Random dyadic tilings of the unit square

Janson, Svante; Randall, Dana; Spencer, Joel

doi:10.1002/rsa.10051

Cited by 9 publications

(21 citation statements)

References 14 publications

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“…Theorem A.23 ([JRS02]). There is an isomorphism between T , the set of dyadic tilings of rank , and the set of HV-trees of depth .…”

Section: A42 Hv-treesmentioning

confidence: 94%

“…(1.5) Turning to the analysis of the restricted chains P i , we present the discovery of a surprising and beautiful connection between valid time configurations of architectures of the form shown in Figure 2 with combinatorial structures known as dyadic tilings [JRS02]. Dyadic tilings are tilings of the unit square by equal-area dyadic rectangles, which are rectangles of the form…”

Section: Proof Sketch For the Spectral Gap Analysismentioning

confidence: 99%

“…There is a natural Markov chain on T called the edge-flip Markov chain [JRS02]. Given a dyadic tiling, there is a distinguished set of edges in the tiling which can be removed and replaced by their perpendicular bisector to obtain another valid dyadic tiling of the same size.…”

Section: A41 Dyadic Tilingsmentioning

confidence: 99%

“…In order to give a more complete understanding of the isomorphism between valid partial configurations of B and T , we introduce an alternate representation of dyadic tilings called HV-trees [JRS02].…”

Section: A42 Hv-treesmentioning

confidence: 99%

“…Janson, Randall, and Spencer showed that the recursive description of a tiling gives an easy isomorphism to a graph called a HV-tree [JRS02]. Consider a complete binary tree of depth where each vertex is labeled either H or V. There is a clear mapping from such trees to dyadic tilings: starting with the unit square and the root of the tree, draw a horizontal (H-cut) bisector or vertical (V-cut) bisector depending on the label of the root.…”

Section: A42 Hv-treesmentioning

confidence: 99%

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Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Bohdanowicz

Crosson

Nirkhe

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute. Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist?We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of [[N, k, d, ε]] approximate QLDPC codes that encode k = Ω(N) logical qubits into N physical qubits with distance d = Ω(N) and approximation infidelity ε = O(1/ polylog(N)). The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in O(polylog N) projectors. We prove the existence of an efficient encoding map and show that the spectral gap of the code Hamiltonian scales as Ω(N −3.09 ). We also show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth.Our family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes (Nirkhe, et al., ICALP 2018) to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes (Brown and Fawzi, ISIT 2013). Then, in order to obtain a code with sparse checks and strong detection of local errors, we use a spacetime circuit Hamiltonian construction in order to take advantage of the parallelism of the Brown-Fawzi circuits.The analysis of the spectral gap of the code Hamiltonian is the main technical contribution of this work. We show that for any depth D quantum circuit on n qubits there is an associated spacetime circuit-to-Hamiltonian construction with spectral gap Ω(n −3.09 D −2 log −6 (n)). To lower bound this gap we use a Markov chain decomposition method to divide the state space of partially completed circuit configurations into overlapping subsets corresponding to uniform circuit segments of depth log n, which are based on bitonic sorting circuits. We use the combinatorial properties of these circuit configurations to show rapid mixing between the subsets, and within the subsets we develop a novel isomorphism between the local update Markov chain on bitonic circuit configurations and the edge-flip Markov chain on equal-area dyadic tilings, whose mixing time was recently shown to be polynomial (Cannon, Levin, and Stauffer, RANDOM 2017). Previous lower bounds on the spectral gap of spacetime circuit Hamiltonians have all been based on a connection to exactly solvable quantum spin chains and applied only to ...

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“…Theorem A.23 ([JRS02]). There is an isomorphism between T , the set of dyadic tilings of rank , and the set of HV-trees of depth .…”

Section: A42 Hv-treesmentioning

confidence: 94%

Section: Proof Sketch For the Spectral Gap Analysismentioning

confidence: 99%

Section: A41 Dyadic Tilingsmentioning

confidence: 99%

Section: A42 Hv-treesmentioning

confidence: 99%

Section: A42 Hv-treesmentioning

confidence: 99%

See 3 more Smart Citations

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Bohdanowicz

Crosson

Nirkhe

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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show abstract

Efficiency test of pseudorandom number generators using random walks

Kang

2005

Journal of Computational and Applied Mathematics

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Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

Cannon

Levin

Stauffer

2018

Combinator. Probab. Comp.

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We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002 [10]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the formfor a, b, s, t ∈ Z ≥0 . The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n 4.09 ), which implies that the mixing time is at most O(n 5.09 ). We complement this by showing that the relaxation time is at least Ω(n 1.38 ), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

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Random dyadic tilings of the unit square

Cited by 9 publications

References 14 publications

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Efficiency test of pseudorandom number generators using random walks

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

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