A first step towards quantum algorithms: Minimizing the guesswork of a
quantum ensemble
In a groundbreaking study, researchers have derived analytical solutions
to the guesswork problem for quantum ensembles

Date:
March 10, 2022
Source:
Waseda University
Summary:
A quantum ensemble -- a set of quantum states with their
corresponding probabilities -- is essential to the encoding of
classical information for transmission over quantum channels. But
receivers must be able to 'guess' the transmitted quantum state,
incurring a cost called 'guesswork.' Recently, researchers have
derived analytical solutions of the guesswork problem for when the
ensemble is subject to a finite set of conditions. The results
constitute a first step towards future algorithms for quantum
software.



FULL STORY ========================================================================== Given the rapid pace at which technology is developing, it comes as
no surprise that quantum technologies will become commonplace within
decades. A big part of ushering in this new age of quantum computing
requires a new understanding of both classical and quantum information
and how the two can be related to each other.


========================================================================== Before one can send classical information across quantum channels, it
needs to be encoded first. This encoding is done by means of quantum
ensembles. A quantum ensemble refers to a set of quantum states,
each with its own probability. To accurately receive the transmitted information, the receiver has to repeatedly 'guess' the state of the information being sent. This constitutes a cost function that is called 'guesswork.' Guesswork refers to the average number of guesses required
to correctly guess the state.

The concept of guesswork has been studied at length in classical
ensembles, but the subject is still new for quantum ensembles. Recently,
a research team from Japan -- consisting of Prof. Takeshi Koshiba of
Waseda University, Michele Dall'Arno from Waseda University and Kyoto University, and Prof. Francesco Buscemi from Nagoya University --
has derived analytical solutions to the guesswork problem subject to
a finite set of conditions. "The guesswork problem is fundamental in
many scientific areas in which machine learning techniques or artificial intelligence are used. Our results trailblaze an algorithmic aspect of
the guesswork problem," says Koshiba. Their findings are published in
IEEE Transactions on Information Theory.

To begin with, the researchers considered a common formalism of quantum circuits that relates the transmitted state of a quantum ensemble ?to the quantum measurement ?. They next introduced the probability distributions
for both the quantum ensemble and the numberings obtained from the
quantum measurement. They then established the guesswork function. The guesswork function maps any pair of ? and ? into the expectation value
of the tthguess (where t refers to the guess number), averaged over the probability distribution of the tthguess being correct. Finally, they
minimized the guesswork function over the elements of ? and used this
result to derive analytical solutions to the guesswork problem subject
to a finite set of conditions.

These solutions included the explicit solution to a qubit ensemble with
a uniform probability distribution. "Previously, results for analytical solutions have been known only for binary and symmetric ensembles. Our calculation for ensembles with a uniform probability distribution extends these," explains Koshiba. The research team also calculated the solutions
for a qubit regular polygonal ensemble, and a qubit regular polyhedral ensemble.

"Guesswork is a very basic scientific problem, but there is very
little research on quantum guesswork and even less on the algorithmic implications of quantum guesswork. Our paper goes a little way towards
filling that gap," concludes Koshiba.

While the consequences of these findings may not be immediately obvious,
in the future they are sure to have a major influence on quantum science,
such as quantum chemistry for drug development and quantum software for
quantum computing.


========================================================================== Story Source: Materials provided by Waseda_University. Note: Content
may be edited for style and length.


========================================================================== Journal Reference:
1. Michele Dall'Arno, Francesco Buscemi, Takeshi Koshiba. Guesswork
of a
quantum ensemble. IEEE Transactions on Information Theory, 2022;
1 DOI: 10.1109/TIT.2022.3146463 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2022/03/220310115109.htm

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