41,428 research outputs found
Efficiently Correcting Matrix Products
We study the problem of efficiently correcting an erroneous product of two
matrices over a ring. Among other things, we provide a randomized
algorithm for correcting a matrix product with at most erroneous entries
running in time and a deterministic -time
algorithm for this problem (where the notation suppresses
polylogarithmic terms in and ).Comment: Fixed invalid reference to figure in v
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
An Adaptive Entanglement Distillation Scheme Using Quantum Low Density Parity Check Codes
Quantum low density parity check (QLDPC) codes are useful primitives for
quantum information processing because they can be encoded and decoded
efficiently. Besides, the error correcting capability of a few QLDPC codes
exceeds the quantum Gilbert-Varshamov bound. Here, we report a numerical
performance analysis of an adaptive entanglement distillation scheme using
QLDPC codes. In particular, we find that the expected yield of our adaptive
distillation scheme to combat depolarization errors exceed that of Leung and
Shor whenever the error probability is less than about 0.07 or greater than
about 0.28. This finding illustrates the effectiveness of using QLDPC codes in
entanglement distillation.Comment: 12 pages, 6 figure
Variations of the McEliece Cryptosystem
Two variations of the McEliece cryptosystem are presented. The first one is
based on a relaxation of the column permutation in the classical McEliece
scrambling process. This is done in such a way that the Hamming weight of the
error, added in the encryption process, can be controlled so that efficient
decryption remains possible. The second variation is based on the use of
spatially coupled moderate-density parity-check codes as secret codes. These
codes are known for their excellent error-correction performance and allow for
a relatively low key size in the cryptosystem. For both variants the security
with respect to known attacks is discussed
Perfect zero knowledge for quantum multiprover interactive proofs
In this work we consider the interplay between multiprover interactive
proofs, quantum entanglement, and zero knowledge proofs - notions that are
central pillars of complexity theory, quantum information and cryptography. In
particular, we study the relationship between the complexity class MIP, the
set of languages decidable by multiprover interactive proofs with quantumly
entangled provers, and the class PZKMIP, which is the set of languages
decidable by MIP protocols that furthermore possess the perfect zero
knowledge property.
Our main result is that the two classes are equal, i.e., MIP
PZKMIP. This result provides a quantum analogue of the celebrated result of
Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP
PZKMIP (in other words, all classical multiprover interactive protocols can be
made zero knowledge). We prove our result by showing that every MIP
protocol can be efficiently transformed into an equivalent zero knowledge
MIP protocol in a manner that preserves the completeness-soundness gap.
Combining our transformation with previous results by Slofstra (Forum of
Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we
obtain the corollary that all co-recursively enumerable languages (which
include undecidable problems as well as all decidable problems) have zero
knowledge MIP protocols with vanishing promise gap
Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes
We present two methods for the construction of quantum circuits for quantum
error-correcting codes (QECC). The underlying quantum systems are tensor
products of subsystems (qudits) of equal dimension which is a prime power. For
a QECC encoding k qudits into n qudits, the resulting quantum circuit has
O(n(n-k)) gates. The running time of the classical algorithm to compute the
quantum circuit is O(n(n-k)^2).Comment: 18 pages, submitted to special issue of IJFC
Quantum Computing with Very Noisy Devices
In theory, quantum computers can efficiently simulate quantum physics, factor
large numbers and estimate integrals, thus solving otherwise intractable
computational problems. In practice, quantum computers must operate with noisy
devices called ``gates'' that tend to destroy the fragile quantum states needed
for computation. The goal of fault-tolerant quantum computing is to compute
accurately even when gates have a high probability of error each time they are
used. Here we give evidence that accurate quantum computing is possible with
error probabilities above 3% per gate, which is significantly higher than what
was previously thought possible. However, the resources required for computing
at such high error probabilities are excessive. Fortunately, they decrease
rapidly with decreasing error probabilities. If we had quantum resources
comparable to the considerable resources available in today's digital
computers, we could implement non-trivial quantum computations at error
probabilities as high as 1% per gate.Comment: 47 page
Kerdock Codes Determine Unitary 2-Designs
The non-linear binary Kerdock codes are known to be Gray images of certain
extended cyclic codes of length over . We show that
exponentiating these -valued codewords by produces stabilizer states, that are quantum states obtained using
only Clifford unitaries. These states are also the common eigenvectors of
commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the
Pauli group. We use this quantum description to simplify the derivation of the
classical weight distribution of Kerdock codes. Next, we organize the
stabilizer states to form mutually unbiased bases and prove that
automorphisms of the Kerdock code permute their corresponding MCS, thereby
forming a subgroup of the Clifford group. When represented as symplectic
matrices, this subgroup is isomorphic to the projective special linear group
PSL(). We show that this automorphism group acts transitively on the Pauli
matrices, which implies that the ensemble is Pauli mixing and hence forms a
unitary -design. The Kerdock design described here was originally discovered
by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is
new which simplifies its description and translation to circuits significantly.
Sampling from the design is straightforward, the translation to circuits uses
only Clifford gates, and the process does not require ancillary qubits.
Finally, we also develop algorithms for optimizing the synthesis of unitary
-designs on encoded qubits, i.e., to construct logical unitary -designs.
Software implementations are available at
https://github.com/nrenga/symplectic-arxiv18a, which we use to provide
empirical gate complexities for up to qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to
2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is
included in the arXiv packag
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