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Computational complexity analysis of the linear combination of unitaries algorithm in the quantum chemistry framework

Defne Tolun

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

Hamiltonian simulation is a fundamental application of quantum computing, particularly in quantum chemistry, where solving the electronic structure problem requires efficiently representing electron repulsion integrals (ERI). This thesis analyzes the computational complexity of Hamiltonian simulation using the Linear Combination ofUnitaries (LCU) algorithm and Tensor Hypercontraction (THC), a factorization method that reduces the cost of ERI representation. By decomposing the Coulomb tensor via Interpolative Separable Density Fitting (ISDF) and mapping the THC representation to an LCU form using the Jordan-Wigner transformation, we achieve a more efficient block encoding of the Hamiltonian in a basis of N molecular orbitals. This enables quantum simulation with an asymptotic Toffoli complexity of O(Nl/e) and a space complexity of O(N), where e = 0.0016 Hartree is the target accuracy and l denotes the 1-norm of the coefficients of the electronic Hamiltonian in the THC representation. Beyond Hamiltonian simulation, we explore how Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) leverage block encoding for eigenvalue transformations and time evolution, extending its utility to a broader class of quantum algorithms. Despite these advancements, large-scale quantum chemistry simulations are far from experimental feasibility, for instance, simulating the FeMoCo molecule in an active space model under surface code constraints would require billions of Toffoli gates, millions of physical qubits and several days of runtime using error-corrected qubits; and that is assuming it ever becomes feasible. This work contributes to the effort of making quantum chemistry simulations more computationally viable by investigating how LCU and THC can be integrated to reduce costs and maintain chemical accuracy, while exploring different methods and their combinations to identify key challenges in quantum chemistry simulations.

