7,808 research outputs found
Postselection threshold against biased noise
The highest current estimates for the amount of noise a quantum computer can
tolerate are based on fault-tolerance schemes relying heavily on postselecting
on no detected errors. However, there has been no proof that these schemes give
even a positive tolerable noise threshold. A technique to prove a positive
threshold, for probabilistic noise models, is presented. The main idea is to
maintain strong control over the distribution of errors in the quantum state at
all times. This distribution has correlations which conceivably could grow out
of control with postselection. But in fact, the error distribution can be
written as a mixture of nearby distributions each satisfying strong
independence properties, so there are no correlations for postselection to
amplify.Comment: 13 pages, FOCS 2006; conference versio
Aspects of Fault-Tolerant Quantum Computation
This thesis is concerned with fault-tolerant quantum information processing using quantum error-correcting codes. It contains two major pieces of work. The first is a study of coherent noise in the context of stabilizer error-correcting codes. The second is a proposed scheme for a universal set of fault-tolerant logical gates in a particular code family built out of the 3D toric code.
Chapter 1 provides an introduction to quantum computation and fault tolerance. Many basic concepts in error-correcting codes are defined. Special attention is paid to the set of code properties that are most likely to determine how easily a given fault-tolerant scheme might be implemented on a physical device. These include the fault-tolerant noise threshold and the overhead.
In Chapters 2 and 3 we study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In Chapter 2 we introduce coherent noise and incoherent noise and a number of methods that are useful for the study of coherent noise. We study the repetition code as a basic example, and we also study a correlated noise model. In Chapter 3 we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.
Chapter 4 is connected to another aspect of fault tolerance, fault-tolerant logical gates. The toric code is a promising candidate for fault-tolerant quantum computation because of its high threshold and low-weight stabilizers. A universal gate set in the toric code generally requires magic state distillation, which can incur a significant qubit overhead. In this work we construct an error-correcting code in three dimensions based on the toric code that features a fault-tolerant T gate with no magic state distillation required. We further describe a subsystem version of our code which supports a universal set of fault-tolerant gates. This code can be converted into the stabilizer version using gauge-fixing. We also argue that our code can be converted to a (2+1)-D protocol, where a 2D lattice undergoes a measurement-based protocol over time. In this way, a fault-tolerant logical T gate can be realized in a 2D toric code structure.</p
Fault-tolerant logical gates in quantum error-correcting codes
Recently, Bravyi and K\"onig have shown that there is a tradeoff between
fault-tolerantly implementable logical gates and geometric locality of
stabilizer codes. They consider locality-preserving operations which are
implemented by a constant depth geometrically local circuit and are thus
fault-tolerant by construction. In particular, they shown that, for local
stabilizer codes in D spatial dimensions, locality preserving gates are
restricted to a set of unitary gates known as the D-th level of the Clifford
hierarchy. In this paper, we elaborate this idea and provide several extensions
and applications of their characterization in various directions. First, we
present a new no-go theorem for self-correcting quantum memory. Namely, we
prove that a three-dimensional stabilizer Hamiltonian with a
locality-preserving implementation of a non-Clifford gate cannot have a
macroscopic energy barrier. Second, we prove that the code distance of a
D-dimensional local stabilizer code with non-trivial locality-preserving m-th
level Clifford logical gate is upper bounded by . For codes with
non-Clifford gates (m>2), this improves the previous best bound by Bravyi and
Terhal. Third we prove that a qubit loss threshold of codes with non-trivial
transversal m-th level Clifford logical gate is upper bounded by 1/m. As such,
no family of fault-tolerant codes with transversal gates in increasing level of
the Clifford hierarchy may exist. This result applies to arbitrary stabilizer
and subsystem codes, and is not restricted to geometrically-local codes. Fourth
we extend the result of Bravyi and K\"onig to subsystem codes. A technical
difficulty is that, unlike stabilizer codes, the so-called union lemma does not
apply to subsystem codes. This problem is avoided by assuming the presence of
error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig
is recovered.Comment: 13 pages, 4 figure
Local Fault-tolerant Quantum Computation
We analyze and study the effects of locality on the fault-tolerance threshold
for quantum computation. We analytically estimate how the threshold will depend
on a scale parameter r which estimates the scale-up in the size of the circuit
due to encoding. We carry out a detailed semi-numerical threshold analysis for
concatenated coding using the 7-qubit CSS code in the local and `nonlocal'
setting. First, we find that the threshold in the local model for the [[7,1,3]]
