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arXiv:2312.09206

Pseudorandomness from Subset States

Tudor Giurgică-Tiron, Adam Bouland

Published Dec 14, 2023• 2 citations
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Abstract

We show it is possible to obtain quantum pseudorandomness and pseudoentanglement from random subset states -- i.e. quantum states which are equal superpositions over (pseudo)random subsets of strings. This answers an open question of Aaronson et al. [arXiv:2211.00747], who devised a similar construction augmented by pseudorandom phases. Our result follows from a direct calculation of the trace distance between $t$ copies of random subset states and the Haar measure, via the representation theory of the symmetric group. We show that the trace distance is negligibly small, as long as the subsets are of an appropriate size which is neither too big nor too small. In particular, we analyze the action of basis permutations on the symmetric subspace, and show that the largest component is described by the Johnson scheme: the double-cosets of the symmetric group $\mathbb{S}_N$ by the subgroup $\mathbb{S}_t \times \mathbb{S}_{N-t}$. The Gelfand pair property of this setting implies that the matrix eigenbasis coincides with the symmetric group irreducible blocks, with the largest eigenblock asymptotically approaching the Haar average. An immediate corollary of our result is that quantum pseudorandom and pseudoentangled state ensembles do not require relative phases.

DOI
10.48550/arxiv.2312.09206

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Pseudorandom states (PRS) are an important primitive in quantum cryptography. In this paper, we show that subset states can be used to construct PRSs. A subset state with respect to SSS, a subset of the computational basis, is 1∣S∣∑i∈S∣i⟩. \frac{1}{\sqrt{|S|}}\sum_{i\in S} |i\rangle. ∣S∣​1​∑i∈S​∣i⟩. As a technical centerpiece, we show that for any fixed subset size ∣S∣=s|S|=s∣S∣=s such that s=2n/ω(poly(n))s = 2^n/ω(\mathrm{poly}(n))s=2n/ω(poly(n)) and s=ω(poly(n))s=ω(\mathrm{poly}(n))s=ω(poly(n)), where nnn is the number of qubits, a random subset state is information-theoretically indistinguishable from a Haar random state even provided with polynomially many copies. This range of parameter is tight. Our work resolves a conjecture by Ji, Liu and Song. Since subset states of small size have small entanglement across all cuts, this construction also illustrates a pseudoentanglement phenomenon.
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There are various notions of quantum pseudorandomness, such as pseudorandom unitaries (PRUs), pseudorandom state generators (PRSGs), pseudorandom function-like state generators (PRFSGs) and quantum-computable PRGs. Unlike the different notions of classical pseudorandomness, which are known to be existentially equivalent to each other, the relations among quantum pseudorandomness have yet to be fully established. We present evidence suggesting that some forms of quantum pseudorandomness are unlikely to be constructed from the others. This indicates that quantum pseudorandomness behaves quite differently from classical pseudorandomness. Our main result is a unitary oracle separation where log-length output PRFSGs exist but quantum-computable pseudorandom generators (QPRGs) with negligible correctness error do not. This result suggests that the inverse-polynomial error in the state-of-the-art construction of QPRGs from log-length PRSGs is inherent. To achieve this, we prove a novel differential geometric barrier theorem for the product Haar measure on quantum states which replaces the usual concentration inequalities by certifying a non-negligible ``gap'' between two large trace-separated sets. Our separation is based on an oracle that outputs Haar random quantum states for each bit string, which can be viewed as a quantum version of the random oracle model, where output strings are replaced by quantum states. Variations of this oracle can be used to study other relationships between quantum cryptographic primitives, and we use it to achieve partial separations, that highlight technical difficulties when dealing with ancillary registers, measurements, and adaptivity in the quantum setting.
Pseudoentangled states are defined by their ability to hide their entanglement structure: they are indistinguishable from random states to any observer with polynomial resources, yet can have much less entanglement than random states. Existing constructions of pseudoentanglement based on phase and/or subset states are limited in the entanglement structures they can hide, e.g., the states may have low entanglement on a single cut, on all cuts at once, or on local cuts in one dimension. Here, we introduce new constructions of pseudoentangled states based on (pseudo)random tensor networks that afford much more flexibility in the achievable entanglement structures. We illustrate our construction with the simplest example of a matrix product state, realizable as a staircase circuit of pseudorandom unitary gates, which exhibits pseudo-area-law scaling of entanglement in one dimension. We then generalize our construction to arbitrary tensor network structures that admit an isometric realization. A notable application of this result is the construction of pseudoentangled "holographic" states whose entanglement entropy obeys a Ryu-Takayanagi "minimum-cut" formula, answering a question posed in Aaronson et al. [arXiv:2211.00747].
We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group (Moore at al., 2005). We next consider the random Fourier method for the hidden subgroup problem (HSP) which consists of Fourier sampling the coset state of the hidden subgroup using random orthonormal bases for the group representations. In cases where every representation of the group has polynomially bounded rank when averaged over the hidden subgroup, the random Fourier method gives a POVM for the HSP operating on one coset state at a time and using totally a polynomial number of coset states. In particular, we get such POVMs whenever the group and the hidden subgroup form a Gel'fand pair, e.g., Abelian, dihedral and Heisenberg groups. This gives a positive counterpart to earlier negative results about random Fourier sampling when the above rank is exponentially large (Grigni et al., 2004), which happens for example in the HSP in the symmetric group. The drawback of random POVMs is that they are not efficient to implement, since measuring in a random basis takes exponential time as can be seen by a counting argument. This leads us to the open question of efficiently implementable pseudorandom measurement bases
Quantum pseudorandomness has found applications in many areas of quantum information, ranging from entanglement theory, to models of scrambling phenomena in chaotic quantum systems, and, more recently, in the foundations of quantum cryptography. Kretschmer (TQC '21) showed that both pseudorandom states and pseudorandom unitaries exist even in a world without classical one-way functions. To this day, however, all known constructions require classical cryptographic building blocks which are themselves synonymous with the existence of one-way functions, and which are also challenging to implement on realistic quantum hardware. In this work, we seek to make progress on both of these fronts simultaneously - by decoupling quantum pseudorandomness from classical cryptography altogether. We introduce a quantum hardness assumption called the Hamiltonian Phase State (HPS) problem, which is the task of decoding output states of a random instantaneous quantum polynomial-time (IQP) circuit. Hamiltonian phase states can be generated very efficiently using only Hadamard gates, single-qubit Z rotations and CNOT circuits. We show that the hardness of our problem reduces to a worst-case version of the problem, and we provide evidence that our assumption is plausibly fully quantum; meaning, it cannot be used to construct one-way functions. We also show information-theoretic hardness when only few copies of HPS are available by proving an approximate t-design property of our ensemble. Finally, we show that our HPS assumption and its variants allow us to efficiently construct many pseudorandom quantum primitives, ranging from pseudorandom states, to quantum pseudoentanglement, to pseudorandom unitaries, and even primitives such as public-key encryption with quantum keys.