important one would be a rigorous comparison between universal spin models [16] and universal Turing machines. (Note that translationally invariant universal classical spin Hamiltonians have also been shown to exist [17]). For example, universal spin models allow for a full characterisation [16,Theorem 15], but universal Turing machines do not, due to Rice's Theoremthus, the two notions of universality do not seem to be equivalent.…”

Section: Discussionmentioning

confidence: 99%

Classical spin Hamiltonians are context-sensitive languages

Sebastian¹,

Drexel²,

Cuevas³

2020

Preprint

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We introduce a new complexity measure of classical spin Hamiltonians in which they are described as automata. Specifically, we associate a classical spin Hamiltonian to the formal language consisting of pairs of spin configurations and the corresponding energy, and classify this language in the Chomsky hierarchy. We prove that the language associated to local one-dimensional (1D) classical spin Hamiltonians is deterministic context-free, and the one associated to the two-dimensional (2D) case is context-sensitive. It follows that the Ising model without fields is easy / hard if defined on a 1D / 2D lattice, in contrast to the computational complexity of its ground state energy problem, which is easy / hard (namely in P / NP-complete) if defined on a planar / non-planar graph. We also prove that only highly non-physical spin Hamiltonians, namely totally unbounded ones, correspond to Turing machines. Our work puts classical spin Hamiltonians at the same level as automata, and paves the road toward a rigorous comparison of universal spin models and universal Turing machines.

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Translationally Invariant Universal Classical Hamiltonians

Cited by 9 publications

References 44 publications

Toy models of holographic duality between local Hamiltonians

Toy models of holographic duality between local Hamiltonians

The 2D Ising model, criticality and AIT

Classical spin Hamiltonians are context-sensitive languages

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