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Using Symmetries for Near-Term Quantum Computing

Eckstein, Timo

Published Jan 1, 2025• 0 citations
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Abstract

Our works [1–7] focus on algorithm development for current and near-term quantum computing platforms. Symmetries, which are mathematically described by groups, divide the full space of possibilities, the Hilbert space, into smaller sectors. If a symmetry is conserved within an application, then one of these smaller symmetry sectors will be sufficient to model it. Therefore, our works incorporate symmetries like translational invariance [1, 2, 5, 7], time-periodicity [3, 4] or electron number conservation [6] to improve algorithms for relevant system classes compared to symmetry-unaware standard methods. In the first direction, we utilise symmetries for Hamiltonian simulation, where one aims to approximate the time evolution of a reproduced quantum system of interest on a quantum computer and predict the time-dependent properties of the system, such as correlations. Concretely, we improve Hamiltonian simulation algorithms for time-periodic systems by taking advantage of high-frequency expansions in works [3, 4]. In Ref. [5], we reduce the approximation error of product formulas by classical variational optimisation, and translational symmetry to scale up the system size n. The second direction is state preparation, where one targets the recreation of a quantum state, typically a ground state, and the prediction of its properties that are too difficult to compute classically. In Ref. [6], we take advantage of the electron number conservation of fermionic systems for a symmetry-aware circuit ansatz and self-consistent configuration recovery. Paired with warm-starting and classical subsystem diagonalisation, it allows us to estimate the Fe(III)-NTA ground state energy within chemical accuracy via samples obtained from an ion-trap quantum computer. The third direction is to use symmetry for the reduction of statistical uncertainty of quantum estimators, which are computed from measured bitstring values at the end of a quantum computation. Concretely, in Refs. [1, 2], we improve the sampling complexity for energy estimation of quantum lattice systems, which can be mapped efficiently to current quantum hardware. Our approach leverages translational invariance to construct more sample-efficient estimators, where their minimal improvement compared to local Pauli bases is guaranteed by an analytical lower bound. Further in Refs. [3, 4], we take advantage of the translational invariance of both the quantum state and observable to boost the number of observed bitstring values per ion trap hardware sample by the number of qubits. [1] T. Eckstein, R. Mansuroglu, S. Wolf, L. Nützel, S. Tasler, M. Kliesch, and M. J. Hartmann, in Quantum Computing Theory in Practice 2025 (Freie Universität Berlin, 2025). [2] T. Eckstein, R. Mansuroglu, S. Wolf, L. Nützel, S. Tasler, M. Kliesch, and M. J. Hartmann, arXiv preprint arXiv:2410.21251 10.48550/arXiv.2410.21251 (2024). [3] T. Eckstein, R. Mansuroglu, P. Czarnik, J.-X. Zhu, M. J. Hartmann, L. Cincio, A. T. Sornborger, and Z. Holmes, in Quantum Computing Theory in Practice 2024 (University of Edinburgh, 2024). [4] T. Eckstein, R. Mansuroglu, P. Czarnik, J.-X. Zhu, M. J. Hartmann, L. Cincio, A. T. Sornborger, and Z. Holmes, npj Quantum Information 10, 84 (2024). [5] R. Mansuroglu, T. Eckstein, L. N¨utzel, S. A. Wilkinson, and M. J. Hartmann, Quantum Science and Technology 8, 025006 (2023). [6] L. Nützel, A. Gresch, L. Hehn, L. Marti, R. Freund, A. Steiner, C. D. Marciniak, T. Eckstein, N. Stockinger, S. Wolf, et al., Quantum Science and Technology 10, 015066 (2025). [7] S. Tasler, J. Old, L. Heunisch, V. Feulner, T. Eckstein, M. Müller, and M. J. Hartmann, arXiv preprint arXiv:2506.09028 10.48550/arXiv.2506.09028 (2025).

