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Register a new userQuantum Filter Diagonalization: Quantum Eigendecomposition without Full Quantum Phase Estimation
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We develop a quantum filter diagonalization method (QFD) that lies somewhere between the variational quantum eigensolver (VQE) and the phase estimation algorithm (PEA) in terms of required quantum circuit resources and conceptual simplicity. QFD uses a set of of time-propagated guess states as a variational basis for approximate diagonalization of a sparse Pauli Hamiltonian. The variational coefficients of the basis functions are determined by the Rayleigh-Ritz procedure by classically solving a generalized eigenvalue problem in the space of time-propagated guess states. The matrix elements of the subspace Hamiltonian and subspace metric matrix are each determined in quantum circuits by a one-ancilla extended swap test, i.e., statistical convergence of a one-ancilla PEA circuit. These matrix elements can be determined by many parallel quantum circuit evaluations, and the final Ritz estimates for the eigenvectors can conceptually be prepared as a linear combination over separate quantum state preparation circuits. The QFD method naturally provides for the computation of ground-state, excited-state, and transition expectation values. We numerically demonstrate the potential of the method by classical simulations of the QFD algorithm for an N=8 octamer of BChl-a chromophores represented by an 8-qubit ab initio exciton model (AIEM) Hamiltonian. Using only a handful of time-displacement points and a coarse, variational Trotter expansion of the time propagation operators, the QFD method recovers an accurate prediction of the absorption spectrum.
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We demonstrate a method that merges the quantum filter diagonalization (QFD) approach for hybrid quantum/classical solution of the time-independent electronic Schrodinger equation with a low-rank double factorization (DF) approach for the representation of the electronic Hamiltonian. In particular, we explore the use of sparse compressed double factorization (C-DF) truncation of the Hamiltonian within the time-propagation elements of QFD, while retaining a similarly compressed but numerically converged double-factorized representation of the Hamiltonian for the operator expectation values needed in the QFD quantum matrix elements. Together with significant circuit reduction optimizations and number-preserving post-selection/echo-sequencing error mitigation strategies, the method is found to provide accurate predictions for low-lying eigenspectra in a number of representative molecular systems, while requiring reasonably short circuit depths and modest measurement costs. The method is demonstrated by experiments on noise-free simulators, decoherence- and shot-noise including simulators, and real quantum hardware.
By using a systematic optimization approach we determine quantum states of light with definite photon number leading to the best possible precision in optical two mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.
Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians $H_{rm I}$ and $H_{rm F}$. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of $H_{rm F}$ from the ground state of $H_{rm I}$. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $tau$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only non-adiabatic but also thermal excitations are likely to occur. It is shown that -- using only local Hamiltonians -- for the 1d quantum Ising model and the cluster model in a transverse field the conventional straight line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.
We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e. the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.
We introduce a new statistical and variational approach to the phase estimation algorithm (PEA). Unlike the traditional and iterative PEAs which return only an eigenphase estimate, the proposed method can determine any unknown eigenstate-eigenphase pair from a given unitary matrix utilizing a simplified version of the hardware intended for the Iterative PEA (IPEA). This is achieved by treating the probabilistic output of an IPEA-like circuit as an eigenstate-eigenphase proximity metric, using this metric to estimate the proximity of the input state and input phase to the nearest eigenstate-eigenphase pair and approaching this pair via a variational process on the input state and phase. This method may search over the entire computational space, or can efficiently search for eigenphases (eigenstates) within some specified range (directions), allowing those with some prior knowledge of their system to search for particular solutions. We show the simulation results of the method with the Qiskit package on the IBM Q platform and on a local computer.
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