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

Cannon, Sarah; Levin, David A.; Stauffer, Alexandre

doi:10.1017/s0963548318000470

Cited by 4 publications

(8 citation statements)

References 27 publications

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“…It remains to lower bound the spectral gaps ∆ P i corresponding to the restricted ("within-block") chains defined in (4.10). By our careful choice of the block decomposition, we demonstrate in Section A.4.2 a one-to-one correspondence between the time configurations in any block Ω i and the equal area dyadic tilings of a unit square, and crucially this correspondence also exactly maps the edge-flip Markov chain moves considered in [CLS17] to the updates which describe the application of a local gate to a valid time configuration. Since the relaxation time of the edge-flip Markov chain is O(n 4.09 ) we have ∆ P i = Ω(n −4.09 ) and so ∆ P = Ω(n −4.09 ), (4.20)…”

Section: Decomposition Of the Circuit Propagation Markov Chainmentioning

confidence: 88%

“…The numbers a count the number of valid partial circuit configurations of B , but they also happen to enumerate a different combinatorial structure: the number of dyadic tilings of the unit square of rank [CLS17]. To facilitate the analysis of the gap of our code Hamiltonian, we will describe an explicit isomorphism between the two sets and the Markov chains defined on them.…”

Section: A41 Dyadic Tilingsmentioning

confidence: 99%

“…Definition A.20 (Edge-flip Markov chain [JRS02,CLS17]). The edge-flip Markov Chain, M on state space T is defined with the following transition rule.…”

Section: A41 Dyadic Tilingsmentioning

confidence: 99%

“…([CLS17]). A flippable edge of a tiling t ∈ T is a c-segment which, when considered alone, forms an entire edge of both dyadic rectangles it borders.…”

mentioning

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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Section: Decomposition Of the Circuit Propagation Markov Chainmentioning

confidence: 88%

Section: A41 Dyadic Tilingsmentioning

confidence: 99%

“…Definition A.20 (Edge-flip Markov chain [JRS02,CLS17]). The edge-flip Markov Chain, M on state space T is defined with the following transition rule.…”

Section: A41 Dyadic Tilingsmentioning

confidence: 99%

“…([CLS17]). A flippable edge of a tiling t ∈ T is a c-segment which, when considered alone, forms an entire edge of both dyadic rectangles it borders.…”

mentioning

confidence: 99%

See 2 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

View full text Add to dashboard Cite

show abstract

“…On the other hand, for many natural chains on Catalan structures, triangulations and related objects not even a polynomial upper bound for the mixing time is known. One such example is that of lattice triangulations, where polynomial bounds are only known for biased versions of the chain [8,9,26]; see also works on rectangular dissections, for which polynomial bounds were obtained very recently [7,6].…”

Section: Introductionmentioning

confidence: 99%

Polynomial mixing time of edge flips on quadrangulations

Caraceni

Stauffer

2019

Probab. Theory Relat. Fields

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We establish the first polynomial upper bound for the mixing time of random edge flips on rooted quadrangulations: we show that the spectral gap of the edge flip Markov chain on quadrangulations with n faces admits, up to constants, an upper bound of n −5/4 and a lower bound of n −11/2 . In order to obtain the lower bound, we also consider a very natural Markov chain on plane trees -or, equivalently, on Dyck paths -and improve the previous lower bound for its spectral gap by Shor and Movassagh.

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A polynomial upper bound for the mixing time of edge rotations on planar maps

Caraceni¹

2020

Electron. J. Probab.

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We consider a natural local dynamic on the set of all rooted planar maps with n edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the n-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this "edge rotation" chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times n −11/2 . In doing so, we provide a partially new proof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulations as defined in [8], which makes it possible to generalise the result of [8] to a variant of the edge flip chain related to edge rotations via Tutte's bijection.

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

Cited by 4 publications

References 27 publications

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Polynomial mixing time of edge flips on quadrangulations

A polynomial upper bound for the mixing time of edge rotations on planar maps

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