Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets

Abstract

Quantum computers can be used to address molecular structure, materials science and condensed matter physics problems, which currently stretch the limits of existing high-performance computing resources [1]. Finding exact numerical solutions to these interacting fermion problems has exponential cost, while Monte Carlo methods are plagued by the fermionic sign problem. These limitations of classical computational methods have made even few-atom molecular structures problems of practical interest for medium-sized quantum computers.Yet, thus far experimental implementations have been restricted to molecules involving only Period I elements [2][3][4][5][6][7][8]. Here, we demonstrate the experimental optimization of up to six-qubit Hamiltonian problems with over a hundred Pauli terms, determining the ground state energy for molecules of increasing size, up to BeH 2 . This is enabled by a hardware-efficient variational quantum eigensolver with trial states specifically tailored to the available interactions in our quantum processor, combined with a compact encoding of fermionic Hamiltonians [9] and a robust stochastic optimization routine [10]. We further demonstrate the flexibility of our approach by applying the technique to a problem of quantum magnetism [11]. Across all studied problems, we find agreement between experiment and numerical simulations with a noisy model of the device. These results help elucidate the requirements for scaling the method to larger systems, and aim at bridging the gap between problems at the forefront of high-performance computing and their implementation on quantum hardware.The fundamental goal of addressing molecular structure problems is to solve for the ground state energy of many-body interacting fermionic Hamiltonians. Solving this problem on a quantum computer relies on a mapping between fermionic and qubit operators [12]. This restates it as a specific instance of a local Hamiltonian problem on a set of qubits. Given a k-local Hamiltonian H, composed of terms that act on at most k qubits, the solution to the local Hamiltonian problem amounts to finding its * These authors contributed equally to this work.smallest eigenvalue E G ,To date, no efficient algorithm is known that can solve this problem in full generality. For k ≥ 2 the problem is known to be QMA-complete [13]. However, it is expected that physical systems have Hamiltonians that do not constitute hard instances of this problem, and can be solved efficiently on a quantum computer, while remaining hard to solve classically. Following Feynman's idea for quantum simulation, a quantum algorithm for the ground state problem of interacting fermions was proposed in [14] and [15]. The approach relies on a good initial state that has a large overlap with the ground state and then solves the problem using the quantum phase estimation algorithm (PEA) [16]. While PEA has been demonstrated to achieve extremely accurate energy estimates for quantum chemistry [2, 3, 5, 8], it applies stringent requirements on quantum coherence.An a...

Analysis of a parametrically driven exchange-type gate and a two-photon excitation gate between superconducting qubits

“…It follows from (12) that the Fourier transform of the coefficients α i,± (t) consists of a series of peaks at integer multiples of the modulation frequency ω Φ with weights given by (13). Inserting the result (12) into (9) and (10) yields analogous expansions for the coupling parameters Ω ± (t) = k Ω ± (k)e ikωΦt and for the detunings ∆ ± (t) = k ∆ ± (k)e ikωΦt (which determine the phases ϕ ± (t)).…”

“…With superconducting circuits, single qubit quantum gates can be carried out with fidelities approaching 99.99% [1-4], while errors in two-qubit operations are typically higher with record fidelities around 99% [3,5]. However, the realization of qubit operations with even higher fidelity is required both for reaching the error threshold for quantum computation [6-9] and for carrying out reliable quantum simulations and optimizations in large arrays of coupled qubits [10][11][12][13]. Moreover, the quest for useful quantum computations before full quantum error correction becomes available may be assisted by efficient, short-depth gate sequences based on two-or multi-qubit gates [14,15] with versatile types of interactions.…”

Electronic Structure Calculations and the Ising Hamiltonian

“…The mapping of Hamiltonian from H to H ′ can be written as: So far we are able to transform our Hamiltonian to a k-local Hamiltonian including only products of σ z terms. In the Supplementary Materials we show the details of the transformation to 2-local spin Hamiltonian but here we present an example of transforming 3-local to a 2-local Ising Hamiltonian [23].Here, we see that by including x 4 , one can show that minimizing 3-local is equivalent to minimizing the sum of 2-local terms.Finally, we succeed in transforming our initial complex electronic structure Hamiltonian from the secondquantization form to an Ising-type Hamiltonian which can be solved using existing quantum computing hardware [14,24,8,25].To illustrate this proposed method (details are in the Supplementary Materials), we present calculations for the Hydrogen molecule H 2 , the Helium dimer He 2 , HeH + diatomic molecule and the LiH molecule. First, we used the Bravyi-Kitaev transformation and the Jordan-Wigner transformation to convert the diatomic molecular Hamiltonian in the minimal basis set (STO-6G) to the spin Hamiltonian of (σ x , σ y , σ z ).…”

“…This demonstrates that one can generally map the electronic ground state energy of a molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on presently available quantum hardware. Moreover, the recent experimental results for simple few electrons diatomic molecules presented by the IBM group have shown that a hardware-efficient optimizer implemented on a 6-qubit superconducting quantum processor is capable of producing the potential energy surfaces of such molecules [24]. The development of efficient quantum hardware and the possibility of mapping the electronic structure problem into an Ising-type Hamiltonian may grant efficient ways to obtain exact solutions to the Schrödinger equation, this being one of the most daunting computational problems present in both chemistry and physics.…”

“…Finally, we succeed in transforming our initial complex electronic structure Hamiltonian from the secondquantization form to an Ising-type Hamiltonian which can be solved using existing quantum computing hardware [14,24,8,25].…”