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arXiv:1812.03607 · Physical Review A

Minimal effective Gibbs ansatz: A simple protocol for extracting an accurate thermal representation for quantum simulation

Jeffrey Cohn, Fan Yang, Khadijeh Najafi, Barbara Jones, James K. Freericks

Published Aug 28, 2020• 20 citations
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Abstract

Quantum Gibbs state sampling algorithms generally suffer from either scaling exponentially with system size or requiring specific knowledge of spectral properties a priori. These algorithms also require a large overhead of bath or scratch or ancilla qubits. We propose a method, termed the minimal effective Gibbs ansatz (MEGA), which uses a quantum computer to determine a small effective ensemble of pure states that accurately reproduce thermal averages of an objective dynamic correlation function. This technique employs properties of correlation functions that can be split into lesser and greater parts; here, we primarily focus on single-particle Green's functions and density-density correlators. When properly measured, these correlation functions provide a simple test to indicate how close a given pure state or ensemble of pure states is to providing accurate thermal expectation values. Further, we show that, when properties such as the eigenstate thermalization hypothesis hold, this approach leads to accurate results with a sparse ensemble of pure states; sometimes only one suffices. We illustrate the ansatz using exact diagonalization simulations on small clusters for the Fermi-Hubbard and Hubbard-like models. Even if the MEGA becomes as computationally complex as other Gibbs state samplers, it still gains an advantage due to its ease of implementation without any a priori information about the Hamiltonian and in the efficient allocation of available qubits by eliminating bath qubits and using a minimal number of ancillas.

DOI
10.1103/physreva.102.022622

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Quantum Gibbs state sampling algorithms generally suffer from either scaling exponentially with system size or requiring specific knowledge of spectral properties a priori. These algorithms also require a large overhead of bath or scratch or ancilla qubits. We propose a method, termed the minimal effective Gibbs ansatz (MEGA), which uses a quantum computer to determine a small effective ensemble of pure states that accurately reproduce thermal averages of an objective dynamic correlation function. This technique employs properties of correlation functions that can be split into lesser and greater parts; here, we primarily focus on single-particle Green's functions and density-density correlators. When properly measured, these correlation functions provide a simple test to indicate how close a given pure state or ensemble of pure states is to providing accurate thermal expectation values. Further, we show that, when properties such as the eigenstate thermalization hypothesis hold, this approach leads to accurate results with a sparse ensemble of pure states; sometimes only one suffices. We illustrate the ansatz using exact diagonalization simulations on small clusters for the Fermi-Hubbard and Hubbard-like models. Even if the MEGA becomes as computationally complex as other Gibbs state samplers, it still gains an advantage due to its ease of implementation without any a priori information about the Hamiltonian and in the efficient allocation of available qubits by eliminating bath qubits and using a minimal number of ancillas.
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We present an efficient protocol for estimating MMM observables to precision ε\varepsilonε from only O(log⁡(M)/ε2)\mathcal{O}(\log (M)/\varepsilon^2)O(log(M)/ε2) copies of a Gibbs state ρ∝exp⁡(−βH)ρ\propto\exp(-βH)ρ∝exp(−βH), given access to its Hamiltonian HHH. The protocol uses single-copy, nonadaptive measurements and a total Hamiltonian simulation time of O~(βM/ε2)\widetilde{\mathcal{O}}(βM/\varepsilon^2)O(βM/ε2); we show that the sample complexity is optimal in a black-box setting. The key idea of our protocol is a new interpretation of quantum Gibbs samplers as detailed-balanced measurement channels: measurements that preserve the Gibbs state when outcomes are marginalized, and thus enable Gibbs samples to be reused for measuring subsequent observables. Consequently, shadow tomography of thermal states admits a general efficient algorithm when controlled Hamiltonian simulation is available, substantially boosting the data acquisition rate in quantum thermal simulation workflows.
Abstract Modeling the dynamics of a quantum system connected to the environment is critical for advancing our understanding of complex quantum processes, as most quantum processes in nature are affected by an environment. Modeling a macroscopic environment on a quantum simulator may be achieved by coupling independent ancilla qubits that facilitate energy exchange in an appropriate manner with the system and mimic an environment. This approach requires a large, and possibly exponential number of ancillary degrees of freedom which is impractical. In contrast, we develop a digital quantum algorithm that simulates interaction with an environment using a small number of ancilla qubits. By combining periodic modulation of the ancilla energies, or spectral combing, with periodic reset operations, we are able to mimic interaction with a large environment and generate thermal states of interacting many-body systems. We evaluate the algorithm by simulating preparation of thermal states of the transverse Ising model. Our algorithm can also be viewed as a quantum Markov chain Monte Carlo process that allows sampling of the Gibbs distribution of a multivariate model. To demonstrate this we evaluate the accuracy of sampling Gibbs distributions of simple probabilistic graphical models using the algorithm.
The preparation of Gibbs thermal states is an important task in quantum computation with applications in quantum simulation, quantum optimization, and quantum machine learning. However, many algorithms for preparing Gibbs states rely on quantum subroutines which are difficult to implement on near-term hardware. Here, we address this by (i) introducing an objective function that, unlike the free energy, is easily measured, and (ii) using dynamically generated, problem-tailored ansätze. This allows for arbitrarily accurate Gibbs state preparation using low-depth circuits. To verify the effectiveness of our approach, we numerically demonstrate that our algorithm can prepare high-fidelity Gibbs states across a broad range of temperatures and for a variety of Hamiltonians.