13 research outputs found
Sparse Blossom: correcting a million errors per core second with minimum-weight matching
In this work, we introduce a fast implementation of the minimum-weight
perfect matching (MWPM) decoder, the most widely used decoder for several
important families of quantum error correcting codes, including surface codes.
Our algorithm, which we call sparse blossom, is a variant of the blossom
algorithm which directly solves the decoding problem relevant to quantum error
correction. Sparse blossom avoids the need for all-to-all Dijkstra searches,
common amongst MWPM decoder implementations. For 0.1% circuit-level
depolarising noise, sparse blossom processes syndrome data in both and
bases of distance-17 surface code circuits in less than one microsecond per
round of syndrome extraction on a single core, which matches the rate at which
syndrome data is generated by superconducting quantum computers. Our
implementation is open-source, and has been released in version 2 of the
PyMatching library.Comment: 34 pages, 14 figure
Fragile boundaries of tailored surface codes
Biased noise is common in physical qubits, and tailoring a quantum code to
the bias by locally modifying stabilizers or changing boundary conditions has
been shown to greatly increase error correction thresholds. In this work, we
explore the challenges of using a specific tailored code, the XY surface code,
for fault-tolerant quantum computation. We introduce efficient and
fault-tolerant decoders, belief-matching and belief-find, which exploit
correlated hyperedge fault mechanisms present in circuit-level noise. Using
belief-matching, we find that the XY surface code has a higher threshold and
lower overhead than the square CSS surface code for moderately biased noise.
However, the rectangular CSS surface code has a lower qubit overhead than the
XY surface code when below threshold. We identify a contributor to the reduced
performance that we call fragile boundary errors. These are string-like errors
that can occur along spatial or temporal boundaries in planar architectures or
during logical state preparation and measurement. While we make partial
progress towards mitigating these errors by deforming the boundaries of the XY
surface code, our work suggests that fragility could remain a significant
obstacle, even for other tailored codes. We expect that our decoders will have
other uses; belief-find has an almost-linear running time, and we show that it
increases the threshold of the surface code to 0.937(2)% in the presence of
circuit-level depolarising noise, compared to 0.817(5)% for the more
computationally expensive minimum-weight perfect matching decoder.Comment: 16 pages, 17 figure
Optimal local unitary encoding circuits for the surface 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 , however the most efficient known method for encoding an
unknown state, introduced by Dennis et al. (2002), has time
complexity. Here, we present an optimal local unitary encoding circuit for the
planar surface code that uses exactly time steps to encode an unknown
state in a distance planar code. We further show how an complexity
local unitary encoder for the toric code can be found by enforcing locality in
the -depth non-local renormalisation encoder. We relate these
techniques by providing an 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.Comment: 15 pages, 13 figure
OpenFermion: The Electronic Structure Package for Quantum Computers
Quantum simulation of chemistry and materials is predicted to be an important
application for both near-term and fault-tolerant quantum devices. However, at
present, developing and studying algorithms for these problems can be difficult
due to the prohibitive amount of domain knowledge required in both the area of
chemistry and quantum algorithms. To help bridge this gap and open the field to
more researchers, we have developed the OpenFermion software package
(www.openfermion.org). OpenFermion is an open-source software library written
largely in Python under an Apache 2.0 license, aimed at enabling the simulation
of fermionic models and quantum chemistry problems on quantum hardware.
Beginning with an interface to common electronic structure packages, it
simplifies the translation between a molecular specification and a quantum
circuit for solving or studying the electronic structure problem on a quantum
computer, minimizing the amount of domain expertise required to enter the
field. The package is designed to be extensible and robust, maintaining high
software standards in documentation and testing. This release paper outlines
the key motivations behind design choices in OpenFermion and discusses some
basic OpenFermion functionality which we believe will aid the community in the
development of better quantum algorithms and tools for this exciting area of
research.Comment: 22 page
Suppressing quantum errors by scaling a surface code logical qubit
Practical quantum computing will require error rates that are well below what
is achievable with physical qubits. Quantum error correction offers a path to
algorithmically-relevant error rates by encoding logical qubits within many
physical qubits, where increasing the number of physical qubits enhances
protection against physical errors. However, introducing more qubits also
increases the number of error sources, so the density of errors must be
sufficiently low in order for logical performance to improve with increasing
code size. Here, we report the measurement of logical qubit performance scaling
across multiple code sizes, and demonstrate that our system of superconducting
qubits has sufficient performance to overcome the additional errors from
increasing qubit number. We find our distance-5 surface code logical qubit
modestly outperforms an ensemble of distance-3 logical qubits on average, both
in terms of logical error probability over 25 cycles and logical error per
cycle ( compared to ). To investigate
damaging, low-probability error sources, we run a distance-25 repetition code
and observe a logical error per round floor set by a single
high-energy event ( when excluding this event). We are able
to accurately model our experiment, and from this model we can extract error
budgets that highlight the biggest challenges for future systems. These results
mark the first experimental demonstration where quantum error correction begins
to improve performance with increasing qubit number, illuminating the path to
reaching the logical error rates required for computation.Comment: Main text: 6 pages, 4 figures. v2: Update author list, references,
Fig. S12, Table I
Improved single-shot decoding of higher dimensional hypergraph product codes
In this work we study the single-shot performance of higher dimensional
hypergraph product codes decoded using belief-propagation and
ordered-statistics decoding [Panteleev and Kalachev, 2019]. We find that
decoding data qubit and syndrome measurement errors together in a single stage
leads to single-shot thresholds that greatly exceed all previously observed
single-shot thresholds for these codes. For the 3D toric code and a
phenomenological noise model, our results are consistent with a sustainable
threshold of 7.1% for errors, compared to the threshold of 2.90% previously
found using a two-stage decoder [Quintavalle et al., 2021]. For the 4D toric
code, for which both and error correction is single-shot, our results
are consistent with a sustainable single-shot threshold of 4.3% which is even
higher than the threshold of 2.93% for the 2D toric code for the same noise
model but using rounds of stabiliser measurement. We also explore the
performance of balanced product and 4D hypergraph product codes which we show
lead to a reduction in qubit overhead compared the surface code for
phenomenological error rates as high as 1%.Comment: 16 pages, 14 figure
Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead
We introduce a technique that uses gauge fixing to significantly improve the
quantum error correcting performance of subsystem codes. By changing the order
in which check operators are measured, valuable additional information can be
gained, and we introduce a new method for decoding which uses this information
to improve performance. Applied to the subsystem toric code with three-qubit
check operators, we increase the threshold under circuit-level depolarising
noise from to . The threshold increases further under a
circuit-level noise model with small finite bias, up to for infinite
bias. Furthermore, we construct families of finite-rate subsystem LDPC codes
with three-qubit check operators and optimal-depth parity-check measurement
schedules. To the best of our knowledge, these finite-rate subsystem codes
outperform all known codes at circuit-level depolarising error rates as high as
, where they have a qubit overhead that is lower than the
most efficient version of the surface code and lower than the
subsystem toric code. Their threshold and pseudo-threshold exceeds for
circuit-level depolarising noise, increasing to under infinite bias
using gauge fixing.Comment: 30 pages, 32 figure