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Register a new userBerry Phase Estimation in Gate-Based Adiabatic Quantum Simulation
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Gate-based quantum computers can in principle simulate the adiabatic dynamics of a large class of Hamiltonians. Here we consider the cyclic adiabatic evolution of a parameter in the Hamiltonian. We propose a quantum algorithm to estimate the Berry phase and use it to classify the topological order of both single-particle and interacting models, highlighting the differences between the two. This algorithm is immediately extensible to any interacting topological system. Our results evidence the potential of near-term quantum hardware for the topological classification of quantum matter.
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We obtain the adiabatic Berry phase by defining a generalised gauge potential whose line integral gives the phase holonomy for arbitrary evolutions of parameters. Keeping in mind that for classical integrable systems it is hardly clear how to obtain open-path Hannay angle, we establish a connection between the open-path Berry phase and Hannay angle by using the parametrised coherent state approach. Using the semiclassical wavefunction we analyse the open-path Berry phase and obtain the open-path Hannay angle. Further, by expressing the adiabatic Berry phase in terms of the commutator of instantaneous projectors with its differential and using Wigner representation of operators we obtain the Poisson bracket between distribution function and its differential. This enables us to talk about the classical limit of the phase holonomy which yields the angle holonomy for open-paths. An operational definition of Hannay angle is provided based on the idea of classical limit of quantum mechanical inner product. A probable application of the open-path Berry phase and Hannay angle to wave-packet revival phenomena is also pointed out.
We present a general theoretical framework for the exact treatment of a hybrid system that is composed of a quantum subsystem and a classical subsystem. When the quantum subsystem is dynamically fast and the classical subsystem is slow, a vector potential is generated with a simple canonical transformation. This vector potential, on one hand, gives rise to the familiar Berry phase in the fast quantum dynamics; on the other hand, it yields a Lorentz-like force in the slow classical dynamics. In this way, the pure phase (Berry phase) of a wavefunction is linked to a physical force.
We revisit quantum phase estimation algorithms for the purpose of obtaining the energy levels of many-body Hamiltonians and pay particular attention to the statistical analysis of their outputs. We introduce the mean phase direction of the parent distribution associated with eigenstate inputs as a new post-processing tool. By connecting it with the unknown phase, we find that if used as its direct estimator, it exceeds the accuracy of the standard majority rule using one less bit of resolution, making evident that it can also be inverted to provide unbiased estimation. Moreover, we show how to directly use this quantity to accurately find the energy levels when the initialized state is an eigenstate of the simulated propagator during the whole time evolution, which allows for shallower algorithms. We then use IBM Q hardware to carry out the digital quantum simulation of three toy models: a two-level system, a two-spin Ising model and a two-site Hubbard model at half-filling. Methodologies are provided to implement Trotterization and reduce the variability of results in noisy intermediate scale quantum computers.
Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems. However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, i.e., quantum simulation. Quantum simulation promises to have applications in the study of many problems in, e.g., condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology. Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct. A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators. This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field.
State preparation is a process encoding the classical data into the quantum systems. Based on quantum phase estimation, we propose the specific quantum circuits for a deterministic state preparation algorithm and a probabilistic state preparation algorithm. To discuss the gate complexity in these algorithms, we decompose the diagonal unitary operators included in the phase estimation algorithms into the basic gates. Thus, we associate the state preparation problem with the decomposition problem of the diagonal unitary operators. We analyse the fidelities in the two algorithms and discuss the success probability in the probabilistic algorithm. In this case, we explain that the efficient decomposition of the corresponding diagonal unitary operators is the sufficient condition for state preparation problems.
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