Efficiently sampling a quantum state that is hard to distinguish from a truly random quantum state is an elementary task in quantum information theory that has both computational and physical uses. This is often referred to as pseudorandom (quantum) state generator, or PRS generator for short.
In existing constructions of PRS generators, security scales with the number of qubits in the states, i.e.\ the (statistical) security parameter for an $n$-qubit PRS is roughly $n$. Perhaps counter-intuitively, $n$-qubit PRS are not known to imply $k$-qubit PRS even for $k<n$. Therefore the question of \emph{scalability} for PRS was thus far open: is it possible to construct $n$-qubit PRS generators with security parameter $λ$ for all $n, λ$. Indeed, we believe that PRS with tiny (even constant) $n$ and large $λ$ can be quite useful.
We resolve the problem in this work, showing that any quantum-secure one-way function implies scalable PRS. We follow the paradigm of first showing a \emph{statistically} secure construction when given oracle access to a random function, and then replacing the random function with a quantum-secure (classical) pseudorandom function to achieve computational security. However, our methods deviate significantly from prior works since scalable pseudorandom states require randomizing the amplitudes of the quantum state, and not just the phase as in all prior works. We show how to achieve this using Gaussian sampling.
Scalable, quantum-accessible, and adaptive pseudorandom quantum state and pseudorandom function-like quantum state generators
Centre for Quantum Technologies, Singapore（新加坡量子技术中心）；Department of Computer Science, National University of Singapore（新加坡国立大学计算机系）；MajuLab, UMI 3654, Singapore（新加坡MajuLab（UMI 3654））
On Limits on the Provable Consequences of Quantum Pseudorandomness
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；KIAS, Korea（韩国高等研究院）；UT Austin, USA（美国德克萨斯大学奥斯汀分校）；KAIST, Daejeon, Korea（韩国科学技术院）；Surf, The Netherlands（荷兰SURF）；De Vinci Higher Education, De Vinci Research Center, Paris, France（法国巴黎德芬奇高等教育与研究中心）
This paper, for the first time, addresses the questions related to the connections between the quantum pseudorandomness and quantum hardware assumptions, specifically quantum physical unclonable functions (qPUFs). Our results show that the efficient pseudorandom quantum states (PRS) are sufficient to construct the challenge set for the universally unforgeable qPUF, improving the previous existing constructions that are based on the Haar-random states. We also show that both the qPUFs and the quantum pseudorandom unitaries (PRUs) can be constructed from each other, providing new ways to obtain …
Pseudorandom states, introduced by Ji, Liu and Song (Crypto'18), are efficiently-computable quantum states that are computationally indistinguishable from Haar-random states. One-way functions imply the existence of pseudorandom states, but Kretschmer (TQC'20) recently constructed an oracle relative to which there are no one-way functions but pseudorandom states still exist. Motivated by this, we study the intriguing possibility of basing interesting cryptographic tasks on pseudorandom states. We construct, assuming the existence of pseudorandom state generators that map a $λ$-bit seed to a $ω…
The key to optical analogy to a multi-particle quantum system is the scalable property. Optical elds modulated with pseudorandom phase sequences is an interesting solution. By utilizing the properties of pseudorandom sequences, mixing multiple optical elds are distinguished by using coherent detection and correlation analysis that are mature methods in optical communication. In this paper, we utilize the methods to investigate optical analogies to multi-particle quantum states. In order to demonstrate the feasibility, numerical simulations are carried out in the paper, which is helpful to the …
In this paper, we introduce an optical analogy to quantum Fourier tanformation based on a pseudorandom phase ensemble. The optical analogy also brings about exponential speedup over classical Fourier tanformation. Using the analogy, we demonstrate three classcial fields to realize Fourier transform similar to three quantum particles.
University of Maryland, College Park（马里兰大学帕克分校）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；IonQ Inc.（IonQ公司）；Technology Innovation Institute（技术创新研究所）；Singapore Institute of Technology（新加坡理工大学）；Agency for Science, Technology and Research (A*STAR)（科学技术研究局（A*STAR））；Singapore University of Technology and Design（新加坡科技设计大学）；California Institute of Technology（加州理工学院）；National University of Singapore（新加坡国立大学）；Nanyang Technological University（南洋理工大学）
In this paper, we introduce a new conception of pseudorandom phase ensemble to simulate a quantum ensemble. A pseudorandom sequence is inseparability and integral that are demonstrated only for a whole sequence not for a single phase unit, which is similar to that of quantum ensembles and a quantum particle. By using the ensemble conception, we demonstrate non-locality properties for classical fields similar to quantum entanglement.
We demonstrate that a tensor product structure could be obtained by introducing pseudorandom phase sequences into classical fields with two orthogonal modes. Using classical fields modulated with pseudorandom phase sequences, we discuss efficient simulation of several typical quantum states, including product state, Bell states, GHZ state, and W state. By performing quadrature demodulation scheme, we could obtain the mode status matrix of the simulating classical fields, based on which we propose a sequence permutation mechanism to reconstruct the simulated quantum states. The research on clas…
Randomness is both a useful way to model natural systems and a useful tool for engineered systems, e.g. in computation, communication and control. Fully random transformations require exponential time for either classical or quantum systems, but in many case pseudorandom operations can emulate certain properties of truly random ones. Indeed in the classical realm there is by now a well-developed theory of such pseudorandom operations. However the construction of such objects turns out to be much harder in the quantum case. Here we show that random quantum circuits are a powerful source of quan…
An effective simulation of quantum entanglement is presented using classical fields modulated with n pseudorandom phase sequences (PPSs) that constitute a n2^n-dimensional Hilbert space with a tensor product structure. Applications to classical fields are examplied by effective simulation of both Bell and GHZ states, and a correlation analysis was performed to characterize the simulation. Results that strictly comply with criteria of quantum entanglement were obtained and the approach was also shown to be applicable to a system consisting of n quantum particles.
