Quantum Error Correction from Complexity in Brownian SYK

• URL: http://arxiv.org/abs/2301.07108v1
• Date: Tue, 17 Jan 2023 19:00:00 GMT
• Title: Quantum Error Correction from Complexity in Brownian SYK
• Authors: Vijay Balasubramanian, Arjun Kar, Cathy Li, Onkar Parrikar, Harshit Rajgadia
• Abstract summary: robustness of error correction by a quantum code is upper bounded by the "mutual purity" of a certain entangled state. We show that when the encoding complexity is small, the mutual purity is $O(1)$ for the erasure of a small number of qubits. We find a hierarchy of complexity scales associated to a tower of subleading contributions to the mutual purity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the robustness of quantum error correction in a one-parameter ensemble of codes generated by the Brownian SYK model, where the parameter quantifies the encoding complexity. The robustness of error correction by a quantum code is upper bounded by the "mutual purity" of a certain entangled state between the code subspace and environment in the isometric extension of the error channel, where the mutual purity of a density matrix $\rho_{AB}$ is the difference $\mathcal{F}_\rho (A:B) \equiv \mathrm{Tr}\;\rho_{AB}^2 - \mathrm{Tr}\;\rho_A^2\;\mathrm{Tr}\;\rho_B^2$. We show that when the encoding complexity is small, the mutual purity is $O(1)$ for the erasure of a small number of qubits (i.e., the encoding is fragile). However, this quantity decays exponentially, becoming $O(1/N)$ for $O(\log N)$ encoding complexity. Further, at polynomial encoding complexity, the mutual purity saturates to a plateau of $O(e^{-N})$. We also find a hierarchy of complexity scales associated to a tower of subleading contributions to the mutual purity that quantitatively, but not qualitatively, adjust our error correction bound as encoding complexity increases. In the AdS/CFT context, our results suggest that any portion of the entanglement wedge of a general boundary subregion $A$ with sufficiently high encoding complexity is robustly protected against low-rank errors acting on $A$ with no prior access to the encoding map. From the bulk point of view, we expect such bulk degrees of freedom to be causally inaccessible from the region $A$ despite being encoded in it.

Related papers

• The closed-branch decoder for quantum LDPC codes [0.0]
  Real-time decoding is a necessity for implementing arbitrary quantum computations on the logical level. We present a new decoder for Quantum Low Density Parity Check (QLDPC) codes, named the closed-branch decoder.
  arXiv  Detail & Related papers  (2024-02-02T16:22:32Z)
• Exact results on finite size corrections for surface codes tailored to biased noise [0.0]
  We study the XY and XZZX surface codes under phase-biased noise. We show that independently estimating thresholds for the $X_L$ (phase-flip), $Y_L$, and $Z_L$ (bit-flip) logical failure rates can give a more confident threshold estimate.
  arXiv  Detail & Related papers  (2024-01-08T16:38:56Z)
• Unitarity estimation for quantum channels [7.323367190336826]
  Unitarity estimation is a basic and important problem in quantum device certification and benchmarking. We provide a unified framework for unitarity estimation, which induces ancilla-efficient algorithms. We show that both the $d$-dependence and $epsilon$-dependence of our algorithms are optimal.
  arXiv  Detail & Related papers  (2022-12-19T09:36:33Z)
• Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder [2.867517731896504]
  We numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with $d=mathcalO(n)$-depth random encoding circuits, can be attained even when the circuit depth is restricted to $d=mathcalO(log n)$ in 1D.
  arXiv  Detail & Related papers  (2022-12-09T19:00:00Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• A lower bound on the space overhead of fault-tolerant quantum computation [51.723084600243716]
  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation. We prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude noise.
  arXiv  Detail & Related papers  (2022-01-31T22:19:49Z)
• Random quantum circuits transform local noise into global white noise [118.18170052022323]
  We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. For local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_textnoisy$ of a generic noisy circuit instance shrink exponentially. If the noise is incoherent, the output distribution approaches the uniform distribution $p_textunif$ at precisely the same rate.
  arXiv  Detail & Related papers  (2021-11-29T19:26:28Z)
• Tight Bound for Estimating Expectation Values from a System of Linear Equations [0.0]
  The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$. The task is to estimate $xdagger Mx$, where $x$ is the solution vector to the equation $Ax = b$. We show that the quantum query complexity for SLEP in this setting is $Theta(alpha/epsilon)$.
  arXiv  Detail & Related papers  (2021-11-20T00:10:51Z)
• Finding the disjointness of stabilizer codes is NP-complete [77.34726150561087]
  We show that the problem of calculating the $c-disjointness, or even approximating it to within a constant multiplicative factor, is NP-complete. We provide bounds on the disjointness for various code families, including the CSS codes,$d codes and hypergraph codes. Our results indicate that finding fault-tolerant logical gates for generic quantum error-correcting codes is a computationally challenging task.
  arXiv  Detail & Related papers  (2021-08-10T15:00:20Z)
• Hybrid Stochastic-Deterministic Minibatch Proximal Gradient: Less-Than-Single-Pass Optimization with Nearly Optimal Generalization [83.80460802169999]
  We show that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model. For loss factors, we prove that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model.
  arXiv  Detail & Related papers  (2020-09-18T02:18:44Z)
• Robustly Learning any Clusterable Mixture of Gaussians [55.41573600814391]
  We study the efficient learnability of high-dimensional Gaussian mixtures in the adversarial-robust setting. We provide an algorithm that learns the components of an $epsilon$-corrupted $k$-mixture within information theoretically near-optimal error proofs of $tildeO(epsilon)$. Our main technical contribution is a new robust identifiability proof clusters from a Gaussian mixture, which can be captured by the constant-degree Sum of Squares proof system.
  arXiv  Detail & Related papers  (2020-05-13T16:44:12Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.