DOI
10.25593/open-fau-2479

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Our works [1–7] focus on algorithm development for current and near-term quantum computing platforms. Symmetries, which are mathematically described by groups, divide the full space of possibilities, the Hilbert space, into smaller sectors. If a symmetry is conserved within an application, then one of these smaller symmetry sectors will be sufficient to model it. Therefore, our works incorporate symmetries like translational invariance [1, 2, 5, 7], time-periodicity [3, 4] or electron number conservation [6] to improve algorithms for relevant system classes compared to symmetry-unaware standard methods. In the first direction, we utilise symmetries for Hamiltonian simulation, where one aims to approximate the time evolution of a reproduced quantum system of interest on a quantum computer and predict the time-dependent properties of the system, such as correlations. Concretely, we improve Hamiltonian simulation algorithms for time-periodic systems by taking advantage of high-frequency expansions in works [3, 4]. In Ref. [5], we reduce the approximation error of product formulas by classical variational optimisation, and translational symmetry to scale up the system size n. The second direction is state preparation, where one targets the recreation of a quantum state, typically a ground state, and the prediction of its properties that are too difficult to compute classically. In Ref. [6], we take advantage of the electron number conservation of fermionic systems for a symmetry-aware circuit ansatz and self-consistent configuration recovery. Paired with warm-starting and classical subsystem diagonalisation, it allows us to estimate the Fe(III)-NTA ground state energy within chemical accuracy via samples obtained from an ion-trap quantum computer. The third direction is to use symmetry for the reduction of statistical uncertainty of quantum estimators, which are computed from measured bitstring values at the end of a quantum computation. Concretely, in Refs. [1, 2], we improve the sampling complexity for energy estimation of quantum lattice systems, which can be mapped efficiently to current quantum hardware. Our approach leverages translational invariance to construct more sample-efficient estimators, where their minimal improvement compared to local Pauli bases is guaranteed by an analytical lower bound. Further in Refs. [3, 4], we take advantage of the translational invariance of both the quantum state and observable to boost the number of observed bitstring values per ion trap hardware sample by the number of qubits. [1] T. Eckstein, R. Mansuroglu, S. Wolf, L. Nützel, S. Tasler, M. Kliesch, and M. J. Hartmann, in Quantum Computing Theory in Practice 2025 (Freie Universität Berlin, 2025). [2] T. Eckstein, R. Mansuroglu, S. Wolf, L. Nützel, S. Tasler, M. Kliesch, and M. J. Hartmann, arXiv preprint arXiv:2410.21251 10.48550/arXiv.2410.21251 (2024). [3] T. Eckstein, R. Mansuroglu, P. Czarnik, J.-X. Zhu, M. J. Hartmann, L. Cincio, A. T. Sornborger, and Z. Holmes, in Quantum Computing Theory in Practice 2024 (University of Edinburgh, 2024). [4] T. Eckstein, R. Mansuroglu, P. Czarnik, J.-X. Zhu, M. J. Hartmann, L. Cincio, A. T. Sornborger, and Z. Holmes, npj Quantum Information 10, 84 (2024). [5] R. Mansuroglu, T. Eckstein, L. N¨utzel, S. A. Wilkinson, and M. J. Hartmann, Quantum Science and Technology 8, 025006 (2023). [6] L. Nützel, A. Gresch, L. Hehn, L. Marti, R. Freund, A. Steiner, C. D. Marciniak, T. Eckstein, N. Stockinger, S. Wolf, et al., Quantum Science and Technology 10, 015066 (2025). [7] S. Tasler, J. Old, L. Heunisch, V. Feulner, T. Eckstein, M. Müller, and M. J. Hartmann, arXiv preprint arXiv:2506.09028 10.48550/arXiv.2506.09028 (2025).
Quantum computers represent a relatively new and promising development in computing technology. One of the applications of quantum computers is modelling quantum many-body systems, which describe interactions between particles on atomic and subatomic scales. Such systems are highly complex and challenging to model with classical computers due to the vast number of possible states and entanglement of the particles. In this dissertation, we examine the extent to which quantum computers can accelerate the modelling of quantum many-body systems compared to classical computers. Specifically, this dissertation presents research on adiabatic state preparation: a quantum algorithmic technique that uses the adiabatic principle from quantum mechanics to approximate eigenstates. We describe three new techniques that fall within this category and, in certain cases, offer advantages over standard methods. First, we consider cases of ground state preparation for fermionic many-body systems, where standard direct interpolation between the initial and final hamiltonian is hindered by level crossings due to discrete symmetries. As an alternative to direct interpolation, we propose adiabatic paths in a higher-dimensional space, which break the relevant symmetries. Next, we present an adiabatic echo verification protocol which mitigates both coherent and incoherent errors, arising from non-adiabatic transitions and hardware noise, respectively. We show that the estimator bias of the observable is reduced when compared to standard adiabatic preparation, achieving up to a quadratic improvement. Finally, we propose a general, fully gate-based and nonvariational quantum algorithm for counterdiabatic driving. We provide a rigorous quantum gate complexity upper bound in terms of the minimum gap Δ\DeltaΔ around this eigenstate. We find that, in the worst case, the algorithm can be run with at most O~(Δ−(3+o(1))ϵ−(1+o(1)))\tilde O(\Delta^{-(3 + o(1))} \epsilon^{-(1 + o(1))})O~(Δ−(3+o(1))ϵ−(1+o(1))) quantum gates such that a target state fidelity of at least 1−ϵ21 - \epsilon^21−ϵ2 is achieved. In certain cases, the gap dependence can be improved to quadratic.
