How Quantum AI could help you

**live to 200**, like a bowhead whale. Google's quantum supremacy news opens exciting doors. A quantum computer completed a task**in 200 seconds**that would take**10,000 years**on today's fastest supercomputer. We explore the special connection between quantum computers and AI. Together, they will change everything.**Quantum supremacy**

In quantum computing,

**quantum supremacy**is the goal of demonstrating that a programmable quantum device can solve a problem that no classical computer can solve in any feasible amount of time (irrespective of the usefulness of the problem). Conceptually, quantum supremacy involves both the engineering task of building a powerful quantum computer and the computational-complexity-theoretic task of finding a problem that can be solved by that quantum computer and has a superpolynomial speedup over the best known or possible classical algorithm for that task. The term was originally popularized by John Preskill, but the concept of a**quantum computational advantage**, specifically for simulating quantum systems, dates back to Yuri Manin's (1980) and Richard Feynman's (1981) proposals of quantum computing. Examples of proposals to demonstrate quantum supremacy include the boson sampling proposal of Aaronson and Arkhipov, D-Wave's specialized frustrated cluster loop problems, and sampling the output of random quantum circuits. However, despite the lofty promises of quantum computation, the likelihood for high error when performing computations with a large number of qubits as well as advances in classical computation that are keeping classical performance comparable to quantum performance encourages skeptics to doubt the importance of demonstrating quantum supremacy at all.**Computational complexity**

Complexity arguments concern how the amount of some resource needed to solve a problem (generally time or memory) scales with the size of the input. In this setting, a problem consists of an inputted problem instance (

**a binary string**) and returned solution (**corresponding output string**), while resources refers to designated elementary operations, memory usage, or communication. A collection of local operations allows for the computer to generate the output string. A circuit model and its corresponding operations are useful in describing both classical and quantum problems; the classical circuit model consists of basic operations such as AND gates, OR gates, and NOT gates while the quantum model consists of classical circuits and the application of unitary operations. Unlike the finite set of classical gates, there are an**infinite amount of quantum gates**due to the continuous nature of unitary operations. In both classical and quantum cases, complexity swells with increasing problem size. As an extension of classical computational complexity theory, quantum complexity theory considers what a theoretical universal quantum computer could accomplish without accounting for the difficulty of building a physical quantum computer or dealing with decoherence and noise. Since quantum information is a generalization of classical information, quantum computers can simulate any classical algorithm.Quantum complexity classes are

**sets of problems**that share a common quantum computational model, with each model containing specified resource constraints. Circuit models are useful in describing quantum complexity classes. The most useful quantum complexity class is BQP (bounded-error quantum polynomial time), the class of decision problems that can be solved in polynomial time by a universal quantum computer.Questions about BQP still remain, such as the connection between BQP and the polynomial-time hierarchy, whether or not BQP contains NP-complete problems, and the exact lower and upper bounds of the BQP class. Not only would answers to these questions reveal the nature of BQP, but they would also answer**difficult classical complexity theory questions**. One strategy for better understanding BQP is by defining related classes, ordering them into a conventional class hierarchy, and then looking for properties that are revealed by their relation to BQP. There are several other quantum complexity classes, such as QMA (quantum Merlin Arthur) and QIP (quantum interactive polynomial time).The difficulty of proving what cannot be done with classical computing is a common problem in definitively demonstrating

**quantum supremacy**. Contrary to decision problems that require yes or no answers, sampling problems ask for samples from probability distributions. If there is a classical algorithm that can efficiently sample from the output of an arbitrary quantum circuit, the polynomial hierarchy would collapse to the third level, which is generally considered to be very unlikely. Boson sampling is a more specific proposal, the classical hardness of which depends upon the intractability of calculating the permanent of a large matrix with complex entries, which is a #P-complete problem. The arguments used to reach this conclusion have also been extended to IQP Sampling,where only the conjecture that the average- and worst-case complexities of the problem are the same is needed.**Proposed experiments**

The following are proposals for demonstrating quantum computational supremacy using current technology, often called NISQ devices. Such proposals include (1)

**a well-defined computational problem**, (2)**a quantum algorithm to solve this problem**, (3)**a comparison best-case classical algorithm to solve the problem**, and (4)**a complexity-theoretic argument that, under a reasonable assumption, no classical algorithm can perform significantly better than current algorithms**(so the quantum algorithm still provides a superpolynomial speedup).**Shor's algorithm for factoring integers**

This algorithm finds the prime

**factorization of an n-bit integer**in time. It can also provide a speedup for any problem that reduces to integer factoring, including the membership problem for matrix groups over fields of odd order.This algorithm is important both practically and historically for quantum computing. It was the first polynomial-time quantum algorithm proposed for a real-world problem that is believed to be hard for classical computers. Namely, it gives a superpolynomial speedup under the reasonable assumption that RSA, today's most common encryption protocol, is secure.

Factoring has some benefit over other supremacy proposals because factoring can be checked quickly with a classical computer just by multiplying integers, even for large instances where factoring algorithms are intractably slow. However, implementing Shor's algorithm for large numbers is infeasible with current technology,so it is not being pursued as a strategy for demonstrating supremacy.

**Boson sampling**

This computing paradigm based upon sending identical photons through a linear-optical network can solve certain sampling and search problems that, assuming a few complexity-theoretical conjectures (that calculating the permanent of Gaussian matrices is #P-Hard and that the polynomial hierarchy does not collapse) are intractable for classical computers. However, it has been shown that boson sampling in a system with large enough loss and noise can be simulated efficiently.

The largest experimental implementation of boson sampling to date had 6 modes so could handle up to 6 photons at a time. The best proposed classical algorithm for simulating boson sampling runs in time for a system with n photons and m output modes. BosonSampling is an open-source implementation in R. The algorithm leads to an estimate of 50 photons required to demonstrate quantum supremacy with boson sampling.

**Sampling the output distribution of random quantum circuits**

The best-known algorithm for simulating an arbitrary random quantum circuit requires an amount of time that scales exponentially with the number of qubits, leading one group to estimate that

**around 50 qubits could be enough to demonstrate quantum supremacy**.Google had announced its intention to demonstrate quantum supremacy by the end of 2017 by constructing and running a 49-qubit chip that would be able to sample distributions inaccessible to any current classical computers in a reasonable amount of time. The largest universal quantum circuit simulator running on classical supercomputers at the time was able to simulate 48 qubits. But for particular kinds of circuits, larger quantum circuit simulations with 56 qubits are possible. This may require increasing the number of qubits to demonstrate quantum supremacy.On October 23, 2019, Google published the results of this quantum supremacy experiment in the Nature article, “**Quantum Supremacy Using a Programmable Superconducting Processor**” in which they**developed a new 53-qubit processor**, named “**Sycamore**”, that is capable of fast, high-fidelity quantum logic gates, in order to perform the benchmark testing. Google claims that their machine performed the target computation in 200 seconds, and estimated that their classical algorithm would take 10,000 years in the world’s fastest supercomputer to solve the same problem. IBM disputed this claim, saying that an improved classical algorithm should be able to solve that problem in two and a half days on that same supercomputer.Thanks to Wikipedia - Quantum supremacy