Integrated optics is an engineering solution proposed for exquisite control of photonic quantum information. Here we use silicon photonics and the linear combination of quantum operators scheme to realise a fully programmable two-qubit quantum processor. The device is fabricated with readily available CMOS based processing and comprises four nonlinear photon-sources, four filters, eightytwo beam splitters and fifty-eight individually addressable phase shifters. To demonstrate performance, we programmed the device to implement ninety-eight various two-qubit unitary operations (with average quantum process fidelity of 93.2±4.5%), a two-qubit quantum approximate optimization algorithm and efficient simulation of Szegedy directed quantum walks. This fosters further use of the linear combination architecture with silicon photonics for future photonic quantum processors.The range and quality of control that a device has over quantum physics determines the extent of quantum information processing (QIP) tasks that it can perform. One device capable of performing any given QIP task is an ultimate goal 1 and silicon quantum photonics 2 has attractive traits to achieve this: photonic qubits are robust to environmental noise 5 , single qubit operations can be performed with high precision 16 , a high density of reconfigurable components have been used to manipulate coherent light 5,6 and established fabrication processes are CMOS compatible. However, quantum control needs to include entangling operations to be relevant to QIPthis is recognised as one of the most challenging tasks for photonics because of the extra resources required for each entangling step 5,6 . Here, we demonstrate a programmable silicon photonics chip that generates two photonic qubits, on which it then performs arbitrary twoqubit untiary operations, including arbitrary entangling operations. This is achieved by using silicon photonics to reach the complexity required to implement an iteration of the linear combination of unitaries architecture 8,9 that we have adapted to realise universal two-qubit processing. The device's performance shows that the design and fabrication techniques used in its implementation work well with the linear combination architecture and can be used to realise larger and more powerful photonic quantum processors.Miniaturisation of quantum-photonic experiments into chip-scale waveguide circuits started 10 from the need to realise many-mode devices with inherent sub-wavelength stability for generalised quantum-interference experi-ments, such as multi-photon quantum walks 11 and boson sampling 12-14 . Universal six-mode linear optics implemented with a silica waveguide chip (coupled to free-space photon sources and fibre-coupled detectors) demonstrated the principle that single photonic devices can be configured to perform any given linear optics task 15 . Silicon waveguides promise even greater capability for large-scale photonic processing, because of their third order nonlinearity that enables photon pair generation within integ...
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise, quantum walks have shown much potential as a framework for developing new quantum algorithms. Here we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs efficiently. We also show that solving the same sampling problem for arbitrary circulant quantum circuits is intractable for a classical computer, assuming conjectures from computational complexity theory. This is a new link between continuous-time quantum walks and computational complexity theory and it indicates a family of tasks that could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle, we experimentally implement the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.
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...
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