Hamiltonian Minimization in the NISQ Era

Yi Hu, Fanxu Meng, Xiaojun Wang +5 more

Abstract Simulating quantum systems is believed to be one of the most important applications of quantum computers. On noisy intermediate-scale quantum (NISQ) devices, the high-level circuit designed by quantum algorithms for Hamiltonian simulation needs to consider hardware limitations such as gate errors and circuit depth before it can be efficiently executed. In this work, we develop a hardware-agnostic circuit optimization algorithm to reduce the overall circuit cost for Hamiltonian simulation problems. Our method employ a novel sub-circuit synthesis in intermediate representation and propose a greedy ordering scheme for gate cancellation to minimize the gate count and circuit depth. To quantify the benefits of this approach, we benchmark proposed algorithm on different Hamiltonian models. Compared with state-of-the-art generic quantum compilers and specific quantum simulation compiler, the benchmarking results of our algorithm show an average reduction in circuit depth by 16.5× (up to 64.1×) and in gate count by 7.8× (up to 23.7×). This significant improvement helps enhance the performance of Hamiltonian simulation in the NISQ era.

Bikrant Bhattacharyya, Gokul Subramanian Ravi

Modern day quantum devices are in the NISQ era, meaning that the effects of size restrictions and noise are essential considerations when developing practical quantum algorithms. Recent developments have demonstrated that Variational Quantum Algorithms (VQAs) are an appropriate choice for current era quantum devices. VQAs make use of classical computation to iteratively optimize parameters of a quantum circuit, referred to as the ansatz. These parameters are usually chosen such that a given objective function is minimized. Generally, the cost function to be minimized is computed on a quantum device, and then the parameters are updated using a classical optimizer. One of the most promising VQAs for practical hardware is the Variational Quantum Eigensolver (VQE). VQE uses the expectation value of a given Hamiltonian as the cost function, and the minimal value of this particular cost function corresponds to the minimal eigenvalue of the Hamiltonian.Because evaluating quantum circuits is currently very noisy, developing classical bootstraps that help minimize the number of times a given circuit has to be evaluated is a powerful technique for improving the practicality of VQE. One possible such bootstrapping method is creating an ansatz which can be efficiently simulated on classical computers for restricted parameter values. Once the optimal set of restricted parameters is determined, they can be used as initial parameters for a VQE optimization which has access to the full parameter space. Stabilizer states are states which are generated by a particular group of operators called the Clifford Group. Because of the underlying structure of these operators, circuits consisting of only Clifford operators can be simulated efficiently on classical computers. Clifford Ansatz For Quantum Algorithms (CAFQA) is a proposed classical bootstrap for VQAs that uses ansatzes which reduce to clifford operators for restricted parameter values [1].CAFQA has been shown to produce fairly accurate initialization for VQE applied to molecular Hamiltonians. Motivated by this result, in this paper we seek to analyze the Clifford states that optimize the cost function for a new type of Hamiltonian, namely Transverse Field Ising Hamiltonians. Our primary result is contained in theorem IV.1 which connects the problem of finding the optimal CAFQA initialization to a submodular minimization problem which in turn can be solved in polynomial time.

Rijul Sachdeva, Vrinda Mehta, Manpreet Singh Jattana +2 more

We study algorithms inspired by quantum annealing that are suited for the NISQ era. First, we analyze approximate quantum annealing (AQA), which employs a discretized annealing ansatz in which the time step and the number of layers are allowed to deviate from a faithful implementation of quantum annealing. Parameter scans identify regimes that reproduce annealing-like behavior with reduced resources, making them more suitable for NISQ devices. The resulting parameters can then be used as an effective warm start for the quantum approximate optimization algorithm (QAOA), improving its performance compared to random initializations. We also introduce evolving Hamiltonian quantum optimization (EHQO), a multistep variational scheme that guides the optimization process through intermediate Hamiltonians derived from the standard annealing Hamiltonian. Numerical simulations on sets of hard 2-SAT instances suggest that quantum annealing-inspired algorithms provide practical strategies for enhancing variational quantum optimization.

Osama M. Raisuddin, Suvranu De

Kevin Thompson