Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeneralized surface codes and packing of logical qubits
75
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a notion of relative homology (and cohomology) for surfaces with two types of boundaries. Using this tool, we study a generalization of Kitaevs code based on surfaces with mixed boundaries. This construction includes both Bravyi and Kitaevs and Freedman and Meyers extension of Kitaevs toric code. We argue that our generalization offers a denser storage of quantum information. In a planar architecture, we obtain a three-fold overhead reduction over the standard architecture consisting of a punctured square lattice.
rate research
Read More
Quantum error-correcting codes are used to protect qubits involved in quantum computation. This process requires logical operators, acting on protected qubits, to be translated into physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in $mathbb{C}^{N times N}$ as a partial $2m times 2m$ binary symplectic matrix, where $N = 2^m$. We state and prove two theorems that use symplectic transvections to efficiently enumerate all binary symplectic matrices that satisfy a system of linear equations. As a corollary of these results, we prove that for an $[![ m,k ]!]$ stabilizer code every logical Clifford operator has $2^{r(r+1)/2}$ symplectic solutions, where $r = m-k$, up to stabilizer degeneracy. The desired physical circuits are then obtained by decomposing each solution into a product of elementary symplectic matrices, that correspond to elementary circuits. This enumeration of all physical realizations enables optimization over the ensemble with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. Our method of circuit synthesis can be applied to any stabilizer code, and this paper discusses a proof of concept synthesis for the $[![ 6,4,2 ]!]$ CSS code. Programs implementing the algorithms in this paper, which includes routines to solve for binary symplectic solutions of general linear systems and our overall LCS (logical circuit synthesis) algorithm, can be found at: https://github.com/nrenga/symplectic-arxiv18a
Surface codes are among the best candidates to ensure the fault-tolerance of a quantum computer. In order to avoid the accumulation of errors during a computation, it is crucial to have at our disposal a fast decoding algorithm to quickly identify and correct errors as soon as they occur. We propose a linear-time maximum likelihood decoder for surface codes over the quantum erasure channel. This decoding algorithm for dealing with qubit loss is optimal both in terms of performance and speed.
We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance $Omega(N^{2/3}/loglog N)$ which can improve Hastingss recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman-Zemor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quantum concatenated codes with parameters ${Q}=[[N,Omega(sqrt{N}),Omega( sqrt{N})]]$ and they also belong to the Bacon-Shor codes. We show that ${Q}$ can be encoded very efficiently by circuits of size $O(N)$ and depth $O(sqrt{N})$, and can correct any adversarial error of weight up to half the minimum distance bound in $O(sqrt{N})$ time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.
We present a quantum error correcting code with dynamically generated logical qubits. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, our measurement patterns generate logical qubits, allowing the code to act as a fault-tolerant quantum memory. Our particular code gives a model very similar to the two-dimensional toric code, but each measurement is a two-qubit Pauli measurement.
We derive one-shot upper bounds for quantum noisy channel codes. We do so by regarding a channel code as a bipartite operation with an encoder belonging to the sender and a decoder belonging to the receiver, and imposing constraints on the bipartite operation. We investigate the power of codes whose bipartite operation is non-signalling from Alice to Bob, positive-partial transpose (PPT) preserving, or both, and derive a simple semidefinite program for the achievable entanglement fidelity. Using the semidefinite program, we show that the non-signalling assisted quantum capacity for memoryless channels is equal to the entanglement-assisted capacity. We also relate our PPT-preserving codes and the PPT-preserving entanglement distillation protocols studied by Rains. Applying these results to a concrete example, the 3-dimensional Werner-Holevo channel, we find that codes that are non-signalling and PPT-preserving can be strictly less powerful than codes satisfying either one of the constraints, and therefore provide a tighter bound for unassisted codes. Furthermore, PPT-preserving non-signalling codes can send one qubit perfectly over two uses of the channel, which has no quantum capacity. We discuss whether this can be interpreted as a form of superactivation of quantum capacity.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