In ab-initio quantum chemistry, solving the electronic structure problem of a molecular system of interest to obtain an approximation of the ground and, in some cases, the excited state energies is a core and central task in various quantum chemistry processes. These include, but certainly not limited to, the calculation of the activation energy–that is the minimal energy required to launch a given chemical reaction–, the dipole and multipole moments which describe the polarity of the molecule, performing the optimization of the geometry of the molecule and searching for transition states. Furthermore, many quantum chemistry applications are predicated, not merely on the approximation of the ground state energy, but also on the corresponding first order derivatives. The body of literature of classical algorithms approximating the ground state energy and its first order derivative is extensively rich with a plethora of methods such as self-consistent mean-field methods, coupled-cluster methods, density matrix renormalization group and many more, each of which has intrinsic set of properties and characteristics that makes it suitable for specific tasks and cases. However, classical methods face a practically insurmountable challenge: the efficient computation of high accuracy approximation of ground state energies of large system sizes. This is fundamentally because the electronic structure problem provably belongs to the complexity class of QMA-hard problems. Which also means that it is out of reach for quantum algorithms as well in general. However, with certain assumptions that are expected to hold molecules of interest–that so far proved to be classically challenging because the required accuracy set by the standard of chemical accuracy threshold (generally held in quantum chemistry)– quantum algorithms can efficiently treat such cases. While originally theorized by Richard Feynmann as a potentially efficient way of simulating some classically intractable many-body systems, quantum computing has since proven to have a stronger and richer potential paving the way for the inception of new algorithms such as Shor’s algorithm for integer factorization, HHL for solving linear system of equation and Grover’s algorithm for constrained binary search, just to name a few. These algorithms provably exhibit an improvement rate that goes respectively exponentially and quadratically in the time complexity. Quantum chemistry is considered a prominent candidate to benefit from quantum computing given the variety of quantum algorithms put forth to solve the electronic structure problem of particular systems of interest. These quantum algorithm proposals can be broadly categorized in three main categories that span the spectrum: from noisy intermediate scale quantum (NISQ) computers to early and full scale fault tolerant quantum ((E)FTQC) computers. Firstly, the variational quantum eigensolver (VQE) family of quantum algorithms which variationally search for the ground state energy through the optimization of the energy of a parametrized state prepared in a quantum computer using an ansatz. Secondly, the quantum Krylov subspace diagonalization (QKSD) methods, which generate a subspace of the original Hilbert space and efficiently produce the lowest energy in said subspace as its output. Lastly, we have the quantum phase estimation (QPE) algorithms, which provably produce the ground state energy in Heisenberg scaling presenting hence a provable efficient solution. In this thesis, we firstly set to work on the problem of efficient Hamiltonian linear combination of unitaries (LCU) representation. This is mainly motivated by the wide-range impact an efficient compression of the Hamiltonian can potentially have on several quantum algorithms such as VQE, QPE...etc. We designed a method that efficiently compress the Hamiltonian representation in a way that is beneficially impactful across the set of the aforementioned quantum algorithms. Namely, we motivate and present in Sections 1.3 and 1.4 respectively, the regularized compressed double-factorization (RC-DF) and show how it reduces the energy measurement complexity both in terms of number of distinct observables and number of samples. Moreover, it also reduces the total depth of the quantum phase estimation circuit thanks to its low normalization factor. Secondly, given the importance of the energy derivatives, in Sections 2.1 and 2.2, we introduce and present an efficient quantum algorithm for the calculation of the analytic gradient of VQE energies of double-factorized Hamiltonians. Furthermore, in Section 2.1, we expand on the utility of RC-DF by presenting a quantum algorithm that determines the analytic gradient of energies measured with respect to RC-DF Hamiltonains. Third, in Section 3.1, we summarize the theoretical results of purification error-mitigation techniques, namely echo verification and virtual distillation. In Section 3.2, we report on the results of the experimental bench mark, of said techniques, performed on on Google’s Sycamore device to validate the theoretical scaling of error suppression and extrapolate the resources needed for classically intractable system sizes. Fourth, in Section 4 we establish the challenges of computing energy gradients in a quantum Krylov frame work. To overcome the problem, we provide, in Section 4.3, an efficient quantum algorithm for calculating energy derivative obtained through QKSD procedures. Our approach shows how the QKSD ground state can be prepared using quantum signal processing (QSP) and reduces the number of observables to O(1) per reduced density matrix (RDM). Lastly, in light of the importance of quantum RAMs due to their utility in a variety of quantum algorithms that require efficient initial state preparation, as explained in Section 5.5, we address the problematic exponential depth of the bucket-brigade QRAM (when decomposed). We present in Section 5.6 a very efficient Clifford+T decomposition of the bucket-brigade QRAM, which preserves the linear scaling of the depth of the circuit without introducing any additional ancilla.
