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arXiv:2306.16580

Quantum Imaginary Time Propagation algorithm for preparing thermal states

Francesco Turro

Published Jun 28, 2023• 0 citations
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Abstract

Calculations at finite temperatures are fundamental in different scientific fields, from nuclear physics to condensed matter. Evolution in imaginary time is a prominent classical technique for preparing thermal states of quantum systems. We propose a new quantum algorithm that prepares thermal states based on the quantum imaginary time propagation method, using a diluted operator with ancilla qubits to overcome the non-unitarity nature of the imaginary time operator. The presented method is the first that allows us to obtain the correct thermal density matrix on a general quantum processor for a generic Hamiltonian. We prove its reliability in the actual quantum hardware computing thermal properties for two and three neutron systems.

DOI
10.48550/arxiv.2306.16580

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Calculations at finite temperatures are fundamental in different scientific fields, from nuclear physics to condensed matter. Evolution in imaginary time is a prominent classical technique for preparing thermal states of quantum systems. We propose a new quantum algorithm that prepares thermal states based on the quantum imaginary time propagation method, using a diluted operator with ancilla qubits to overcome the non-unitarity nature of the imaginary time operator. The presented method is the first that allows us to obtain the correct thermal density matrix on a general quantum processor for a generic Hamiltonian. We prove its reliability in the actual quantum hardware computing thermal properties for two and three neutron systems.
Evolution in imaginary time is a prominent technique for finding the ground state of quantum many-body systems, and the heart of a number of numerical methods that have been used with great success in quantum chemistry, condensed matter, and nuclear physics. We propose an algorithm to implement imaginary time propagation on a quantum computer. Our algorithm is devised in the context of an efficient encoding into an optimized gate, drawing on the underlying characteristics of the quantum device of a unitary operation in an extended Hilbert space. However, we prove that for simple problems it can also be successfully applied to standard digital quantum machines. This work paves the way for porting quantum many-body methods based on imaginary-time propagation to near-term quantum devices, enabling the future quantum simulation of the ground states of a broad class of microscopic systems.
Abstract Imaginary-time evolution plays an important role in algorithms for computing ground-state and thermal equilibrium properties of quantum systems, but can be challenging to simulate on classical computers. Many quantum algorithms for imaginary-time evolution have resource requirements that are prohibitive for current quantum devices and face performance issues due to noise. Here, we propose a new algorithm for computing imaginary-time evolved expectation values on quantum computers, inspired by probabilistic error cancellation, an error-mitigation technique. Our algorithm works by decomposing a Trotterization of imaginary-time evolution into a probabilistic linear combination of operations, each of which is then implemented on a quantum computer. The measurement data is then classically post-processed to obtain the expectation value of the imaginary-time evolved state. Our algorithm requires no ancillary qubits and can be made noise-resilient without additional error-mitigation. It is well-suited for estimating thermal expectation values by making use of the notion of a thermal pure quantum state. We demonstrate our algorithm by performing numerical simulations of thermal pure quantum state preparation for the 1D Heisenberg Hamiltonian on 8 qubits, and by using an IBM quantum computer to estimate the energy of the same Hamiltonian on 2 qubits. We observe promising results compared to the exact values, illustrating the potential of our algorithm for probing the physics of quantum many-body systems on current hardware.
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Nature is governed by precise physical laws, which can inspire the discovery of new computer-run simulation algorithms. Thermal states are the most ubiquitous for they are the equilibrium states of matter. Simulating thermal states of quantum matter has applications ranging from quantum machine learning to better understanding of high-temperature superconductivity and quantum chemistry. The computational complexity of this task is hopelessly hard for classical computers. The existing quantum algorithms come with caveats: most either require quantum phase estimation rendering them impractical for current noisy hardware, or are variational which face obstacles such as initialization, barren plateaus, and a general lack of provable guarantee. We provide two quantum algorithms with provable guarantees to prepare thermal states on (near-term) quantum computers that avoid these drawbacks. The first algorithm is inspired by the natural thermalization process where the ancilla qubits act as the infinite thermal bath. This algorithm can potentially run in polynomial time to sample thermal distributions of ergodic systems -- the vast class of physical systems that equilibrate in isolation with respect to local observables. The second algorithm works for any system and in general runs in exponential time. However, it requires significantly smaller quantum resources than previous such algorithms. In addition, we provide an error mitigation technique for both algorithms to fight back decoherence, which enables us to run our algorithms on the near-term quantum devices. To illustration, we simulate the thermal state of the hardcore Bose-Hubbard model on the latest generation of available quantum computers.
Abstract Imaginary-time evolution is fundamental for analyzing quantum many-body systems, with applications spanning quantum chemistry, condensed matter physics, and quantum field theory, yet classical simulation requires exponentially growing resources in both system size and evolution time. While quantum approaches reduce the system-size scaling, existing methods rely on heuristic techniques with measurement precision or success probability that deteriorates as evolution time increases. We present a quantum algorithm that prepares normalized imaginary-time evolved states using an adaptive normalization factor to maintain a stable success probability over long imaginary-time intervals. Our algorithm approximates the target state with error polynomially small in the inverse imaginary time using a polynomial number of elementary quantum gates and a single ancilla qubit, with success probability close to one. When the initial state has reasonable overlap with the ground state, this algorithm also achieves polynomial query complexity in the system size. To our knowledge, this is the first quantum algorithm for imaginary-time evolution with provably polynomial resource scaling in evolution time. Numerical experiments validate our theoretical analysis for evolution time up to 50, demonstrating the algorithm’s effectiveness for long-time evolution. Building on this technique, we further develop imaginary-time-evolution-based algorithms for ground-state-related problems and for simulating open quantum systems. These algorithms can reduce circuit depth in certain regimes compared with existing methods, at the expense of higher total query complexity, advancing the practical feasibility of quantum simulation on early fault-tolerant devices.
Imaginary-time evolution is a powerful tool in the study of quantum physics. However, existing classical algorithms for simulating imaginary-time evolution suffer from high computational complexity as the dimension of the quantum system increases. In this study we propose a quantum algorithm for implementing imaginary-time evolution using nonunitary quantum circuits with one ancillary qubit. The success probability of our algorithm is a polynomial function of the output error and can be enhanced by reorganizing the terms of the Hamiltonian. To illustrate the practicality of our algorithm on current quantum devices, we conduct a demonstration on superconducting and trapped-ion quantum processors to calculate the ground-state energy and determine the most stable molecular structure of H2{\mathrm{H}}_{2}H2​. Additionally, we validate the feasibility of our algorithm by numerically simulating the ground-state energies of LiH molecules and the quantum Ising chain. In contrast to existing algorithms, our method provides a systematic approach to construct the required nonunitary circuits using universal quantum gates, making it suitable for experimental implementation. Our algorithm opens up possibilities for exploring other physical phenomena such as finite-temperature properties and non-Hermitian systems.