Quantum Kernel Function Expansion for Thermal Quantum Ensemble

Abstract

Simulating quantum many-body systems is one major application of quantum computing, having a huge potential to impact the fields of computational physics and quantum chemistry. To fulfill the potential impacts, it is crucial to design quantum algorithms that efficiently employ the computation power of the quantum computing devices. Here, we introduce a quantum kernel function expansion algorithm for determining the thermodynamic quantities of quantum many-body systems, including local observables, free energy, and thermal entropy, which has been inspired by the kernel polynomial method in classical Hamiltonian simulation. We design quantum circuits to measure Fourier moments which constitute the thermal ensemble average as an analytic function of energy density via kernel Fourier expansion. Implementing this algorithm on a quantum computer has exponential quantum advantage in the cost of time and space, as compared to its classical analogue. In computing finite temperature properties of a quantum system our quantum algorithm has an overall polynomial time complexity, provided that the corresponding Hamiltonian ground state problem belongs to BQP. We demonstrate its efficiency with applications to one and two-dimensional quantum spin models, and a fermionic lattice. By analyzing the realization on digital and analogue quantum devices, we show the quantum algorithm is accessible to near-term quantum technology.

Calculating the many-body density of states on a digital quantum computer

“…Using pseudorandom states to stochastically evaluate the trace has been proposed in various papers [35,[49][50][51][52], and recently used on quantum hardware to extract hightemperature transport exponents [11]. In order to generate pseudo-random states, one can use a circuit composed of alternating layers of 2-qubit gates and layers with random single qubit rotations, as suggested by Ref.…”

“…Block encoding of a Hamiltonian is deeply connected with the Chebyshev polynomials [30], implementing the Hamiltonian as a quantum walk as exploited in the context of the KPM in [31] and more generally to estimate physical properties in [32,33]. An alternative is to compute the Chebyshev moments iteratively in a variational quantum algorithm [34] or otherwise overcoming the problem of implementing the Chebyshev polynomials using suitably defined Fourier ones [35,36].…”

A Near-Term Quantum Algorithm for Computing Molecular and Materials Properties based on Recursive Variational Series Methods

“…Alternatively, one can circumvent the problem by switching to a Fourier basis, as shown in Ref. [38]. Lanczos recursion is closely related to KPM, but instead of using moment expansion it performs iterative construction of the continued fraction, representing a spectral function.…”