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Mathematics may have caught up with Google’s claim of quantum supremacy

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Enlarge / Google’s Sycamore processor.

In 2019, Word was excluded that Google-built quantum computers performed calculations they claimed to be virtually impossible to replicate on supercomputing hardware. It turned out to be completely incorrect, as Google neglected to consider the storage available on the supercomputer. If it was included, the quantum computer lead would shrink in just a few days.

However, adding just a handful of qubits will reestablish a large lead in quantum computers. But recently, a draft manuscript pointing out important facts was published on arXiv. Google’s claim relied on a comparison with a very specific approach to performing calculations on standard computing hardware. There are other ways to perform calculations, but this paper suggests that one of them can actually be used by a supercomputer to stay ahead of its quantum competitors.

Multiple roads to random

The calculations performed by Google were specially designed to be difficult to simulate on a regular computer. Set the 54 qubits of the Sycamore processor to a random state so that quantum interference between adjacent qubits affects the passage of time in the system. After a short interval, the hardware began repeatedly measuring the state of the qubit. A series of random bits were generated for each individual measurement, making Sycamore a very expensive random number generator. However, with sufficient measurements, the specific pattern produced by quantum interference becomes apparent.

Now that we understand this law of quantum interference, we can also calculate the patterns that should be found in random numbers generated by Sycamore. However, performing these calculations is very computationally expensive, and each time you add a qubit to your system, it is costly. Google estimates that it will take an unrealistically long time to perform these calculations on the world’s most advanced supercomputers.

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One flaw in this discussion pointed out early in the process was that Google didn’t consider storage connected to the world’s largest supercomputer. However, the reality remains that these calculations are extremely difficult with traditional computing hardware.

The new manuscript focuses on these calculations, which are an important aspect of the phrase. Google has chosen a very specific method of calculating the expected behavior of the processor, but there are other ways to do the equivalent. In the meantime, several options have been considered to improve performance. Currently, Feng Pan, Keyang Chen, and Pan Zhang describe specific ways to enable GPU-based clusters to produce comparable output in just 15 hours. They estimate that they perform better than the Sycamore quantum processor when run on major supercomputers.

Some brief math explanations

There are several ways to see what Pan, Chen and Zhang have achieved. Try three of them and gradually deepen your math.

The easiest way to display this is in terms of the output provided by Sycamore. Measuring the qubit state of the Sycamore processor individually produced a truly random string of 1s and 0s, but a sufficient measurement of the single initial state of the processor reveals the pattern. Setting up a classical calculation that summarizes Sycamore’s physics yields the same level of randomness and the same pattern.

The new paper describes how to trade off some of the calculated fidelity of processor behavior, but the process gets much faster computations. In other words, the new calculations do not accurately reproduce Sycamore’s behavior, but they still generate patterns and underlying randomness and can be completed much faster.

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That is one of the explanations. Option 2 to understand this involves considering how the start state of the Sycamore processor changes to the state at the time of measurement. Multiple possible paths reach it, and since this is a quantum system, we will explore them all. To get an accurate model of the output of a Sycamore processor, you need to look at every path, which is very computationally expensive. Pan, Chen, and Zhang have found a way to limit the paths displayed to make the computation easier to handle while achieving comparable output.

Those who want to avoid math should skip to the next section header. In the actual calculation method, the interaction of qubits in Sycamore is described as a 3D tensor network, and the tensor determines the relationship between the properties of the qubits. The algorithm then simplifies this by disconnecting some of the network connections. Researchers describe this as doing the equivalent of drilling a three-dimensional hole in a network.

Each time you drill a hole, the fidelity of the calculation is halved. However, this makes the fidelity fully adjustable. Just by limiting the number of holes to drill, you can fully reproduce the behavior of Sycamore. The mathematics of where these holes were made in the network was determined by the physical structure of the Sycamore chip itself.

The resulting contract tensor network was very easy to model, but researchers had to break it down into subtasks that could be stored on the system at work. They then used an algorithm to model the behavior of a smaller qubit network on Sycamore and show that it produces accurate results within the limits of fidelity.

Sources

1/ https://Google.com/

2/ https://arstechnica.com/science/2021/11/math-may-have-caught-up-with-googles-quantum-supremacy-claims/

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