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Hamiltonian simulation is a fundamental application of quantum computing, particularly in quantum chemistry, where solving the electronic structure problem requires efficiently representing electron repulsion integrals (ERI). This thesis analyzes the computational complexity of Hamiltonian simulation using the Linear Combination ofUnitaries (LCU) algorithm and Tensor Hypercontraction (THC), a factorization method that reduces the cost of ERI representation. By decomposing the Coulomb tensor via Interpolative Separable Density Fitting (ISDF) and mapping the THC representation to an LCU form using the Jordan-Wigner transformation, we achieve a more efficient block encoding of the Hamiltonian in a basis of N molecular orbitals. This enables quantum simulation with an asymptotic Toffoli complexity of O(Nl/e) and a space complexity of O(N), where e = 0.0016 Hartree is the target accuracy and l denotes the 1-norm of the coefficients of the electronic Hamiltonian in the THC representation. Beyond Hamiltonian simulation, we explore how Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) leverage block encoding for eigenvalue transformations and time evolution, extending its utility to a broader class of quantum algorithms. Despite these advancements, large-scale quantum chemistry simulations are far from experimental feasibility, for instance, simulating the FeMoCo molecule in an active space model under surface code constraints would require billions of Toffoli gates, millions of physical qubits and several days of runtime using error-corrected qubits; and that is assuming it ever becomes feasible. This work contributes to the effort of making quantum chemistry simulations more computationally viable by investigating how LCU and THC can be integrated to reduce costs and maintain chemical accuracy, while exploring different methods and their combinations to identify key challenges in quantum chemistry simulations.
We describe quantum circuits with only O(N ) Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of N arbitrary (e.g., molecular) orbitals.With O(λ/ ) repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where λ is the 1-norm of Hamiltonian coefficients and is the target precision.This is the lowest complexity shown for quantum computations of chemistry within an arbitrary basis.Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can work only with orthogonal-basis functions diagonalizing the Coloumb operator (e.g., the plane-wave dual basis).Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a nonorthogonal basis defined by the THC factors.We then use qubitization to simulate the nonorthogonal THC Hamiltonian, in a fashion that avoids most complications of the nonorthogonal basis.We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work.In addition to having lower asymptotic scaling space-time volume, compilation of our algorithm for challenging finitesized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach.By laying out and optimizing the surface-code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under 4 days of runtime, assuming 1-μs cycle times and physical gate-error rates no worse than 0.1%.
Simulation of molecular vibrations is emerging as a promising application of quantum computing in quantum chemistry. In this work, we develop new strategies for block-encoding the second-quantized nnn-mode representation of the vibrational Hamiltonian, whose algebraic structure closely parallels that of the second-quantized electronic Hamiltonian. We derive linear-combination-of-unitaries (LCU) representations of this Hamiltonian based on Pauli decomposition and tensor hypercontraction (THC) and construct efficient circuits for the corresponding block encodings. We also extend the Block-Invariant Symmetry Shift (BLISS) technique to vibrational Hamiltonians, yielding a lower LCU 1-norm and, consequently, a lower overall circuit cost. For the molecules considered here, BLISS-THC reduces the Toffoli cost of quantum phase estimation by more than two orders of magnitude relative to existing block-encoding approaches. In comparison with recent Trotter-based methods, it shows up to a ninefold reduction in the T-gate count, with the relative savings tending to increase with system size. Asymptotically, and up to polylogarithmic factors, both the Pauli and THC LCUs trade additional ancillas for a quadratic improvement in scaling with the number of modes, while THC provides an additional quadratic improvement in scaling with the number of modals. These results highlight the importance of Hamiltonian factorizations and Toffoli-qubit tradeoffs for reducing the computational cost of quantum simulations of molecular vibrations.
Reducing the complexity of quantum algorithms to treat quantum chemistry problems is essential to demonstrate an eventual quantum advantage of noisy-intermediate scale quantum devices over their classical counterpart. Significant improvements have been made recently to simulate the time-evolution operator $U(t)={e}^{i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}t}$, where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}$ is the electronic structure Hamiltonian, or to simulate $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}$ directly (when written as a linear combination of unitaries) by using block encoding or qubitization techniques. A fundamental measure quantifying the practical implementation complexity of these quantum algorithms is the so-called 1-norm of the qubit representation of the Hamiltonian, which can be reduced by writing the Hamiltonian in factorized or tensor-hypercontracted forms, for instance. In this paper, we investigate the effect of classical preoptimization of the electronic structure Hamiltonian representation, via single-particle basis transformation, on the 1-norm. Specifically, we employ several localization schemes and benchmark the 1-norm of several systems of different sizes (number of atoms and active space sizes). We also derive a formula for the 1-norm as a function of the electronic integrals and use this quantity as a cost function for an orbital-optimization scheme that improves over localization schemes. This paper gives more insights about the importance of the 1-norm in quantum computing for quantum chemistry and provides simple ways of decreasing its value to reduce the complexity of quantum algorithms.
Accurately characterizing molecular properties and reactivity often requires electronic-structure calculations across an ensemble of molecular geometries. Quantum algorithms that process such ensembles coherently could address this challenge, but they require Hamiltonian block encoding circuits that combine advanced factorization techniques with sublinear scaling in ensemble size. Here, we introduce SVD-THC, which extends tensor hypercontraction to sets of Hamiltonians defined at different nuclear geometries through a singular value decomposition of the ensemble. We use SVD-THC to construct a single multiplexed block encoding circuit for all geometries in an ensemble. We then evaluate the approach on ensembles generated by varying internal rotational coordinates in 1-bromo-2-chloroethane as well as in glycine and alanine dipeptides, corresponding to established model systems for peptide conformational landscapes, with up to 5,184 geometries. Across several resolutions of the internal-coordinate grids, the Toffoli complexity of the resulting circuits grows only weakly with the number of geometries, changing by at most 12\% over grids that differ in size by up to 576×\times×. In an energy-thresholding circuit, this yields 35--200×\times× fewer Toffoli gates than slicing together single-geometry block encodings. This approach could help make coherent processing of molecular geometry ensembles a useful primitive for fault-tolerant quantum computing.
Electronic structure calculations of molecular systems are among the most promising applications for fault-tolerant quantum computing (FTQC) in quantum chemistry and drug design. However, while recent algorithmic advancements such as qubitization and tensor hypercontraction (THC) have significantly reduced the complexity of such calculations, they do not yet achieve computational runtimes short enough to be practical for industrially relevant use cases. In this work, we combine several advances to electronic structure calculation for molecular systems, resulting in a 2-orders-of-magnitude speedup of estimated runtimes over prior-art algorithms run on comparable quantum devices. One of these advances is a novel framework for block-invariant symmetry-shifted tensor hypercontraction, with which we achieve the tightest Hamiltonian factorizations reported to date. We compile our algorithm for an active volume (AV) architecture, a technical layout that has recently been proposed for fusion-based photonic quantum hardware. AV compilation contributes towards a lower runtime of our computation by eliminating overheads stemming from connectivity issues in the underlying surface code. We present a detailed benchmark of our approach, focusing primarily on the computationally challenging benchmark molecule P450. Leveraging a number of hardware trade-offs in interleaving-based photonic FTQC, we estimate runtimes for the electronic structure calculation of P450 as a function of the device footprint.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.