Self-Organized Error Correction in Random Unitary Circuits with Measurement

Ruihua Fan; Sagar Vijay; Ashvin Vishwanath; Yi-Zhuang You

doi:10.1103/PhysRevB.103.174309

paper

Self-Organized Error Correction in Random Unitary Circuits with Measurement

arXiv:2002.12385 · doi:10.1103/PhysRevB.103.174309

Abstract

Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a sub-thermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. Here we quantify these notions by identifying a universal, subleading logarithmic contribution to the volume law entanglement entropy: $S^{(2)} (A) = κ L_{A} + \frac{3}{2} lo g L_{A}$ which bounds the mutual information between a qudit inside region $A$ and the rest of the system. Specifically, we find the power law decay of the mutual information $I ({x} : A ˉ) \propto x^{- 3/2}$ with distance $x$ from the region's boundary, which implies that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$ . We obtain these results by mapping the entanglement dynamics to the imaginary time evolution of an Ising model, to which we can apply field-theoretic and matrix-product-state techniques. Finally, exploiting the error-correction viewpoint, we assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code to obtain a bound on the critical measurement strength $p_{c}$ as a function of the qudit dimension $d$ : $p_{c} lo g [(d^{2} - 1) (p_{c}^{- 1} - 1)] \leq lo g [(1 - p_{c}) d]$ . The bound is saturated at $p_{c} (d \to \infty) = 1/2$ and provides a reasonable estimate for the qubit transition: $p_{c} (d = 2) \leq 0.1893$ .

26 pages, 10 figures

Self-Organized Error Correction in Random Unitary Circuits with Measurement

Abstract

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