Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

Abstract

Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop a new algorithmic framework "Quantum phase processing" that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled-unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigen-information of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase-related. As a notable application, we propose a new quantum phase estimation algorithm without quantum Fourier transform, which requires the least ancilla qubits and matches the best performance so far. We further exploit the power of our QPP framework by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropies estimation, demonstrating improvements or optimality for almost all cases.

Ground state preparation with shallow variational warm-start

“…The core of our approaches has two steps: firstly, prepare an initial state with a considerable overlap, such as states generated by shallow-depth VQE, Gibbs-VQE [46], or Hartree-Fock states (which will be discussed in numerical experiments); secondly, use a quantum algorithm to project out the ground state from the initial state with high probability. The projection can be realised using the quantum algorithmic tool, namely the quantum phase search algorithm (QPS) [43], which can classify the eigenvectors of a unitary operator. Using QPS, the probability of sampling the ground state is highly approximate to the initial overlap.…”

“…The phase factors φ, θ, and ω are chosen such that the circuit implements a trigonometric polynomial transformation to each phase of the unitary, i.e., f : λ → L j=−L c j e ijλ , where each c j ∈ C and satisfies j |c j | ≤ 1. For a certain trigonometric polynomial, the corresponding factors can be classically computed beforehand [43,48]. The action of running the circuit to an eigenvector of the unitary U is presented in the lemma below.…”

“…To solve ground-state problems in physics and chemistry, we notice the demand for overcoming limitations set by the current algorithms, such as the prior information of the ground-state energy and an efficient initial state preparation. This paper addresses these issues using shallow-depth VQE and the recent advances in quantum phase estimation [43]. Intuitively, one would expect to prepare the ground state with very high fidelity when estimating the ground state energy.…”

“…However, it cannot be realised using the conventional phase estimation due to the imperfect residual states, as discussed in [25,26]. Fortunately, this expectation can be fulfilled by the algorithm of [43], which uses only one ancilla qubit and can apply to block encoding and Hamiltonian evolution input models. We show that measuring the ancilla qubit can estimate the ground state energy up to the desired precision with a probability ≈ γ 2 .…”

“…The structures of QSVT and QET-U are similar to ours but use block encoding and Hamiltonian evolution, respectively. Rotation angles, usually referred to as the phase factors, are chosen so that the circuit can simulate a specific function, such as sign function [28,43] and shifted sign function [30]. The number of phase factors is crucial for preparing a high-fidelity ground state.…”