More than a century after their appearance, the ideas of quantum mechanics continue to generate surprising results. Among them, the quantum computing, which promises to give rise to the most powerful computers known, occupies a prominent place. But to be used effectively, these designs face a serious difficulty: their operation is extremely sensitive to noise, to the point that it makes it impossible to execute complex calculations. Mathematics makes it possible to describe this phenomenon and improve current quantum algorithms, according to recent results.
Noise appears due to the interaction of the system with the environment, in which temperature causes atoms to move randomly. It has the effect of corrupt the states of the qubits, the basic information processing unit of the quantum computer. This introduces errors that propagate when complex algorithms are executed, ruining all the computing potential that a priori could provide this new form of computing.
Faced with this problem, the most obvious options are to lower the operating temperature to the limit of what is possible or to build quantum processors with the highest noise immunity. However, despite the efforts made in these two avenues, current technology seems to have bottomed out in these approaches. This is where mathematics, and specifically complex algebra, plays a crucial role. Firstly, they allow us to model the processes that take place in quantum computers and analyze the noise. Furthermore, the conclusions of these studies allow us to design more robust algorithms that give acceptable results even in the presence of noise.
The noise is modeled through quantum channelsthat They describe how information propagates in a quantum algorithm as it runs on a computer. These channels are a set of mathematical operations that interact with the quantum states of the system, modifying them. For example, him amplitude channel produces a decrease in energy in a quantum state – similar to the friction that slows down any movement – while the phase channel represents the loss of information caused by the change of one combination of states into a different one. The result of this formulation is a set of equations that describe how errors propagate and accumulate in a quantum system. It also allows us to identify noise sources and design quantum algorithms. error correction in real time.
The error-correcting codes act analogously to the letter that appears at the end of the Spanish national identity document, which allows the appearance of an incorrect digit to be detected and corrected. In the context of quantum noise, these codes are described in terms of unitary Pauli matrices. These transformations give a redundant, encoded structure to the initial quantum information by distributing copies of that information across multiple qubits, commonly referred to as code blocks. These encodings enable error detection by identifying discrepancies between the original information and that stored in the code blocks. Furthermore, once an error is detected in the quantum state, it is possible to apply specific correction operations based on Pauli operators. These operations help reverse the effects of quantum errors and immediately restore the qubits to their correct state.
Apart from attempting to mitigate or correct for quantum noise, methods have recently been proposed to take advantage of the noise beneficially, that is, they offer superior results in noisy environments. Specific, has been shown that certain quantum machine learning algorithms, such as the so-called quantum reservoir computingare capable of taking advantage of noise strategically.
This algorithm uses random quantum circuits to unravel the relevant information that is hidden in an input data set. This information is then processed by a machine learning algorithm responsible for generating accurate predictions. The quantum circuit in question, which is generally composed of a sequence of randomly interconnected quantum gates and qubits, functions as a temporary memory. These circuits make use of intrinsic properties of quantum mechanics, such as superposition and entanglement, to simultaneously explore multiple potential quantum transformations. This process facilitates the efficient extraction of relevant information, essential for the success of the algorithm.
The advantage of this quantum algorithm lies in its ability to strategically exploit the inherent noise in the circuit. In this context, noise introduces greater variability into the algorithm, enriching its ability to extract complex information from the input data. As a result, the quality and accuracy of the final predictions generated is significantly improved. This approach, developed in various jobshas proven effective in a wide range of applications, including molecular calculationsprediction temporal series and the design of new drugs.
Laia Domingo She is a predoctoral researcher at Institute of Mathematical Sciences (ICMAT)
Florentino Borondo He is a professor of the Autonomous University of Madrid
Gabriel Carlos He is a researcher in the National Atomic Energy Commission in Buenos Aires, Argentina
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Editing and coordination: Ágata Timón García-Longoria. Is coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences (ICMAT)
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