scholarly journals Dynamically Generated Logical Qubits

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 564
Author(s):  
Matthew B. Hastings ◽  
Jeongwan Haah
Keyword(s):  
Fault Tolerant ◽  
Error Correcting Code ◽  
Quantum Memory ◽  
Two Dimensional ◽  
Logical Qubits ◽  
Quantum Error ◽  
Toric Code

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.

scholarly journals Exponential suppression of bit or phase errors with cyclic error correction

Nature ◽  
2021 ◽  
Vol 595 (7867) ◽  
pp. 383-387
Author(s):  
◽  
Zijun Chen ◽  
Kevin J. Satzinger ◽  
Juan Atalaya ◽  
Alexander N. Korotkov ◽  
...  
Keyword(s):  
Error Correction ◽  
Error Detection ◽  
Fault Tolerant ◽  
Error Rates ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Two Dimensional ◽  
Logical Error ◽  
Quantum Error ◽  
Logical Qubit

AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

scholarly journals Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code

2012 ◽  
Vol 12 (11&12) ◽  
pp. 1034-1080
Author(s):  
Adam Paetznick ◽  
Ben W. Reichardt
Keyword(s):  
Error Propagation ◽  
Fault Tolerant ◽  
Error Correcting Code ◽  
Block Size ◽  
Golay Code ◽  
Important Target ◽  
Noise Threshold ◽  
The Cost ◽  
Quantum Error ◽  
Logical Qubit

In fault-tolerant quantum computing schemes, the overhead is often dominated by the cost of preparing codewords reliably. This cost generally increases quadratically with the block size of the underlying quantum error-correcting code. In consequence, large codes that are otherwise very efficient have found limited fault-tolerance applications. Fault-tolerant preparation circuits therefore are an important target for optimization. We study the Golay code, a $23$-qubit quantum error-correcting code that protects the logical qubit to a distance of seven. In simulations, even using a na{\"i}ve ancilla preparation procedure, the Golay code is competitive with other codes both in terms of overhead and the tolerable noise threshold. We provide two simplified circuits for fault-tolerant preparation of Golay code-encoded ancillas. The new circuits minimize error propagation, reducing the overhead by roughly a factor of four compared to standard encoding circuits. By adapting the malignant set counting technique to depolarizing noise, we further prove a threshold above $\threshOverlap$ noise per gate.

scholarly journals Magic state distillation with the ternary Golay code

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200187 ◽  
Author(s):  
Shiroman Prakash
Keyword(s):  
Fourier Transform ◽  
Quantum Computing ◽  
Fault Tolerant ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
High Threshold ◽  
Golay Code ◽  
Quadratic Error ◽  
Quantum Error ◽  
Error Suppression

The ternary Golay code—one of the first and most beautiful classical error-correcting codes discovered—naturally gives rise to an 11-qutrit quantum error correcting code. We apply this code to magic state distillation, a leading approach to fault-tolerant quantum computing. We find that the 11-qutrit Golay code can distil the ‘most magic’ qutrit state—an eigenstate of the qutrit Fourier transform known as the strange state —with cubic error suppression and a remarkably high threshold. It also distils the ‘second-most magic’ qutrit state, the Norell state, with quadratic error suppression and an equally high threshold to depolarizing noise.

scholarly journals Deterministic generation of a two-dimensional cluster state

Science ◽  
2019 ◽  
Vol 366 (6463) ◽  
pp. 369-372 ◽  
Author(s):  
Mikkel V. Larsen ◽  
Xueshi Guo ◽  
Casper R. Breum ◽  
Jonas S. Neergaard-Nielsen ◽  
Ulrik L. Andersen
Keyword(s):  
Quantum Computation ◽  
Quantum Information Processing ◽  
Fault Tolerant ◽  
Cluster State ◽  
Two Dimensional ◽  
Cluster States ◽  
Squeezed Light ◽  
Universal Quantum ◽  
Speed Up ◽  
Quantum Error

Measurement-based quantum computation offers exponential computational speed-up through simple measurements on a large entangled cluster state. We propose and demonstrate a scalable scheme for the generation of photonic cluster states suitable for universal measurement-based quantum computation. We exploit temporal multiplexing of squeezed light modes, delay loops, and beam-splitter transformations to deterministically generate a cylindrical cluster state with a two-dimensional (2D) topological structure as required for universal quantum information processing. The generated state consists of more than 30,000 entangled modes arranged in a cylindrical lattice with 24 modes on the circumference, defining the input register, and a length of 1250 modes, defining the computation depth. Our demonstrated source of two-dimensional cluster states can be combined with quantum error correction to enable fault-tolerant quantum computation.

