Fault tolerant quantum simulation via the phase estimation algorithm and qubitization has a T-gate count that scales proportionally to the 1-norm of the Hamiltonian, the cost of block encoding the Hamiltonian, and inversely proportionally to the desired accuracy. Tensor factorization methods have been successfully used to reduce T-gate counts in the ground state electronic structure problem. Here we introduce the use of tensor factorization methods to reduce the T-gate count of quantum phase estimation. In particular, we show how Canonical Polyadic and Tucker decompositions of the tensors representing the vibrational and vibronic Hamiltonians can be utilized to rewrite the Hamiltonian in terms of linear combination of bosonic position operators representing nuclear vibrations. We demonstrate the use of these factorization methods on the water and monodeutered methane molecules.
Fault tolerant quantum simulation via the phase estimation algorithm and qubitization has a T-gate count that scales proportionally to the 1-norm of the Hamiltonian, the cost of block encoding the Hamiltonian, and inversely proportionally to the desired accuracy. Tensor factorization methods have been successfully used to reduce T-gate counts in the ground state electronic structure problem. Here we introduce the use of tensor factorization methods to reduce the T-gate count of quantum phase estimation. In particular, we show how Canonical Polyadic and Tucker decompositions of the tensors representing the vibrational and vibronic Hamiltonians can be utilized to rewrite the Hamiltonian in terms of linear combination of bosonic position operators representing nuclear vibrations. We demonstrate the use of these factorization methods on the water and monodeutered methane molecules.
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 n-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.
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous best techniques for industrially and biologically relevant molecules, including the nitrogenase cofactor (FeMoCo) and cytochrome P450. Benchmark calculations comparing our method with the original double-factorization method show a 75% reduction in the block-encoding scaling constant. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the journey to the feasibility of practical Hamiltonian simulations.
Quantum computation of vibrational properties of molecules is a promising platform to obtain computational advantages for computational chemistry. However, fault-tolerant quantum computations of vibrational properties remain a relatively unexplored field in quantum computing. In this work, we present different algorithms for efficient encodings of vibrational Hamiltonians using qubitization. Specifically, we investigate different encoding representations, high-order tensor decomposition to obtain low rank approximations for the vibrational Hamiltonian, rectilinear, and polyspherical coordinate systems, parallelization, and grouping algorithms. To investigate the performance of the different methods, we perform benchmark computations for both small and large molecules with more than one hundred vibrational modes with two- and three-mode coupled Hamiltonian. We present the first resource estimates for qubitization using multimode vibrational Hamiltonians with 1 cm –1 spectral resolution for molecules with up to one hundred vibrational modes. Additionally, we show that tensor decomposition and operator parallelization effectively reduce the T gate depth of the quantum algorithm.
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%.
Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a CDF of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizable reduction of the Toffoli gate count in comparison to other variants of DF or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.
The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectral amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectral amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor Λ to just 2 Λ E gap , where E gap ≪ Λ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes—namely, double factorization and tensor hypercontraction—that when combined with spectral amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of iron-sulfur complexes and a CO 2 -fixation catalyst.