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Register a new userSimulating quantum dynamics: Evolution of algorithms in the HPC context
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Due to complexity of the systems and processes it addresses, the development of computational quantum physics is influenced by the progress in computing technology. Here we overview the evolution, from the late 1980s to the current year 2020, of the algorithms used to simulate dynamics of quantum systems. We put the emphasis on implementation aspects and computational resource scaling with the model size and propagation time. Our mini-review is based on a literature survey and our experience in implementing different types of algorithms.
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We consider the natural generalization of the Schr{o}dinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an $n$-qubit system for time $t$ within precision $epsilon$. If the Lindbladian consists of $mathrm{poly}(n)$ operators that can each be expressed as a linear combination of $mathrm{poly}(n)$ tensor products of Pauli operators then the gate cost of our algorithm is $O(t, mathrm{polylog}(t/epsilon)mathrm{poly}(n))$. We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schr{o}dinger evolution on a larger system and tracing out the ancillary system: the cost of such a textit{reduction} incurs an efficiency overhead of $O(t^2/epsilon)$ even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the linear combinations of unitaries construction that pertains to channels.
We present a suite of holographic quantum algorithms for efficient ground-state preparation and dynamical evolution of correlated spin-systems, which require far-fewer qubits than the number of spins being simulated. The algorithms exploit the equivalence between matrix-product states (MPS) and quantum channels, along with partial measurement and qubit re-use, in order to simulate a $D$-dimensional spin system using only a ($D$-1)-dimensional subset of qubits along with an ancillary qubit register whose size scales logarithmically in the amount of entanglement present in the simulated state. Ground states can either be directly prepared from a known MPS representation, or obtained via a holographic variational quantum eigensolver (holoVQE). Dynamics of MPS under local Hamiltonians for time $t$ can also be simulated with an additional (multiplicative) ${rm poly}(t)$ overhead in qubit resources. These techniques open the door to efficient quantum simulation of MPS with exponentially large bond-dimension, including ground-states of 2D and 3D systems, or thermalizing dynamics with rapid entanglement growth. As a demonstration of the potential resource savings, we implement a holoVQE simulation of the antiferromagnetic Heisenberg chain on a trapped-ion quantum computer, achieving within $10(3)%$ of the exact ground-state energy of an infinite chain using only a pair of qubits.
We present efficient quantum algorithms for simulating time-dependent Hamiltonian evolution of general input states using an oracular model of a quantum computer. Our algorithms use either constant or adaptively chosen time steps and are significant because they are the first to have time-complexities that are comparable to the best known methods for simulating time-independent Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian are satisfied. We provide a thorough cost analysis of these algorithms that considers discretizion errors in both the time and the representation of the Hamiltonian. In addition, we provide the first upper bounds for the error in Lie-Trotter-Suzuki approximations to unitary evolution operators, that use adaptively chosen time steps.
The quantum Rabi model describing the fundamental interaction between light and matter is a cornerstone of quantum physics. It predicts exotic phenomena like quantum phase transitions and ground-state entanglement in the ultrastrong-coupling (USC) regime, where coupling strengths are comparable to subsystem energies. Despite progress in many experimental platforms, the few experiments reaching USC have been limited to spectroscopy: demonstrating USC dynamics remains an outstanding challenge. Here, we employ a circuit QED chip with moderate coupling between a resonator and transmon qubit to realise accurate digital quantum simulation of USC dynamics. We advance the state of the art in solid-state digital quantum simulation by using up to 90 second-order Trotter steps and probing both subsystems in a combined Hilbert space dimension $sim80$, demonstrating the Schrodinger-cat like entanglement and build-up of large photon numbers characteristic of deep USC. This work opens the door to exploring extreme USC regimes, quantum phase transitions and many-body effects in the Dicke model.
The space of density matrices is embedded in a Euclidean space to deduce the dynamical equation satisfied by the state of an open quantum system. The Euclidean norm is used to obtain an explicit expression for the speed of the evolution of the state. The unitary contribution to the evolution speed is given by the modified skew information of the Hamiltonian, while the radial component of the evolution speed, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. An open-system analogue of the quantum navigation problem is posed, and a perturbative analysis is presented to identify the amount of change on the speed. Properties of the evolution speed are examined further through example systems, showing that the evolution speed need not be a decreasing function of time.
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