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Time Evolution Operators for Pauli-String based Quantum Hamiltonians

Huang, Qunsheng, Christian B. Mendl, Technische Universität München

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

Quantum computing promises a paradigm shift in computational capabilities, but its realization demands problem representations fundamentally different from classical methods. To harness quantum advantages, problems must be encoded in forms native to quantum systems—commonly via the Pauli basis, which maps operations to rotations on the Bloch sphere. This dissertation investigates the bottlenecks limiting the implementation of such encodings across quantum, hybrid, and classical contexts.

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Quantum computing promises a paradigm shift in computational capabilities, but its realization demands problem representations fundamentally different from classical methods. To harness quantum advantages, problems must be encoded in forms native to quantum systems—commonly via the Pauli basis, which maps operations to rotations on the Bloch sphere. This dissertation investigates the bottlenecks limiting the implementation of such encodings across quantum, hybrid, and classical contexts.
Pauli matrices and Pauli strings are widely used in quantum computing. These mathematical objects are useful to describe or manipulate the quantum state of qubits. They offer a convenient basis to express operators and observables used in different problem instances such as molecular simulation and combinatorial optimization. Therefore, it is important to have a well-rounded, versatile and efficient tool to handle a large number of Pauli strings and operators expressed in this basis. This is the objective behind the development of the PauliArray library presented in this work. This library introduces data structures to represent arrays of Pauli strings and operators as well as various methods to modify and combine them. Built using NumPy, PauliArray offers fast operations and the ability to use broadcasting to easily carry out otherwise cumbersome manipulations. Applications to the fermion-to-qubit mapping, to the estimation of expectation values and to the computation of commutators are considered to illustrate how PauliArray can simplify some relevant tasks and accomplish them faster than current libraries.
As quantum computers are based on the laws of quantum mechanics, they are capable of solving certain problems faster than their classical counterparts. However, often algorithms which a theoretical speed-up assume that data can be loaded efficiently. In general, the runtime complexity of the loading routine depends on (i) the data encoding that defines how the data is represented and (ii) the data itself. In some cases, loading the data requires at least exponential time which destroys a potential speed-up. And especially for the first generation of devices that are currently available, the resources (qubits and operations) needed to encode the data are limited. Therefore, understanding the consequences of a particular data encoding is crucial. To capture knowledge about different encodings, we present two data encoding patterns that extend our previous collection of encoding patterns [1].
Quantum chemistry and materials science are among the most promising areas for demonstrating algorithmic quantum advantage and quantum utility due to their inherent quantum mechanical nature. Still, large-scale simulations of quantum circuits are essential for determining the problem size at which quantum solutions outperform classical methods. In this work, we present a novel hybrid simulation approach, forming a hybrid of a fullstate and a Clifford simulator, specifically designed to address the computational challenges associated with the time evolution of quantum chemistry Hamiltonians. Our method focuses on the efficient emulation of multi-qubit rotations, a critical component of Trotterized Hamiltonian evolution. By optimizing the representation and execution of multi-qubit operations leveraging the Pauli frame, our approach significantly reduces the computational cost of simulating quantum circuits, enabling more efficient simulations. Beyond its impact on chemistry applications, our emulation strategy has broad implications for any computational workload that relies heavily on multi-qubit rotations. By increasing the efficiency of quantum simulations, our method facilitates more accurate and cost-effective studies of complex quantum systems. We quantify the performance improvements and computational savings for this emulation strategy, and we obtain a speedup of a factor ≈18\approx 18≈18 (≈22\approx 22≈22 with MPI) for our evaluated chemistry Hamiltonians with 24 qubits. Thus, we evaluate our integration of this emulation strategy into the Intel Quantum SDK, further bridging the gap between theoretical algorithm development and practical quantum software implementations.
The complexity of simulating the out-of-equilibrium evolution of local operators in the Heisenberg picture is governed by the operator entanglement, which grows linearly in time for generic nonintegrable systems, leading to an exponential increase in computational resources. A promising approach to simplify this challenge involves discarding parts of the operator and focusing on a subspace formed by ``light'' Pauli strings---strings with few Pauli matrices---as proposed by Rakovszki et al. [Phys. Rev. B 105, 075131 (2022)]. In this work, we investigate whether this strategy can be applied to quenches starting from homogeneous product states. For ergodic dynamics, these initial states grant access to a wide range of equilibration temperatures. By concentrating on the desired matrix elements and retaining only the portion of the operator that contains Pauli strings parallel to the initial state, we uncover a complex scenario. In some cases, the light Pauli strings suffice to describe the dynamics, enabling efficient simulation with current algorithms. However, in other cases, heavier strings become necessary, pushing computational demands beyond our current capabilities. We analyze this behavior by introducing a measure of complexity, the operator weight entropy, which we compute for different operators across most points on the Bloch sphere.
Decomposing classical matrices into linear combinations of Pauli strings is a major bottleneck for end-to-end implementations of near-term quantum algorithms. In this work, we consider a promise version of this Pauli decomposition problem in which the matrix is guaranteed to have support on only k=poly(n)k = \mathsf{poly}(n)k=poly(n) Pauli strings and is given through classical sparse query access. Existing Pauli decomposition algorithms are designed for the generic, dense problem and do not inherently take advantage of this promised sparsity, so these approaches take time that is exponential in nnn. We present a randomized classical algorithm that does take advantage of this sparsity and recovers the exact Pauli decomposition with success probability at least 1−δ1 - δ1−δ, for any δδδ. Under the stated access model, the algorithm executes with query and runtime complexity that is polynomial in nnn, kkk, and log⁡(1/δ)\log(1/δ)log(1/δ). These results show that, even though finding the Pauli decomposition is exponentially hard for general matrices, it becomes efficiently solvable for matrices that are known to be sparse in the Pauli basis, a regime that is relevant to near-term quantum algorithms operating on structured classical input.