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Block Lanczos method for excited states on a quantum computer
TL;DR: In this paper, the authors extended the quantum Lanczos recursion method to solve for multiple excitations on the quantum computer, and the error of the ground state energy based on the accuracy of the Lanczos coefficients was investigated.
Abstract: The method of quantum Lanczos recursion is extended to solve for multiple excitations on the quantum computer. While quantum Lanczos recursion is in principle capable of obtaining excitations, the extension to a block Lanczos routine can resolve degeneracies with better precision and only costs $\mathcal{O}(d^2)$ for $d$ excitations on top of the previously introduced quantum Lanczos recursion method. The formal complexity in applying all operators to the system at once with oblivious amplitude amplification is exponential, but this cost can be kept small to obtain the ground state by incrementally adding operators. The error of the ground state energy based on the accuracy of the Lanczos coefficients is investigated and the error of the ground state energy. It is demonstrated to scale linearly with the uncertainty of the Lanczos coefficients. Extension to non-Hermitian operators is also discussed.
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TL;DR: In this article, the authors explore the preparation of specific nuclear states on gate-based quantum hardware using variational algorithms, and find that often one needs to minimize over a large number of parameters, using a number of entanglements that makes challenging the application on existing hardware.
Abstract: We explore the preparation of specific nuclear states on gate-based quantum hardware using variational algorithms. Large scale classical diagonalization of the nuclear shell model have reached sizes of $10^9 - 10^{10}$ basis states, but are still severely limited by computational resources. Quantum computing can, in principle, solve such systems exactly with exponentially fewer resources than classical computing. Exact solutions for large systems require many qubits and large gate depth, but variational approaches can effectively limit the required gate depth.
We use the unitary coupled cluster approach to construct approximations of the ground-state vectors, later to be used in dynamics calculations. The testing ground is the phenomenological shell model space, which allows us to mimic the complexity of the inter-nucleon interactions. We find that often one needs to minimize over a large number of parameters, using a large number of entanglements that makes challenging the application on existing hardware. Prospects for rapid improvements with more capable hardware are, however, very encouraging.
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Book•
01 Jan 2000
TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Abstract: Part I Fundamental Concepts: 1 Introduction and overview 2 Introduction to quantum mechanics 3 Introduction to computer science Part II Quantum Computation: 4 Quantum circuits 5 The quantum Fourier transform and its application 6 Quantum search algorithms 7 Quantum computers: physical realization Part III Quantum Information: 8 Quantum noise and quantum operations 9 Distance measures for quantum information 10 Quantum error-correction 11 Entropy and information 12 Quantum information theory Appendices References Index
25,929 citations
01 Jul 1996
TL;DR: In this paper, it was shown that a quantum mechanical computer can solve integer factorization problem in a finite power of O(log n) time, where n is the number of elements in a given integer.
Abstract: were proposed in the early 1980’s [Benioff80] and shown to be at least as powerful as classical computers an important but not surprising result, since classical computers, at the deepest level, ultimately follow the laws of quantum mechanics. The description of quantum mechanical computers was formalized in the late 80’s and early 90’s [Deutsch85][BB92] [BV93] [Yao93] and they were shown to be more powerful than classical computers on various specialized problems. In early 1994, [Shor94] demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers. This is the problem of integer factorization, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) . ----------------------------------------------
6,335 citations
IBM1
TL;DR: The experimental optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms is demonstrated, determining the ground-state energy for molecules of increasing size, up to BeH2.
Abstract: The ground-state energy of small molecules is determined efficiently using six qubits of a superconducting quantum processor. Quantum simulation is currently the most promising application of quantum computers. However, only a few quantum simulations of very small systems have been performed experimentally. Here, researchers from IBM present quantum simulations of larger systems using a variational quantum eigenvalue solver (or eigensolver), a previously suggested method for quantum optimization. They perform quantum chemical calculations of LiH and BeH2 and an energy minimization procedure on a four-qubit Heisenberg model. Their application of the variational quantum eigensolver is hardware-efficient, which means that it is optimized on the given architecture. Noise is a big problem in this implementation, but quantum error correction could eventually help this experimental set-up to yield a quantum simulation of chemically interesting systems on a quantum computer. Quantum computers can be used to address electronic-structure problems and problems in materials science and condensed matter physics that can be formulated as interacting fermionic problems, problems which stretch the limits of existing high-performance computers1. Finding exact solutions to such problems numerically has a computational cost that scales exponentially with the size of the system, and Monte Carlo methods are unsuitable owing to the fermionic sign problem. These limitations of classical computational methods have made solving even few-atom electronic-structure problems interesting for implementation using medium-sized quantum computers. Yet experimental implementations have so far been restricted to molecules involving only hydrogen and helium2,3,4,5,6,7,8. Here we demonstrate the experimental optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms, determining the ground-state energy for molecules of increasing size, up to BeH2. We achieve this result by using a variational quantum eigenvalue solver (eigensolver) with efficiently prepared trial states that are tailored specifically to the interactions that are available in our quantum processor, combined with a compact encoding of fermionic Hamiltonians9 and a robust stochastic optimization routine10. We demonstrate the flexibility of our approach by applying it to a problem of quantum magnetism, an antiferromagnetic Heisenberg model in an external magnetic field. In all cases, we find agreement between our experiments and numerical simulations using a model of the device with noise. Our results help to elucidate the requirements for scaling the method to larger systems and for bridging the gap between key problems in high-performance computing and their implementation on quantum hardware.
2,348 citations
Book•
01 Jan 2000TL;DR: Second Quantization Spin in Second Quantization Orbital Rotations Exact and Approximate Wave Functions The Standard Models Atomic Basis Functions Short-range Interactions and Orbital Expansions Gaussian Basis Sets Molecular Integral Evaluation Hartree-Fock Theory Configuration-Interaction Theory Multiconfigurational Self-Consistent Field Theory Coupled-Cluster Theory Perturbation Theory Calibration of the Electronic-Structure Models List of Acronyms Index
Abstract: Second Quantization Spin in Second Quantization Orbital Rotations Exact and Approximate Wave Functions The Standard Models Atomic Basis Functions Short-Range Interactions and Orbital Expansions Gaussian Basis Sets Molecular Integral Evaluation Hartree-Fock Theory Configuration-Interaction Theory Multiconfigurational Self-Consistent Field Theory Coupled-Cluster Theory Perturbation Theory Calibration of the Electronic-Structure Models List of Acronyms Index
1,740 citations
Book•
01 Jan 1987
TL;DR: This book discusses iterative projection methods for solving Eigenproblems, and some of the techniques used to solve these problems came from the literature on Hermitian Eigenvalue.
Abstract: List of symbols and acronyms List of iterative algorithm templates List of direct algorithms List of figures List of tables 1: Introduction 2: A brief tour of Eigenproblems 3: An introduction to iterative projection methods 4: Hermitian Eigenvalue problems 5: Generalized Hermitian Eigenvalue problems 6: Singular Value Decomposition 7: Non-Hermitian Eigenvalue problems 8: Generalized Non-Hermitian Eigenvalue problems 9: Nonlinear Eigenvalue problems 10: Common issues 11: Preconditioning techniques Appendix: of things not treated Bibliography Index .
1,418 citations