256-bit Elliptic Curve Logarithm is demonstrated to compute 126 133 cat qubits according to Alice & Bob

256-Bit Elliptic Curve Logarithm Is Demonstrated To Compute 126 133 Cat Qubits According To Alice &Amp; Bob

Alice & Bob, a major developer of quantum computing technology based in Paris, France, announced a remarkable discovery in its most recent research publication. The findings, acquired in partnership with the CEA (France’s Atomic Energy Commission) and other University researchers from France, their study revealed that quantum computers using “cat qubits” like those developed by Alice & Bob would be able to run Shor’s algorithm with as little as 126 133 qubits to compute a 256-bit logarithm in 9 hours, 60 times less than the qubits required by certain other quantum computing methods to do the same task.

Quantum computers use qubits, which can be challenging to keep coherent (i.e., error-free) even for short periods of time. Scaling the number of qubits in a quantum system to a level where that system can give a quantum advantage to consumers while maintaining coherence over such a huge number of qubits is one of the industry’s biggest barriers. Many qubits in these systems are used up in “error-correcting” to ensure coherence. Because a considerable fraction of the qubits is employed for purposes other than calculation, this imposes significant qubit hardware overhead.

Thus, this paved the way for Alice & Bob to create error-free, universal quantum computers to assist corporations and researchers in solving the most difficult challenges. The company is working on self-correcting quantum bits known as cat qubits, enabling fault-tolerant quantum computing and the execution of any quantum algorithm.

Alice & Bob’s experiment resulted in 60 times less than the qubits required by certain other quantum computing methods to do the same task. As a result, employing cat qubits, a usable quantum computer could be developed years sooner than previously anticipated.

“The hardware efficiency that Alice & Bob’s technology offers can help get us into the quantum era sooner than expected before. Reducing the number of qubits required to achieve this performance means that scaling these machines to a size where they are useful for a wide range of applications can happen sooner than expected.”

Theau Peronnin, CEO at Alice & Bob.

“Our results highlight the benefit of using Alice and Bob’s platform and more broadly, of focusing on any platform exhibiting a noise bias to implement a large-scale quantum computer. This significantly reduces the number of qubits, by
up to two orders of magnitude, to run large scale algorithms.”

Nicolas Sangouard, researcher at CEA added

According to the study, Cat qubits are intriguing quantum computer-building elements. They have a configurable noise bias, resulting in an exponential suppression of bit-flips with the average photon number. A simple repetition code can defend against residual phase errors.

Quantum Computing Design using Shor’s Algorithm

Primarily, the study proposes a generic tool for analyzing the performance of quantum computing architectures using Shor’s algorithm for computing discrete logarithms on elliptic curves over prime fields – a hard classical problem at the heart of crypto-systems widely used for key exchange and digital signatures.

The security of these crypto-systems against classical attacks is based on exact knowledge about the performance of classical algorithms – knowledge that can be used to observe a quantum advantage. The study also presents a concrete architecture based on cat qubits located at the nodes of a 2D grid and have only physical links to their surrounding qubits. Routing qubits ensure all-to-all connectivity between logical qubits. Lattice surgery is used to build two-qubit gates, and gate teleportation is used to obtain Toffoli gates using an offline fault-tolerant magic state preparation based on projective measurements.

The study quantifies the cost of a repeating code by doing a performance analysis based on the computation of discrete logarithms on an elliptic curve with Shor’s algorithm. It provides significant information for selecting a large-scale architecture using cat qubits. They propose implementing two-qubit gates using lattice surgery and Toffoli gates with off-line fault-tolerant preparation of magic states via projective measurements and subsequent gate teleportations by focusing on a 2D grid of cat qubits with nearby connectivity.

Circuit-level Error Model

The circuit-level error model, which covers every conceivable location for an error to occur, is used to evaluate the logical failure of a circuit from physical errors. The researchers described enhanced quantum computing of discrete logarithms on elliptic curves, using the one used to secure signatures in Bitcoin transactions as an example.

In conclusion, the study found that implementing Shor’s algorithm for the factorization of 2 048-RSA integers would take 349 133 cat qubits and four days to demonstrate the benefits of employing this technique for performance analysis. To demonstrate the benefit of employing a 1D code, this cost estimate reveals that a 2D grid of superconducting qubits using a typical surface code would require 20 million qubits and 8 hours to realize the same factorization. They discovered that it takes 126 133 qubits to compute a 256-bit logarithm in 9 hours.

It should also be noted that parallelization was not used in this work. This might greatly cut runtime, especially since magical state preparation is resource-efficient. Increasing the number of their factories would not significantly increase the total amount of qubits, allowing acceptable usage of the look-ahead adder.

Read the full research article here.