56 research outputs found
Homological Product Codes
Quantum codes with low-weight stabilizers known as LDPC codes have been
actively studied recently due to their simple syndrome readout circuits and
potential applications in fault-tolerant quantum computing. However, all
families 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 families of LDPC codes are known
that combine constant encoding rate and linear distance. Here we propose the
first family of good quantum codes with low-weight stabilizers. The new codes
have a constant encoding rate, linear distance, and stabilizers acting on at
most qubits, where is the code length. For comparison, all
previously known families of good quantum codes have stabilizers of linear
weight. Our 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 stabilizer codes with
stabilizer weight for any . Finally, we apply the homological
product to construct new small codes with low-weight stabilizers.Comment: 49 page
Fault-tolerant gates via homological product codes
A method for the implementation of a universal set of fault-tolerant logical
gates is presented using homological product codes. In particular, it is shown
that one can fault-tolerantly map between different encoded representations of
a given logical state, enabling the application of different classes of
transversal gates belonging to the underlying quantum codes. This allows for
the circumvention of no-go results pertaining to universal sets of transversal
gates and provides a general scheme for fault-tolerant computation while
keeping the stabilizer generators of the code sparse.Comment: 11 pages, 3 figures. v2 (published version): quantumarticle
documentclass, expanded discussion on the conditions for a fault tolerance
threshol
Single-shot error correction of three-dimensional homological product codes
Single-shot error correction corrects data noise using only a single round of noisy measurements on the data qubits, removing the need for intensive measurement repetition. We introduce a general concept of confinement for quantum codes, which roughly stipulates qubit errors cannot grow without triggering more measurement syndromes. We prove confinement is sufficient for single-shot decoding of adversarial errors and linear confinement is sufficient for single-shot decoding of local stochastic errors. Further to this, we prove that all three-dimensional homological product codes exhibit confinement in their X components and are therefore single shot for adversarial phase-flip noise. For local stochastic phase-flip noise, we numerically explore these codes and again find evidence of single-shot protection. Our Monte Carlo simulations indicate sustainable thresholds of 3.08(4)% and 2.90(2)% for three-dimensional (3D) surface and toric codes, respectively, the highest observed single-shot thresholds to date. To demonstrate single-shot error correction beyond the class of topological codes, we also run simulations on a randomly constructed family of 3D homological product codes
Single-shot error correction of three-dimensional homological product codes
Single-shot error correction corrects data noise using only a single round of noisy measurements on the data qubits, removing the need for intensive measurement repetition. We introduce a general concept of confinement for quantum codes, which roughly stipulates qubit errors cannot grow without triggering more measurement syndromes. We prove confinement is sufficient for single-shot decoding of adversarial errors and linear confinement is sufficient for single-shot decoding of local stochastic errors. Further to this, we prove that all three-dimensional homological product codes exhibit confinement in their X components and are therefore single shot for adversarial phase-flip noise. For local stochastic phase-flip noise, we numerically explore these codes and again find evidence of single-shot protection. Our Monte Carlo simulations indicate sustainable thresholds of 3.08(4)% and 2.90(2)% for three-dimensional (3D) surface and toric codes, respectively, the highest observed single-shot thresholds to date. To demonstrate single-shot error correction beyond the class of topological codes, we also run simulations on a randomly constructed family of 3D homological product codes
A theory of single-shot error correction for adversarial noise
Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code's measurement checks. Good code soundness in topological (or low-density parity check, LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking
Numerical Techniques for Finding the Distances of Quantum Codes
We survey the existing techniques for calculating code distances of classical
codes and apply these techniques to generic quantum codes. For classical and
quantum LDPC codes, we also present a new linked-cluster technique. It reduces
complexity exponent of all existing deterministic techniques designed for codes
with small relative distances (which include all known families of quantum LDPC
codes), and also surpasses the probabilistic technique for sufficiently high
code rates.Comment: 5 pages, 1 figure, to appear in Proceedings of ISIT 2014 - IEEE
International Symposium on Information Theory, Honolul
A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)
We propose a family of local CSS stabilizer codes as possible candidates for
self-correcting quantum memories in 3D. The construction is inspired by the
classical Ising model on a Sierpinski carpet fractal, which acts as a classical
self-correcting memory. Our models are naturally defined on fractal subsets of
a 4D hypercubic lattice with Hausdorff dimension less than 3. Though this does
not imply that these models can be realised with local interactions in 3D
Euclidean space, we also discuss this possibility. The X and Z sectors of the
code are dual to one another, and we show that there exists a finite
temperature phase transition associated with each of these sectors, providing
evidence that the system may robustly store quantum information at finite
temperature.Comment: 16 pages, 6 figures. In v2, erroneous argument about embeddability
into R3 was removed. In v3, minor changes to match journal versio
- …