We analyze the complexity of quantum state verification in the context of solving systems of linear equations of the form $A \vec x = \vec b$. We show that any quantum operation that verifies whether a given quantum state is within a constant distance from the solution of the quantum linear systems problem requires $q=Ω(κ)$ uses of a unitary that prepares a quantum state $\left| b \right>$, proportional to $\vec b$, and its inverse in the worst case. Here, $κ$ is the condition number of the matrix $A$. For typical instances, we show that $q=Ω(\sqrt κ)$ with high probability. These lower bounds are almost achieved if quantum state verification is performed using known quantum algorithms for the quantum linear systems problem. We also analyze the number of copies of $\left| b \right>$ required by verification procedures of the prepare and measure type. In this case, the lower bounds are quadratically worse, being $Ω(κ^2)$ in the worst case and $Ω(κ)$ in typical instances with high probability. We discuss the implications of our results to known variational and related approaches to this problem, where state preparation, gate, and measurement errors will need to decrease rapidly with $κ$ for worst-case and typical instances if error correction is not used, and present some open problems.
Quantum subspace verification for error correction codes
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普里茨克分子工程学院）；Department of Computer Science, University of Hong Kong（香港大学计算机科学系）；Key Laboratory for Information Science of Electromagnetic Waves (Ministry of Education), Fudan University（复旦大学电磁波信息科学教育部重点实验室）
Quantum computing is seeking to realize hardware-optimized algorithms for application-related computational tasks. NP (nondeterministic-polynomial-time) is a complexity class containing many important but intractable problems like the satisfiability of potentially conflict constraints (SAT). According to the well-founded exponential time hypothesis, verifying an SAT instance of size $n$ requires generally the complete solution in an $O(n)$-bit proof. In contrast, quantum verification algorithms, which encode the solution into quantum bits rather than classical bit strings, can perform the veri…
Verifying the correct functioning of quantum gates is a crucial step towards reliable quantum information processing, but it becomes an overwhelming challenge as the system size grows due to the dimensionality curse. Recent theoretical breakthroughs show that it is possible to verify various important quantum gates with the optimal sample complexity of $O(1/ε)$ using local operations only, where $ε$ is the estimation precision. In this work, we propose a variant of quantum gate verification (QGV) which is robust to practical gate imperfections, and experimentally realize efficient QGV on a two…
We aim to devise feasible, efficient verification schemes for bosonic channels. To this end, we construct an average-fidelity witness that yields a tight lower bound for average fidelity plus a general framework for verifying optimal quantum channels. For both multi-mode unitary Gaussian channels and single-mode amplification channels, we present experimentally feasible average-fidelity witnesses and reliable verification schemes, for which sample complexity scales polynomially with respect to all channel specification parameters. Our verification scheme provides an approach to benchmark the p…
We present two verification protocols where the correctness of a "target" computation is checked by means of "trap" computations that can be efficiently simulated on a classical computer. Our protocols rely on a minimal set of noise-free operations (preparation of eight single-qubit states or measurement of four observables, both on a single plane of the Bloch sphere) and achieve linear overhead. To the best of our knowledge, our protocols are the least demanding techniques able to achieve linear overhead. They represent a step towards further reducing the quantum requirements for verification…
I present a quick and sound method for the robustness verification of a sort of quantum classifiers who are Linear Sound. Since quantum machine learning has been put into practice in relevant fields and Linear Sound Property, LSP is a pervasive property, the method could be universally applied. I implemented my method with a Quantum Convolutional Neural Network, QCNN using MindQuantum, Huawei and successfully verified its robustness when classifying MNIST dataset.
Quantum state verification (QSV) and quantum data hiding (QDH) have so far been studied separately. QSV decides whether a given quantum state is close to the ideal one, with significantly lower sample complexity compared with direct application of quantum tomography. On the other hand, QDH is a cryptographic protocol that encodes secret classical information in multipartite quantum states, providing stronger security guarantees than conventional classical secret-sharing schemes. Here, we consider two fundamental quantities of a pure quantum state, determining the sample complexity needed for Q…
Quantum computers promise to efficiently solve not only problems believed to be intractable for classical computers, but also problems for which verifying the solution is also considered intractable. This raises the question of how one can check whether quantum computers are indeed producing correct results. This task, known as quantum verification, has been highlighted as a significant challenge on the road to scalable quantum computing technology. We review the most significant approaches to quantum verification and compare them in terms of structure, complexity and required resources. We al…
Randomized encoding is a powerful cryptographic primitive with various applications such as secure multiparty computation, verifiable computation, parallel cryptography, and complexity lower-bounds. Intuitively, randomized encoding $\hat{f}$ of a function $f$ is another function such that $f(x)$ can be recovered from $\hat{f}(x)$, and nothing except for $f(x)$ is leaked from $\hat{f}(x)$. Its quantum version, quantum randomized encoding, has been introduced recently [Brakerski and Yuen, arXiv:2006.01085]. Intuitively, quantum randomized encoding $\hat{F}$ of a quantum operation $F$ is another …
