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Simulating the time evolution of large, sparse quantum states

Tanul Gupta

Published Feb 10, 2021• 0 citations
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Abstract

Master's thesis, Institute for Theoretical Physics, University of Cologne (submitted 10 February 2021). Supervisors: Prof. Dr. Tommaso Calarco and Dr. Robert Zeier. This thesis implements in Julia an algorithm, motivated by work of R. Zeier, for simulating the time evolution of large, sparse mixed quantum states in the product-operator basis. Instead of constructing the exponentially large unitary, the evolution is expanded in nested Lie commutators, which reduces the problem to sparse-state storage and fast commutator evaluation. For storage, the thesis introduces ClassicalMemory, a cache-aware associative array based on Judy arrays that supports essentially unbounded integer keys. It is benchmarked against SortedDict for up to 10^7 entries and key widths up to 640 bits. For commutators, the structure of the commutator and anticommutator tables is encoded in bitwise logic, following ideas and a prototype by R. Zeier, making commutator evaluation roughly 10^3–10^4 times faster than a table-based iterative approach. Example simulations include a 250-qubit Heisenberg chain, a 512-qubit XY chain, and a 5×5 XY lattice.

DOI
10.5281/zenodo.23234099

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Master's thesis, Institute for Theoretical Physics, University of Cologne (submitted 10 February 2021). Supervisors: Prof. Dr. Tommaso Calarco and Dr. Robert Zeier. This thesis implements in Julia an algorithm, motivated by work of R. Zeier, for simulating the time evolution of large, sparse mixed quantum states in the product-operator basis. Instead of constructing the exponentially large unitary, the evolution is expanded in nested Lie commutators, which reduces the problem to sparse-state storage and fast commutator evaluation. For storage, the thesis introduces ClassicalMemory, a cache-aware associative array based on Judy arrays that supports essentially unbounded integer keys. It is benchmarked against SortedDict for up to 10^7 entries and key widths up to 640 bits. For commutators, the structure of the commutator and anticommutator tables is encoded in bitwise logic, following ideas and a prototype by R. Zeier, making commutator evaluation roughly 10^3–10^4 times faster than a table-based iterative approach. Example simulations include a 250-qubit Heisenberg chain, a 512-qubit XY chain, and a 5×5 XY lattice.
We present the Julia package PauliStrings.jl for quantum many-body simulations, which performs fast operations on the Pauli group by encoding Pauli strings in binary. All of the Pauli string algebra is encoded into low-level logic operations on integers, and is made efficient by various truncation methods which allow for systematic extrapolation of the results. We illustrate the effectiveness of our package by (i) performing Heisenberg time evolution through direct numerical integration and (ii) by constructing a Liouvillian Krylov space. We benchmark the results against tensor network methods, and we find our package performs favorably. In addition, we show that this representation allows for easy encoding of any geometry. We present results for chaotic and integrable spin systems in 1D as well as some examples in 2D. Currently, the main limitations are the inefficiency of representing non-trivial pure states (or other low-rank operators), as well as the need to introduce dissipation to probe long-time dynamics.
Classical simulation of quantum operations is essential for algorithm design, noise characterisation, and benchmarking of quantum hardware. The most general physically realisable operation can be described by a positive linear map acting on a hermitian operator, representing either a density matrix or an observable. Established simulators vectorise the density matrix on an nnn-qubit Hilbert space and reuse state-vector kernels, storing all 22n2^{2n}22n elements and forgoing the benefits of hermitian symmetry. In this work, I introduce \emph{Orkan}, a simulation library that uses a tiled memory layout storing only the lower triangle of the hermitian matrix at tile granularity, roughly halving both the memory footprint and the wall time to simulate the evolution of quantum states under generic quantum operations. The implementation treats any hermitian operator uniformly and is agnostic to whether the Schrödinger or Heisenberg picture is used. Dedicated kkk-local conjugation algorithms update all entries of the hermitian matrix in a single pass. Benchmarks against Qiskit Aer, QuEST, and Qulacs show consistent wall-clock speedups of 222-4×4{\times}4× partly attributable to the reduced memory footprint.
The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. \n \nIn the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.
Efficient manipulation of Pauli-algebraic objects is a key bottleneck in the classical emulation and benchmarking of quantum algorithms for chemistry and many-body physics. This bottleneck appears in Hamiltonian construction, variational ansatz preparation, expectation-value and gradient evaluation, and real-time propagation, all of which require repeated Pauli-algebra operations. Here, we present a high-performance Pauli-algebra framework tailored to quantum many-body and quantum-chemical simulations. The framework combines compact binary symplectic encoding, canonical coefficient reduction, and grouped sparse operator representations that exploit shared bit-flip patterns among Pauli strings. The resulting Julia/C\texttt{++} implementation accelerates Pauli multiplication, Hamiltonian construction, and operator--state multiplication in sparse and symmetry-adapted many-electron spaces. Benchmarks demonstrate efficient Hamiltonian construction, large-active-space VQE and ADAPT-VQE calculations, and real-time variational dynamics on modern multicore CPU and GPU architectures. These results show that structure-aware Pauli-algebra engines provide a scalable classical backend for developing and benchmarking quantum algorithms in quantum chemistry and many-body simulation.
We present the Julia package PauliStrings.jl for quantum many-body simulations, which performs fast operations on the Pauli group by encoding Pauli strings in binary. All of the Pauli string algebra is encoded into low-level logic operations on integers, and is made efficient by various truncation methods which allow for systematic extrapolation of the results. We illustrate the effectiveness of our package by (i) performing Heisenberg time evolution through direct numerical integration and (ii) by constructing a Liouvillian Krylov space. We benchmark the results against tensor network methods, and we find our package performs favorably. In addition, we show that this representation allows for easy encoding of any geometry. We present results for chaotic and integrable spin systems in 1D as well as some examples in 2D. Currently, the main limitations are the inefficiency of representing non-trivial pure states (or other low-rank operators), as well as the need to introduce dissipation to probe long-time dynamics.
We propose an efficient algorithm based on shadow Hamiltonian simulation to approximately simulate the real-time dynamics of observables under time-independent Hamiltonians. Shadow Hamiltonian simulation works at the level of the operator algebra generated by the observables through commutators with the Hamiltonian. Exactly encoding the quantum state in this picture is generally inefficient for interacting systems due to the exponential growth of the operator algebra. Our algorithm overcomes this bottleneck by systematically identifying the elements of the algebra most relevant to the target observables. This targeted approach is a controlled approximation that yields a highly efficient quantum state encoding that substantially reduces the size of the qubit register required to perform the time evolution using the shadow Hamiltonian. We propose two main pruning schemes, one based on a predefined operator basis and another on a constructed Krylov basis. We also present a hybrid scheme that builds a Krylov basis within a pruned algebra in the predefined basis. We benchmark our algorithm using lattice spin systems in one and two dimensions, for both one- and higher-point correlators as observables.