Quantum key distribution (QKD) offers a way for establishing information-theoretically secure communications. An important part of QKD technology is a high-quality random number generator (RNG) for quantum states preparation and for post-processing procedures. In the present work, we consider a novel class of prepare-and-measure QKD protocols, utilizing additional pseudorandomness in the preparation of quantum states. We study one of such protocols and analyze its security against the intercept-resend attack. We demonstrate that, for single-photon sources, the considered protocol gives better …
We demonstrate that a single-photon field modulated with n different pseudorandom phase sequences (PPSs) can constitute a 2^n-dimensional Hilbert space that contains tensor product structure. By using the single photon field modulated with PPSs, we discuss effective simulation of Bell states and GHZ state, and apply both correlation analysis and von Neumann entropy to characterize the simulation. We obtain similar results with the cases in quantum mechanics and find that the conclusions can be easily generalized to n quantum particles. The research on simulation of quantum entanglement may be …
We prove a quantum information-theoretic conjecture due to Ji, Liu and Song (CRYPTO 2018) which suggested that a uniform superposition with random \emph{binary} phase is statistically indistinguishable from a Haar random state. That is, any polynomial number of copies of the aforementioned state is within exponentially small trace distance from the same number of copies of a Haar random state. As a consequence, we get a provable elementary construction of \emph{pseudorandom} quantum states from post-quantum pseudorandom functions. Generating pseduorandom quantum states is desirable for physica…
By modeling quantum chaotic dynamics with ensembles of random operators, we explore howmachine learning learning algorithms can be used to detect pseudorandom behavior in qubit systems.We analyze samples consisting of pieces of correlation functions and find that machine learningalgorithms are capable of determining the degree of pseudorandomness which a system is subjectto in a precise sense. This is done without computing any correlators explicitly. Interestingly,even samples drawn from two-point functions are found to be sufficient to solve this classificationproblem. This presents the poss…
Measurement based (MB) quantum computation allows for universal quantum computing by measuring individual qubits prepared in entangled multipartite states, known as graph states. Unless corrected for, the randomness of the measurements leads to the generation of ensembles of random unitaries, where each random unitary is identified with a string of possible measurement results. We show that repeating an MB scheme an efficient number of times, on a simple graph state, with measurements at fixed angles and no feed-forward corrections, produces a random unitary ensemble that is an ε-approximate t…
In the classical world, the existence of commitments is equivalent to the existence of one-way functions. In the quantum setting, on the other hand, commitments are not known to imply one-way functions, but all known constructions of quantum commitments use at least one-way functions. Are one-way functions really necessary for commitments in the quantum world? In this work, we show that non-interactive quantum commitments (for classical messages) with computational hiding and statistical binding exist if pseudorandom quantum states exist. Pseudorandom quantum states are sets of quantum states …
We show that quantum algorithms of time $T$ and space $S\ge \log T$ with unitary operations and intermediate measurements can be simulated by quantum algorithms of time $T \cdot \mathrm{poly} (S)$ and space $ {O}(S\cdot \log T)$ with unitary operations and without intermediate measurements. The best results prior to this work required either $Ω(T)$ space (by the deferred measurement principle) or $\mathrm{poly}(2^S)$ time [FR21,GRZ21]. Our result is thus a time-efficient and space-efficient simulation of algorithms with unitary operations and intermediate measurements by algorithms with unitar…
We demonstrate that a tensor product structure and optical analogy of quantum entanglement can be obtained by introducing pseudorandom phase sequences into classical fields with two orthogonal modes. Using the classical analogy, we discuss efficient simulation of several typical quantum states, including product state, Bell states, GHZ state, and W state. By performing quadrature demodulation scheme, we propose a sequence permutation mechanism to simulate certain quantum states and a generalized gate array model to simulate quantum algorithm, such as Shor's algorithm and Grover's algorithm. Th…
Random dynamics in isolated quantum systems is of practical use in quantum information and is of theoretical interest in fundamental physics. Despite a large number of theoretical studies, it has not been addressed how random dynamics can be verified from experimental data. In this paper, based on an information-theoretic formulation of random dynamics, i.e., unitary $t$-designs, we propose a method for verifying random dynamics from the data that is experimentally easy-to-access. More specifically, we use measurement probabilities estimated by a finite number of measurements of quantum states…
On the Feasibility of Simulation-Secure Quantum Functional Encryption
Université de Montréal (DIRO)（蒙特利尔大学）；University of Ottawa（渥太华大学）；CISPA Helmholtz Center for Information Security（CISPA赫尔姆霍兹信息安全中心）；Saarland University（萨尔兰大学）
