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arXiv:2103.07429 · Materials Theory

Constant-depth circuits for dynamic simulations of materials on quantum computers

Lindsay Bassman Oftelie, Roel Van Beeumen, Ed Younis, Ethan Smith, Costin Iancu, Wibe Albert de Jong

Published Mar 7, 2022• 38 citations
arXivHTMLPDF

Abstract

Abstract Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a specific subset of one-dimensional (1D) materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the constant-depth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many time-steps. For an N -spin system, the constant-depth circuit contains only $\mathcal {O}(N^{2})$ O ( N 2 ) CNOT gates. Such compact circuits enable us to successfully execute long-time dynamic simulation of ubiquitous models, such as the transverse field Ising and XY models, on current quantum hardware for systems of up to 5 qubits without the need for complex error mitigation techniques. Aside from enabling long-time dynamic simulations with minimal Trotter error for a specific subset of 1D Hamiltonians, our constant-depth circuits can advance materials simulations on quantum computers more broadly in a number of indirect ways.

DOI
10.1186/s41313-022-00043-x

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Abstract Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a specific subset of one-dimensional (1D) materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the constant-depth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many time-steps. For an N -spin system, the constant-depth circuit contains only O(N2)\mathcal {O}(N^{2})O(N2) O ( N 2 ) CNOT gates. Such compact circuits enable us to successfully execute long-time dynamic simulation of ubiquitous models, such as the transverse field Ising and XY models, on current quantum hardware for systems of up to 5 qubits without the need for complex error mitigation techniques. Aside from enabling long-time dynamic simulations with minimal Trotter error for a specific subset of 1D Hamiltonians, our constant-depth circuits can advance materials simulations on quantum computers more broadly in a number of indirect ways.
Abstract Determining the ground and excited state properties of materials is considered one of the most promising applications of quantum computers. On near-term hardware, the limiting constraint on such simulations is the requisite circuit depths and qubit numbers, which currently lie well beyond near-term capabilities. Here we develop a quantum algorithm which reduces the estimated cost of material simulations. For example, we obtain a circuit depth improvement by up to 6 orders of magnitude for a Trotter layer of time-dynamics simulation in the transition-metal oxide SrVO3compared with the best previous quantum algorithms. We achieve this by introducing a collection of connected techniques, including highly localised and physically compact representations of materials Hamiltonians in the Wannier basis, a hybrid fermion-to-qubit mapping, and an efficient circuit compiler. Combined together, these methods leverage locality of materials Hamiltonians and result in a design that generates quantum circuits with depth independent of the system’s size. Although the requisite resources for the quantum simulation of materials are still beyond current hardware, our results show that realistic simulation of specific properties may be feasible without necessarily requiring fully scalable, fault-tolerant quantum computers, providing quantum algorithm design incorporates deeper understanding of the target materials and applications.
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The quantum circuit model is the de-facto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In this work, we develop quantum algorithms for Hamiltonian simulation "one level below" the circuit model, exploiting the underlying control over qubit interactions available in most quantum hardware and deriving analytic circuit identities for synthesising multi-qubit evolutions from two-qubit interactions. We then analyse the impact of these techniques under the standard error model where errors occur per gate, and an error model with a constant error rate per unit time. To quantify the benefits of this approach, we apply it to time-dynamics simulation of the 2D spin Fermi-Hubbard model. Combined with new error bounds for Trotter product formulas tailored to the non-asymptotic regime and an analysis of error propagation, we find that e.g. for a 5 × 5 Fermi-Hubbard lattice we reduce the circuit depth from 1, 243, 586 using the best previous fermion encoding and error bounds in the literature, to 3, 209 in the per-gate error model, or the circuit-depth-equivalent to 259 in the per-time error model. This brings Hamiltonian simulation, previously beyond reach of current hardware for non-trivial examples, significantly closer to being feasible in the NISQ era.
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.
The increasing depth of quantum circuits presents a major limitation for the execution of quantum algorithms, as the limited coherence time of physical qubits leads to noise that manifests as errors during computation. In this work, we focus on CNOT ladder circuits, which find applications in several quantum computing tasks, including the preparation of GHZ states, the implementation of fan-out and long-range CNOT gates, fermionic simulations, and the construction of ansatz circuits for variational quantum computing. The linearly increasing depth of a CNOT ladder circuit can be exchanged for constant CNOT depth at the expense of wider circuits that rely on mid-circuit measurements and classically controlled operations. Our error analysis shows that the choice between these two constructions depends on the relative difference between CNOT and idling error rates. Overall, the technique developed in this work enables low-depth implementations of circuits that are ubiquitous in quantum computing algorithms.
Dynamic quantum simulation is a leading application for achieving quantum advantage. However, high circuit depths remain a limiting factor on near-term quantum hardware. We present a compilation algorithm based on Matrix Product Operators for generating compressed circuits enabling real-time simulation on digital quantum computers, that for a given depth are more accurate than all Trotterizations of the same depth. By the efficient use of environment tensors, the algorithm is scalable in depth far beyond prior work, and we present circuit compilations of up to 64 layers of SU(4) gates. Surpassing only 1D circuits, our approach can flexibly target a particular quasi-2D gate topology. We demonstrate this by compiling a 52-qubit 2D Transverse-Field Ising propagator onto the IBM Heavy-Hex topology. For all circuit depths and widths tested, we produce circuits with smaller errors than all equivalent depth Trotter unitaries, corresponding to reductions in error by up to 4 orders of magnitude and circuit depth compressions with a factor of over 6.