Recent advances on quantum computing hardware have pushed quantum computing to the verge of quantum supremacy. Here we bring together many-body quantum physics and quantum computing by using a method for strongly interacting two-dimensional systems, the Projected Entangled-Pair States, to realize an effective general-purpose simulator of quantum algorithms. We apply our method to study random quantum circuits, which are outstanding candidates to demonstrate quantum supremacy on quantum computers that supports nearest-neighbour gate operations on a two-dimensional configuration. Our approach allows to quantify precisely the memory usage and the time requirements of random quantum circuits, thus showing the frontier of quantum supremacy. Applying this general quantum circuit simulator we measured amplitudes for a 7 × 7 lattice of qubits with depth (1 + 40 + 1) and double-precision numbers in 31 minutes using less than 93 TB memory on the Tianhe-2 supercomputer. Our analytic complexity bounds also show that simulating a 8 × l circuit (l > 8) with depth (1 + 40 + 1), or a 10 × l (l > 10) circuit with depth (1 + 32 + 1) is within reach of current supercomputers.Quantum computers offer the promise of efficiently solving certain problems that are intractable for classical computers, most famously factorizing large numbers [1][2][3]. With the rapid progress of various quantum systems towards Noisy Intermediate-Scale Quantum computing devices [4][5][6][7][8][9][10][11], we are now on the verge of quantum supremacy [12], i.e. demonstrating that an actual quantum computer has the ability to do a computation that no classical computers can tackle, an important milestone in the field of computer science. Various candidates have been suggested to demonstrate quantum supremacy, such as BosonSampling [13,14], the instantaneous quantum polynomial protocol [15,16] and random quantum circuits (RQCs) [3,17] which demand less physical resources and are easier to implement compared to, for instance, factorization. The central aspect for all these near-term supremacy proof-of-principle computations, which poses fundamental limitations to classical computations, is that the quantum states produced, and from which we wish to sample configurations, live in a Hilbert space that grows exponentially with the system size.In view of recent progresses in quantum computing hardware, it is important to find effective ways to simulate accurately quantum algorithms on classical computers. While the quantum circuit simulator we present can tackle generic circuits, in the following we focus on RQCs. They consist of a series of single and two-qubit gates which are applied to different qubits in a particular order. A group of commuting gates which can be applied simultaneously constitute one layer of the circuit, and the more groups of operations that do not commute, the deeper the circuit is. The qualification of random circuit comes from the fact that the single-qubit gates applied are chosen at random from a small set of them (for more details about...

Applications of quantum walks can depend on the number, exchange symmetry and indistinguishability of the particles involved, and the underlying graph structures where they move. Here, we show that silicon photonics, by exploiting an entanglement-driven scheme, can realize quantum walks with full control over all these properties in one device. The device we realize implements entangled two-photon quantum walks on any five-vertex graph, with continuously tunable particle exchange symmetry and indistinguishability. We show how this simulates single-particle walks on larger graphs, with size and geometry controlled by tuning the properties of the composite quantum walkers. We apply the device to quantum walk algorithms for searching vertices in graphs and testing for graph isomorphisms. In doing so, we implement up to 100 sampled time steps of quantum walk evolution on each of 292 different graphs. This opens the way to large-scale, programmable quantum walk processors for classically intractable applications.

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