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arXiv:2308.08912

A Bottom-up Approach to Constructing Symmetric Variational Quantum Circuits

Babatunde M. Ayeni

Published Aug 17, 2023• 0 citations
arXivHTMLPDF

Abstract

In the age of noisy quantum processors, the exploitation of quantum symmetries can be quite beneficial in the efficient preparation of trial states, an important part of the variational quantum eigensolver algorithm. The benefits include building quantum circuits which are more compact, with lesser number of paramaters, and more robust to noise, than their non-symmetric counterparts. Leveraging on ideas from representation theory we show how to construct symmetric quantum circuits. Similar ideas have been previously used in the field of tensor networks to construct symmetric tensor networks. We focus on the specific case of particle number conservation, that is systems with U(1) symmetry. Based on the representation theory of U(1), we show how to derive the particle-conserving exchange gates, which are commonly used in constructing hardware-efficient quantum circuits for fermionic systems, like in quantum chemistry, material science, and condensed-matter physics. We tested the effectiveness of our circuits with the Heisenberg XXZ model.

DOI
10.48550/arxiv.2308.08912

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In the age of noisy quantum processors, the exploitation of quantum symmetries can be quite beneficial in the efficient preparation of trial states, an important part of the variational quantum eigensolver algorithm. The benefits include building quantum circuits which are more compact, with lesser number of paramaters, and more robust to noise, than their non-symmetric counterparts. Leveraging on ideas from representation theory we show how to construct symmetric quantum circuits. Similar ideas have been previously used in the field of tensor networks to construct symmetric tensor networks. We focus on the specific case of particle number conservation, that is systems with U(1) symmetry. Based on the representation theory of U(1), we show how to derive the particle-conserving exchange gates, which are commonly used in constructing hardware-efficient quantum circuits for fermionic systems, like in quantum chemistry, material science, and condensed-matter physics. We tested the effectiveness of our circuits with the Heisenberg XXZ model.
The variational quantum simulation technique on gate-based quantum processors can be made more efficient if symmetries of the Hamiltonian system of interest are factored into the design construction of the underlying parameterized quantum circuits (PQC) needed by the technique. In the first part of this work, we present an easy and practical approach that can be used to construct symmetric quantum circuits. The method we present can directly be adapted for any symmetry, though in the present work we focus explicitly on the examples of Z 2 , Z 3 , and particle number conservation. We show how to construct symmetric gates that respects the aforementioned symmetries. In the second part, we focus on numerical analysis of parameterized particle-conserving gates. In variational quantum simulation, for example, the variational quantum eigensolver (VQE), the symmetric gates of the PQC are parameterized. Since there are often many possible inequivalent parametrizations, it is often not immediately obvious which gate parametrization would be most efficient in general for a wide range of Hamiltonians that respect the symmetry. We streamline our analysis to particle-conserving circuits and Hamiltonians. We numerically analyze the efficiency of different particle-conserving gate parametrizations by studying two examples, namely, a physics problem of learning the ground state of Heisenberg spin chains, and an artificial problem of learning the random arbitary states sampled within the particle number conserved subspace.
Abstract The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and time-reversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the H2{H}_{2}H2​ H 2 and LiHLiHLiH L i H molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.
Variational quantum algorithms are among the most promising approaches for simulating interacting quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. However, the practical success of variational quantum eigensolvers (VQE) critically depends on the structure of the chosen variational ansatz. In this work, we investigate the ground-state properties of the one-dimensional antiferromagnetic Heisenberg spin-1/2 chain using both generic hardware-efficient ansatz and physics-informed, symmetry-preserving variational circuits. We benchmark variational results against exact diagonalization and noiseless simulations, and subsequently validate the approach on real IQM Garnet quantum hardware. Our results demonstrate that incorporating physical symmetries into the circuit design leads to significantly improved energy estimates, enhanced robustness against hardware noise, and clearer convergence behavior when compared to hardware-efficient ansatz under identical resource constraints. These findings highlight the importance of problem specific ansatz construction for reliable quantum simulations in the NISQ era.
Quantum circuits, fundamental to quantum computation, represent quantum states through a sequence of unitary transformations. The exponential growth in the Hilbert space dimension with increasing circuit size presents a significant challenge to simulation and analysis. This work introduces a novel tensor network approach to formally model and manipulate quantum circuits, specifically designed to mitigate this exponential scaling issue. The core idea is to represent the circuit's state vector as a tensor network, leveraging techniques for efficient compression and expansion. We explore the application of variational tensor networks (VTNs) to approximate the dynamics of quantum circuits, offering a pathway towards more scalable simulations. The effectiveness of this method is demonstrated through theoretical analysis and conceptual design, focusing on key aspects of tensor network construction and manipulation tailored to quantum circuit representations. This approach aims to provide a more scalable representation compared to existing methods.
An efficient representation of a quantum circuit is of great importance in achieving a quantum advantage on current noisy intermediate scale quantum (NISQ) devices and the classical simulation of quantum manybody systems.The quantum circuits are playing the key ingredient in the performance of variational quantum algorithms and quantum dynamics in problems of physics and chemistry.In this paper, we study the role of the network structure of a quantum circuit in its performance.We discuss the variational optimization of quantum circuit (a unitary tensor-network circuit) with different network structures.The ansatz is performed based on a generalization of well-developed multiscale entanglement renormalization algorithm and also the conjugategradient method with an effective line search.We present the benchmarking calculations for different network structures by studying the Heisenberg model in a strongly disordered magnetic field and a tensor-network QR decomposition.Our work can contribute to achieve the most out of NISQ hardware and to classically develop isometric tensor network states.
The variational quantum eigensolver (VQE) is considered one of the most powerful alternative approaches for near-term noisy intermediate-scale quantum devices. However, the successful implementation of the VQE is quite challenging, as the ansatz must be sufficiently expressive, yet, if it is overly expressive, it may hinder the optimization process of the VQE in finding the global minimum. To address this challenge of the VQE, a commonly used approach is to apply the symmetries of the system, which reduces the dimensionality of the parameter space for the VQE. Here, we propose a quantum circuit that can be used for systems with mirror symmetry and occupation-number-inversion symmetry. We demonstrate its usefulness by applying it to find the eigenenergies of the Anderson impurity model, a representative fundamental model for various quantum simulations. In addition, we demonstrate the usability of our quantum circuit by applying it to a quantum algorithm for Green's function, which plays a crucial role in understanding quantum systems of condensed matter physics.