Faster quantum mixing of Markov chains in non-regular graph with fewer qubits

Sampling from the stationary distribution is one of the fundamental tasks of Markov chain-based algorithms and has important applications in machine learning, combinatorial optimization and network science. For the quantum case, qsampling from Markov chains can be constructed as preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. In this paper, a new qsampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks and works without any limit compared with existing results. In detail, we build a qsampling algorithm that not only accelerates non-regular graphs but also keeps the speed-up of existing quantum algorithms for regular graphs. In non-regular graphs, the invocation of the quantum fast-forward algorithm accelerates existing state-of-the-art qsampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms we reduce log n, where n is the number of graph vertices. In regular graphs, our result matches other quantum algorithms, and our reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared to the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm is the first algorithm to achieve complete quadratic acceleration (without log factor) over the classical case without any limit. To enlarge success probability amplitude amplification is introduced. We construct a new reflection on stationary state with fewer ancilla qubits and think it may have independent application.

💡 Research Summary

The paper presents a universal quantum sampling (qsampling) algorithm for any reversible Markov chain, achieving quadratic speed‑up over classical mixing while substantially reducing the number of ancilla qubits required. The authors combine two recent quantum techniques: quantum interpolated walks and quantum fast‑forwarding. In the interpolated walk framework, an interpolation parameter s is chosen so that both the initial basis state |g⟩ (corresponding to a randomly selected vertex) and the target stationary state |π⟩ have constant overlap with the 1‑eigenvector of the walk operator W(s). Traditionally, phase estimation is used to isolate this eigenvector, but it incurs a dependence on the error parameter ε and demands many ancilla qubits.

Quantum fast‑forwarding replaces phase estimation. It simulates t steps of the classical random walk using only √t steps of the quantum walk, thereby distinguishing the 1‑eigenvector from the rest of the spectrum with far fewer resources. This substitution reduces the overall query complexity to Θ(√HT·log ε⁻¹) (where HT is the classical hitting time) and lowers the ancilla depth to Θ(log √HT + log log ε).

For non‑regular graphs, the stationary probabilities πg are not known a priori. The authors devise a new estimation routine that narrows the possible range of πg using only O(log(π_min⁻¹/n)) additional queries, where π_min is the smallest stationary probability and n is the number of vertices. In many sparse graphs (π_min = Θ(1/n)) this overhead becomes constant. For regular graphs, πg is uniform and no estimation is needed.

After preparing an intermediate state |ψ⟩ with constant overlap with |π⟩, the algorithm constructs an approximate reflection ˜R_π on the stationary state using fast‑forwarding rather than phase estimation. This reflection requires far fewer ancilla qubits than prior constructions (e.g., those in