code has a 1/r dependence, which is in correspondence with our analytical
estimate. Second, the threshold, beyond the 1/r dependence, does not depend too
strongly on the noise levels for transporting qubits. Beyond these results, we
find that it is important to look at more than one level of concatenation in
order to estimate the threshold and that it may be beneficial in certain
places, like in the transportation of qubits, to do error correction only
infrequently.Comment: REVTeX, 44 pages, 19 figures, to appear in Physical Review
Flag fault-tolerant error correction with arbitrary distance codes
In this paper we introduce a general fault-tolerant quantum error correction
protocol using flag circuits for measuring stabilizers of arbitrary distance
codes. In addition to extending flag error correction beyond distance-three
codes for the first time, our protocol also applies to a broader class of
distance-three codes than was previously known. Flag circuits use extra ancilla
qubits to signal when errors resulting from faults in the circuit have
weight greater than . The flag error correction protocol is applicable to
stabilizer codes of arbitrary distance which satisfy a set of conditions and
uses fewer qubits than other schemes such as Shor, Steane and Knill error
correction. We give examples of infinite code families which satisfy these
conditions and analyze the behaviour of distance-three and -five examples
numerically. Requiring fewer resources than Shor error correction, flag error
correction could potentially be used in low-overhead fault-tolerant error
correction protocols using low density parity check quantum codes of large code
length.Comment: 29 pages (18 pages main text), 22 figures, 7 tables. Comments
welcome! V3 represents the version accepted to quantu
The Small Stellated Dodecahedron Code and Friends
We explore a distance-3 homological CSS quantum code, namely the small
stellated dodecahedron code, for dense storage of quantum information and we
compare its performance with the distance-3 surface code. The data and ancilla
qubits of the small stellated dodecahedron code can be located on the edges
resp. vertices of a small stellated dodecahedron, making this code suitable for
3D connectivity. This code encodes 8 logical qubits into 30 physical qubits
(plus 22 ancilla qubits for parity check measurements) as compared to 1 logical
qubit into 9 physical qubits (plus 8 ancilla qubits) for the surface code. We
develop fault-tolerant parity check circuits and a decoder for this code,
allowing us to numerically assess the circuit-based pseudo-threshold.Comment: 19 pages, 14 figures, comments welcome! v2 includes updates which
conforms with the journal versio
Bounding quantum gate error rate based on reported average fidelity
Remarkable experimental advances in quantum computing are exemplified by
recent announcements of impressive average gate fidelities exceeding 99.9% for
single-qubit gates and 99% for two-qubit gates. Although these high numbers
engender optimism that fault-tolerant quantum computing is within reach, the
connection of average gate fidelity with fault-tolerance requirements is not
direct. Here we use reported average gate fidelity to determine an upper bound
on the quantum-gate error rate, which is the appropriate metric for assessing
progress towards fault-tolerant quantum computation, and we demonstrate that
this bound is asymptotically tight for general noise. Although this bound is
unlikely to be saturated by experimental noise, we demonstrate using explicit
examples that the bound indicates a realistic deviation between the true error
rate and the reported average fidelity. We introduce the Pauli distance as a
measure of this deviation, and we show that knowledge of the Pauli distance
enables tighter estimates of the error rate of quantum gates.Comment: New Journal of Physics Fast Track Communication. Gold open access
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Universal fault-tolerant gates on concatenated stabilizer codes
It is an oft-cited fact that no quantum code can support a set of
fault-tolerant logical gates that is both universal and transversal. This no-go
theorem is generally responsible for the interest in alternative universality
constructions including magic state distillation. Widely overlooked, however,
is the possibility of non-transversal, yet still fault-tolerant, gates that
work directly on small quantum codes. Here we demonstrate precisely the
existence of such gates. In particular, we show how the limits of
non-transversality can be overcome by performing rounds of intermediate
error-correction to create logical gates on stabilizer codes that use no
ancillas other than those required for syndrome measurement. Moreover, the
logical gates we construct, the most prominent examples being Toffoli and
controlled-controlled-Z, often complete universal gate sets on their codes. We
detail such universal constructions for the smallest quantum codes, the 5-qubit
and 7-qubit codes, and then proceed to generalize the approach. One remarkable
result of this generalization is that any nondegenerate stabilizer code with a
complete set of fault-tolerant single-qubit Clifford gates has a universal set
of fault-tolerant gates. Another is the interaction of logical qubits across
different stabilizer codes, which, for instance, implies a broadly applicable
method of code switching.Comment: 18 pages + 5 pages appendix, 12 figure
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