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arXiv:2112.04892

On Lagrangian Formalism of Quantum Computation

Jue Xu

Published Dec 7, 2021• 0 citations
arXivHTMLPDF

Abstract

We reformulate quantum computation in terms of Lagrangian (sum-over-path) formalism, in contrast to the widely used Hamiltonian (unitary gate) formulation. We exemplify this formalism with some widely-studied models, including the standard quantum circuit model, quantum optimization heuristics, and quantum random walks. The meanings of Lagrangian (action) are interpreted in various contexts of quantum computation, such as complexity analysis. Furthermore, an analog quantum simulation scheme is suggested where the Lagrangian serves as the starting point and the sum-over-path method is applied.

DOI
10.48550/arxiv.2112.04892

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We reformulate quantum computation in terms of Lagrangian (sum-over-path) formalism, in contrast to the widely used Hamiltonian (unitary gate) formulation. We exemplify this formalism with some widely-studied models, including the standard quantum circuit model, quantum optimization heuristics, and quantum random walks. The meanings of Lagrangian (action) are interpreted in various contexts of quantum computation, such as complexity analysis. Furthermore, an analog quantum simulation scheme is suggested where the Lagrangian serves as the starting point and the sum-over-path method is applied.
Abstract We provide a new paradigm for quantum simulation based on path integration that allows quantum speedups to be observed for problems that are more naturally expressed using the path integral formalism rather than the conventional Hamiltonian formalism. We provide two novel quantum algorithms based on Hamiltonian versions of the path integral formulation and another for Lagrangians of the form m 2 x ˙ 2 − V ( x ) , while avoiding the sign problems that preclude efficient Monte-Carlo algorithms for such problems. This Lagrangian path integral algorithm is based on a new rigorous derivation of a discrete version of the Lagrangian path integral. Our first Hamiltonian path integral method breaks up the paths into short timesteps. It is efficient under appropriate sparsity assumptions and requires a number of queries to oracles that give the eigenvalues and overlaps between the eigenvectors of the Hamiltonian terms that scales as t o ( 1 ) / ϵ o ( 1 ) for simulation time t and error ϵ . The second approach uses long-time path integrals for near-adiabatic systems and has query complexity that scales as O ( 1 / ϵ ) if the energy eigenvalue gaps and simulation time is sufficiently long. Finally, we show that our Lagrangian simulation algorithm requires a number of queries to an oracle that computes the discrete Lagrangian that scales for a system with η particles in D + 1 dimensions as O ~ ( η D t 2 / ϵ ) if V ( x ) is bounded and finite and the wavefunction obeys appropriate position and momentum cutoffs. This shows that Lagrangian dynamics can be efficiently simulated on quantum computers and opens up the possibility for quantum field theories for which the Hamiltonian is unknown to be efficiently simulated on quantum computers.
The classical random walk formalism plays an important role in a wide range of applications. Its quantum counterpart, the quantum walk, is proposed as an important theoretical model for quantum computing. By exploiting quantum effects such as superposition, interference, and entanglement, quantum walks and their variations have been extensively studied for achieving computing power beyond that of classical computing and have been broadly used in designing quantum algorithms for algebraic and optimization problems, graph and network analysis, and quantum Hamiltonian and biochemical process simulations. Moreover, quantum walk models have been proven capable of universal quantum computation. Unlike conventional quantum circuit models, quantum walks provide a feasible path for implementing application-specific quantum computing, particularly in the noisy intermediate-scale quantum era. Recently, remarkable progress has been achieved in implementing a wide variety of quantum walks and quantum walk applications, which demonstrates the great potential of quantum walks. In this review, we provide a thorough summary of quantum walks and quantum walk computing, including theories and characteristics, physical implementations, and applications. We also discuss the challenges facing quantum walk computing, which aims to realize a practical quantum computer in the near future.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
In this work is discussed possibility and actuality of Lagrangian approach to quantum computations. Finite-dimensional Hilbert spaces used in this area provide some challenge for such consideration. The model discussed here can be considered as an analogue of Weyl quantization of field theory via path integral in L. D. Faddeev's approach. Weyl quantization is possible to use also in finite-dimensional case, and some formulas may be simply rewritten with change of integrals to finite sums. On the other hand, there are specific difficulties relevant to finite case. This work has some allusions with phase space models of quantum computations developed last time by different authors.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
This article proposes a formalism which unifies Hamiltonian simulation techniques from different fields. This formalism leads to a competitive method to construct the Hamiltonian simulation with a comprehensible, simple-to-implement circuit generation technique. It leads to a gate decomposition and a scaling different from the usual strategy based on a Linear Combination of Unitaries (LCU) reformulation of the problem. It can significantly reduce the quantum circuit number of rotational gates, multi-qubit gates, and the circuit depth. This method leads to one exact Hamiltonian simulation for each summed term and Trotter step. Each of these Hamiltonian simulation unitary matrices also allows the construction of the non-exponential terms with a maximum of six unitary matrices to be Block-encoded.The formalism is easy to apply to the widely studied High-order Unconstrained Binary Optimization (HUBO) problems, fermionic transition Hamiltonian, and basic finite difference method instances. For the HUBO problems, our implementation exponentially reduces the number of gates for high-order cost functions with respect to the HUBO problem order. The individual electronic transitions are implemented without error for the second-quantization Fermionic Hamiltonian. Finite difference proposed matrix decompositions are straightforward, very versatile, and scale as the state-of-the-art proposals.