Haar-random unitaries provide a canonical model of generic random quantum evolution, but they typically have exponential description and circuit complexity. Although pseudorandom unitaries efficiently emulate a single Haar-random unitary, many tasks require an entire reusable family of independently random-looking operations. We introduce pseudorandom function-like unitaries (PRFU), which generate such a family indexed by public labels using only a single short key. We distinguish classical- and coherent-label access and establish security under adaptive quantum queries. For classical labels, we provide a generic construction from a post-quantum pseudorandom function and a pseudorandom unitary. For coherent labels, we use an indexed path-recording framework to analyze interference across labels and prove security of a construction based on quantum-secure function and permutation primitives. We further show that PRFUs generate pseudorandom channels and quantum combs with private memory, which are secure against adaptive interventions and concurrent sessions. Combining PRFUs with unitary gluing yields a one-key family of pseudorandom unitaries whose supported register widths can be chosen after key generation. Further applications include nonce-resolved quantum authentication, coherently masked QRAM queries, and efficient emulation of random multi-time dynamics. Our results extend quantum pseudorandomness from individual operations to efficiently generated families of random-looking quantum dynamics.
Haar-random unitaries provide a canonical model of generic random quantum evolution, but they typically have exponential description and circuit complexity. Although pseudorandom unitaries efficiently emulate a single Haar-random unitary, many tasks require an entire reusable family of independently random-looking operations. We introduce pseudorandom function-like unitaries (PRFU), which generate such a family indexed by public labels using only a single short key. We distinguish classical- and coherent-label access and establish security under adaptive quantum queries. For classical labels, we provide a generic construction from a post-quantum pseudorandom function and a pseudorandom unitary. For coherent labels, we use an indexed path-recording framework to analyze interference across labels and prove security of a construction based on quantum-secure function and permutation primitives. We further show that PRFUs generate pseudorandom channels and quantum combs with private memory, which are secure against adaptive interventions and concurrent sessions. Combining PRFUs with unitary gluing yields a one-key family of pseudorandom unitaries whose supported register widths can be chosen after key generation. Further applications include nonce-resolved quantum authentication, coherently masked QRAM queries, and efficient emulation of random multi-time dynamics. Our results extend quantum pseudorandomness from individual operations to efficiently generated families of random-looking quantum dynamics.
Pseudorandom unitaries (PRUs) are ensembles of efficiently implementable unitary operators that cannot be distinguished from Haar random unitaries by any quantum polynomial-time algorithm with query access to the unitary. We present a simple PRU construction that is a concatenation of a random Clifford unitary, a pseudorandom binary phase operator, and a pseudorandom permutation operator. We prove that this PRU construction is secure against non-adaptive distinguishers assuming the existence of quantum-secure one-way functions. This means that no efficient quantum query algorithm that is allowed a single application of U⊗poly(n) can distinguish whether an n-qubit unitary U was drawn from the Haar measure or our PRU ensemble. We conjecture that our PRU construction remains secure against adaptive distinguishers, i.e. secure against distinguishers that can query the unitary polynomially many times in sequence, not just in parallel.
We provide the first evidence relative to a classical oracle that pseudorandom unitaries (PRUs) can exist without quantum-computable one-way functions.
To obtain this result, we first prove that five independent random diagonal phase layers interleaved with Hadamard transforms U=F5HF4HF3HF2HF1 form a strong PRU with security under adaptive, controlled access to the unitary, its inverse, transpose, and complex conjugate.
Our main result is that, when the phase layers are implemented using a classical random oracle O, the construction remains secure against uniform BQP adversaries with coherent access to suitable Boolean completions of classical-input $QMA^{PH}^{O}}$ decision problems. Such an adversary can invert (even quantum-computable) one-way functions and solve efficiently verifiable one-way puzzles defined relative to O. Our result therefore provides classical-oracle evidence that PRUs do not imply quantum-computable one-way functions, in the form of a black-box separation.
We substantially strengthen the classical-oracle result of Kretschmer, Qian, and Tal (STOC 2025) by obtaining PRUs secure against these QMA-aided adversaries. Our construction also consists of a simple, efficient circuit that queries a random function, suggesting a route to concrete implementation via the random-oracle heuristic.
The existence of pseudorandom unitaries (PRUs) -- efficient quantum circuits that are computationally indistinguishable from Haar-random unitaries -- has been a central open question, with significant implications for cryptography, complexity theory, and fundamental physics. In this work, we close this question by proving that PRUs exist, assuming that any quantum-secure one-way function exists. We establish this result for both (1) the standard notion of PRUs, which are secure against any efficient adversary that makes queries to the unitary U, and (2) a stronger notion of PRUs, which are secure even against adversaries that can query both the unitary U and its inverse U†. In the process, we prove that any algorithm that makes queries to a Haar-random unitary can be efficiently simulated on a quantum computer, up to inverse-exponential trace distance.
Quantum computation gives rise to new pseudorandom primitives for states and unitaries, including pseudorandom state generators (PRSGs), pseudorandom function-like state generators (PRFSGs), and pseudorandom unitaries (PRUs). In this paper, we show a full unitary oracle separation between PRFSGs and PRUs.
The separation holds between the strongest state notion and the weakest unitary notion: even adaptively secure, quantum-accessible PRFSGs do not imply non-adaptively secure, forward-only PRUs, even when their implementations are allowed to be non-unitary and use an arbitrary number of ancillary qubits. This reveals a fundamental distinction between pseudorandomness for quantum states and for quantum unitaries.
Our main technical idea is to view a candidate PRU construction with access to state generation oracles as a map from the underlying oracle states to implemented unitaries, and to study the derivatives of this map. These derivatives are inherently low rank, and we exploit this low-rank structure to distinguish the resulting unitaries from truly random ones. We believe this differential perspective may be useful for studying other structural questions about quantum states and unitaries.
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.
We introduce the pseudorandom quantum authentication scheme (PQAS), an efficient method for encrypting quantum states that relies solely on the existence of pseudorandom unitaries (PRUs). The scheme guarantees that for any eavesdropper with quantum polynomial-time (QPT) computational power, the encrypted states are indistinguishable from the maximally mixed state. Furthermore, the receiver can verify that the state has not been tampered with and recover the original state with asymptotically unit fidelity. Our scheme is cost-effective, requiring only polylogarithmic circuit depth and a single shared key to encrypt a polynomial number of states. Notably, the PQAS can potentially exist even without quantum-secure one-way functions, requiring fundamentally weaker computational assumptions than semantic classical cryptography. Additionally, PQAS is secure against attacks that plague protocols based on QPT indistinguishability from Haar random states, such as chosen-plaintext attacks (CPAs) and attacks that reveal meta-information such as quantum resources. We relate the amount of meta-information that is leaked to quantum pseudoresources, giving the concept a practical meaning. As an application, we construct important cryptographic primitives, such as verifiable pseudorandom density matrices (VPRDMs), which are QPT-indistinguishable from random mixed states while being efficiently verifiable via a secret key, as well as verifiable noise-robust EFI pairs and one-way state generators (OWSGs). Our results establish a new paradigm of quantum information processing with weaker computational assumptions.