A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requirement is typical of chaotic systems. We extend previous results in computational complexity to argue more formally that this sampling task must take exponential time in a classical computer. We study the convergence to the chaotic regime using extensive supercomputer simulations, modeling circuits with up to 42 qubits -the largest quantum circuits simulated to date for a computational task that approaches quantum supremacy. We argue that while chaotic states are extremely sensitive to errors, quantum supremacy can be achieved in the near-term with approximately fifty superconducting qubits. We introduce cross entropy as a useful benchmark of quantum circuits which approximates the circuit fidelity. We show that the cross entropy can be efficiently measured when circuit simulations are available. Beyond the classically tractable regime, the cross entropy can be extrapolated and compared with theoretical estimates of circuit fidelity to define a practical quantum supremacy test. arXiv:1608.00263v3 [quant-ph]

We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically, we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, either exactly in total variation distance or even approximately up to 41 per cent multiplicative error in the probabilities, then the infinite tower of classical complexity classes known as the polynomial hierarchy would collapse to its third level. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only O(log n) lines then it may, in fact, be classically efficiently sampled.

We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complextemperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture.Quantum computers are conjectured to outperform classical computers for a variety of important tasks ranging from integer factorisation [1] to the simulation of quantum mechanics [2]. However, to date there is relatively little rigorous evidence for this conjecture. It is well established that quantum computers can yield an exponential advantage in the query and communication complexity models. But in the more physically meaningful model of time complexity, there are no proven separations known between quantum and classical computation.This can be seen as a consequence of the extreme difficulty of proving bounds on the power of classical computing models, such as the famous P vs. NP problem. Given this difficulty, the most we can reasonably hope for is to show that quantum computations cannot be simulated efficiently classically, assuming some widely believed complexity-theoretic conjecture. For example, any set of quantum circuits that can implement Shor's algorithm [1] provides a canonical example, with the unlikely consequence of efficient classical simulation of this class of quantum circuits being the existence of an efficient classical factoring algorithm. However, one could hope for the existence of other examples that have wider-reaching complexity-theoretic consequences.With this in mind, in both [3] and [4] it was shown that the existence of an efficient classical sampler from a distribution that is close to the output distribution of an arbitrary quantum circuit, to within a small multiplicative error in each output probability, would imply that post-selected classical computation is equivalent to post-selected quantum computation. This consequence is considered very unlikely as it would collapse the infinite tower of complexity classes known as the Polynomial Hierarchy [5] to its third level. In both works this was proven even for non-universal quantum circuit families: commuting quantum circuits in the case of [3], and linear-optical networks in [4]. These non-universal families are of physical interest because they are simpler to implement, and easier to analyse because of the elegant mathematical structures on which they are based. * michael.bremner@uts.edu.au Unfortu...

We examine theoretic architectures and an abstract model for a restricted class of quantum computation, called here temporally unstructured ('instantaneous') quantum computation because it allows for essentially no temporal structure within the quantum dynamics. Using the theory of binary matroids, we argue that the paradigm is rich enough to enable sampling from probability distributions that cannot, classically, be sampled efficiently and accurately. This paradigm also admits simple interactive proof games that may convince a sceptic of the existence of truly quantum effects. Furthermore, these effects can be created using significantly fewer qubits than are required for running Shor's algorithm.

How useful is a quantum dynamical operation for quantum information processing? Motivated by this question, we investigate several strength measures quantifying the resources intrinsic to a quantum operation. We develop a general theory of such strength measures, based on axiomatic considerations independent of state-based resources. The power of this theory is demonstrated with applications to quantum communication complexity, quantum computational complexity, and entanglement generation by unitary operations.

What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? We provide an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling n-qubit Hamiltonian and local unitaries. It follows that universal quantum computation can be performed using any entangling interaction and local unitary operations.

Which gates are universal for quantum computation? Although it is well known that certain gates on two-level quantum systems (qubits), such as the controlled-not, are universal when assisted by arbitrary one-qubit gates, it has only recently become clear precisely what class of two-qubit gates is universal in this sense. We present an elementary proof that any entangling two-qubit gate is universal for quantum computation, when assisted by one-qubit gates. A proof of this result for systems of arbitrary finite dimension has been provided by Brylinski and Brylinski; however, their proof relies on a long argument using advanced mathematics. In contrast, our proof provides a simple constructive procedure which is close to optimal and experimentally practical.

The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O( √ n log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.