30 research outputs found
Magic state distillation with low overhead
We propose a new family of error detecting stabilizer codes with an encoding
rate 1/3 that permit a transversal implementation of the pi/8-rotation on
all logical qubits. The new codes are used to construct protocols for
distilling high-quality `magic' states by Clifford group gates and Pauli
measurements. The distillation overhead has a poly-logarithmic scaling as a
function of the output accuracy, where the degree of the polynomial is
. To construct the desired family of codes, we introduce
the notion of a triorthogonal matrix --- a binary matrix in which any pair and
any triple of rows have even overlap. Any triorthogonal matrix gives rise to a
stabilizer code with a transversal -gate on all logical qubits, possibly
augmented by Clifford gates. A powerful numerical method for generating
triorthogonal matrices is proposed. Our techniques lead to a two-fold overhead
reduction for distilling magic states with output accuracy compared
with the best previously known protocol.Comment: 11 pages, 3 figure
Structure of 2D Topological Stabilizer Codes
We provide a detailed study of the general structure of two-dimensional
topological stabilizer quantum error correcting codes, including subsystem
codes. Under the sole assumption of translational invariance, we show that all
such codes can be understood in terms of the homology of string operators that
carry a certain topological charge. In the case of subspace codes, we prove
that two codes are equivalent under a suitable set of local transformations if
and only they have equivalent topological charges. Our approach emphasizes
local properties of the codes over global ones.Comment: 54 pages, 11 figures, version accepted in journal, improved
presentation and result
Disorder-assisted error correction in Majorana chains
It was recently realized that quenched disorder may enhance the reliability
of topological qubits by reducing the mobility of anyons at zero temperature.
Here we compute storage times with and without disorder for quantum chains with
unpaired Majorana fermions - the simplest toy model of a quantum memory.
Disorder takes the form of a random site-dependent chemical potential. The
corresponding one-particle problem is a one-dimensional Anderson model with
disorder in the hopping amplitudes. We focus on the zero-temperature storage of
a qubit encoded in the ground state of the Majorana chain. Storage and
retrieval are modeled by a unitary evolution under the memory Hamiltonian with
an unknown weak perturbation followed by an error-correction step. Assuming
dynamical localization of the one-particle problem, we show that the storage
time grows exponentially with the system size. We give supporting evidence for
the required localization property by estimating Lyapunov exponents of the
one-particle eigenfunctions. We also simulate the storage process for chains
with a few hundred sites. Our numerical results indicate that in the absence of
disorder, the storage time grows only as a logarithm of the system size. We
provide numerical evidence for the beneficial effect of disorder on storage
times and show that suitably chosen pseudorandom potentials can outperform
random ones.Comment: 50 pages, 7 figure
Quantum entanglement growth under random unitary dynamics
Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the “entanglement tsunami” in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar-Parisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1 = 3 and are spatially correlated over a distance ∝ ðtimeÞ 2 = 3 . We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a “minimal cut” picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the “velocity” of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder
Limits on the storage of Quantum information in a volume of space
We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits , the number of encoded qubits , the code distance , the accuracy parameter that quantifies how well the erasure channel can be reversed, and the locality parameter that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy that is exponentially small in , which is the case for perturbations of local commuting projector codes, our bound reads for codes on -dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed if . As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance and the size of a minimal region that can support all approximate logical operators satisfies , where the logical operators are accurate up to in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types