More Like this

1. Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure

   https://doi.org/10.3390/e21121218  

   Tranter, Andrew; Love, Peter J.; Mintert, Florian; Wiebe, Nathan; Coveney, Peter V. (December 2019, Entropy)

   Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian. 

   more » « less
2. Time-dependent unbounded Hamiltonian simulation with vector norm scaling

   https://doi.org/10.22331/q-2021-05-26-459  

   An, Dong; Fang, Di; Lin, Lin (May 2021, Quantum)

   null (Ed.)

   The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations. 

   more » « less
3. Faster Kernel Matrix Algebra via Density Estimation.

   Backurs, Arturs; Indyk, Piotr; Musco, Cameron; Wagner, Tal. (January 2021, International Conference on Machine Learning (ICML))

   We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix K∈ R^{n*n} corresponding to n points x1,…,xn∈R^d. In particular, we consider estimating the sum of kernel matrix entries, along with its top eigenvalue and eigenvector. We show that the sum of matrix entries can be estimated to 1+ϵ relative error in time sublinear in n and linear in d for many popular kernels, including the Gaussian, exponential, and rational quadratic kernels. For these kernels, we also show that the top eigenvalue (and an approximate eigenvector) can be approximated to 1+ϵ relative error in time subquadratic in n and linear in d. Our algorithms represent significant advances in the best known runtimes for these problems. They leverage the positive definiteness of the kernel matrix, along with a recent line of work on efficient kernel density estimation. 

   more » « less
4. Pauli error estimation via Population Recovery

   https://doi.org/10.22331/q-2021-09-23-549  

   Flammia, Steven T.; O'Donnell, Ryan (September 2021, Quantum)

   Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n -qubit channel to precision ϵ in ℓ ∞ using just O ( 1 / ϵ 2 ) log ⁡ ( n / ϵ ) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O ( 1 / ϵ ) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤ 1 / 4 .We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1 − η . In the regime of small η we extend our algorithm to achieve multiplicative precision 1 ± ϵ (i.e., additive precision ϵ η ) using just O ( 1 ϵ 2 η ) log ⁡ ( n / ϵ ) applications of the channel. 

   more » « less
5. Demonstration of the rodeo algorithm on a quantum computer

   https://doi.org/10.1140/epja/s10050-024-01373-9  

   Qian, Zhengrong; Watkins, Jacob; Given, Gabriel; Bonitati, Joey; Choi, Kenneth; Lee, Dean (July 2024, The European Physical Journal A)

   The rodeo algorithm is an efficient algorithm for eigenstate preparation and eigenvalue estimation for any observable on a quantum computer. This makes it a promising tool for studying the spectrum and structure of atomic nuclei as well as other fields of quantum many-body physics. The only requirement is that the initial state has sufficient overlap probability with the desired eigenstate. While it is exponentially faster than well-known algorithms such as phase estimation and adiabatic evolution for eigenstate preparation, it has yet to be implemented on an actual quantum device. In this work, we apply the rodeo algorithm to determine the energy levels of a random one-qubit Hamiltonian, resulting in a relative error of 0.08% using mid-circuit measurements on the IBM Q device Casablanca. This surpasses the accuracy of directly-prepared eigenvector expectation values using the same quantum device. We take advantage of the high-accuracy energy determination and use the Hellmann-Feynman theorem to compute eigenvector expectation values for a different random one-qubit observable. For the Hellmann-Feynman calculations, we find a relative error of 0.7%. We conclude by discussing possible future applications of the rodeo algorithm for multi-qubit Hamiltonians. 

   more » « less