[Paper Review] Szegedy Walk Unitaries for Quantum Maps

Paweł Wocjan, Kristan Temme|arXiv (Cornell University)|Jul 15, 2021

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a quantum algorithm that generalizes Szegedy's quantization of classical random walks to quantum maps, specifically detailed balanced Lindbladians and quantum channels. By constructing a walk unitary whose eigenphase gap is quadratically amplified relative to the spectral gap of the Lindbladian, the method enables quantum speedups for simulating open quantum systems, with efficient implementation via block-encoding and energy estimation techniques.

ABSTRACT

Szegedy developed a generic method for quantizing classical algorithms based on random walks [Proceedings of FOCS, 2004, pp. 32-41]. A major contribution of his work was the construction of a walk unitary for any reversible random walk. Such unitary posses two crucial properties: its eigenvector with eigenphase $0$ is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of the random walk. It was an open question if it is possible to generalize Szegedy's quantization method for stochastic maps to quantum maps. We answer this in the affirmative by presenting an explicit construction of a Szegedy walk unitary for detailed balanced Lindbladians -- generators of quantum Markov semigroups -- and detailed balanced quantum channels. We prove that our Szegedy walk unitary has a purification of the fixed point of the Lindbladian as eigenvector with eigenphase $0$ and that its eigenphase gap is quadratically larger than the spectral gap of the Lindbladian. To construct the walk unitary we leverage a canonical form for detailed balanced Lindbladians showing that they are structurally related to Davies generators. We also explain how the quantization method for Lindbladians can be applied to quantum channels. We give an efficient quantum algorithm for quantizing Davies generators that describe many important open-system dynamics, for instance, the relaxation of a quantum system coupled to a bath. Our algorithm extends known techniques for simulating quantum systems on a quantum computer.

Motivation & Objective

• To extend Szegedy's quantization framework from classical reversible Markov chains to quantum maps, particularly detailed balanced Lindbladians and quantum channels.
• To construct a walk unitary whose eigenvector with eigenphase 0 is a purification of the fixed point state of the quantum map.
• To achieve quadratic eigenphase gap amplification relative to the spectral gap of the Lindbladian, enabling quantum speedups in open-system dynamics simulation.
• To provide an efficient quantum algorithm for quantizing Davies generators—commonly used in open quantum systems—using block-encoding and energy estimation.

Proposed method

• Leverages a canonical form for detailed balanced Lindbladians, showing they are structurally equivalent to Davies generators.
• Introduces a quantum discriminate Q via a similarity transformation with respect to the fixed point σ, which is independent of σ and invariant under different formulations of detailed balance.
• Constructs a walk unitary W(L) such that its eigenstate with eigenphase 0 is the purification |σ^{1/2}}⟩ = (σ^{1/2} ⊗ I)|Ω⟩ of the fixed point σ.
• Employs block-encoding techniques to efficiently implement the walk unitary, using controlled operations and phase estimation to encode the Bohr frequencies of the Hamiltonian.
• Applies filter rotations and controlled cyclic shifts to coherently implement sub-isometries T0 and T1, which are combined into a full isometry T for the walk unitary.
• Reduces the quantization of general detailed balanced Lindbladians to the special case of Davies generators by embedding them in a larger Hilbert space with reflection-based coupling operators.

Experimental results

Research questions

• RQ1 Can Szegedy's quantization method for classical random walks be generalized to quantum Markov processes governed by Lindbladians and quantum channels?
• RQ2 What is the structure of detailed balanced Lindbladians, and can they be systematically transformed into a form amenable to quantization?
• RQ3 How can a walk unitary be constructed such that its eigenstate with eigenphase 0 corresponds to a purification of the fixed point of a quantum map?
• RQ4 Can the eigenphase gap of the walk unitary be quadratically amplified relative to the spectral gap of the Lindbladian, as in the classical case?
• RQ5 What efficient quantum algorithms exist for implementing such walk unitaries, particularly for Davies generators with known thermal fixed points?

Key findings

• The proposed walk unitary W(L) has an eigenvector with eigenphase 0 that is the purification |σ^{1/2}}⟩ of the fixed point σ of the detailed balanced Lindbladian L.
• The eigenphase gap of W(L) is quadratically larger than the spectral gap of L, enabling quantum speedups analogous to those in classical Szegedy walks.
• The quantum discriminate Q, derived from the Lindbladian via a similarity transformation, is independent of the fixed point σ and retains the same structure across different formulations of detailed balance.
• The method reduces the quantization of general detailed balanced Lindbladians to the case of Davies generators, which are efficiently quantizable using block-encoding and energy estimation.
• An efficient quantum algorithm is constructed for Davies generators with thermal Gibbs fixed points, provided the Hamiltonian energies satisfy a rounding promise and can be resolved via energy estimation.
• The full walk unitary is implemented via a controlled isometry T, combining sub-isometries T0 and T1 using filter functions and phase estimation, with a final reflection R that enables coherent superposition over extended Bohr frequencies.