58 research outputs found
Partial Syndrome Measurement for Hypergraph Product Codes
Hypergraph product codes are a promising avenue to achieving fault-tolerant
quantum computation with constant overhead. When embedding these and other
constant-rate qLDPC codes into 2D, a significant number of nonlocal connections
are required, posing difficulties for some quantum computing architectures. In
this work, we introduce a fault-tolerance scheme that aims to alleviate the
effects of implementing this nonlocality by measuring generators acting on
spatially distant qubits less frequently than those which do not. We
investigate the performance of a simplified version of this scheme, where the
measured generators are randomly selected. When applied to hypergraph product
codes and a modified small-set-flip decoding algorithm, we prove that for a
sufficiently high percentage of generators being measured, a threshold still
exists. We also find numerical evidence that the logical error rate is
exponentially suppressed even when a large constant fraction of generators are
not measured.Comment: 10 pages, 4 figure
Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute.
Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist?
We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of [[N,k,d,Δ]] approximate QLDPC codes that encode k = Ω(N) logical qubits into N physical qubits with distance d = Ω(N) and approximation infidelity Δ = 1/(N). The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in N projectors. We prove the existence of an efficient encoding map and show that the spectral gap of the code Hamiltonian scales as Ω(N^(â3.09)). We also show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth.
Our family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes (Nirkhe, et al., ICALP 2018) to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes (Brown and Fawzi, ISIT 2013). Then, in order to obtain a code with sparse checks and strong detection of local errors, we use a spacetime circuit-to-Hamiltonian construction in order to take advantage of the parallelism of the Brown-Fawzi circuits. Because of this, we call our codes spacetime codes.
The analysis of the spectral gap of the code Hamiltonian is the main technical contribution of this work. We show that for any depth D quantum circuit on n qubits there is an associated spacetime circuit-to-Hamiltonian construction with spectral gap Ω(n^(â3.09)Dâ»ÂČ logâ»â¶ (n)). To lower bound this gap we use a Markov chain decomposition method to divide the state space of partially completed circuit configurations into overlapping subsets corresponding to uniform circuit segments of depth logn, which are based on bitonic sorting circuits. We use the combinatorial properties of these circuit configurations to show rapid mixing between the subsets, and within the subsets we develop a novel isomorphism between the local update Markov chain on bitonic circuit configurations and the edge-flip Markov chain on equal-area dyadic tilings, whose mixing time was recently shown to be polynomial (Cannon, Levin, and Stauffer, RANDOM 2017). Previous lower bounds on the spectral gap of spacetime circuit Hamiltonians have all been based on a connection to exactly solvable quantum spin chains and applied only to 1+1 dimensional nearest-neighbor quantum circuits with at least linear depth
A Converse for Fault-tolerant Quantum Computation
With improvements in achievable redundancy for fault-tolerant quantum
computing, it is natural to ask: what is the minimum required redundancy? In
this paper, we obtain a lower bound on the minimum redundancy required for
-accurate implementation of a large class of operations, which
includes unitary operators. For the practically relevant case of
sub-exponential (in input size) depth and sub-linear gate size, our bound on
redundancy is tighter than the best known lower bound in \cite{FawziMS2022}. We
obtain this bound by connecting fault-tolerant computation with a set of finite
blocklength quantum communication problems whose accuracy requirements satisfy
a joint constraint. This bound gives a strictly lower noise threshold for
non-degradable noise and captures its dependence on gate size. This bound
directly extends to the case where noise at the outputs of a gate are
correlated but noise across gates are independent.Comment: 10 page
Stabilizer Inactivation for Message-Passing Decoding of Quantum LDPC Codes
We propose a post-processing method for message-passing (MP) decoding of CSS
quantum LDPC codes, called stabilizer-inactivation (SI). It relies on
inactivating a set of qubits, supporting a check in the dual code, and then
running the MP decoding again. This allows MP decoding to converge outside the
inactivated set of qubits, while the error on these is determined by solving a
small, constant size, linear system. Compared to the state of the art
post-processing method based on ordered statistics decoding (OSD), we show
through numerical simulations that MP-SI outperforms MP-OSD for different
quantum LDPC code constructions, different MP decoding algorithms, and
different MP scheduling strategies, while having a significantly reduced
complexity
Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
We study approximate quantum low-density parity-check (QLDPC) codes, which
are approximate quantum error-correcting codes specified as the ground space of
a frustration-free local Hamiltonian, whose terms do not necessarily commute.
Such codes generalize stabilizer QLDPC codes, which are exact quantum
error-correcting codes with sparse, low-weight stabilizer generators (i.e. each
stabilizer generator acts on a few qubits, and each qubit participates in a few
stabilizer generators). Our investigation is motivated by an important question
in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes
with constant rate, linear distance, and constant-weight stabilizers exist?
We show that obtaining such optimal scaling of parameters (modulo
polylogarithmic corrections) is possible if we go beyond stabilizer codes: we
prove the existence of a family of approximate QLDPC
codes that encode logical qubits into physical
qubits with distance and approximation infidelity
. The code space is
stabilized by a set of 10-local noncommuting projectors, with each physical
qubit only participating in projectors. We
prove the existence of an efficient encoding map, and we show that arbitrary
Pauli errors can be locally detected by circuits of polylogarithmic depth.
Finally, we show that the spectral gap of the code Hamiltonian is
by analyzing a spacetime circuit-to-Hamiltonian
construction for a bitonic sorting network architecture that is spatially local
in dimensions.Comment: 51 pages, 13 figure
Improved rate-distance trade-offs for quantum codes with restricted connectivity
For quantum error-correcting codes to be realizable, it is important that the
qubits subject to the code constraints exhibit some form of limited
connectivity. The works of Bravyi & Terhal (BT) and Bravyi, Poulin & Terhal
(BPT) established that geometric locality constrains code properties -- for
instance quantum codes defined by local checks on the
-dimensional lattice must obey . Baspin and Krishna
studied the more general question of how the connectivity graph associated with
a quantum code constrains the code parameters. These trade-offs apply to a
richer class of codes compared to the BPT and BT bounds, which only capture
geometrically-local codes. We extend and improve this work, establishing a
tighter dimension-distance trade-off as a function of the size of separators in
the connectivity graph. We also obtain a distance bound that covers all
stabilizer codes with a particular separation profile, rather than only LDPC
codes.Comment: 15 pages, 2 figure
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