Quantum Phase Processing and its Applications in Estimating Phase and Entropies

TL;DR

The paper introduces Quantum Phase Processing (QPP), a framework that generalizes quantum signal processing to apply arbitrary trigonometrical transformations to eigenphases of a unitary, enabling direct phase manipulation and indirect eigen-information extraction via a single ancilla qubit. By constructing multi-qubit QPP circuits from controlled-U and controlled-U† layers, the authors show how any trigonometric polynomial on eigenphases can be implemented, and they derive powerful results for quantum phase estimation without the quantum Fourier transform, as well as for Hamiltonian simulation and entropy estimation using block encoding and qubitization. The key contributions include the formal development of QPP, a phase-search based PIS algorithm for efficient eigenphase estimation, and comprehensive complexity analyses for von Neumann, quantum relative, and Rényi entropy estimation within this framework, often achieving favorable rank-dependent or single-qubit measurement advantages. Collectively, QPP broadens the quantum algorithm toolbox, offering practical, phase-centric methods that complement QSVT and hold promise for near-term quantum hardware and applications in quantum information, physics, and chemistry.

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 toolbox "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 fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy and quantum entropies estimation, demonstrating improvements or optimality for almost all cases.

Figures (10)

• Figure 1: General circuit for quantum phase processing $V_{\omega,\bm\theta,\bm\phi}^L(U)$, here the number of layers $L$ is an even integer.
• Figure 2: Illustration of the capabilities of quantum phase processing. A trigonometric transformation $F(x) = \frac{1}{2} \left(e^{\sin x} - e^{\cos x} \right)$ of eigenphases of an $n$-qubit unitary $U$ can be represented by a QPP circuit $V(U)$ in two different ways. (a) One approach is to directly simulate the target function. By post-selecting the ancilla qubit to be $\ket{0}$, the function transformations of phases are encoded into the amplitudes of eigenvectors,. (b) The other approach is to retrieve the target function by the expectation value of measuring a Pauli-$Z$ observable on the ancilla qubit. More intuitively, if the input state of circuit $V(U)$ is a maximally mixed state, then the sum of eigenphase transformations is approximately the square difference between two shaded areas in (b) multiplied by a constant. (c-e) Three examples for the QPP circuits to approximate functions near a discontinuity, which will be useful in applications of quantum phase estimation and quantum entropies estimation. In these three subfigures, the quantity "deg" refers to the degree of trigonometric polynomials, where every increase of such degree takes an extra layer in a QPP circuit.
• Figure 3: The QPP circuit in Theorem \ref{['thm:state entropy estimation']} for quantum entropies estimation.
• Figure : Phase Interval Search (PIS)
• Figure : Quantum Phase Search Algorithm (QPS)

Theorems & Definitions (34)

• Lemma 1: Trigonometric quantum signal processing yu2022power
• Corollary 2
• Corollary 3
• Lemma 4: Eigenspace Decomposition of QPP
• Theorem 5: Quantum phase evolution
• Theorem 6: Quantum phase evaluation
• Lemma 7: Phase classification
• Theorem 8: Complexity of Quantum Phase Search
• Theorem 9
• Theorem 10: von Neumann entropy estimation