Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansätze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry—calculating the ground-state molecular energy for He–H+. The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future.

Quantum walks of correlated particles offer the possibility to study large-scale quantum interference, simulate biological, chemical and physical systems, and a route to universal quantum computation. Here we demonstrate quantum walks of two identical photons in an array of 21 continuously evanescently-coupled waveguides in a SiOxNy chip. We observe quantum correlations, violating a classical limit by 76 standard deviations, and find that they depend critically on the input state of the quantum walk. These results open the way to a powerful approach to quantum walks using correlated particles to encode information in an exponentially larger state space.With origins dating back to observations by Lucretius in 60BC and Brown in the 1800's, random walks are a powerful tool used in a broad range of fields from genetics to economics [1]. The quantum mechanical analoguequantum walks [2, 3]-corresponds to the tunnelling of quantum particles into several possible sites, generating large coherent superposition states and allowing massive parallelism in exploring multiple trajectories through a given connected graph (eg. Fig. 1). This quantum state evolution is a reversible (unitary) process and so requires low noise (decoherence) systems for observation. In contrast to the diffusive behaviour of (classical) random walks, which tend towards a steady state, the wave function in a quantum walk propagates ballistically (Fig. 2(c)). These features are at the heart of new algorithms for database-search [4], random graph navigation, models for quantum communication using spin chains [5], universal quantum computation [6] and quantum simulation [7].Quantum walks have been demonstrated using nuclear magnetic resonance [8,9], phase [10,11] and position [12] space of trapped ions, the frequency space of an optical resonator [13], single photons in bulk [14] and fibre [15] optics and the scattering of light in coupled waveguide arrays [16]. However, to date, all realisations have been limited to single particle quantum walks, which have an exact mapping to classical wave phenomena [17], and therefore cannot provide any advantage from quantum effects (note that the quantum walk with two trapped ions [11] encodes in the centre of mass mode and is therefore effectively a single particle quantum walk on a line). Indeed single particle quantum walks have been observed using classical light [16,18]. In contrast, for quantum walks of more than one indistinguishable particle, classical theory no longer provides a sufficient description-quantum theory predicts that probability amplitudes interfere leading to distinctly non-classical correlations [19,20]. This quantum behaviour gives rise to a computational advantage in quantum walks of two identical particles, which can be used to solve the graph isomorphism problem for example [21]. The major challenge associated with realising quantum walks of correlated particles is the need for a low decoherence system that preserves their non-classical features.The intrinsically low decoherence properti...

Graph states are special kinds of multipartite entangled states that correspond to mathematical graphs where the vertices take the role of quantum spin systems and the edges represent interactions. They not only provide an efficient model to study multiparticle entanglement, but also find wide applications in quantum error correction, multi-party quantum communication and most prominently, serve as the central resource in one-way quantum computation. Here we report the creation of two special instances of graph states, the six-photon Greenberger-Horne-Zeilinger states -- the largest photonic Schr\"{o}dinger cat, and the six-photon cluster states-- a state-of-the-art one-way quantum computer. Flexibly, slight modifications of our method allow creation of many other graph states. Thus we have demonstrated the ability of entangling six photons and engineering multiqubit graph states, and created a test-bed for investigations of one-way quantum computation and studies of multiparticle entanglement as well as foundational issues such as nonlocality and decoherence

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...

We introduce the concept of an eigenstate witness and use it to find energies of quantum systems with quantum computers.