Quantum computers have improved and recent experiments have claimed quantum advantage -- completion of a computational task that is evidently hard for any conventional computer. The problem solved is that of obtaining samples, by quantum measurement, from a certain probability distribution. This review shows in what precise way quantum random sampling can be seen as a computation. It explains what that computation solves, in what way it outperforms classical computations, and what methods of verification are available. Quantum random sampling becomes a first test of the presumed exponential computational advantage of quantum computers over classical ones.
Quantum computers have improved and recent experiments have claimed quantum advantage -- completion of a computational task that is evidently hard for any conventional computer. The problem solved is that of obtaining samples, by quantum measurement, from a certain probability distribution. This review shows in what precise way quantum random sampling can be seen as a computation. It explains what that computation solves, in what way it outperforms classical computations, and what methods of verification are available. Quantum random sampling becomes a first test of the presumed exponential computational advantage of quantum computers over classical ones.
Abstract Demonstrating quantum advantage on noisy quantum devices is one of the most important tasks of quantum computing research in the near future. Random quantum circuit sampling is proposed as the most promising approach to the task and has been implemented in experiments for achieving quantum advantage. A series of encouraging computational complexity results, based on some plausible assumptions, show that this task is impossible to complete efficiently by classical computers. However, in practical experiments on noisy quantum devices, the approximate average‐case hardness and the effects of the noise need to be further checked. The competition between the classical simulation algorithm (mainly based on tensor network algorithms) and noisy quantum devices is the explicit way to understand quantum advantage in practice. This review briefly overviews the computational complexity arguments for the hardness of classical simulation of random quantum circuits sampling, then focus on various methods of classical simulation of quantum circuits based on tensor network method.
Quantum computing is rapidly becoming one of the largest research areas in quantum physics and yet we are still unable to prove the superiority of quantum computing over classical with the exception of a few contrived and unrealistic scenarios. Understanding the areas in which quantum computing provides an advantage and, more importantly, where that advantage comes from is the key to making progress in the field. We certainly do not solve this mystery in this thesis but do explore some unconventional routes through which answers may be found in the future. To this end we explore quantum advantage in a different computational setting. Rather than working with Turing Machines as most research does, we focus on the Bernoulli Factory: a randomness processing scheme studied in the fields of statistics and computer science. The Bernoulli Factory takes an infinite string of random bits generated with a fixed unknown bias as input and outputs a single random bit, the bias of which is a function of the input bias. This can be seen more intuitively as generating one biased coin from another, given a desired relationship between the two biases. The Bernoulli Factory is inherently probabilistic, like quantum mechanics, and so, it leads us to questions about the fundamental nature of quantum randomness and quantum sampling. We find, as we hoped, that the Bernoulli Factory is more tractable than the Turing Machine and that we are able to prove quantum superiority in both resource efficiency and classes of computable functions. We first give the quantum analogue of the classical Keane-O'’Brien theorem which establishes the increased class of computable functions for the quantum case. We then adjust the problem in a variety of ways, such as introducing multiple coins, errors in state preparation, phases and restriction on the bias or operations allowed. These lead to additional results which lend insight into the problems faced in quantum computing as well as the nature of quantum randomness.
Abstract The achievement of quantum computational advantage, also known as quantum supremacy, is a major milestone at which a quantum computer can solve a problem significantly faster than the world's most powerful classical computers. Two tasks, boson sampling and random quantum circuit sampling, have experimentally exhibited quantum advantages on photonic and superconducting platforms respectively. Classical benchmarking is essential, yet challenging, because these tasks are intractable for classical computers. This study reviews models, algorithms and large‐scale simulations of these two sampling tasks. These approaches continue to hold substantial significance for research in both current noisy intermediate‐scale quantum (NISQ) systems and future fault‐tolerant quantum computing.
Consider a fixed universe of N=2^{n} elements and the uniform distribution over elements of some subset of size K. Given samples from this distribution, the task of complement sampling is to provide a sample from the complementary subset. We give a simple quantum algorithm that uses only a single quantum sample-a single copy of the uniform superposition over elements of the subset. When K=N/2, we show that the quantum algorithm succeeds with probability 1, whereas any classical algorithm that succeeds with bounded probability of error requires a number of samples of the order of N. This shows that in a sample-to-sample setting, quantum computation can achieve the largest possible separation over classical computation. We show that the same bound can be lifted to prove average-case hardness, paving the way for demonstrations on noisy intermediate-scale quantum (NISQ) computers. It follows that under the assumption of the existence of one-way functions, complement sampling gives provable, verifiable and NISQable quantum advantage in a sample complexity setting.
Testing the predictions of quantum mechanics has been one of the main experimental endeavors for decades. Recent advancements in technology led to a number of demonstrations which test non-classicality via specific computational tasks. Limitations of these experiments include dependence on complexity theory assumptions, susceptibility to hardware noise and inefficient verification, raising questions about their scalability. We propose to test non-classicality using a game based on complement sampling, an efficiently verifiable problem that achieves the largest possible separation between quantum and classical computation when both input and output represent samples from probability distributions. When restricting the input to instances inspired by the Bernstein-Vazirani problem, our game admits an exponentially large violation of classicality unconditionally, i.e. without relying on computational hardness assumptions. We demonstrate the game on Quantinuum System Model H2 trapped-ion quantum computers, with experiments consisting of thousands of different circuits on up to 55 qubits. The observed scores can be explained by a systematic adoption of a quantum strategy, further corroborating the quantum nature of the hardware in an efficient and scalable way.
Quantum advantage is when quantum computers can outperform classical computers.One of the first largescale demonstrations of quantum advantage was recently done by Google running random multi-qubit quantum circuits.While Google's quantum computer took 200 seconds, calculations show that the then-fastest classical supercomputer would have taken 10000 years.Since then, different groups have claimed quantum advantage through other quantum systems.This paper will cover why quantum computers can be faster than classical computers by covering the primary differences between classical computing and quantum computing.Then, we will investigate the Deutsch-Jozsa quantum algorithm and Google's claim of quantum advantage.We will review current research regarding "quantum advantage," tying these events to the use cases covered.Additionally, different quantum algorithms that are theoretically faster than any current quantum algorithms will be analyzed, and we will delve into why this is the case.The covered examples will show methods used to improve quantum computers to provide a significant time difference to achieve a quantum advantage.After that, we make predictions concerning two critical quantum hardware limitations.Finally, we will investigate what the future of quantum advantage entails and some future cases of quantum advantage.