Conventional computers use bits, represented by zeros and ones, to transmit information, while quantum computers use quantum bits (qubits) instead. Like bits, qubits have two primary states or values: 0 and 1. However, unlike a bit, a qubit can exist in both states at the same time. While this may seem like a baffling paradox, it can be explained through a simple coin analogy. A classic bit can be represented as a coin with heads or tails (one or zero) facing up, while a qubit can be thought of as a spinning coin, which also has heads and tails, but either heads or tails can only be determined once it stops spinning, i.e. loses its original state. When a spinning coin stops, it can serve as an analogy for a quantum measurement, whereby one of two qubit states is selected. In quantum computing, several qubits must be related to each other, for example, the 0 (1) states of one qubit must be uniquely related to the 0 (1) states of another qubit. When the quantum states of two or more objects become entangled, this is called quantum entanglement.

The main difficulty with quantum computing is that qubits are surrounded by and interact with an environment. This interaction can cause the quantum entanglement of the qubits to degrade, causing them to become untangled from each other. An analogy with two coins can help to understand this concept. If two identical coins are tossed at the same time and then stopped after a short time, they can both end up with the same side up, heads or tails. This synchronicity between spinning coins can be compared to quantum entanglement. However, if the coins keep spinning for a longer period, they will eventually lose synchronicity and no longer end up with the same side (heads or tails) facing up. Loss of synchronicity occurs because the spinning coins gradually lose energy, mainly due to friction with the table, and each coin does so in a unique way. In the quantum world, friction, or the loss of energy due to interaction with the environment, ultimately leads to quantum decoherence, or a loss of synchronicity between qubits. This results in a qubit shift, in which the phase of the quantum state (represented by the rotation angle of the coin) changes randomly over time, causing a loss of quantum information and making quantum computation impossible.

A key challenge faced by many researchers today is to preserve quantum coherence for longer periods. This can be achieved by accurately describing the evolution of the quantum state over time, also known as quantum dynamics.

Scientists at the MIEM HSE Center for Quantum Metamaterials, in collaboration with German and British colleagues, have proposed an algorithm called Automated Compression of Arbitrary Environments (ACE) as a solution for studying the interaction of qubits with their environment and the resulting changes in their quantum state in time.

“The nearly infinite number of vibrational modes or degrees of freedom in the environment makes calculating quantum dynamics particularly challenging. In effect, this task involves calculating the dynamics of a single quantum system while surrounded by trillions of others. Direct calculation is impossible in this case, as no computer can handle it. However, not all changes in the environment are equally important: those that occur at a sufficient distance from our quantum system are unable to affect its dynamics substantially. The division into ‘relevant’ and ‘irrelevant’ environmental degrees of freedom is at the basis of our method’, he says **Alexei Vagov**co-author of the article, Director of the MIEM HSE Center for Quantum Metamaterials.

According to the interpretation of quantum mechanics proposed by the famous American physicist Richard Feynman, calculating the quantum state of a system involves calculating the sum of all possible ways in which the state can be achieved. This interpretation assumes that a quantum particle (system) can move in all possible directions, including forward or backward, right or left, and even backwards in time. The quantum probabilities of all these trajectories must be added together to calculate the final state of the particle.

‘The problem is that there are too many possible trajectories for even one particle, let alone the entire environment. Our algorithm allows to consider only the trajectories that contribute significantly to the dynamics of the qubit, discarding those with negligible contributions. In our method, the evolution of a qubit and its environment is captured by tensors, which are matrices or tables of numbers describing the state of the entire system at different times. We therefore select only those portions of the tensors that are relevant to the dynamics of the system,’ he explains **Alexei Vagov**.

The researchers point out that the Automated Compression of Arbitrary Environments algorithm is publicly available and implemented as computer code. According to the authors, it opens up entirely new possibilities for precisely calculating the dynamics of multiple quantum systems. In particular, this method makes it possible to estimate the time until pairs of entangled photons in quantum telephone lines will unravel, the distance a quantum particle can be “teleported”, or how long it can take for the qubits of a computer quantum lose coherence.

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