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arXiv:2011.08942

Efficient algorithm for generating Pauli coordinates for an arbitrary linear operator

Daniel Gunlycke, Mark C. Palenik, Alex R. Emmert, Sean A. Fischer

Published Nov 17, 2020• 7 citations
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Abstract

Several linear algebra routines for quantum computing use a basis of tensor products of identity and Pauli operators to describe linear operators, and obtaining the coordinates for any given linear operator from its matrix representation requires a basis transformation, which for an $\mathrm N\times\mathrm N$ matrix generally involves $\mathcal O(\mathrm N^4)$ arithmetic operations. Herein, we present an efficient algorithm that for our particular basis transformation only involves $\mathcal O(\mathrm N^2\log_2\mathrm N)$ operations. Because this algorithm requires fewer than $\mathcal O(\mathrm N^3)$ operations, for large $\mathrm N$, it could be used as a preprocessing step for quantum computing algorithms for certain applications. As a demonstration, we apply our algorithm to a Hamiltonian describing a system of relativistic interacting spin-zero bosons and calculate the ground-state energy using the variational quantum eigensolver algorithm on a quantum computer.

DOI
10.48550/arxiv.2011.08942

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Several linear algebra routines for quantum computing use a basis of tensor products of identity and Pauli operators to describe linear operators, and obtaining the coordinates for any given linear operator from its matrix representation requires a basis transformation, which for an N×N\mathrm N\times\mathrm NN×N matrix generally involves O(N4)\mathcal O(\mathrm N^4)O(N4) arithmetic operations. Herein, we present an efficient algorithm that for our particular basis transformation only involves O(N2log⁡2N)\mathcal O(\mathrm N^2\log_2\mathrm N)O(N2log2​N) operations. Because this algorithm requires fewer than O(N3)\mathcal O(\mathrm N^3)O(N3) operations, for large N\mathrm NN, it could be used as a preprocessing step for quantum computing algorithms for certain applications. As a demonstration, we apply our algorithm to a Hamiltonian describing a system of relativistic interacting spin-zero bosons and calculate the ground-state energy using the variational quantum eigensolver algorithm on a quantum computer.
Abstract Analysis of quantum processes, especially in the context of noise, errors, and decoherence is essential for the improvement of quantum devices. An intuitive representation of those processes modeled by quantum channels are Pauli transfer matrices. They display the action of a linear map in the n-qubit Pauli basis in a way, that is more intuitive, since Pauli strings are more tangible objects than the standard basis matrices. We set out to investigate classical algorithms that convert the various representations into Pauli transfer matrices. We propose new algorithms that make explicit use of the tensor product structure of the Pauli basis. They convert a quantum channel in a given representation (Chi or process matrix, Choi matrix, superoperator, or Kraus operators) to the corresponding Pauli transfer matrix. Moreover, the underlying principle can also be used to calculate the Pauli transfer matrix of other linear operations over n-qubit matrices such as left-, right-, and sandwich multiplication as well as forming the (anti-)commutator with a given operator. Finally, we investigate the runtime of these algorithms, derive their asymptotic scaling and demonstrate improved performance using instances with up to seven qubits.
We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. Our algorithms start from a classical data structure in which the matrix of interest is specified in the Pauli basis. For N×NN\times NN×N Hermitian matrices, the space cost is log⁡(N)+1\log(N)+1log(N)+1 qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size O(N2)O(N^2)O(N2), when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.
Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We provide a decomposition of a fermionic second-quantized Hamiltonian into a number of one-sparse, self-inverse, Hermitian terms linear in the number of ladder operator monomials in the second-quantized representation. We provide a decomposition of a general d-sparse Hamiltonian into O(d^{2}) such terms. In both cases, a single sample of any term can be obtained using two ansatz state preparations and at most six oracle queries. The number of samples required to estimate the expectation value to precision ε scales as ε^{-2} as for Pauli-based VQE. This widens the domain of applicability of VQE to systems whose Hamiltonian and other observables are most efficiently described in terms of sparse matrices.
Quantum computing is being extensively used in quantum chemistry, especially in simulating simple molecules and evaluating properties like the ground state energy, dipole moment, etc. The transformation of a molecular Hamiltonian from the fermionic space to the qubit space provides us with a series of Pauli strings and the energy calculation involves the evaluation of the expectation values of all these individual strings. This introduces a major bottleneck for applications of VQEs in quantum chemistry. Unlike the fermionic Hamiltonian, the terms in a qubit Hamiltonian are additive and the present paper exploits this property to describe a new approach for extracting information from the partial qubit Hamiltonian of simple molecules to design more efficient variational quantum eigensolvers. In the partial (qubit) Hamiltonian approach (PHA), the qubit Hamiltonian is studied term-by-term to understand their relative contributions to the overall energy, and a partial Hamiltonian is constructed with fewer Pauli strings that can resolve the entire Hamiltonian. With PHA, we can simulate molecules at a much lower computational cost with a truncated Hamiltonian. Additionally, the outcomes of the measurements with PHA quench the error due to noise introduced by the quantum circuits. We have also demonstrated the application of PHA as an initialization technique, where the simple partial Hamiltonian can be used to find an optimal starting point for a more complex system.
This article deals with quantum computations in the Pauli basis whose elements are usually identified with the Pauli strings. This approach allows us to represent quantum states, observables, and unitary operators in the unified form of linear combination of Pauli strings, so that all operations can be reduced to the string compositions. Nevertheless a formal justification of the Pauli basis for quantum computations should be based on the strong results of complex linear algebra and the theory of Hilbert spaces. We briefly review the main features of Pauli strings for quantum states and unitary operators, and also the key operations with them, including an algorithm for string compositions and a transformation algorithm from the standard basis to the Pauli basis.