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arXiv:2312.11637

Strategies for simulating time evolution of Hamiltonian lattice field theories

Siddharth Hariprakash, Neel S. Modi, Michael Kreshchuk, Christopher F. Kane, C. Bauer

Published Dec 18, 2023• 0 citations
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Abstract

Simulating the time evolution of quantum field theories given some Hamiltonian $H$ requires developing algorithms for implementing the unitary operator e^{-iHt}. A variety of techniques exist that accomplish this task, with the most common technique used so far being Trotterization, which is a special case of the application of a product formula. However, other techniques exist that promise better asymptotic scaling in certain parameters of the theory being simulated, the most efficient of which are based on the concept of block encoding. In this work we study the performance of such algorithms in simulating lattice field theories. We derive and compare the asymptotic gate complexities of several commonly used simulation techniques in application to Hamiltonian Lattice Field Theories. Using the scalar ϕ^4 theory as a test, we also perform numerical studies and compare the gate costs required by Product Formulas and Signal Processing based techniques to simulate time evolution. For the latter, we use the the Linear Combination of Unitaries construction augmented with the Quantum Fourier Transform circuit to switch between the field and momentum eigenbases, which leads to immediate order-of-magnitude improvement in the cost of preparing the block encoding. The paper also includes a pedagogical review of utilized techniques, in particular Product Formulas, LCU, Qubitization, QSP, as well as a technique we call HHKL based on its inventors' names.

