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Preparation of excited states on a quantum computer

Alessandro Roggero, Chenyi Gu, Alessandro Baroni, T. Papenbrock

Published Sep 28, 2020• 0 citations
OpenAlex

Abstract

We study two different methods to prepare excited states on a quantum computer, a key initial step to study dynamics within linear response theory. The first method uses unitary evolution for a short time $T=\mathcal{O}(\sqrt{1-F})$ to approximate the action of an excitation operator $\hat{O}$ with fidelity $F$ and success probability $P\approx1-F$. The second method probabilistically applies the excitation operator using the Linear Combination of Unitaries (LCU) algorithm. We benchmark these techniques on emulated and real quantum devices, using a toy model for thermal neutron-proton capture. Despite its larger memory footprint, the LCU-based method is efficient even on current generation noisy devices and can be implemented at a lower gate cost than a naive analysis would suggest. These findings show that quantum techniques designed to achieve good asymptotic scaling on fault tolerant quantum devices might also provide practical benefits on devices with limited connectivity and gate fidelity.

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We study two different methods to prepare excited states on a quantum computer, a key initial step to study dynamics within linear response theory. The first method uses unitary evolution for a short time T=O(1−F)T=\mathcal{O}(\sqrt{1-F})T=O(1−F​) to approximate the action of an excitation operator O^\hat{O}O^ with fidelity FFF and success probability P≈1−FP\approx1-FP≈1−F. The second method probabilistically applies the excitation operator using the Linear Combination of Unitaries (LCU) algorithm. We benchmark these techniques on emulated and real quantum devices, using a toy model for thermal neutron-proton capture. Despite its larger memory footprint, the LCU-based method is efficient even on current generation noisy devices and can be implemented at a lower gate cost than a naive analysis would suggest. These findings show that quantum techniques designed to achieve good asymptotic scaling on fault tolerant quantum devices might also provide practical benefits on devices with limited connectivity and gate fidelity.
Quantum mechanical problems are among the hardest to simulate and, in some cases, remain intractable even for the most powerful computers. Quantum computing has emerged as a new technological platform to address such challenges, with rapid advances in recent years. Yet, current quantum devices remain noisy and limited in scale. Hence, it is essential to identify classically hard problems of practical interest and tractable with existing quantum devices. Among potential applications, the real-time simulation of quantum systems is one of the most promising to deliver an early, practical quantum advantage. This doctoral thesis is therefore centered around simulating quantum dynamics on quantum devices. We first present an overview of the most relevant quantum algorithms for quantum dynamics, highlighting respective advantages and limitations. Further, we identify relevant problems within quantum dynamics that could benefit from quantum simulation in the near future. Second, we propose a method for benchmarking hardware and error mitigation algorithms that is based on well-understood theoretical results and suffers from no scaling issues. The resulting quality metric is intuitive and transferable to other applications. We successfully implement the scheme on up to 133 qubits, demonstrating coherent evolution up to a two-qubit gate depth of 28, featuring 1396 two-qubit gates, before noise becomes prevalent. Third, we propose a novel variant of probabilistic error amplification to implement open quantum dynamics, relying on characterizing and altering hardware noise to mimic the system-environment interaction under study. Lastly, we present two studies on state preparation and phase classification: first, a hybrid algorithm to prepare ground states of electron-phonon systems; second, a quantum machine learning-based approach to distinguish phases in previously prepared quantum states.
We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method ( Single-Ancilla LCU ) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach ( Analog LCU ) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method ( Ancilla-free LCU ) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.
Diagonal unitary operators are commonly found in many quantum algorithms. They find application as analytical potential operators for quantum simulation, as well as for complex oracles used in quantum searches. However, in order to implement a quantum algorithm on a given quantum device, each operator must be decomposed into a sequence of fault-tolerant, device-level instructions. In general, to implement an nnn-qubit diagonal unitary {\\em exactly} on a quantum computer generally requires 2n+1−32^{n+1}-32n+1−3 one- and two-qubit gates. However, for most practical implementations of diagonal unitaries, some degree of approximation will be necessary if the circuit is to be efficient. In this thesis we develop two complementary methods for the approximate synthesis of quantum circuits for diagonal unitaries. We show how to apply these techniques to real-space quantum simulation and show how efficient high fidelity quantum simulations can be implemented with low-depth quantum circuits.
Electronic excited states are central to a vast array of physical and chemical phenomena, yet accurate and efficient methods for preparing them on quantum devices remain challenging and comparatively underexplored. We introduce a general dissipative algorithm for selectively preparing ab initio electronic excited states. The key idea is to recast excited-state preparation as an effective ground-state problem by suitably modifying the underlying Lindblad dynamics so that the target excited state becomes the unique steady state of a designed quantum channel. We develop three complementary strategies, tailored to different types of prior information about the excited state, such as symmetry and approximate energy. We demonstrate the effectiveness and versatility of these schemes through numerical simulations of atomic and molecular spectra, including valence excitations in prototypical planar conjugated molecules and transition-metal complexes. Taken together, these results provide a new pathway for advancing quantum simulation methods for realistic strongly correlated electronic systems.
Abstract The linear combination of unitaries (LCU) method has proven to scale better than existing product formulas in simulating long time Hamiltonian dynamics. However, given the number of multi-control gate operations in the standard prepare-select-unprepare architecture of LCU, it is still resource-intensive to implement on the current quantum computers. In this work, we demonstrate LCU implementations on an ion trap quantum computer for calculating squared overlaps | ⟨ ψ ( t = 0 ) | ψ ( t > 0 ) ⟩ | 2 of time-evolved states. This is achieved by an optimized LCU method, based on pre-selecting relevant unitaries, coupled with a compilation strategy which makes use of quantum multiplexor gates, leading to a significant reduction in the depth and number of two-qubit(2Q) gates in circuits. For L Pauli strings in a Taylor series expanded n -qubit-mapped time evolution operator, we find a 2Q gate count of 2 ⌈ log 2 ( L ) ⌉ ( 2 n + 1 ) − n − 2 . We test this approach by simulating a Rabi–Hubbard Hamiltonian.