arXiv subjects

Mingnan Zhao

Publications and source records attributed to Mingnan Zhao.

7 recordsLinked to original sources

arXiv2026

A Tale of Two Walks: From Kac to KMP and Back

We reveal an unexpected connection between two interacting particle systems: Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. We first establish a mixing-time analogue of Aldous's conjecture for the KMP process on general graphs: the mixing time of the $t$-particle process on any connected $d$-vertex graph is at most $O(\log d)$ times the corresponding single-particle mixing time, uniformly in $t$. As an application, we improve the mixing-time bound for coordinate hit-and-run on the $n$-dimensional standard simplex from $\widetilde{O}(n^3)$ [KV26] to $O(n\log n)$, while removing the dependence on the initial distribution. We then show that the twirling channel induced by the parallel Kac's walk on the symmetric subspace can be exactly encoded by a Markov chain on partitions, which naturally lifts to a parallel KMP chain on the complete graph. We prove that $O(\log d+\log(1/\varepsilon))$ repetitions of the parallel Kac's walk suffice to approximate Haar twirling on the symmetric subspace of $(\mathbb{C}^d)^{\otimes t}$ to error $\varepsilon$, uniformly in all $t$. Our main technical contribution is a new structural property of KMP-type dynamics: conditional product structure, which holds for both the standard KMP processes on general graphs and the parallel KMP processes on complete graphs. Conditioned on suitable auxiliary randomness, the evolution of the labeled KMP process factorizes into independent single-particle updates. Combining this structure with an exact coupling, we remove the dependence on the number of particles from the mixing bounds for both unlabeled KMP models. This bypasses the $\log(π_{\min}^{-1})$ dependency from spectral-gap arguments, which is often prohibitively large for many-particle systems. Using this framework, we prove mixing bounds that are sharp up to logarithmic factors for both unlabeled KMP models.

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arXiv2026

unitaryQIP = unitaryPSPACE

Unitary synthesis complexity studies the computational cost of applying a target unitary to an unknown quantum state, extending complexity theory beyond its traditional focus on problems with classical inputs and outputs. Rosenthal and Yuen (ITCS 2022) introduced two complexity classes for unitary synthesis: $\mathsf{unitaryQIP}$ and $\mathsf{unitaryPSPACE}$ for the unitary families implementable via interactive proofs and in polynomial space, respectively. In this work, we prove the equality $\mathsf{unitaryQIP}=\mathsf{unitaryPSPACE}$ and thereby establish a unitary-complexity analogue of $\mathsf{QIP} = \mathsf{PSPACE}$ (Jain, Ji, Upadhyay, and Watrous, J. ACM 2011) and $\mathsf{stateQIP}=\mathsf{statePSPACE}$ (Rosenthal and Yuen, ITCS 2022; Metger and Yuen, FOCS 2023). To prove $\mathsf{unitaryQIP} \subseteq \mathsf{unitaryPSPACE}$, we use an acceptance rigidity lemma derived from the soundness condition to show that the polynomial-space prover constructed for a fixed input in state synthesis (Metger and Yuen, FOCS 2023) implements the target unitary on every input, leading to a polynomial-space simulation of the protocol. For the reverse inclusion, we build on OpenAI's recently announced solution to the unitary synthesis problem (Aaronson and Kuperberg, Theory Comput. 2007), and show that $\mathsf{unitaryPSPACE}$ families can be synthesized in polynomial time with a $\mathsf{PSPACE}$ oracle. We then implement the oracle queries using auxiliary states prepared via the $\mathsf{statePSPACE} \subseteq \mathsf{stateQIP}$ protocol (Rosenthal and Yuen, ITCS 2022). Beyond the equality, we show that our protocol for $\mathsf{unitaryPSPACE}$ can be parallelized to constant rounds when the verifier receives polynomially many copies of the input, and we also obtain a complexity-theoretic barrier to proving the existence of $\mathsf{EFI}$ pairs.

