<!--HTML-->Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entangled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit-based to measurement-based schemes. For each scheme, we provide an application in high energy physics, namely the Schwinger model and Z_2 lattice gauge theory.
<!--HTML-->Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entangled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit-based to measurement-based schemes. For each scheme, we provide an application in high energy physics, namely the Schwinger model and Z_2 lattice gauge theory.
Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entangled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit- to measurement-based schemes. Both schemes offer problem-specific advantages in terms of the required resources and coherence times.
Variational methods have long been used to study strongly-correlated quantum systems. Despite the successes of classical protocols, these are limited in the study of quantum systems, either due to numerical errors like the sign problem or the massive memory requirement to represent large quantum systems. Variational quantum eigensolvers (VQEs) overcome these problems and as a result have been an exciting area of research for the past decade. VQEs use a hybrid classical-quantum set-up, where a cost function (usually the expectation value of an operator) is produced by the quantum processor and then optimized on a classical computer. VQEs have been applied to fields such as quantum chemistry, condensed matter physics, high-energy physics, and classical optimization problems. In this thesis, we investigate VQEs to study gauge theories, which describe fundamental physical interactions and are hard to simulate classically. We use two different approaches in implementing a VQE to study two classes of problems. In the first half of the thesis, we design a VQE to study the U(1) Higgs gauge theory in one spatial dimension. This model is particularly interesting due to the presence of a topological term, which is a candidate to explain the observed CP violation in the universe. In order to perform a quantum simulation of the U(1) Higgs model, the theory is cast on a lattice. In the limit where the lattice size is large and the lattice spacing is small, the continuum limit can be reached and the results correctly resemble the original theory. However, when the lattice is too small or the spacing too big, results are spoiled by finite size effects. In this thesis, we investigate the effects of a finite lattice and determine the parameters to correctly reproduce a first-order phase transition of the continuous theory. Importantly, to resemble a realistic experiment, we include the statistical and intrinsic quantum noise in our VQE. We design the protocol to be applied on a microwave-photonic platform, which is well-suited due to the bosonic nature of the theory's electrical modes. \nIn the second half of the thesis, we design a new class of VQEs, based on a measurement-based quantum computer (MBQC) instead of the typical circuit-based set-up. Rather than performing gates on an initial state, a MBQC transforms an initial state by entangling it with a number of auxiliary qubits which are subsequently measured, resulting in a modification of the initial state. In this part of the thesis, we compare the standard, circuit-based VQE with our newly-designed measurement-based VQE (MB-VQE) through two meaningful examples. First, we introduce a novel way to construct variational state families, using measurement-based techniques, that is more costly to access with circuit-based VQEs. We apply this technique to find the ground state of a two-dimensional, periodic Z2 matter-free lattice gauge theory. In the second approach, we directly translate \nthe circuit-based VQE to a MB-VQE and apply this to the U(1) Schwinger lattice gauge theory, which is used as a benchmarking problem for VQEs. These examples demonstrate that there are specific problems which are better suited for a MB-VQE.
Abstract Measurement-based quantum computing (MBQC) is a promising approach to reducing circuit depth in noisy intermediate-scale quantum algorithms such as the variational quantum eigensolver (VQE). Unlike gate-based computing, MBQC employs local measurements on a preprepared resource state, offering a trade-off between circuit depth and qubit count. Ensuring determinism is crucial to MBQC, particularly in the VQE context, as a lack of flow in measurement patterns leads to evaluating the cost function at irrelevant locations. This study introduces MBVQE-ansätze that respect determinism and resemble the widely used problem-agnostic hardware-efficient VQE ansatz. We evaluate our approach using ideal simulations on the Schwinger Hamiltonian and XY-model and perform experiments on IBM hardware with an adaptive measurement capability. In our use case, we find that ensuring determinism works better via postselection than by adaptive measurements at the expense of increased sampling cost. Additionally, we propose an efficient MBQC-inspired method to prepare the resource state, specifically the cluster state, on hardware with heavy-hex connectivity, requiring a single measurement round, and implement this scheme on quantum computers with 27 and 127 qubits. We observe notable improvements for larger cluster states, although direct gate-based implementation achieves higher fidelity for smaller instances.
Measurement-based quantum computing (MBQC) is a promising approach to reducing circuit depth in noisy intermediate-scale quantum algorithms such as the Variational Quantum Eigensolver (VQE). Unlike gate-based computing, MBQC employs local measurements on a preprepared resource state, offering a trade-off between circuit depth and qubit count. Ensuring determinism is crucial to MBQC, particularly in the VQE context, as a lack of flow in measurement patterns leads to evaluating the cost function at irrelevant locations. This study introduces MBVQE-ansätze that respect determinism and resemble the widely used problem-agnostic hardware-efficient VQE ansatz. We evaluate our approach using ideal simulations on the Schwinger Hamiltonian and XY-model and perform experiments on IBM hardware with an adaptive measurement capability. In our use case, we find that ensuring determinism works better via postselection than by adaptive measurements at the expense of increased sampling cost. Additionally, we propose an efficient MBQC-inspired method to prepare the resource state, specifically the cluster state, on hardware with heavy-hex connectivity, requiring a single measurement round, and implement this scheme on quantum computers with 27 and 127 qubits. We observe notable improvements for larger cluster states, although direct gate-based implementation achieves higher fidelity for smaller instances.
Variational quantum eigensolvers (VQEs) are successful algorithms for studying physical systems on quantum computers. Recently, they were extended to the measurement-based model of quantum computing, bringing resource graph states and their advantages into the realm of quantum simulation. In this Letter, we incorporate such ideas into traditional VQE circuits. This enables novel problem-informed designs and versatile implementations of many-body Hamiltonians. We showcase our approach on real superconducting quantum computers by performing VQE simulations of testbed systems including the perturbed planar code, Z_{2} lattice gauge theory, 1D quantum chromodynamics, and the LiH molecule.
High Resolution Image Download MS PowerPoint Slide We propose a cost-efficient measurement scheme of the variational quantum eigensolver (VQE) for atomistic simulations of electronic structures based on a tight-binding (TB) theory. Leveraging the lattice geometry of a material domain, the sparse TB Hamiltonian is constructed in a bottom-up manner and is represented as a linear combination of the standard-basis (SB) operators. The cost function is evaluated with an extended version of the Bell measurement circuit that can simultaneously measure multiple SB operators and therefore reduces the number of circuits required by the evaluation process. The proposed VQE scheme is applied to find band gap energies of metal-halide-perovskite supercells that have finite dimensions with closed boundaries and are described with an sp 3 TB model. Experimental results confirm that the proposed scheme gives solutions that follow well the accurate ones but, more importantly, has computing efficiency that is obviously superior to the commutativity-based Pauli grouping methods. Extending the application scope of VQE to three-dimensional confined atomic structures, this work can serve as a practical guideline for handling TB simulations and, more generally, for calculations of sparse Hermitian matrices in the noise-intermediate-scale quantum devices.