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arXiv:quant-ph/9903055 · Physical Review A

Higher-order methods for simulations on quantum computers

Andrew T. Sornborger, Ewan D. Stewart

Published Sep 1, 1999• 94 citations
arXivHTMLPDF

Abstract

To implement many-qubit gates for use in quantum simulations on quantum computers efficiently, we develop and present methods reexpressing $\mathrm{exp}[\ensuremath{-}{i(H}_{1}{+H}_{2}+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot})\ensuremath{\Delta}t]$ as a product of factors $\mathrm{exp}[\ensuremath{-}{\mathrm{iH}}_{1}\ensuremath{\Delta}t],$ $\mathrm{exp}[\ensuremath{-}{\mathrm{iH}}_{2}\ensuremath{\Delta}t],\dots{}$, which is accurate to third or fourth order in $\ensuremath{\Delta}t$. The methods we derive are an extended form of the symplectic method, and can also be used for an integration of classical Hamiltonians on classical computers. We derive both integral and irrational methods, and find the most efficient methods in both cases.

DOI
10.1103/physreva.60.1956

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To implement many-qubit gates for use in quantum simulations on quantum computers efficiently, we develop and present methods reexpressing exp[\ensuremath−i(H1+H2+\ensuremath⋅\ensuremath⋅\ensuremath⋅)\ensuremathΔt]\mathrm{exp}[\ensuremath{-}{i(H}_{1}{+H}_{2}+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot})\ensuremath{\Delta}t]exp[\ensuremath−i(H1​+H2​+\ensuremath⋅\ensuremath⋅\ensuremath⋅)\ensuremathΔt] as a product of factors exp[\ensuremath−iH1\ensuremathΔt],\mathrm{exp}[\ensuremath{-}{\mathrm{iH}}_{1}\ensuremath{\Delta}t],exp[\ensuremath−iH1​\ensuremathΔt], exp[\ensuremath−iH2\ensuremathΔt],…\mathrm{exp}[\ensuremath{-}{\mathrm{iH}}_{2}\ensuremath{\Delta}t],\dots{}exp[\ensuremath−iH2​\ensuremathΔt],…, which is accurate to third or fourth order in \ensuremathΔt\ensuremath{\Delta}t\ensuremathΔt. The methods we derive are an extended form of the symplectic method, and can also be used for an integration of classical Hamiltonians on classical computers. We derive both integral and irrational methods, and find the most efficient methods in both cases.
Quantum simulation involves using a quantum system to study another quantum system. The proposal to simulate quantum physics using quantum mechanical devices, as outlined by Richard Feynman in 1982, stems from classical computers' challenges in simulating quantum systems. The pursuit of simulating complex quantum systems has led to the intersection of quantum mechanics and quantum computing. This offers new opportunities to understand and manipulate the quantum world. Central to describing quantum dynamics is the Hamiltonian operator H^\hat{H}H^, which governs the time evolution of physical systems and serves as the mathematical foundation for quantum simulations. This thesis explores the methodologies and advancements in simulating Hamiltonian operators on quantum computers, focusing on decomposing these operators into quantum gates and the efficiency gains over classical computational approaches. Initially, the thesis explains the fundamental theory and basics of quantum computing. Further it dives into the foundations of the Hamiltonian operator within quantum mechanics, highlighting its role in describing the dynamics of quantum systems. The complexity and high-dimensional nature of Hamiltonian matrices necessitate using quantum computers to simulate quantum phenomena accurately and efficiently. The process of simulating the Hamiltonian operator on a quantum computer involves a detailed examination of the strategies for decomposing the operator into a sequence of quantum gates. This decomposition is crucial for translating the abstract mathematical description of quantum systems into the physical operations of a quantum computer. By utilizing the principles of quantum mechanics, such as superposition and entanglement, quantum gates manipulate qubits to mimic the evolution of quantum states as the Hamiltonian operator dictates. Latter portions of the thesis are dedicated to comparing the efficiency of quantum simulations against traditional classical simulations and inspecting efficient quantum simulations. Quantum computers offer a polynomial or an exponential speedup for certain classes of problems, such as those involving highly entangled quantum states or requiring the simulation of complex quantum dynamics. This advantage is due to the quantum computer's ability to naturally represent and process quantum information - a task for which classical computers require exponential resources as the system size increases. The efficiency of simulating the Hamiltonian operator on a quantum computer is further enhanced through optimization techniques and algorithmic improvements. These include error mitigation strategies and the development of quantum algorithms tailored to specific types of Hamiltonian operators. The thesis presents a comprehensive analysis of these methods, demonstrating how they make quantum simulations more practical and accessible for scientific research and technological development. In conclusion this thesis is a literature review that examines the transformative potential of quantum simulations. By efficiently simulating the Hamiltonian operator on quantum computers researchers can explore quantum phenomena with precision, paving the way for breakthroughs in quantum physics, materials science, and beyond. The advancements in quantum computing technology and algorithmic approaches discussed here highlight the current capabilities and future explorations in the quantum realm.
