International Workshop on Quantum LDPC Codes

Homological product codes

Sergey Bravyi
IBM (United States)

July 16, 2014

PIRSA:14070013
Measuring the overhead of a quantum error correcting code

Austin Fowler
Institute for Quantum Computing (IQC)

July 16, 2014

PIRSA:14070014
Spatially coupled quantum LDPC codes

Jean-Pierre Tillich

July 16, 2014

PIRSA:14070015

Homological product codes

Sergey Bravyi
IBM (United States)

July 16, 2014

PIRSA:14070013

All examples of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good LDPC codes are known that combine constant encoding rate and linear distance. In this talk I will describe the first family of good quantum "almost LDPC" codes. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most square root of n qubits, where n is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. The proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good quantum codes with stabilizer weight n^a for any a>0. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.

This is a joint work with Matthew Hastings
Preprint: arXiv:1311.0885
Measuring the overhead of a quantum error correcting code

Austin Fowler
Institute for Quantum Computing (IQC)

July 16, 2014

PIRSA:14070014

If one's goal is large-scale quantum computation, ultimately one wishes to minimize the amount of time, number of qubits, and qubit connectivity required to outperform a classical system, all while assuming some physically reasonable gate error rate. We present two examples of such an overhead study, focusing on the surface code with and without long-range interactions.
Spatially coupled quantum LDPC codes

Jean-Pierre Tillich

July 16, 2014

PIRSA:14070015

Spatially coupled LDPC were introduced by Felström and Zigangirov in 1999. They might be viewed in the following way, take several several instances of a certain LDPC code family, arrange them in a row and then mix the edges of the codes randomly among neighboring layers. Moreover fix the bits of the first and last layers to zero. It has soon been found out that iterative decoding behaves much better for this code than for the original LDPC code. A breakthrough occurred when it was proved by Kudekar, Richardson and Urbanke that these codes attain the capacity of all binary input memoryless output-symmetric channels.

All these nice features of classical spatially coupled LDPC codes suggest to study whether they have a quantum analogue. The fact that spatially coupled LDPC codes may afford to have large degrees and still perform well under iterative decoding would be quite interesting in the quantum setting, since by the very nature of the quantum construction of stabilizer codes the rows of the parity-check matrix of the quantum code have to belong to the code which is decoded by the iterative decoder. This implies that we should have rather large row weights to avoid severe error-floor phenomena and/or oscillatory behavior of iterative decoding which degrades significantly its performance.

With Andriyanova and Maurice, I showed last year that it is possible to come up with coupled versions of quantum LDPC codes that perform excellently under iterative decoding. For instance we have constructed a spatially coupled LDPC code family of rate $\approx \frac{1}{4}$ which performs well under iterative decoding even for noise values close to the hashing bound $p \approx 0.127$.
This represents a tremendous improvement over all previous known families of quantum LDPC codes of the same rate.
I will discuss in this talk what can be expected from this approach when these spatially coupled LDPC codes are used for performing fault tolerant computation.

Title	Speaker(s)	Date	Talk Type	Info
Homological product codes	Sergey Bravyi	2014-07-16	Conference	View details
Measuring the overhead of a quantum error correcting code	Austin Fowler	2014-07-16	Conference	View details
Spatially coupled quantum LDPC codes	Jean-Pierre Tillich	2014-07-16	Conference	View details

International Workshop on Quantum LDPC Codes

Format results

Homological product codes

Measuring the overhead of a quantum error correcting code

Spatially coupled quantum LDPC codes

Homological product codes

Measuring the overhead of a quantum error correcting code

Spatially coupled quantum LDPC codes