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arXiv2026

Rapid Mixing of Parallel Kac's Walk: From Spheres to Stiefel Manifolds

Kac's walk is a classical local random walk whose action on a single real unit vector in dimension $d$ mixes in total variation in $Θ(d\log d)$ sequential steps~\cite{PS17}. Lu, Qin, Song, Yao, and Zhao introduced a parallel version of Kac's walk that mixes a single quantum state in $O(\log d)$ rounds~\cite{LQSY+26}. After discretizing the randomness and replacing it by suitable pseudorandom primitives, this parallel walk gives rise to pseudorandom state scramblers, and was subsequently shown to yield pseudorandom unitaries~\cite{LQSY+25}. We study what happens when the parallel Kac's walk acts simultaneously on $k$ orthonormal quantum states. We prove that, for any $1\leq k < d$, after $O\!\left((k+\log d)\log(d/\varepsilon)\right)$ steps, the joint distribution of the $k$ output states is $\varepsilon$-close, in both Wasserstein and total variation distance, to that obtained by applying a common Haar-random unitary to the same inputs. This generalizes the dispersing property of the parallel Kac's walk from a single quantum state to multiple orthonormal quantum states. Equivalently, viewing an ordered collection of $k$ orthonormal states as a point on the complex Stiefel manifold $V_{d,k}=\{X\in\mathbb C^{d\times k}:X^\dagger X=I_k\}$, we show that the parallel Kac's walk mixes rapidly on $V_{d,k}$, with both Wasserstein and total variation mixing times bounded by $O\!\left((k+\log d)\log(d/\varepsilon)\right)$. This extends the Wasserstein mixing result of Pillai, Smith, and Vaikuntanathan for the standard Kac's walk on real Stiefel manifolds~\cite{PSV26}.

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arXiv2026

Fooling Thresholds of Halfspaces

We initiate the study of constructing explicit pseudorandom generators for thresholds of halfspaces with seed length polylogarithmic in the number of halfspaces. This class of functions lies at the frontier of circuit complexity [CTW26]. We show that the generator designed by O'Donnell, Servedio, and Tan for polytopes [OST22] also fools this broader class. To analyze the generator, we develop a threshold-specific smooth approximation framework based on a Bentkus-type mollifier. We prove derivative bounds for this mollifier and also establish a Boolean anticoncentration theorem for thresholds of halfspaces via a random thinning argument. These ingredients imply that the generator $δ$-fools every $k$-out-of-$m$ threshold of $m$ halfspaces over $\{-1,1\}^n$ with seed length $\widetilde{O}(κ^{6+2\varepsilon}\log^{6+2\varepsilon}\!m\cdotδ^{-(2+2\varepsilon)}\log n)$, for any arbitrarily small constant $\varepsilon>0$, where $κ=\min\{k,m-k+1\}$. The random thinning argument also yields bounds on the noise sensitivity and Gaussian surface area for thresholds of halfspaces, leading to learning algorithms under both the uniform and Gaussian distributions.

cs.CC↗

arXiv2026

Parallel Kac's Walk Generates PRU

Ma and Huang recently proved that the PFC construction, introduced by Metger, Poremba, Sinha and Yuen [MPSY24], gives an adaptive-secure pseudorandom unitary family PRU. Their proof developed a new path recording technique [MH25]. In this work, we show that a linear number of sequential repetitions of the parallel Kac's Walk, introduced by Lu, Qin, Song, Yao and Zhao [LQS+26], also forms an adaptive-secure PRU, confirming a conjecture therein. Moreover, it additionally satisfies strong security against adversaries making inverse queries. This gives an alternative PRU construction, and provides another instance demonstrating the power of the path recording technique. We also discuss some further simplifications and implications.

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arXiv2025

A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions

Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-$d$ polynomial threshold functions (PTFs) over Gaussian space. Our main result is an explicit construction of PRG with seed length $\mathrm{poly}(k,d,1/ε)\cdot\log n$ that can fool any function of $k$ degree-$d$ PTFs with probability at least $1-\varepsilon$. More specifically, we show that the summation of $L$ independent $R$-moment-matching Gaussian vectors $ε$-fools functions of $k$ degree-$d$ PTFs, where $L=\mathrm{poly}( k, d, \frac{1}ε)$ and $R = O({\log \frac{kd}ε})$. The PRG is then obtained by applying an appropriate discretization to Gaussian vectors with bounded independence.

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arXiv2024

Quantum Pseudorandom Scramblers

Quantum pseudorandom state generators (PRSGs) have stimulated exciting developments in recent years. A PRSG, on a fixed initial (e.g., all-zero) state, produces an output state that is computationally indistinguishable from a Haar random state. However, pseudorandomness of the output state is not guaranteed on other initial states. In fact, known PRSG constructions provably fail on some initial states. In this work, we propose and construct quantum Pseudorandom State Scramblers (PRSSs), which can produce a pseudorandom state on an arbitrary initial state. In the information-theoretical setting, we obtain a scrambler which maps an arbitrary initial state to a distribution of quantum states that is close to Haar random in total variation distance. As a result, our scrambler exhibits a dispersing property. Loosely, it can span an $ε$-net of the state space. This significantly strengthens what standard PRSGs can induce, as they may only concentrate on a small region of the state space provided that average output state approximates a Haar random state. Our PRSS construction develops a parallel extension of the famous Kac's walk, and we show that it mixes exponentially faster than the standard Kac's walk. This constitutes the core of our proof. We also describe a few applications of PRSSs. While our PRSS construction assumes a post-quantum one-way function, PRSSs are potentially a weaker primitive and can be separated from one-way functions in a relativized world similar to standard PRSGs.

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