8 research outputs found
On Steane-Enlargement of Quantum Codes from Cartesian Product Point Sets
In this work, we study quantum error-correcting codes obtained by using
Steane-enlargement. We apply this technique to certain codes defined from
Cartesian products previously considered by Galindo et al. in [4]. We give
bounds on the dimension increase obtained via enlargement, and additionally
give an algorithm to compute the true increase. A number of examples of codes
are provided, and their parameters are compared to relevant codes in the
literature, which shows that the parameters of the enlarged codes are
advantageous. Furthermore, comparison with the Gilbert-Varshamov bound for
stabilizer quantum codes shows that several of the enlarged codes match or
exceed the parameters promised by the bound.Comment: 12 page
Linear programming bounds for quantum amplitude damping codes
Given that approximate quantum error-correcting (AQEC) codes have a
potentially better performance than perfect quantum error correction codes, it
is pertinent to quantify their performance. While quantum weight enumerators
establish some of the best upper bounds on the minimum distance of quantum
error-correcting codes, these bounds do not directly apply to AQEC codes.
Herein, we introduce quantum weight enumerators for amplitude damping (AD)
errors and work within the framework of approximate quantum error correction.
In particular, we introduce an auxiliary exact weight enumerator that is
intrinsic to a code space and moreover, we establish a linear relationship
between the quantum weight enumerators for AD errors and this auxiliary exact
weight enumerator. This allows us to establish a linear program that is
infeasible only when AQEC AD codes with corresponding parameters do not exist.
To illustrate our linear program, we numerically rule out the existence of
three-qubit AD codes that are capable of correcting an arbitrary AD error.Comment: 5 page
Weight Distribution of Classical Codes Influences Robust Quantum Metrology
Quantum metrology (QM) is expected to be a prominent use-case of quantum
technologies. However, noise easily degrades these quantum probe states, and
negates the quantum advantage they would have offered in a noiseless setting.
Although quantum error correction (QEC) can help tackle noise, fault-tolerant
methods are too resource intensive for near-term use. Hence, a strategy for
(near-term) robust QM that is easily adaptable to future QEC-based QM is
desirable. Here, we propose such an architecture by studying the performance of
quantum probe states that are constructed from binary block codes of
minimum distance . Such states can be interpreted as a logical
state of a CSS code whose logical group is defined by the aforesaid binary
code. When a constant, , number of qubits of the quantum probe state are
erased, using the quantum Fisher information (QFI) we show that the resultant
noisy probe can give an estimate of the magnetic field with a precision that
scales inversely with the variances of the weight distributions of the
corresponding shortened codes. If is any code concatenated with inner
repetition codes of length linear in , a quantum advantage in QM is
possible. Hence, given any CSS code of constant length, concatenation with
repetition codes of length linear in is asymptotically optimal for QM with
a constant number of erasure errors. We also explicitly construct an observable
that when measured on such noisy code-inspired probe states, yields a precision
on the magnetic field strength that also exhibits a quantum advantage in the
limit of vanishing magnetic field strength. We emphasize that, despite the use
of coding-theoretic methods, our results do not involve syndrome measurements
or error correction. We complement our results with examples of probe states
constructed from Reed-Muller codes.Comment: 21 pages, 3 figure
Characterization and mass formulas of symplectic self-orthogonal and LCD codes and their application
The object of this paper is to study two very important classes of codes in
coding theory, namely self-orthogonal (SO) and linear complementary dual (LCD)
codes under the symplectic inner product, involving characterization,
constructions, and their application. Using such a characterization, we
determine the mass formulas of symplectic SO and LCD codes by considering the
action of the symplectic group, and further obtain some asymptotic results.
Finally, under the Hamming distance, we obtain some symplectic SO (resp. LCD)
codes with improved parameters directly compared with Euclidean SO (resp. LCD)
codes. Under the symplectic distance, we obtain some additive SO (resp.
additive complementary dual) codes with improved parameters directly compared
with Hermitian SO (resp. LCD) codes. Further, we also construct many good
additive codes outperform the best-known linear codes in Grassl's code table.