Computational methods are central to modern physics, yet many problems, especially those involving Quantum Many-Body Systems (QMBSs), remain challenging at scale. The core difficulty of simulating QMBSs is the exponential growth of the solution space with system size. This renders exact treatments intractable and pushes approximate methods to their limits when tackling two- and higher-dimensional systems and long-time dynamics. Quantum computers offer a quantum-native platform for representing and manipulating such systems. Recent algorithmic and technological advancements have enabled experiments on quantum processors in the quantum utility regime, a scale at which exact classical reproduction is no longer feasible. This represents the first step towards realizing quantum advantage, where quantum computers solve specific, practically relevant problems more efficiently than any classical approach. While advancing towards the goal of quantum advantage, useful computations on quantum processors are expected to be realised through hybrid quantum-classical algorithms, which combine quantum experiments and classical computational resources. A prominent example are variational quantum algorithms, where a Parametrized Quantum Circuit (PQC) prepares a trial quantum state whose parameters are iteratively updated by a classical optimisation routine to minimize a target cost function (e.g., the energy of a target Hamiltonian). However, the optimisation of such PQCs is affected by trainability issues, such as barren plateaus in the parameter landscape, which can render the optimisation increasingly difficult for larger system sizes. This motivates the development of novel hybrid approaches overcoming the limits of PQCs. In this thesis, we introduce a new class of approaches based on hybrid Tensor Networks (TNs). A hybrid TN is obtained by integrating quantum states prepared on a quantum processor within TNs -- one of the state-of-the-art approaches for simulating quantum systems. We first show how hybrid TNs can be used for ground-state searches within a variational setting. Then, we further extend the framework to non-variational hybrid TN ansätze for which the quantum tensors are prepared via quantum annealing and subsequently post-processed by classical TNs. In the context of quantum machine learning, which also relies on PQCs, we further investigate heuristic best practices that mitigate the same exponential concentration effects in kernel-based models for anomaly detection. Another class of hybrid algorithms are quantum error-mitigation workflows, where classical post-processing is used to reconstruct expectation values from noise-affected circuit executions. Such methods generally feature a trade-off between the bias and the variance: the price to pay for mitigating noise in a bias-free fashion is that the number of measurements usually grows exponentially with the depth of the circuit in order to achieve a fixed accuracy. In this thesis, we develop a low-resource error-mitigation strategy that reduces the measurement overhead in practical digital Hamiltonian-simulation workflows. We apply this method to Trotterized dynamics in a lattice-gauge-theory model, enabling the extraction of expectation values with improved accuracy at time-scales that allow the observation of post-collision scattering dynamics.
Many important physical and mathematical problems are difficult to solve on classical computers, resulting in difficulty in finding new advanced materials, simulating quantum field theories and solving large combinatorial optimisation tasks. Quantum computers have shown great promise in being able to solve these (and many other) problems faster than their classical counterparts, and therefore should be thoroughly investigated as a significant tool for enabling the acceleration of scientific progress. This work develops methods to tackle several problems across the spectrum of quantum algorithms, providing methods suitable for the near-term hardware, to the future where large fault-tolerant quantum computers exist. Therefore, the works in this thesis range from efficiently making use of limited quantum hardware to taking advantage of hardware benefits (e.g. excess qubits) as quantum computation advances. Covered topics include: an alternative paradigm for the optimisation of variational circuits and quantum neural networks; techniques for improving energy estimation of quantum states through the collection and processing of large amounts of randomised measurement data; the use of entanglement in guiding the automatic discovery of quantum circuits; investigating the role of symmetries in adiabatic state preparation of molecules and the large scale parallelisation of a common fault-tolerant subroutine, potentially producing large runtime speed-ups for the simulation of chemistry on quantum computers. Although there are many challenges remaining from both hardware and algorithms perspectives before the large-scale utility of quantum computers can be demonstrated, the contributions of this work make steps towards the efficient utilisation of quantum hardware that will be available in the coming years.
