The fundamental problem faced in quantum chemistry is the calculation of molecular properties, which are of practical importance in fields ranging from materials science to biochemistry. Within chemical precision, the total energy of a molecule as well as most other properties, can be calculated by solving the Schrödinger equation. However, the computational resources required to obtain exact solutions on a conventional computer generally increase exponentially with the number of atoms involved 1,2 . This renders such calculations intractable for all but the smallest of systems. Recently, an efficient algorithm has been proposed enabling a quantum computer to overcome this problem by achieving only a polynomial resource scaling with system size 2,3,4 . Such a tool would therefore provide an extremely powerful tool for new science and technology. Here we present a photonic implementation for the smallest problem: obtaining the energies of H 2 , the hydrogen molecule in a minimal basis. We perform a key algorithmic step-the iterative phase estimation algorithm 5,6,7,8 -in full, achieving a high level of precision and robustness to error. We implement other algorithmic steps with assistance from a classical computer and explain how this non-scalable approach could be avoided. Finally, we provide new theoretical results which lay the foundations for the next generation of simulation experiments using quantum computers. We have made early experimental progress towards the long-term goal of exploiting quantum information to speed up quantum chemistry calculations.Experimentalists are just beginning to command the level of control over quantum systems required to explore their information processing capabilities. An important long-term application is to simulate and calculate properties of other many-body quantum systems. Pioneering experiments were first performed using nuclear-magnetic-resonance-based systems to simulate quantum oscillators 9 , leading up to recent simulations of a pairing Hamiltonian 7,10 . Very recently the phase transitions of a two-spin quantum magnet were simulated 11 using an ion-trap system. Here we simulate a quantum chemical system and calculate its energy spectrum, using a photonic system. Molecular energies are represented as the eigenvalues of an associated time-independent HamiltonianĤ and can be efficiently obtained to fixed accuracy, using a quantum algorithm with three distinct steps 6 : encoding a molecular wavefunction into qubits; simulating its time evolution using quantum logic gates; and extracting the approximate energy using the phase estimation algorithm 3,12 . The latter is a general-purpose quantum algorithm for evaluating the eigenvalues of arbitrary Hermitian or unitary operators. The algorithm estimates the phase, φ, accumulated by a molecular eigenstate, |Ψ , under the action of the time-evolution operator,Û =e −iĤt/ , i.e.,where E is the energy eigenvalue of |Ψ . Therefore, estimating the phase for each eigenstate amounts to estimating the eigenvalues of the Hamiltonia...
By weakly measuring the polarization of a photon between two strong polarization measurements, we experimentally investigate the correlation between the appearance of anomalous values in quantum weak measurements and the violation of realism and nonintrusiveness of measurements. A quantitative formulation of the latter concept is expressed in terms of a Leggett-Garg inequality for the outcomes of subsequent measurements of an individual quantum system. We experimentally violate the Leggett-Garg inequality for several measurement strengths. Furthermore, we experimentally demonstrate that there is a one-to-one correlation between achieving strange weak values and violating the LeggettGarg inequality.here has been much debate in quantum physics over the question of whether measurable quantities have definite values prior to their measurement. Key ideas addressing this question include the Bell inequality, which considers correlations between measurements on components of a composite system that are space-like separated (1, 2) and contextuality tests, which examine whether identical experiments produce results in different "classically equivalent" contexts (3, 4). A conceptually elegant extension to these ideas is the Leggett-Garg inequality (LGI) (5), which is an inequality constructed from the correlation functions of a series of three consecutive measurements on a single system. Leggett and Garg derive limits based on the joint assumptions of (i) macroscopic realism: An observable for a system will have a definite value at all times; and (ii) noninvasive measurement: It is possible to determine this value with arbitrarily small disturbance on the subsequent evolution of the system. The limits on the value of the inequality derived from these assumptions differ from the predictions of quantum mechanics. Thus the LGI tests the limits of measurement and macroscopic realism.Here we present an experimental test of a generalized LGI using weak measurements (6-9) of the polarization of single photons and measure violations by up to 14 standard deviations. Additionally, we experimentally demonstrate a one-to-one relation (10, 11) between LGI violations and strange weak-valued measurements (6-8), which also arise from the inability to assign values to physical quantities between an earlier and a later measurement.Testing the LGI requires monitoring the system without projecting it into a specific state. For a quantum system a quantum nondemolition (QND) experiment (12-14) would be one way to do this. But QND measurements are not the only way to perform a noninvasive measurement. A generalization of the QND measurement is the so-called weak measurement (6). A weak measurement is one for which it is possible to adjust the strength of the measurement and, in principle, to reduce the back action on the system to an arbitrarily small amount. In other words, a weak measurement is one for which the level of "invasivness" can be controlled.
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