As an application, we construct a number of record-breaking
(entanglement-assisted) quantum error-correcting codes, which improve Grassl's
code table
Describing quantum metrology with erasure errors using weight distributions of classical codes
Quantum sensors are expected to be a prominent use-case of quantum
technologies, but in practice, noise easily degrades their performance. Quantum
sensors can for instance be afflicted with erasure errors. Here, we consider
using quantum probe states with a structure that corresponds to classical
binary block codes of minimum distance . We obtain bounds
on the ultimate precision that these probe states can give for estimating the
unknown magnitude of a classical field after at most qubits of the quantum
probe state are erased. We show that the quantum Fisher information is
proportional to the variances of the weight distributions of the corresponding
shortened codes. If the shortened codes of a fixed code with
have a non-trivial weight distribution, then the probe states obtained by
concatenating this code with repetition codes of increasing length enable
asymptotically optimal field-sensing that passively tolerates up to erasure
errors
Transmitting Quantum Information Reliably across Various Quantum Channels
Transmitting quantum information across quantum channels is an important task. However quantum information is delicate, and is easily corrupted. We address the task of protecting quantum information from an information theoretic perspective -- we encode some message qudits into a quantum code, send the encoded quantum information across the noisy quantum channel, then recover the message qudits by decoding. In this dissertation, we discuss the coding problem from several perspectives.}
The noisy quantum channel is one of the central aspects of the quantum coding problem, and hence quantifying the noisy quantum channel from the physical model is an important problem.
We work with an explicit physical model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set. We thereby derive the matrix elements of the Choi-Jamiolkowski operator of the corresponding truncated channel, which are truncated transition amplitudes. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically.
In the context of truncated noisy channels, we revisit the notion of approximate error correction of finite dimension codes. We derive a computationally simple lower bound on the worst case entanglement fidelity of a quantum code, when the truncated recovery map of Leung et. al. is rescaled. As an application, we apply our bound to construct a family of multi-error correcting amplitude damping codes that are permutation-invariant. This demonstrates an explicit example where the specific structure of the noisy channel allows code design out of the stabilizer formalism via purely algebraic means.
We study lower bounds on the quantum capacity of adversarial channels, where we restrict the selection of quantum codes to the set of concatenated quantum codes.
The adversarial channel is a quantum channel where an adversary corrupts a fixed fraction of qudits sent across a quantum channel in the most malicious way possible. The best known rates of communicating over adversarial channels are given by the quantum Gilbert-Varshamov (GV) bound, that is known to be attainable with random quantum codes. We generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region.
We next study upper bounds on the quantum capacity of some low dimension quantum channels.
The quantum capacity of a quantum channel is the maximum rate at which quantum information can be transmitted reliably across it, given arbitrarily many uses of it. While it is known that random quantum codes can be used to attain the quantum capacity, the quantum capacity of many classes of channels is undetermined, even for channels of low input and output dimension. For example, depolarizing channels are
important quantum channels, but do not have tight numerical bounds.
We obtain upper bounds on the quantum capacity of some unital and non-unital channels
-- two-qubit Pauli channels, two-qubit depolarizing channels, two-qubit locally symmetric channels,
shifted qubit depolarizing channels, and shifted two-qubit Pauli channels --
using the coherent information of some degradable channels. We use the notion
of twirling quantum channels, and Smith and Smolin's method of constructing
degradable extensions of quantum channels extensively. The degradable channels we
introduce, study and use are two-qubit amplitude damping channels. Exploiting the
notion of covariant quantum channels, we give sufficient conditions for the quantum
capacity of a degradable channel to be the optimal value of a concave program with
linear constraints, and show that our two-qubit degradable amplitude damping channels have this property