Random access codes via quantum contextual redundancy
By: Giancarlo Gatti, Daniel Huerga, Enrique Solano, and Mikel Sanz
2023–01–13
In this paper, the authors propose a new type of quantum random access code (QRAC) that uses quantum contextual redundancy to achieve better retrieval success probability than previous QRACs.
A QRAC is a scheme that compresses an initial amount of data into a smaller amount, such that any initial bit can be recovered with high probability. In the classical setting, a QRAC can be implemented using a simple parity check matrix. However, in the quantum setting, parity check matrices cannot be used because they do not take into account the quantum correlations between the encoded bits.
The authors of the paper propose to use a set of mutually biased bases (MBBs) to encode the data in the QRAC. MBBs are a set of bases such that each basis is orthogonal to a subset of the other bases. This means that if a measurement is performed in one basis, the outcome of the measurement will provide information about the outcomes of the measurements in the other bases.
Introduction
The introduction provides an overview of random access codes (RACs) and quantum random access codes (QRACs). RACs are a type of data storage scheme that allows for random access to any bit of data with high probability. QRACs are a quantum version of RACs that take advantage of quantum correlations to achieve better performance than classical RACs.
Preliminaries
The preliminaries section introduces some of the basic concepts and notation that will be used in the rest of the paper. This includes definitions of QRACs, mutually unbiased bases (MUBs), and mutually biased bases (MBBs).
Random access codes (RACs)
A RAC is a scheme that allows for random access to any bit of data with high probability. In a classical RAC, the data is encoded into a sequence of bits, and each bit is associated with a unique measurement basis. To retrieve a bit of data, a measurement is performed in the corresponding basis. If the outcome of the measurement is the same as the value of the bit, then the bit has been successfully retrieved.
The retrieval success probability of a RAC is the probability that a randomly chosen bit can be successfully retrieved. The higher the retrieval success probability, the better the RAC.
Quantum random access codes (QRACs)
A QRAC is a quantum version of a RAC that takes advantage of quantum correlations to achieve better performance than classical RACs. In a QRAC, the data is encoded into a quantum state, and each bit of data is associated with a unique measurement operator. To retrieve a bit of data, a measurement is performed in the corresponding measurement operator. If the outcome of the measurement is the same as the value of the bit, then the bit has been successfully retrieved.
The retrieval success probability of a QRAC can be higher than that of a classical RAC because of quantum entanglement. Quantum entanglement is a property of quantum systems that allows them to be correlated even if they are separated by a distance. This correlation can be used to improve the retrieval success probability of a QRAC.
Mutually unbiased bases (MUBs)
MUBs are a set of bases such that any two bases in the set are orthogonal to each other. This means that if a measurement is performed in one basis, the outcome of the measurement will provide no information about the outcome of the measurement in the other bases.
MUBs are important for quantum information processing because they can be used to construct QRACs with high retrieval success probability.
Mutually biased bases (MBBs)
MBBs are a set of bases such that each basis is orthogonal to a subset of the other bases. This means that if a measurement is performed in one basis, the outcome of the measurement will provide some information about the outcomes of the measurements in the other bases.
MBBs are less powerful than MUBs, but they can be used to construct QRACs with lower complexity.
Contextual QRAC
The authors propose a new type of QRAC that uses quantum contextual redundancy to achieve better retrieval success probability than previous QRACs. The contextual QRAC works by encoding the data into a quantum state that is a superposition of the eigenstates of a set of mutually biased bases (MBBs). To retrieve a bit of data, a measurement is performed in a randomly chosen MBB. The outcome of the measurement provides information about the value of the bit.
The authors show that the retrieval success probability of the contextual QRAC can be higher than that of previous QRACs that use MUBs. This is because the MBBs have more overlap than the MUBs, which means that the outcomes of the measurements in the MBBs are more correlated.
Counting contexts
A context is a set of MBBs such that each bit of data is encoded into a unique superposition of the eigenstates of the MBBs in the context.
The number of contexts depends on the number of bits and the overlap between the MBBs. The more overlap there is between the MBBs, the fewer contexts are needed.
The authors show that the number of contexts grows exponentially with the number of bits. However, the rate of growth is slower than for QRACs that use MUBs. This means that the contextual QRAC can encode more bits into a given number of qubits than a QRAC that uses MUBs.
The authors also show that the number of contexts can be reduced by using a technique called symmetry reduction. Symmetry reduction is a method of finding a subset of contexts that is equivalent to the full set of contexts. This can be done by finding a set of transformations that leave the retrieval success probability of the QRAC unchanged.
The authors show that symmetry reduction can be used to reduce the number of contexts by a factor of up to 2^n, where n is the number of bits. This means that the contextual QRAC can encode even more bits into a given number of qubits.
Implementation
The implementation section describes how to implement the contextual QRAC in a quantum circuit. The circuit uses a series of Hadamard gates and Pauli measurements to encode and decode the data.
The contextual QRAC can be implemented using a relatively simple quantum circuit. The circuit consists of the following steps:
Statistical extrapolation
The statistical extrapolation section discusses how to estimate the retrieval success probability of the contextual QRAC from a finite number of trials. The authors use a statistical method called bootstrapping to estimate the success probability.
Wrapping up the QRAC
The wrapping up the QRAC section summarizes the main results of the paper. The authors show that their contextual QRAC has better retrieval success probability than previous QRACs for a certain number of bits. They also show that the QRAC can be amplified into a nearly-lossless compression protocol.
Comparison
The comparison section compares the contextual QRAC to other QRACs. The authors show that their QRAC has better retrieval success probability than other QRACs for a certain number of bits.
Applications
The applications section discusses some of the potential applications of the contextual QRAC. The authors suggest that the QRAC could be used to store large datasets, compress data, or implement decision trees.
Conclusion and outlook
The conclusion and outlook section summarizes the paper and discusses future directions. The authors conclude that the contextual QRAC is a promising new approach to QRACs. They suggest that future work could focus on improving the retrieval success probability of the QRAC and on developing new applications for the QRAC.
The authors show that by using MBBs, they can encode a number of bits into a number of qubits that is smaller than the number of bits that would be required to store the data classically. They also show that the retrieval success probability of their QRAC is greater than that of previous QRACs for a certain number of bits.
The authors’ QRAC has a number of potential applications. For example, it could be used to store large datasets that need to be accessed randomly, such as decision trees. It could also be used to compress data, such as images or videos.
Originally published at http://quantumsumm.wordpress.com on July 4, 2023.