Simulating the Hamiltonian dynamics of quantum systems is one of the most promising applications of digital quantum computers. In this dissertation, we develop an understanding of quantum simulation algorithms concerning their design, analysis, implementation, and application. We implement three leading simulation algorithms, employing diverse techniques to tighten their error analyses and optimize circuit implementations. We produce concrete resource estimates for simulating a Heisenberg spin system, a problem arising in condensed matter physics that is otherwise difficult to solve on a classical computer. The resulting circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry, suggesting the simulation of spin systems as a promising candidate for an early demonstration of practical quantum computation. We design new simulation algorithms by using classical randomness. We show that by simply randomizing how the terms in the Hamiltonian are ordered, one can prove stronger bounds for product formulas and thereby give more efficient quantum simulations. We also develop a classical sampler for time-dependent Hamiltonians, using which we give a simulation algorithm that substantially improves over previous approaches when the Hamiltonian varies significantly with time. We propose a general theory to analyzing product formulas, an approach to quantum simulation widely used in experimental demonstrations but whose error scaling was poorly understood. Our approach directly exploits the commutativity of Hamiltonian, overcoming the limitations of prior error analyses. We prove new speedups of product formulas for simulating many quantum systems, including simulations of nearest-neighbor lattice systems, second-quantized plane-wave electronic structure, kkk-local Hamiltonians, rapidly decaying power-law interactions, and clustered Hamiltonians, nearly matching or even outperforming the best previous results in quantum simulation. We accompany our analysis with numerical calculation, which suggests that the bounds also have nearly tight constant prefactors. We identify applications of quantum simulation to designing other quantum algorithms and improving quantum Monte Carlo methods. We develop an algorithmic framework ``quantum singular value transformation'' using techniques from quantum simulation and apply it to implement principal component regression. We also apply our new analysis of product formulas and obtain improved quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets.
Quantum chemistry and materials science are among the most promising areas for demonstrating algorithmic quantum advantage and quantum utility due to their inherent quantum mechanical nature. Still, large-scale simulations of quantum circuits are essential for determining the problem size at which quantum solutions outperform classical methods. In this work, we present a novel hybrid simulation approach, forming a hybrid of a fullstate and a Clifford simulator, specifically designed to address the computational challenges associated with the time evolution of quantum chemistry Hamiltonians. Our method focuses on the efficient emulation of multi-qubit rotations, a critical component of Trotterized Hamiltonian evolution. By optimizing the representation and execution of multi-qubit operations leveraging the Pauli frame, our approach significantly reduces the computational cost of simulating quantum circuits, enabling more efficient simulations. Beyond its impact on chemistry applications, our emulation strategy has broad implications for any computational workload that relies heavily on multi-qubit rotations. By increasing the efficiency of quantum simulations, our method facilitates more accurate and cost-effective studies of complex quantum systems. We quantify the performance improvements and computational savings for this emulation strategy, and we obtain a speedup of a factor ≈18\approx 18≈18 (≈22\approx 22≈22 with MPI) for our evaluated chemistry Hamiltonians with 24 qubits. Thus, we evaluate our integration of this emulation strategy into the Intel Quantum SDK, further bridging the gap between theoretical algorithm development and practical quantum software implementations.
The Gottesman-Knill theorem says that a stabilizer circuit---that is, a quantum circuit consisting solely of controlled-NOT (CNOT), Hadamard, and phase gates---can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. First, by removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor of 2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely available program called CHP (CNOT-Hadamard-phase), which can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer circuits is complete for the classical complexity class $\ensuremath{\bigoplus}\mathsf{L}$, which means that stabilizer circuits are probably not even universal for classical computation. Third, we give efficient algorithms for computing the inner product between two stabilizer states, putting any nnn-qubit stabilizer circuit into a ``canonical form'' that requires at most O(n2∕logn)O({n}^{2}∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}n)O(n2∕logn) gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.