Variational quantum simulation of general processes

Variational quantum algorithms have been proposed to solve static and dynamic problems of closed many-body quantum systems. Here we investigate variational quantum simulation of three general types of tasks---generalised time evolution with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum system dynamics. The algorithm for generalised time evolution provides a unified framework for variational quantum simulation. In particular, we show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalised time evolution. Meanwhile, assuming a tensor product structure of the matrices, we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution. Finally, we introduce variational quantum simulation for open system dynamics. We variationally implement the stochastic Schr\"odinger equation, which consists of dissipative evolution and stochastic jump processes. We numerically test the algorithm with a six-qubit 2D transverse field Ising model under dissipation.Comment: 18 page

Quantum simulation of quantum field theories as quantum chemistry

Conformal truncation is a powerful numerical method for solving generic strongly-coupled quantum field theories based on purely field-theoretic technics without introducing lattice regularization. We discuss possible speedups for performing those computations using quantum devices, with the help of near-term and future quantum algorithms. We show that this construction is very similar to quantum simulation problems appearing in quantum chemistry (which are widely investigated in quantum information science), and the renormalization group theory provides a field theory interpretation of conformal truncation simulation. Taking two-dimensional Quantum Chromodynamics (QCD) as an example, we give various explicit calculations of variational and digital quantum simulations in the level of theories, classical trials, or quantum simulators from IBM, including adiabatic state preparation, variational quantum eigensolver, imaginary time evolution, and quantum Lanczos algorithm. Our work shows that quantum computation could not only help us understand fundamental physics in the lattice approximation, but also simulate quantum field theory methods directly, which are widely used in particle and nuclear physics, sharpening the statement of the quantum Church-Turing Thesis.Comment: 58 pages, many figures, some simulations. v2, v3, v4, v5, v6: small changes on errors and discussions of existing algorithms. Hamiltonians are generated using the code https://github.com/andrewliamfitz/LCT, associated with the paper arXiv:2005.1354

Measurement cost of metric-aware variational quantum algorithms

Variational quantum algorithms are promising tools for near-term quantum computers as their shallow circuits are robust to experimental imperfections. Their practical applicability, however, strongly depends on how many times their circuits need to be executed for sufficiently reducing shot-noise. We consider metric-aware quantum algorithms: variational algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object. For example, the recently introduced quantum natural gradient approach uses the quantum Fisher information matrix as a metric tensor to correct the gradient vector for the co-dependence of the circuit parameters. We rigorously characterise and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries. Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.Comment: 17 pages, 3 figure

Revealing quantum chaos with machine learning

Understanding properties of quantum matter is an outstanding challenge in science. In this paper, we demonstrate how machine-learning methods can be successfully applied for the classification of various regimes in single-particle and many-body systems. We realize neural network algorithms that perform a classification between regular and chaotic behavior in quantum billiard models with remarkably high accuracy. We use the variational autoencoder for autosupervised classification of regular/chaotic wave functions, as well as demonstrating that variational autoencoders could be used as a tool for detection of anomalous quantum states, such as quantum scars. By taking this method further, we show that machine learning techniques allow us to pin down the transition from integrability to many-body quantum chaos in Heisenberg XXZ spin chains. For both cases, we confirm the existence of universal W shapes that characterize the transition. Our results pave the way for exploring the power of machine learning tools for revealing exotic phenomena in quantum many-body systems.Comment: 12 pages, 12 figure