@article {2772, title = {On the Impossibility of Post-Quantum Black-Box Zero-Knowledge in Constant Rounds}, year = {2021}, month = {3/20/2021}, abstract = {

We investigate the existence of constant-round post-quantum black-box zero-knowledge protocols for NP. As a main result, we show that there is no constant-round post-quantum black-box zero-knowledge argument for NP unless NP\⊆BQP. As constant-round black-box zero-knowledge arguments for NP exist in the classical setting, our main result points out a fundamental difference between post-quantum and classical zero-knowledge protocols. Combining previous results, we conclude that unless NP\⊆BQP, constant-round post-quantum zero-knowledge protocols for NP exist if and only if we use non-black-box techniques or relax certain security requirements such as relaxing standard zero-knowledge to ϵ-zero-knowledge. Additionally, we also prove that three-round and public-coin constant-round post-quantum black-box ϵ-zero-knowledge arguments for NP do not exist unless NP\⊆BQP.

}, url = {https://arxiv.org/abs/2103.11244}, author = {Nai-Hui Chia and Kai-Min Chung and Qipeng Liu and Takashi Yamakawa} } @article {2826, title = {Quantum Meets the Minimum Circuit Size Problem}, year = {2021}, month = {8/6/2021}, abstract = {

In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory---its hardness is mysterious, and a better understanding of its hardness can have surprising implications to many fields in computer science.
We first define and investigate the basic complexity-theoretic properties of minimum quantum circuit size problems for three natural objects: Boolean functions, unitaries, and quantum states. We show that these problems are not trivially in NP but in QCMA (or have QCMA protocols). Next, we explore the relations between the three quantum MCSPs and their variants. We discover that some reductions that are not known for classical MCSP exist for quantum MCSPs for unitaries and states, e.g., search-to-decision reduction and self-reduction. Finally, we systematically generalize results known for classical MCSP to the quantum setting (including quantum cryptography, quantum learning theory, quantum circuit lower bounds, and quantum fine-grained complexity) and also find new connections to tomography and quantum gravity. Due to the fundamental differences between classical and quantum circuits, most of our results require extra care and reveal properties and phenomena unique to the quantum setting. Our findings could be of interest for future studies, and we post several open problems for further exploration along this direction.

}, url = {https://arxiv.org/abs/2108.03171}, author = {Nai-Hui Chia and Chi-Ning Chou and Jiayu Zhang and Ruizhe Zhang} } @article {2728, title = {A Black-Box Approach to Post-Quantum Zero-Knowledge in Constant Rounds}, year = {2020}, month = {11/5/2020}, abstract = {

In a recent seminal work, Bitansky and Shmueli (STOC \&$\#$39;20) gave the first construction of a constant round zero-knowledge argument for NP secure against quantum attacks. However, their construction has several drawbacks compared to the classical counterparts. Specifically, their construction only achieves computational soundness, requires strong assumptions of quantum hardness of learning with errors (QLWE assumption) and the existence of quantum fully homomorphic encryption (QFHE), and relies on non-black-box simulation. In this paper, we resolve these issues at the cost of weakening the notion of zero-knowledge to what is called ε-zero-knowledge. Concretely, we construct the following protocols: - We construct a constant round interactive proof for NP that satisfies statistical soundness and black-box ε-zero-knowledge against quantum attacks assuming the existence of collapsing hash functions, which is a quantum counterpart of collision-resistant hash functions. Interestingly, this construction is just an adapted version of the classical protocol by Goldreich and Kahan (JoC \&$\#$39;96) though the proof of ε-zero-knowledge property against quantum adversaries requires novel ideas. - We construct a constant round interactive argument for NP that satisfies computational soundness and black-box ε-zero-knowledge against quantum attacks only assuming the existence of post-quantum one-way functions. At the heart of our results is a new quantum rewinding technique that enables a simulator to extract a committed message of a malicious verifier while simulating verifier\&$\#$39;s internal state in an appropriate sense.

}, url = {https://arxiv.org/abs/2011.02670}, author = {Nai-Hui Chia and Kai-Min Chung and Takashi Yamakawa} } @article {2501, title = {Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning}, journal = {to appear in Proceedings of STOC 2020}, year = {2020}, month = {6/18/2020}, abstract = {

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang\&$\#$39;s breakthrough quantum-inspired algorithm for recommendation systems [STOC\&$\#$39;19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gily{\'e}n et al. [STOC\&$\#$39;19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: l2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

}, doi = {https://doi.org/10.1145/3357713.3384314}, url = {https://arxiv.org/abs/1910.06151}, author = {Nai-Hui Chia and Andras Gilyen and Tongyang Li and Han-Hsuan Lin and Ewin Tang and Chunhao Wang} } @article {2338, title = {Quantum-inspired classical sublinear-time algorithm for solving low-rank semidefinite programming via sampling approaches}, year = {2019}, abstract = {

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. Recently, quantum algorithms with superpolynomial speedups for solving SDPs have been proposed assuming access to its constraint matrices in quantum superposition. Mutually inspired by both classical and quantum SDP solvers, in this paper we present a sublinear classical algorithm for solving low-rank SDPs which is asymptotically as good as existing quantum algorithms. Specifically, given an SDP with m constraint matrices, each of dimension n and rank poly(logn), our algorithm gives a succinct description and any entry of the solution matrix in time O(m\⋅poly(logn,1/ε)) given access to a sample-based low-overhead data structure of the constraint matrices, where ε is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with both the SDP solvers based on the matrix multiplicative weight (MMW) framework and the recent studies of quantum-inspired machine learning algorithms. The cost of solving SDPs by MMW mainly comes from the exponentiation of Hermitian matrices, and we propose two new technical ingredients (compared to previous sample-based algorithms) for this task that may be of independent interest: . Weighted sampling: assuming sampling access to each individual constraint matrix A1,\…,Aτ, we propose a procedure that gives a good approximation of A=A1+...+Aτ. . Symmetric approximation: we propose a sampling procedure that gives low-rank spectral decomposition of a Hermitian matrix A. This improves upon previous sampling procedures that only give low-rank singular value decompositions, losing the signs of eigenvalues.

}, url = {https://arxiv.org/abs/1901.03254}, author = {Nai-Hui Chia and Tongyang Li and Han-Hsuan Lin and Chunhao Wang} }