Recent breakthroughs have opened the possibility of intermediate-scale quantum computing with tens to hundreds of qubits, and shown the potential for solving classical challenging problems, such as in chemistry and condensed matter physics. However, the high accuracy needed to surpass classical computers poses a critical demand on the circuit depth, which is severely limited by the non-negligible gate infidelity, currently around 0.1%-1%. The limited circuit depth places restrictions on the performance of variational quantum algorithms (VQA) and prevents VQAs from exploring desired nontrivial quantum states. To resolve this problem, we propose a paradigm of Schrödinger-Heisenberg variational quantum algorithms (SHVQA). Using SHVQA, the expectation values of operators on states that require very deep circuits to prepare can now be efficiently measured by rather shallow circuits. The idea is to incorporate a virtual Heisenberg circuit, which acts effectively on the measurement observables, into a real shallow Schrödinger circuit, which is implemented realistically on the quantum hardware. We choose a Clifford virtual circuit, whose effect on the Hamiltonian can be seen as efficient classical processing. Yet, it greatly enlarges the state's expressivity, realizing much larger unitary t designs. Our method enables accurate quantum simulation and computation that otherwise are only achievable with much deeper circuits or more accurate operations conventionally. This has been verified in our numerical experiments for a better approximation of random states, higher-fidelity solutions to the XXZ model, and the electronic structure Hamiltonians of small molecules. Thus, together with effective quantum error mitigation, our work paves the way for realizing accurate quantum computing algorithms with near-term quantum devices.
Recent breakthroughs have opened the possibility of intermediate-scale quantum computing with tens to hundreds of qubits, and shown the potential for solving classical challenging problems, such as in chemistry and condensed matter physics. However, the high accuracy needed to surpass classical computers poses a critical demand on the circuit depth, which is severely limited by the non-negligible gate infidelity, currently around 0.1%-1%. The limited circuit depth places restrictions on the performance of variational quantum algorithms (VQA) and prevents VQAs from exploring desired nontrivial quantum states. To resolve this problem, we propose a paradigm of Schrödinger-Heisenberg variational quantum algorithms (SHVQA). Using SHVQA, the expectation values of operators on states that require very deep circuits to prepare can now be efficiently measured by rather shallow circuits. The idea is to incorporate a virtual Heisenberg circuit, which acts effectively on the measurement observables, into a real shallow Schrödinger circuit, which is implemented realistically on the quantum hardware. We choose a Clifford virtual circuit, whose effect on the Hamiltonian can be seen as efficient classical processing. Yet, it greatly enlarges the state's expressivity, realizing much larger unitary t designs. Our method enables accurate quantum simulation and computation that otherwise are only achievable with much deeper circuits or more accurate operations conventionally. This has been verified in our numerical experiments for a better approximation of random states, higher-fidelity solutions to the XXZ model, and the electronic structure Hamiltonians of small molecules. Thus, together with effective quantum error mitigation, our work paves the way for realizing accurate quantum computing algorithms with near-term quantum devices.
Achieving chemical accuracy with shallow quantum circuits is a significant challenge in quantum computational chemistry, particularly for near-term quantum devices. In this work, we present a Clifford-based Hamiltonian engineering algorithm, namely CHEM, that addresses the trade-off between circuit depth and accuracy. Based on a variational quantum eigensolver and hardware-efficient ansatz, our method designs the Clifford-based Hamiltonian transformation that (1) ensures a set of initial circuit parameters corresponding to the Hartree–Fock energy can be generated, (2) effectively maximizes the initial energy gradient with respect to circuit parameters, (3) imposes negligible overhead for classical processing and does not require additional quantum resources, and (4) is compatible with any circuit topology. We demonstrate the efficacy of our approach using a quantum hardware emulator, achieving chemical accuracy for systems as large as 12 qubits with fewer than 30 two-qubit gates. Our Clifford-based Hamiltonian engineering approach offers a promising avenue for practical quantum computational chemistry on near-term quantum devices.
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.
Quantum mechanics has introduced a new theoretical framework for the study of molecules, enabling the prediction of properties and dynamics through the solution of the Schrödinger equation applied to these systems. However, solving this equation is computationally expensive, which has led to the development of various mathematical frameworks and computational methods designed to balance the available resources with the desired level of accuracy. In particular, quantum computers have emerged as a promising technology with the potential to address these problems more efficiently in the coming decades, whether through reductions in memory, time, and energy consumption, i. e., reductions in computational complexity or by enhancing precision. This research field is known as Quantum Simulation. Given the current technological limitations of quantum computers, Variational Quantum Algorithms (VQAs), especially the Variational Quantum Eigensolver (VQE), stand out as a feasible approach to demonstrating advantages over classical methods in the near term. This feasibility arises from their lower demand for quantum gates and the reduced depth of the circuits required for their implementation. In this work, we present the application of quantum mechanics to the study of molecules, provide an introduction to the fundamentals of quantum computing, and explore the integration of these fields by employing the VQE in molecular simulations. Finally, we discuss the spatial and temporal complexity associated with the algorithm, highlighting its implications and challenges.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
Abstract Efficient computation of molecular energies is an exciting application of quantum computing for quantum chemistry, but current noisy intermediate-scale quantum (NISQ) devices can only execute shallow circuits, limiting existing variational quantum algorithms, which require deep entangling quantum circuit ansatzes to capture correlations, to small molecules. Here we demonstrate a variational NISQ-friendly algorithm that generates a set of Hartree–Fock ansatzes using multiple shallow circuits with depth linear in the number of qubits to estimate electronic correlation energies via perturbation theory up to the second order. We tested the algorithm on several small molecules, both with classical simulations including noise models and on cloud quantum processors, showing that it not only reproduces the equilibrium molecular energies but it also captures the perturbative electronic correlation effects at longer bond distances. The number of shallow circuits required for our algorithms scales O ( N 5 ) , with N being the system size, thus enabling the study of larger molecules compared to other approaches, where circuit depth or number of qubits required become prohibitive for NISQ devices.
Dynamical simulation of quantum many-body systems is a central task in quantum chemistry and requires efficient wavefunction representations. Although tensor-network and neural-network quantum states have achieved considerable success in ground-state calculations, entanglement growth hinders their application to quantum dynamics. Quantum computing may offer a route to quantum advantage, but algorithms such as Trotterization generally require fault-tolerant quantum computers, whereas near-term hardware supports only limited circuit sizes. Variational quantum simulation (VQS) instead represents the evolving state with a single shallow, fixed-size parameterized quantum circuit (PQC). Here, we introduce nonorthogonal variational quantum simulation (NOVQS), which applies linear combinations of parameterized quantum states to real- and imaginary-time evolutions. We design a shallow, hardware-friendly ansatz tailored to second-quantized electronic-structure Hamiltonians, together with resource-efficient protocols for measuring the matrices and vectors in the parameter equations of motion. Error analysis and resource estimation are also provided. Numerical simulations of hydrogen chains and the nitrogen molecule demonstrate that a collection of shallow, or even single-layer, PQCs can match or outperform a much deeper PQC in VQS. In particular, we identify a trade-off between circuit number and depth. NOVQS therefore provides a flexible route to enhancing wavefunction expressivity under circuit-depth constraints, making it a promising framework for quantum dynamics simulations on near-term quantum processors.