Abstract. Quantum pseudorandom state generators ( PRSG s) 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 ( PRSS s), 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 [Formula: see text]-net of the state space. This significantly strengthens what standard PRSG s can induce, as they may only concentrate on a small region of the state space provided that the 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 PRSS s. While our PRSS construction assumes a postquantum one-way function, PRSS s are potentially a weaker primitive and can be separated from one-way functions in a relativized world similar to standard PRSG s.
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(dlogd) 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(logd) 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≤k<d, after
O((k+logd)log(d/ε)) steps, the joint distribution of the
k output states is
ε-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
Vd,k={X∈Cd×k:X†X=Ik}, we show that the parallel Kac's walk mixes rapidly on
Vd,k, with both Wasserstein and total variation mixing times bounded by
O((k+logd)log(d/ε)). This extends the Wasserstein mixing result of Pillai, Smith, and Vaikuntanathan for the standard Kac's walk on real Stiefel manifolds~\cite{PSV26}.
Quantum pseudorandomness is an emerging research area. Ji, Liu, and Song defined pseudorandom states (PRSs) and pseudorandom unitaries (PRUs) as quantum analogs of pseudorandom generators and pseudorandom functions. A unitary oracle separation result between one-way functions and PRSs/PRUs, established by Kretschmer, suggests that certain quantum primitives may remain secure even if classical cryptography is compromised. This insight has spurred extensive work on quantum pseudorandomness and its applications in quantum cryptography. Many constructions of PRSs have been established under standard assumptions, yet building a secure PRU was a long-standing open problem. This dissertation aims to narrow the gap between PRSs and PRUs and presents results that go beyond PRSs. We introduce Pseudorandom State Scramblers (PRSSs), a new primitive that lies between PRSs and PRUs. A PRSS maps any pure state to a pseudorandom state, a property shared with PRUs but not with PRSs. We present a construction of PRSSs inspired by the well-known Kac’s walk, and in particular, we develop a parallel variant that significantly accelerates the mixing time, enabling an efficient construction. PRSSs support cryptographic tasks not known to be achievable from PRSs alone, including a quantum encryption scheme and a succinct quantum state commitment. Additionally, when suitable classical randomness is provided, our construction exhibits a special dispersing property not known to be satisfied by any existing construction of quantum pseudorandom primitives. Our subsequent work shows that, without asymptotically increasing the number of steps, our construction based on the parallel Kac’s walk yields PRUs with standard or even strong security. The proof builds on a recently developed technique for establishing adaptive security, known as the path-recording method. This result provides an alternative construction of PRUs and further showcases the power of this proof technique. In addition, this dissertation includes two side projects. The first revisits the Hidden Subgroup Problem over ℤn, providing a simplified analysis of a known quantum algorithm using elementary lattice tools. The second establishes a quantum analogue of a classical impossibility result for statistical non-interactive zero-knowledge arguments, showing limitations of black-box reductions under classical-query quantum adversaries.
Pseudorandom quantum states (PRSs) and pseudorandom unitaries (PRUs) possess the dual nature of being efficiently constructible while appearing completely random to any efficient quantum algorithm. In this study, we establish fundamental bounds on pseudorandomness. We show that PRSs and PRUs exist only when the probability that an error occurs is negligible, ruling out their generation on noisy intermediate-scale and early fault-tolerant quantum computers. Further, we show that PRUs need imaginarity while PRS do not have this restriction. This implies that quantum randomness requires in general a complex-valued formalism of quantum mechanics, while for random quantum states real numbers suffice. Additionally, we derive lower bounds on the coherence of PRSs and PRUs, ruling out the existence of sparse PRUs and PRSs. We also show that the notions of PRS, PRUs and pseudorandom scramblers (PRSSs) are distinct in terms of resource requirements. We introduce the concept of pseudoresources, where states which contain a low amount of a given resource masquerade as high-resource states. We define pseudocoherence, pseudopurity and pseudoimaginarity, and identify three distinct types of pseudoresources in terms of their masquerading capabilities. Our work also establishes rigorous bounds on the efficiency of property testing, demonstrating the exponential complexity in distinguishing real quantum states from imaginary ones, in contrast to the efficient measurability of unitary imaginarity. Further, we show an exponential advantage in imaginarity testing when having access to the complex conjugate of the state. Lastly, we show that the transformation from a complex to a real model of quantum computation is inefficient, in contrast to the reverse process, which is efficient. Our results establish fundamental limits on property testing and provide valuable insights into quantum pseudorandomness.