Pseudorandom states (PRSs) are state ensembles that cannot be efficiently distinguished from Haar-random states. However, the definition of PRSs has been limited to pure states and lacks robustness against noise. In this work, we introduce pseudorandom density matrices (PRDMs), ensembles of n -qubit states that are computationally indistinguishable from the generalized Hilbert-Schmidt ensemble (GHSE), which is constructed from ( n + m ) -qubit Haar random states with m qubits traced out. For m = 0 , PRDMs are equivalent to PRSs, whereas for m = ω ( log n ) , PRDMs are computationally indistinguishable from the maximally mixed state. PRDMs with m = ω ( log n ) are robust to unital noise channels and separated in terms of security from PRS. PRDMs can disguise valuable quantum resources as trivial states. In particular, we construct pseudoresource state ensembles, which possess near-maximal entanglement, magic and coherence, but are computationally indistinguishable from resource-free states. PRDMs exhibit a pseudoresource gap of Θ ( n ) vs 0, surpassing previously found gaps. We also render EFI pairs, a fundamental cryptographic primitive, robust to strong mixed unitary noise. Our work has major implications on quantum resource theory. We show that entanglement, magic, and coherence cannot be efficiently tested, and that black-box resource distillation requires a superpolynomial number of copies. We also establish lower bounds on the purity needed for efficient testing and black-box distillation. Finally, we introduce memoryless PRSs, a noise-robust notion of PRS, which are indistinguishable to Haar random states for efficient algorithms without quantum memory, as well as noise-robust quantum money. Our work provides a comprehensive framework of pseudorandomness for mixed states, which yields powerful quantum cryptographic primitives and fundamental bounds on quantum resource theories.
Pseudorandom states (PRSs) are state ensembles that cannot be efficiently distinguished from Haar-random states. However, the definition of PRSs has been limited to pure states and lacks robustness against noise. In this work, we introduce pseudorandom density matrices (PRDMs), ensembles of n -qubit states that are computationally indistinguishable from the generalized Hilbert-Schmidt ensemble (GHSE), which is constructed from ( n + m ) -qubit Haar random states with m qubits traced out. For m = 0 , PRDMs are equivalent to PRSs, whereas for m = ω ( log n ) , PRDMs are computationally indistinguishable from the maximally mixed state. PRDMs with m = ω ( log n ) are robust to unital noise channels and separated in terms of security from PRS. PRDMs can disguise valuable quantum resources as trivial states. In particular, we construct pseudoresource state ensembles, which possess near-maximal entanglement, magic and coherence, but are computationally indistinguishable from resource-free states. PRDMs exhibit a pseudoresource gap of Θ ( n ) vs 0, surpassing previously found gaps. We also render EFI pairs, a fundamental cryptographic primitive, robust to strong mixed unitary noise. Our work has major implications on quantum resource theory. We show that entanglement, magic, and coherence cannot be efficiently tested, and that black-box resource distillation requires a superpolynomial number of copies. We also establish lower bounds on the purity needed for efficient testing and black-box distillation. Finally, we introduce memoryless PRSs, a noise-robust notion of PRS, which are indistinguishable to Haar random states for efficient algorithms without quantum memory, as well as noise-robust quantum money. Our work provides a comprehensive framework of pseudorandomness for mixed states, which yields powerful quantum cryptographic primitives and fundamental bounds on quantum resource theories.
A pseudorandom quantum state (PRS) is an ensemble of quantum states indistinguishable from Haar-random states to observers with efficient quantum computers. It allows one to substitute the costly Haar-random state with efficiently preparable PRS as a resource for cryptographic protocols, while also finding applications in quantum learning theory, black hole physics, many-body thermalization, quantum foundations, and quantum chaos. All existing constructions of PRS equate the notion of efficiency to quantum computers which runtime is bounded by a polynomial in its input size. In this work, we relax the notion of efficiency for PRS with respect to observers with near-term quantum computers implementing algorithms with runtime that scales slower than polynomial-time. We introduce the T-PRS which is indistinguishable to quantum algorithms with runtime T(n) that grows slower than polynomials in the input size n. We give a set of reasonable conditions that a T-PRS must satisfy and give two constructions by using quantum-secure pseudorandom functions and pseudorandom functions. For T(n) being linearithmic, linear, polylogarithmic, and logarithmic function, we characterize the amount of quantum resources a T-PRS must possess, particularly on its coherence, entanglement, and magic. Our quantum resource characterization applies generally to any two state ensembles that are indistinguishable to observers with computational power T(n), giving a general necessary condition of whether a low-resource ensemble can mimic a high-resource ensemble, forming a T-pseudoresource pair. We demonstate how the necessary amount of resource decreases as the observer's computational power is more restricted, giving a T-pseudoresource pair with larger resource gap for more computationally limited observers.
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.
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.
Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement'', a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states -- first defined by Ji, Liu and Song -- which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to logn across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously.
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 error-correcting codes (PRCs), introduced by Christ and Gunn [CRYPTO 2024], are classical error-correcting codes whose codewords are computationally indistinguishable from uniformly random strings. We initiate the study of quantum analogues of PRCs. We define quantum pseudorandom error-correcting codes (QPRCs). Assuming that Learning Parity with Noise (LPN) is hard for 2O(n)-time quantum algorithms, we construct QPRCs whose encodings are indistinguishable from Haar-random isometries. We call these pseudorandom isometric error-correcting codes (PRICs), and our construction is robust to all o(nlognloglogn)-local quantum noise, where n is the number of physical qubits. Under the same assumption, we also construct QPRCs whose encodings are indistinguishable from the completely depolarizing channel and that are robust to all αn-local quantum noise for some constant α>0. The latter QPRCs give a direct quantum analogue of classical PRCs.
To construct PRICs, we develop two technical ingredients of independent interest. First, we introduce a new classical cryptographic primitive which we call pseudorandom functional error-correcting codes (PRFCs) and construct them under the same LPN assumption. Second, we develop a new efficient decoding procedure within the codeword-stabilized (CWS) framework introduced in [Cross, Smith, Smolin, and Zeng, IEEE ISIT 2008], which is a generic framework for constructing quantum error-correcting codes by combining (possibly nonlinear) classical error-correcting codes and graphs. This resolves the open problem of finding a general method for efficiently decoding such CWS codes based on nonlinear classical codes, raised in [Li, Dumer, Grassl, and Pryadko, Physical Review A 2010].