10 pages, requires fullpage,amsthm,amsfonts,amsmath; To appear in Algorithmica; earlier version appeared in ICALP 2005; corrects minor typos, results are unchanged
We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in O (k^{2/3}). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Omega(k^{2/3}), we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.
Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each other or not with as few queries to the oracle as possible. In order to solve this problem, we use a theorem of Mario Szegedy (quant-ph/0401053) that relates a hitting time of a classical random walk to that of a quantum walk. We also take a look at another method of quantum walk by Andris Ambainis (quant-ph/0311001). We apply both walks into triangle findi…
We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain speed-ups over the best known quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an $n$-input $m$-output boolean function is balanced or $ε$-far from balanced; and (c) give a sub-lin…
The complexity of semidefinite programs for testing $k$-block-positivity
Mathematics Research Center, School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理学院科研中心）；Université de Lyon, Inria, ENS de Lyon, UCBL, LIP（里昂大学、法国国家信息与自动化研究所、里昂高等师范学校、里昂第一大学、里昂数学研究所）；QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院量子信息计算倡议中心）；Kyoto University, Mathematics department（京都大学数学系）
There has been a surge of progress in recent years in developing algorithms for testing and learning quantum states that achieve optimal copy complexity. Unfortunately, they require the use of entangled measurements across many copies of the underlying state and thus remain outside the realm of what is currently experimentally feasible. A natural question is whether one can match the copy complexity of such algorithms using only independent---but possibly adaptively chosen---measurements on individual copies. We answer this in the negative for arguably the most basic quantum testing problem: d…
In the $k$-junta testing problem, a tester has to efficiently decide whether a given function $f:\{0,1\}^n\rightarrow \{0,1\}$ is a $k$-junta (i.e., depends on at most $k$ of its input bits) or is $ε$-far from any $k$-junta. Our main result is a quantum algorithm for this problem with query complexity $\tilde O(\sqrt{k/ε})$ and time complexity $\tilde O(n\sqrt{k/ε})$. This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing pr…
We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the dist…
We construct an efficient autonomous quantum-circuit design algorithm for creating efficient quantum circuits to simulate Hamiltonian many-body quantum dynamics for arbitrary input states. The resultant quantum circuits have optimal space complexity and employ a sequence of gates that is close to optimal with respect to time complexity. We also devise an algorithm that exploits commutativity to optimize the circuits for parallel execution. As examples, we show how our autonomous algorithm constructs circuits for simulating the dynamics of Kitaev's honeycomb model and the Bardeen-Cooper-Schrief…
In this paper, we show that $Θ(\mathrm{poly}(n)\cdot\frac{4^n}{ε^2})$ is the sample complexity of testing whether two $n$-qubit quantum states $ρ$ and $σ$ are identical or $ε$-far in trace distance using two-outcome Pauli measurements.
We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether $S$ is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group. We present quantum algorithms for these problems that improve the best known classical complexity bounds. In particular, we give the first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha that improves the previous bounds by Ambainis and Szegedy. We also present several lower bounds for testin…
One of the main challenges in the Variational Quantum Eigensolver (VQE) framework is construction of the unitary transformation. The dimensionality of the space for unitary rotations of $N$ qubits is $4^N-1$, which makes the choice of a polynomial subset of generators exponentially difficult process. Moreover, due to non-commutativity of generators, the order in which they are used strongly affects results. Choosing the optimal order in a particular subset of generators requires testing the factorial number of combinations. We propose an approach based on the Lie algebra - Lie group connection…
We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging…
The theory of random walks on finite graphs is well developed with numerous applications. In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. They have been successfully applied in the development of quantum algorithms. In particular, to solve problems that can be mapped to searching or property testing on some specific graph. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cay…
In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We present a rather detailed description of a somewhat simplified version of the MNRS algorithm. Finally, in the query complexity model, we show how quantum walks can be applied to the following search problems: Element Distinctness, Matrix Product Verification, Restricted Range Associativity, Triangle, and Group Comm…
Suppose one has access to oracles generating samples from two unknown probability distributions P and Q on some N-element set. How many samples does one need to test whether the two distributions are close or far from each other in the L_1-norm ? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L_1-distance between P and Q can be estimated with a constant precision using approximately N^{1/2} queries in the quantum settin…
Sample complexity of quantum resource testing via one-shot quantum blurring
QuSoft；Institute for Logic, Language and Computation, University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Scuola Normale Superiore（比萨高等师范学院）
Cryptographic Conditions for Efficient Testing of Distributions and Quantum States
University of Oxford（牛津大学）；New York University（纽约大学）；Hon-Hai Research Institute（鸿海研究院）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
