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arXiv:2203.02521 · Quantum

Quantum algorithms for grid-based variational time evolution

Pauline J. Ollitrault, Sven Jandura, Alexander Miessen, Irène Burghardt, Rocco Martinazzo, Francesco Tacchino, Ivano Tavernelli

Published Oct 12, 2023• 24 citations
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Abstract

The simulation of quantum dynamics calls for quantum algorithms working in first quantized grid encodings. Here, we propose a variational quantum algorithm for performing quantum dynamics in first quantization. In addition to the usual reduction in circuit depth conferred by variational approaches, this algorithm also enjoys several advantages compared to previously proposed ones. For instance, variational approaches suffer from the need for a large number of measurements. However, the grid encoding of first quantized Hamiltonians only requires measuring in position and momentum bases, irrespective of the system size. Their combination with variational approaches is therefore particularly attractive. Moreover, heuristic variational forms can be employed to overcome the limitation of the hard decomposition of Trotterized first quantized Hamiltonians into quantum gates. We apply this quantum algorithm to the dynamics of several systems in one and two dimensions. Our simulations exhibit the previously observed numerical instabilities of variational time propagation approaches. We show how they can be significantly attenuated through subspace diagonalization at a cost of an additional O(MN2) 2-qubit gates where M is the number of dimensions and NM is the total number of grid points.

DOI
10.22331/q-2023-10-12-1139

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The simulation of quantum dynamics calls for quantum algorithms working in first quantized grid encodings. Here, we propose a variational quantum algorithm for performing quantum dynamics in first quantization. In addition to the usual reduction in circuit depth conferred by variational approaches, this algorithm also enjoys several advantages compared to previously proposed ones. For instance, variational approaches suffer from the need for a large number of measurements. However, the grid encoding of first quantized Hamiltonians only requires measuring in position and momentum bases, irrespective of the system size. Their combination with variational approaches is therefore particularly attractive. Moreover, heuristic variational forms can be employed to overcome the limitation of the hard decomposition of Trotterized first quantized Hamiltonians into quantum gates. We apply this quantum algorithm to the dynamics of several systems in one and two dimensions. Our simulations exhibit the previously observed numerical instabilities of variational time propagation approaches. We show how they can be significantly attenuated through subspace diagonalization at a cost of an additional O(MN2) 2-qubit gates where M is the number of dimensions and NM is the total number of grid points.
Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter approximation of the time-evolution operator. This approach typically relies on deep circuits and is therefore hampered by the substantial limitations of available noisy and near-term quantum hardware. On the other hand, variational quantum algorithms (VQAs) have become an indispensable alternative, enabling small-scale simulations on present-day hardware. However, despite the recent development of VQAs for quantum dynamics, a detailed assessment of their efficiency and scalability is yet to be presented. To fill this gap, we applied a VQA based on McLachlan's principle to simulate the dynamics of a spin-boson model subject to varying levels of realistic hardware noise as well as in different physical regimes, and discuss the algorithm's accuracy and scaling behavior as a function of system size. We observe a good performance of the variational approach used in combination with a general, physically motivated wave function ansatz, and compare it to the conventional first-order Trotter evolution. Finally, based on this, we make scaling predictions for the simulation of a classically intractable system. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage for the solution of time-dependent problems.
Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.
Simulating the time evolution of quantum systems is a classically challenging and often intractable task due to the exponential scaling of its constituent operators. This scaling makes it a promising application for quantum computation, especially for correlated systems. Constructing an approximate time evolution operator can be done using the Trotter-Suzuki decomposition with an error that scales at most polynomially in the target simulation time and can be decreased at the cost of longer circuit depth. Variational approaches to classically optimize this approximation have shown significant improvement in accuracy for large quantum systems in both gate count and accuracy by several orders of magnitude. However, it is unclear if a similar improvement can be obtained for strongly correlated systems. We extend this variational approach to two example systems with strong correlations: a one-dimensional Fermi-Hubbard chain and an extended Fermi-Hubbard chain. The encoding between qubits and fermionic modes in these systems is generally nonlocal and so we propose an encoding scheme for these Hamiltonians with minimal overall operator distance between nearest neighbor connected qubits. We demonstrate that this variational method grants a similarly strong improvement in both gate count and accuracy and examine the performance of these optimized gate sequences using realistic noise models to estimate their performance on noisy quantum hardware. Finally, we discuss an approach to combine the optimized nonlocal gates of small chains to larger systems without the need for additional computational resources.
We propose an adaptive quantum algorithm to prepare accurate variational time evolved wave functions. The method is based on the projected variational quantum dynamics (pVQD) algorithm, that performs a global optimization with linear scaling in the number of variational parameters. Instead of fixing a variational ansatz at the beginning of the simulation, the circuit is grown systematically during the time evolution. Moreover, the adaptive step does not require auxiliary qubits and the gate search can be performed in parallel on different quantum devices. We apply the algorithm, named adaptive pVQD, to the simulation of driven spin models and fermionic systems, where it shows an advantage when compared to both Trotterized circuits and nonadaptive variational methods. Finally, we use the shallower circuits prepared using the adaptive pVQD algorithm to obtain more accurate measurements of physical properties of quantum systems on hardware. Published by the American Physical Society 2024
Simulating time evolution of quantum systems is one of the most promising applications of quantum computing and also appears as a subroutine in many applications such as Green's function methods. In the current era of NISQ machines we assess the state of algorithms for simulating time dynamics with limited resources. We propose the Jaynes-Cummings model and extensions to it as useful toy models to investigate time evolution algorithms on near-term quantum computers. Using these simple models, direct Trotterisation of the time evolution operator produces deep circuits, requiring coherence times out of reach on current NISQ hardware. Therefore we test two alternative responses to this problem: variational compilation of the time evolution operator, and variational quantum simulation of the wavefunction ansatz. We demonstrate numerically to what extent these methods are successful in time evolving this system. The costs in terms of circuit depth and number of measurements are compared quantitatively, along with other drawbacks and advantages of each method. We find that computational requirements for both methods make them suitable for performing time evolution simulations of our models on NISQ hardware. Our results also indicate that variational quantum compilation produces more accurate results than variational quantum simulation, at the cost of a larger number of measurements.
Dynamical quantum simulation may be one of the first applications to see quantum advantage. However, the circuit depth of standard Trotterization methods can rapidly exceed the coherence time of noisy quantum computers. This has led to recent proposals for variational approaches to dynamical simulation. In this work, we aim to make variational dynamical simulation even more practical and near-term. We propose a new algorithm called Variational Hamiltonian Diagonalization (VHD), which approximately transforms a given Hamiltonian into a diagonal form that can be easily exponentiated. VHD allows for fast forwarding, i.e., simulation beyond the coherence time of the quantum computer with a fixed-depth quantum circuit. It also removes Trotterization error and allows simulation of the entire Hilbert space. We prove an operational meaning for the VHD cost function in terms of the average simulation fidelity. Moreover, we prove that the VHD cost function does not exhibit a shallow-depth barren plateau, i.e., its gradient does not vanish exponentially. Our proof relies on locality of the Hamiltonian, and hence we connect locality to trainability. Our numerical simulations verify that VHD can be used for fast-forwarding dynamics.