The accurate and efficient simulation of many body systems has been a long-standing challenge for quantum chemists and physicists. At the heart of this challenge lies the correlation between quantum particles, which is crucial in understanding various physical phenomena, but is numerically hard to calculate. Over the decades, there has been remarkable progress in simplifying this problem aided by mathematical and chemical approximations and the availability of powerful computers. However, an exact solution to the problem still remains intractable as it requires exponential re- sources with respect to the system size. Recent advances in the field of quantum computing have shown promise in Feynman’s idea of simulating quantum mechanics on quantum computers. It has been proposed that an ideal fault-tolerant quantum computer can reduce the scaling of these sim- ulations from exponential to polynomial. However, initial quantum algorithms have shown that there is a large gap between the capabilities of current hardware and the resources required for simulating quantum systems of interest. The work of this thesis focuses on reducing this gap by introducing two novel algorithms for quantum xiv chemistry simulations, which are suitable for near-term hardware. First, we intro- duce a hybrid algorithm to decrease the total number of operations required for the quantum simulations. We use a classical computer to generate an effective Hamilto- nian containing O(n2) terms compared to the O(n4) terms of the full Hamiltonian, where n is the number of orbitals. This sparse Hamiltonian can then be used with a high-level method to recover the ground state energies. We demonstrate that this sparsification of the Hamiltonian reduces the number of quantum operations required for the simulation by an order of magnitude, thus making it accessible for the near- term Hardware. Our second algorithm aims to make the calculation of dynamic correlation functions (Green’s functions) more feasible on these quantum machines. This algorithm is designed with the motive of avoiding the use of time-evolution and reduced use of two-qubit gates. We demonstrate that we can reproduce the Green’s function within reasonable error limits using this approach. We have also discussed the use of re-sampling techniques for proper error-propagation of the stochastic data obtained from the quantum machines. The general structure of this thesis is as follows. We begin by motivating this work in Chapter 1. Chapters 2 and 3 provide a general introduction of quantum computing and Green’s function based methods, respectively. In chapter 4, the development of effective Hamiltonian is discussed in detail. Chapter 5 demonstrates the use of these effective Hamiltonians on the quantum machine for time evolution. In chapter 6, we have discussed the algorithm to calculate Green’s functions on a quantum machine. This is followed by conclusions and future directions in Chapters 7 and 8, respectively.
Computer simulation is central to understanding the behaviour of matter. Yet, more than a century after the development of quantum theory, first-principles simulation of chemistry remains a daunting challenge, owing to the exponential growth in memory required to represent interacting quantum particles such as electrons in molecules. Quantum computing offers, in principle, a fundamentally more efficient paradigm for simulating quantum systems, with quantum chemistry often cited as a flagship application. However, many quantum algorithms face intrinsic limitations, and any genuine quantum advantage must surpass the strong baseline set by decades of progress in classical algorithms and hardware. While classical methods often suffice for simulating weakly correlated systems, systems with many strongly correlated electrons remain a challenge for both classical and quantum approaches. In this thesis, we develop quantum algorithms for solving the time-independent Schrödinger equation, with a focus on molecular systems with strongly correlated electrons. We introduce a framework for identifying a compact set of states that parametrise the many-body wavefunction by capturing the correlation due to spin coupling between localised electrons. This representation requires very few variational parameters but incurs a very high memory cost on classical computers. To overcome this, we design quantum circuits that efficiently prepare such states by exploiting their symmetry, encoding the dominant entanglement structure of electronic wavefunctions into the initial qubit states. Through theoretical analysis and classical simulations of quantum computers, we show that this substantially reduces the quantum resources required to accurately compute ground- and excitedstate properties. Finally, we develop an approach for approximating the optimal ansatz within the variational quantum eigensolver, an algorithm designed for near-term quantum devices. Our method constructs wavefunctions from products of symmetrypreserving unitary operators and couples discrete and continuous optimisation to yield highly compact representations of strongly correlated electrons using shallow quantum circuits. Together, these results establish complementary directions for embedding problemspecific structure into quantum simulation algorithms, advancing the practical simulation of strongly correlated electrons.