390 research outputs found
Efficient Computations of Encodings for Quantum Error Correction
We show how, given any set of generators of the stabilizer of a quantum code,
an efficient gate array that computes the codewords can be constructed. For an
n-qubit code whose stabilizer has d generators, the resulting gate array
consists of O(n d) operations, and converts k-qubit data (where k = n - d) into
n-qubit codewords.Comment: 16 pages, REVTeX, 3 figures within the tex
Degenerate Viterbi decoding
We present a decoding algorithm for quantum convolutional codes that finds
the class of degenerate errors with the largest probability conditioned on a
given error syndrome. The algorithm runs in time linear with the number of
qubits. Previous decoding algorithms for quantum convolutional codes optimized
the probability over individual errors instead of classes of degenerate errors.
Using Monte Carlo simulations, we show that this modification to the decoding
algorithm results in a significantly lower block error rate
Algebraic Quantum Error-Correction Codes
Based on the group structure of a unitary Lie algebra, a scheme is provided
to systematically and exhaustively generate quantum error correction codes,
including the additive and nonadditive codes. The syndromes in the process of
error-correction distinguished by different orthogonal vector subspaces, the
coset subspaces. Moreover, the generated codes can be classified into four
types with respect to the spinors in the unitary Lie algebra and a chosen
initial quantum state
Codes for Simultaneous Transmission of Quantum and Classical Information
We consider the characterization as well as the construction of quantum codes
that allow to transmit both quantum and classical information, which we refer
to as `hybrid codes'. We construct hybrid codes with
length and distance , that simultaneously transmit qudits and
symbols from a classical alphabet of size . Many good codes such as
, , ,
, , ,
, , ,
, , ,
have been found. All these codes have better parameters
than hybrid codes obtained from the best known stabilizer quantum codes.Comment: 6 page
Decoding Schemes for Foliated Sparse Quantum Error Correcting Codes
Foliated quantum codes are a resource for fault-tolerant measurement-based
quantum error correction for quantum repeaters and for quantum computation.
They represent a general approach to integrating a range of possible quantum
error correcting codes into larger fault-tolerant networks. Here we present an
efficient heuristic decoding scheme for foliated quantum codes, based on
message passing between primal and dual code 'sheets'. We test this decoder on
two different families of sparse quantum error correcting code: turbo codes and
bicycle codes, and show reasonably high numerical performance thresholds. We
also present a construction schedule for building such code states.Comment: 23 pages, 15 figures, accepted for publication in Phys. Rev.
The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing
and protecting fragile qubits against the undesirable effects of quantum
decoherence. Similar to classical codes, hashing bound approaching QECCs may be
designed by exploiting a concatenated code structure, which invokes iterative
decoding. Therefore, in this paper we provide an extensive step-by-step
tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided
concatenated quantum codes based on the underlying quantum-to-classical
isomorphism. These design lessons are then exemplified in the context of our
proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the
outer component of a concatenated quantum code. The proposed QIRCC can be
dynamically adapted to match any given inner code using EXIT charts, hence
achieving a performance close to the hashing bound. It is demonstrated that our
QIRCC-based optimized design is capable of operating within 0.4 dB of the noise
limit
Geometric Approach to Digital Quantum Information
We present geometric methods for uniformly discretizing the continuous
N-qubit Hilbert space. When considered as the vertices of a geometrical figure,
the resulting states form the equivalent of a Platonic solid. The
discretization technique inherently describes a class of pi/2 rotations that
connect neighboring states in the set, i.e. that leave the geometrical figures
invariant. These rotations are shown to generate the Clifford group, a general
group of discrete transformations on N qubits. Discretizing the N-qubit Hilbert
space allows us to define its digital quantum information content, and we show
that this information content grows as N^2. While we believe the discrete sets
are interesting because they allow extra-classical behavior--such as quantum
entanglement and quantum parallelism--to be explored while circumventing the
continuity of Hilbert space, we also show how they may be a useful tool for
problems in traditional quantum computation. We describe in detail the discrete
sets for one and two qubits.Comment: Introduction rewritten; 'Sample Application' section added. To appear
in J. of Quantum Information Processin
Quantum Lego and XP Stabilizer Codes
We apply the recent graphical framework of ''quantum lego'' to XP stabilizer
codes where the stabilizer group is generally non-abelian. We show that the
idea of operator matching continues to hold for such codes and is sufficient
for generating all their XP symmetries provided the resulting code is XP. We
provide an efficient classical algorithm for tracking these symmetries under
tensor contraction or conjoining. This constitutes a partial extension of the
algorithm implied by Gottesman-Knill theorem beyond Pauli stabilizer states and
Clifford operations. Because conjoining transformations generate quantum
operations that are universal, the XP symmetries obtained from these algorithms
do not uniquely identify the resulting tensors in general. Using this extended
framework, we provide a novel XP stabilizer code with higher distance and a
code with fault-tolerant gate. For XP regular codes, we also
construct a tensor-network-based the maximum likelihood decoder for any i.i.d.
