4 research outputs found
Bhattacharyya parameter of monomials codes for the Binary Erasure Channel: from pointwise to average reliability
Monomial codes were recently equipped with partial order relations, fact that
allowed researchers to discover structural properties and efficient algorithm
for constructing polar codes. Here, we refine the existing order relations in
the particular case of Binary Erasure Channel. The new order relation takes us
closer to the ultimate order relation induced by the pointwise evaluation of
the Bhattacharyya parameter of the synthetic channels. The best we can hope for
is still a partial order relation. To overcome this issue we appeal to related
technique from network theory. Reliability network theory was recently used in
the context of polar coding and more generally in connection with decreasing
monomial codes. In this article, we investigate how the concept of average
reliability is applied for polar codes designed for the binary erasure channel.
Instead of minimizing the error probability of the synthetic channels, for a
particular value of the erasure parameter p, our codes minimize the average
error probability of the synthetic channels. By means of basic network theory
results we determine a closed formula for the average reliability of a
particular synthetic channel, that recently gain the attention of researchers.Comment: 21 pages, 5 figures, 3 tables. Submitted for possible publicatio
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
A New Family of Fault Tolerant Quantum Reed-Muller Codes
Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.
We prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual
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