scholarly journals A Holographic Quantum Code

2019 ◽  
Vol 20 (2) ◽  
Author(s):  
Grant Elliot
Keyword(s):  
Error Correction ◽  
Error Correcting Code ◽  
Quantum Error Correction ◽  
Quantum Code ◽  
Tensor Networks ◽  
Logical Qubits ◽  
Quantum Error ◽  
The Way

Abstract: It was shown by [2] how bulk operators in the AdS/CFT correspondence can be represented on the boundary analogously to the way logical qubits are represented in an encoded subspace in quantum error correction. Then in [1]  holographic tensor networks that serve as toy models of the bulk boundary. This paper reviews some of the developments of [1] and [2]. Then it is demonstrated explicitly how to construct perfect tensors, which are essential to the tensor networks mentioned in [2]. Lastly a new example of a holographic quantum error-correcting code based on an eight index perfect tensor is presented.

scholarly journals Fault-Tolerant quantum computation with constant overhead

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1339-1371
Author(s):  
Daniel Gottesman
Keyword(s):  
Fault Tolerant ◽  
Quantum Circuit ◽  
Error Correcting Codes ◽  
Parity Check ◽  
Low Density Parity Check ◽  
The Family ◽  
Minimum Number ◽  
Logical Qubits ◽  
Quantum Error ◽  
Parity Check Codes

What is the minimum number of extra qubits needed to perform a large fault-tolerant quantum circuit? Working in a common model of fault-tolerance, I show that in the asymptotic limit of large circuits, the ratio of physical qubits to logical qubits can be a constant. The construction makes use of quantum low-density parity check codes, and the asymptotic overhead of the protocol is equal to that of the family of quantum error-correcting codes underlying the fault-tolerant protocol.

scholarly journals A fault-tolerant non-Clifford gate for the surface code in two dimensions

2020 ◽  
Vol 6 (21) ◽  
pp. eaay4929 ◽  
Author(s):  
Benjamin J. Brown
Keyword(s):  
Fault Tolerant ◽  
Error Correcting Code ◽  
Three Dimensional ◽  
Logic Gates ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Decoding Algorithms ◽  
Future Technology ◽  
Surface Code ◽  
Three Dimensional Models

Fault-tolerant logic gates will consume a large proportion of the resources of a two-dimensional quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum error-correcting code now under intensive development. This alleviates the need for distillation or higher-dimensional components to complete a universal gate set. The operation uses both local transversal gates and code deformations over a time that scales with the size of the qubit array. An important component of the gate is a just-in-time decoder. These decoding algorithms allow us to draw upon the advantages of three-dimensional models using only a two-dimensional array of live qubits. Our gate is completed using parity checks of weight no greater than four. We therefore expect it to be amenable with near-future technology. As the gate circumvents the need for magic-state distillation, it may reduce the resource overhead of surface-code quantum computation considerably.

scholarly journals Optimal local unitary encoding circuits for the surface code

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 517
Author(s):  
Oscar Higgott ◽  
Matthew Wilson ◽  
James Hefford ◽  
James Dborin ◽  
Farhan Hanif ◽  
...  
Keyword(s):  
Error Correcting Code ◽  
High Threshold ◽  
Planar Surface ◽  
Compact Mapping ◽  
Lattice Size ◽  
Unknown State ◽  
Non Local ◽  
Surface Code ◽  
Quantum Error ◽  
Toric Code

The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size L, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has O(L2) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(log⁡L)-depth non-local renormalisation encoder. We relate these techniques by providing an O(L) local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.

scholarly journals Hypermap-homology quantum codes

2014 ◽  
Vol 12 (01) ◽  
pp. 1430001 ◽  
Author(s):  
Martin Leslie
Keyword(s):  
Quantum Information ◽  
Error Correcting Code ◽  
Chain Complex ◽  
Error Correcting Codes ◽  
Quantum Computers ◽  
Quantum Codes ◽  
Straightforward Generalization ◽  
New Type ◽  
Quantum Error ◽  
Toric Code

We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m2, 2, m] code as compared to the toric code which is a [2m2, 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

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