A common situation in quantum many-body physics is that the underlying theories are known but too complicated to solve efficiently. In such cases one usually builds simpler effective theories as low-energy or large-scale alternatives to the original theories. Here the central tasks are finding the optimal effective theories and proving their equivalence to the original theories. Recently quantum computing has shown the potential of solving quantum many-body systems by exploiting its inherent parallelism. It is thus an interesting topic to discuss the emergence of effective theories and design efficient tools for finding them based on the results from quantum computing. As the first step towards this direction, in this paper, we propose two approaches that apply quantum computing to find the optimal effective theory of a quantum many-body system given its full Hamiltonian. The first algorithm searches the space of effective Hamiltonians by quantum phase estimation and amplitude amplification. The second algorithm is based on a variational approach that is promising for near-future applications.
A common situation in quantum many-body physics is that the underlying theories are known but too complicated to solve efficiently. In such cases one usually builds simpler effective theories as low-energy or large-scale alternatives to the original theories. Here the central tasks are finding the optimal effective theories and proving their equivalence to the original theories. Recently quantum computing has shown the potential of solving quantum many-body systems by exploiting its inherent parallelism. It is thus an interesting topic to discuss the emergence of effective theories and design efficient tools for finding them based on the results from quantum computing. As the first step towards this direction, in this paper, we propose two approaches that apply quantum computing to find the optimal effective theory of a quantum many-body system given its full Hamiltonian. The first algorithm searches the space of effective Hamiltonians by quantum phase estimation and amplitude amplification. The second algorithm is based on a variational approach that is promising for near-future applications.
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 simulation of quantum physical systems is expected to be an important application for quantum computers. The work presented in this dissertation aims to improve the resource requirements of quantum computers for solving simulation problems, by providing both novel quantum algorithms and improved implementations of existing ones. I present three main results that cover diverse aspects of simulation including equilibrium physics, the preparation of useful quantum states, and simulations based on classical stochastic processes. The results rely on established quantum algorithms and other recent techniques which I review. My first original contribution is a new quantum algorithm to sample from the thermal Gibbs state of quantum Hamiltonians. The complexity of this algorithm has an exponentially improved dependence on precision compared to an earlier quantum algorithm, and also provides polynomial improvements in other parameters. My second contribution is an improved algorithm to implement reflections about the eigenstates of unitary operators and Hamiltonians, an operation that can be used to prepare ground states of many-body Hamiltonians by quantum amplitude amplification. The improved algorithm for implementing reflections significantly reduces the number of ancilla qubits required over earlier implementations. The improvement relies in part on a novel quantum algorithm for efficiently preparing states with Gaussian-like amplitudes, which may have broad applications. I also prove a lower bound for implementing reflections, which shows that our algorithm is almost optimal. My third and final contribution regards quantum algorithms for classical stochastic processes, which are of importance in statistical physics. I present a quantum algorithm that estimates the average running time of a stochastic process, and provides a quadratic speed-up over a comparable classical algorithm.
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.
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.
Quantum computers are expected to significantly outperform classical computers for certain computational tasks. One of the most promising applications is the simulation of many-body systems such as complex molecules or materials. In this thesis, we present several quantum algorithms for simulating many-body systems that can provide considerably better scaling than prior approaches. First, we describe a fully coherent quantum algorithm for simulating molecular dynamics within the Born-Oppenheimer approximation that leads to super-polynomially better scaling with respect to the error tolerance than previous work. Our algorithm utilizes the Koopman-von Neumann formulation of classical mechanics. This allows us to express the classical equations of motion as unitary dynamics which can be efficiently simulated on a quantum computer. We demonstrate that these simulations can be performed either at constant energy or at constant temperature. The latter allows us to estimate thermodynamic properties such as the Helmholtz free energy of the system. Next, we present two closely related quantum algorithms for solving continuous optimization problems such as finding the equilibrium configurations of molecules. These algorithms do not require any explicit gradient estimations. Instead, we encode the optimization problem into the dynamics of a quantum or classical system with friction. The dynamics are governed by a carefully chosen time-dependent Hamiltonian and thus can be simulated using efficient Hamiltonian simulation algorithms. We provide evidence that our fully coherent quantum algorithms can, in certain cases, provide exponentially better asymptotic scaling than approaches based on gradient descent. Lastly, we present a quantum algorithm for preparing the ground state of a set of interacting systems that can circumvent the exponential scaling suggested by the Van Vleck catastrophe. Our algorithm utilizes a divide-and-conquer strategy that repeatedly uses phase estimation to merge ground states of increasingly larger subsystems. We provide rigorous bounds for this approach and show that, under certain assumptions, our algorithm scales only quasi-polynomially with the number of systems.
Finding eigenstates of a given many-body Hamiltonian is a long-standing challenge due to the perceived computational complexity. Leveraging on the hardware of a quantum computer accommodating the exponential growth of the Hilbert space size with the number of qubits, more quantum algorithms to find the eigenstates of many-body Hamiltonians will be of wide interest with profound implications and applications. In this work, we advocate a quantum algorithm to find the ground state and excited states of many-body systems, without any penalty functions, variational steps or hybrid quantum-classical steps. Our fully quantum algorithm will be an important addition to the quantum computational toolbox to tackle problems intractable on classical machines.