DOI
10.48550/arxiv.2312.11637

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Simulating the time evolution of quantum field theories given some Hamiltonian HHH requires developing algorithms for implementing the unitary operator e^{-iHt}. A variety of techniques exist that accomplish this task, with the most common technique used so far being Trotterization, which is a special case of the application of a product formula. However, other techniques exist that promise better asymptotic scaling in certain parameters of the theory being simulated, the most efficient of which are based on the concept of block encoding. In this work we study the performance of such algorithms in simulating lattice field theories. We derive and compare the asymptotic gate complexities of several commonly used simulation techniques in application to Hamiltonian Lattice Field Theories. Using the scalar ϕ^4 theory as a test, we also perform numerical studies and compare the gate costs required by Product Formulas and Signal Processing based techniques to simulate time evolution. For the latter, we use the the Linear Combination of Unitaries construction augmented with the Quantum Fourier Transform circuit to switch between the field and momentum eigenbases, which leads to immediate order-of-magnitude improvement in the cost of preparing the block encoding. The paper also includes a pedagogical review of utilized techniques, in particular Product Formulas, LCU, Qubitization, QSP, as well as a technique we call HHKL based on its inventors' names.
Simulating the time evolution of quantum field theories given some Hamiltonian HHH requires developing algorithms for implementing the unitary operator ${\mathrm{e}}^{\ensuremath{-}iHt}$. A variety of techniques exist that accomplish this task, with the most common technique used so far being Trotterization, which is a special case of the application of a product formula. However, other techniques exist that promise better asymptotic scaling in certain parameters of the theory being simulated, the most efficient of which are based on the concept of block encoding. In this work we study the performance of such algorithms in simulating lattice field theories. We derive and compare the asymptotic gate complexities of several commonly used simulation techniques in application to Hamiltonian lattice field theories. Using the scalar ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varphi}}}^{4}$ theory as a test, we also perform numerical studies and compare the gate costs required by product formulas and signal-processing-based techniques to simulate time evolution. For the latter, we use the linear combination of unitaries (LCU) construction augmented with the quantum Fourier transform circuit to switch between the field and momentum eigenbases, which leads to immediate order-of-magnitude improvement in the cost of preparing the block encoding. This paper also includes a pedagogical review of the techniques used, in particular product formulas, LCU, qubitization, quantum signal processing, as well as the technique for simulating geometrically-local Hamiltonians developed by Haah, Hastings, Kothari, and Low.
In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.
Quantum simulation of lattice gauge theories offers a path to first-principles calculations of real-time dynamics and finite-density phenomena that are inaccessible to classical Euclidean Monte Carlo methods. Realizing this potential requires a pipeline of controlled approximations---lattice discretization, truncation, digitization, state preparation, time evolution, measurement, and continuum extrapolation---each contributing errors and computational costs. General algorithmic asymptotic complexity arguments capture only part of the cost picture: the practical resources required for end-to-end simulation depend on numerical pre-factors, the cost of the subroutines that various algorithms call, and structural features of the lattice theory, all of which the asymptotic scaling leaves unspecified. This thesis develops contributions that aim to make such dependencies explicit and quantitative. We begin with a rigorous analysis of the asymptotic gate complexity of product formulas and Quantum Signal Processing (QSP) for the time evolution of general classes of lattice Hamiltonians, identifying the parameter regimes in which each method is favorable and estimating the associated pre-factors numerically. Since the cost of QSP-based simulation is dominated (in regimes of practical interest) by the block-encoding subroutine it calls, we also develop several novel block-encoding constructions tailored to bosonic lattice field theories and show that signal-processing techniques can be repurposed as highly efficient block-encoding subroutines. We then turn to the problem of truncating the Kogut--Susskind electric-field Hilbert space. Existing error bounds rely on energy-conservation arguments and require the truncation to grow with simulation time. By exploiting Hilbert space fragmentation we derive the tightest error bounds known to date: they decay factorially in the truncation parameter and are independent of simulation time. We validate them numerically for U(1) pure gauge theory and the Schwinger model. Finally, we address the continuum limit in the presence of approximate time evolution. For product formulas, we show that Trotter errors correspond to irrelevant operators that vanish in the continuum limit, which yields a renormalization trajectory independent of the Trotter step size. More generally, we introduce the Statistically-Bounded Time Evolution (SBTE) protocol, an algorithm-independent framework that treats approximation errors as a systematic uncertainty to be driven below the working statistical precision. This decouples the continuum-limit prescription from the choice of time-evolution algorithm and provides a basis for end-to-end cost comparisons across simulation methods.
Block Encoding (BE) is a crucial subroutine in many modern quantum algorithms, including those with near-optimal scaling for simulating quantum many-body systems, which often rely on Quantum Signal Processing (QSP). Currently, the primary methods for constructing BEs are the Linear Combination of Unitaries (LCU) and the sparse oracle approach. In this work, we demonstrate that QSP-based techniques, such as Quantum Singular Value Transformation (QSVT) and Quantum Eigenvalue Transformation for Unitary Matrices (QETU), can themselves be efficiently utilized for BE implementation. Specifically, we present several examples of using QSVT and QETU algorithms, along with their combinations, to block encode Hamiltonians for lattice bosons, an essential ingredient in simulations of high-energy physics. We also introduce a straightforward approach to BE based on the exact implementation of Linear Operators Via Exponentiation and LCU (LOVE-LCU). We find that, while using QSVT for BE results in the best asymptotic gate count scaling with the number of qubits per site, LOVE-LCU outperforms all other methods for operators acting on up to ≲ 11 qubits, highlighting the importance of concrete circuit constructions over mere comparisons of asymptotic scalings. Using LOVE-LCU to implement the BE, we simulate the time evolution of single-site and two-site systems in the lattice φ 4 theory using the Generalized QSP algorithm and compare the gate counts to those required for Trotter simulation.
We present the problem of approximating the time-evolution operator e-iHt to error ϵ, where the Hamiltonian H = ((G|⊗I)U (|Gi⊗I) is the projection of a unitary oracle U onto the state |Gi created by another unitary oracle. Our algorithm solves this with a query complexity O ( t + log(1/ϵ) ) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H is a density matrix. A key technical result is 'qubitization', which uses the controlled version of these oracles to embed any H in an invariant SU(2) subspace. A large class of operator functions of H can then be computed with optimal query complexity, of which e-iHt is a special case.
The time evolution operator plays a crucial role in the precise computation of chemical experiments on quantum computers and holds immense promise for advancing the fields of physical and computer sciences, with applications spanning quantum simulation and machine learning. However, the construction of large-scale quantum computers poses significant challenges, prompting the need for innovative and resource-efficient strategies. Traditional methods like phase estimation or variational algorithms come with certain limitations such as the use of classical optimization or complex quantum circuitry. One successful method is the Trotterization technique used for quantum simulation, specifically in atomic structure problems with a gate complexity of approximately O(n^2) for an n-qubit realization. In this work, we have proposed a new encoding method, namely quantum approximate time evolution (QATE) for the quantum implementation of the kinetic energy operator as a diagonal unitary operator considering the first quantization level. The theoretical foundations of our approach are discussed, and experimental results are obtained on an IBM quantum machine. Our proposed method offers gate complexity in sub-quadratic polynomial with qubit size nnn which is an improvement over previous work. Further, the fidelity improvement for the time evolution of the Gaussian wave packet has also been demonstrated.