Nai-Hui Chia

Quantum state learning is a fundamental problem in physics and computer science. As near-term quantum devices are error-prone, it is important to design error-resistant algorithms. Apart from device errors, other unexpected factors could also affect the algorithm, such as careless human read-out error, or even a malicious hacker deliberately altering the measurement results. Thus, we want our algorithm to work even in the worst case when things go against our favor. We consider the practical setting of single-copy measurements and propose the $\gamma$-adversarial corruption model where an imaginary adversary can arbitrarily change $\gamma$-fraction of the measurement outcomes. This is stronger than the $\gamma$-bounded SPAM noise model, where the post-measurement state changes by at most $\gamma$ in trace distance. Under our stronger model of corruption, we design an algorithm using non-adaptive measurements that can learn an unknown rank-$r$ state up to $\tilde{O}(\gamma\sqrt{r})$ in trace distance, provided that the number of copies is sufficiently large. We further prove an information-theoretic lower bound of $\Omega(\gamma\sqrt{r})$ for non-adaptive measurements, demonstrating the optimality of our algorithm. Our upper and lower bounds also hold for quantum state testing, where the goal is to test whether an unknown state is equal to a given state or far from it. Our results are intriguingly optimistic and pessimistic at the same time. For general states, the error is dimension-dependent and $\gamma\sqrt{d}$ in the worst case, meaning that only corrupting a very small fraction ($1/\sqrt{d}$) of the outcomes could totally destroy any non-adaptive learning algorithm. However, for constant-rank states that are useful in many quantum algorithms, it is possible to achieve dimension-independent error, even in the worst-case adversarial setting.

We establish connections between state tomography, pseudorandomness, quantum state synthesis, and circuit lower bounds. In particular, let C be a family of non-uniform quantum circuits of polynomial size and suppose that there exists an algorithm that, given copies of |ψ⟩, distinguishes whether |ψ⟩ is produced by C or is Haar random, promised one of these is the case. For arbitrary fixed constant c, we show that if the algorithm uses at most O(2^n^c) time and 2^n^0.99 samples then stateBQE⊄stateC. Here stateBQE:=stateBQTIME[2^O(n)] and stateC are state synthesis complexity classes as introduced by Rosenthal and Yuen (ITCS 2022), which capture problems with classical inputs but quantum output. Note that efficient tomography implies a similarly efficient distinguishing algorithm against Haar random states, even for nearly exponential-time algorithms. Because every state produced by a polynomial-size circuit can be learned with 2O(n) samples and time, or O(n^ω(1)) samples and 2^O(n^ω(1)) time, we show that even slightly non-trivial quantum state tomography algorithms would lead to new statements about quantum state synthesis. Finally, a slight modification of our proof shows that distinguishing algorithms for quantum states can imply circuit lower bounds for decision problems as well. This help sheds light on why time-efficient tomography algorithms for non-uniform quantum circuit classes has only had limited and partial progress. Our work parallels results by Arunachalam et al. (FOCS 2021) that revealed a similar connection between quantum learning of Boolean functions and circuit lower bounds for classical circuit classes, but modified for the purposes of state tomography and state synthesis.

In a recent seminal work, Bitansky and Shmueli (STOC '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 $\epsilon$-zero-knowledge. Concretely, we construct the following protocols: - We construct a constant round interactive proof for NP that satisfies statistical soundness and black-box $\epsilon$-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 '96) though the proof of $\epsilon$-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 $\epsilon$-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's internal state in an appropriate sense.

We investigate the existence of constant-round post-quantum black-box zero-knowledge protocols for $\mathbf{NP}$. As a main result, we show that there is no constant-round post-quantum black-box zero-knowledge argument for $\mathbf{NP}$ unless $\mathbf{NP}\subseteq \mathbf{BQP}$. As constant-round black-box zero-knowledge arguments for $\mathbf{NP}$ 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 $\mathbf{NP}\subseteq \mathbf{BQP}$, constant-round post-quantum zero-knowledge protocols for $\mathbf{NP}$ exist if and only if we use non-black-box techniques or relax certain security requirements such as relaxing standard zero-knowledge to $\epsilon$-zero-knowledge. Additionally, we also prove that three-round and public-coin constant-round post-quantum black-box $\epsilon$-zero-knowledge arguments for $\mathbf{NP}$ do not exist unless $\mathbf{NP}\subseteq \mathbf{BQP}$.

The closest pair problem is a fundamental problem of computational geometry: given a set of $n$ points in a $d$-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in $O(n\log n)$ time in constant dimensions (i.e., when $d=O(1)$). This paper asks and answers the question of the problem’s quantum {time} complexity. Specifically, we give an $\tilde{O}(n^{2/3})$ algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference. In $\text{polylog}(n)$ dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover’s algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the \emph{Quantum Strong Exponential Time Hypothesis (QSETH)}, which is based on the assumption that Grover’s algorithm is optimal for \textsf{CNF-SAT} when the clause width is large. We show that the na\”{i}ve Grover approach to closest pair in higher dimensions is optimal up to an $n^{o(1)}$ factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.