single qubit error channel.Comment: 18 pages, 6 figure
Tailoring surface codes for highly biased noise
The surface code, with a simple modification, exhibits ultra-high error
correction thresholds when the noise is biased towards dephasing. Here, we
identify features of the surface code responsible for these ultra-high
thresholds. We provide strong evidence that the threshold error rate of the
surface code tracks the hashing bound exactly for all biases, and show how to
exploit these features to achieve significant improvement in logical failure
rate. First, we consider the infinite bias limit, meaning pure dephasing. We
prove that the error threshold of the modified surface code for pure dephasing
noise is , i.e., that all qubits are fully dephased, and this threshold
can be achieved by a polynomial time decoding algorithm. We demonstrate that
the sub-threshold behavior of the code depends critically on the precise shape
and boundary conditions of the code. That is, for rectangular surface codes
with standard rough/smooth open boundaries, it is controlled by the parameter
, where and are dimensions of the surface code lattice. We
demonstrate a significant improvement in logical failure rate with pure
dephasing for co-prime codes that have , and closely-related rotated
codes, which have a modified boundary. The effect is dramatic: the same logical
failure rate achievable with a square surface code and physical qubits can
be obtained with a co-prime or rotated surface code using only
physical qubits. Finally, we use approximate maximum likelihood decoding to
demonstrate that this improvement persists for a general Pauli noise biased
towards dephasing. In particular, comparing with a square surface code, we
observe a significant improvement in logical failure rate against biased noise
using a rotated surface code with approximately half the number of physical
qubits.Comment: 18+4 pages, 24 figures; v2 includes additional coauthor (ASD) and new
results on the performance of surface codes in the finite-bias regime,
obtained with beveled surface codes and an improved tensor network decoder;
v3 published versio
Quantum Copy-Protection and Quantum Money
Forty years ago, Wiesner proposed using quantum states to create money that
is physically impossible to counterfeit, something that cannot be done in the
classical world. However, Wiesner's scheme required a central bank to verify
the money, and the question of whether there can be unclonable quantum money
that anyone can verify has remained open since. One can also ask a related
question, which seems to be new: can quantum states be used as copy-protected
programs, which let the user evaluate some function f, but not create more
programs for f? This paper tackles both questions using the arsenal of modern
computational complexity. Our main result is that there exist quantum oracles
relative to which publicly-verifiable quantum money is possible, and any family
of functions that cannot be efficiently learned from its input-output behavior
can be quantumly copy-protected. This provides the first formal evidence that
these tasks are achievable. The technical core of our result is a
"Complexity-Theoretic No-Cloning Theorem," which generalizes both the standard
No-Cloning Theorem and the optimality of Grover search, and might be of
independent interest. Our security argument also requires explicit
constructions of quantum t-designs. Moving beyond the oracle world, we also
present an explicit candidate scheme for publicly-verifiable quantum money,
based on random stabilizer states; as well as two explicit schemes for
copy-protecting the family of point functions. We do not know how to base the
security of these schemes on any existing cryptographic assumption. (Note that
without an oracle, we can only hope for security under some computational
assumption.)Comment: 14-page conference abstract; full version hasn't appeared and will
never appear. Being posted to arXiv mostly for archaeological purposes.
Explicit money scheme has since been broken by Lutomirski et al
(arXiv:0912.3825). Other quantum money material has been superseded by
results of Aaronson and Christiano (coming soon). Quantum copy-protection
ideas will hopefully be developed in separate wor
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