Publications
Click on a title to see the abstract. The authors of papers in theoretical computer science are listed by the alphabetical order of the last name. Unless explicitly described, the authors of the following publications are listed in this way.
 Quantitative Robustness Analysis of Quantum Programs

Abstract:
Quantum computation is a topic of significant recent interest, with practical advances coming from both
research and industry. A major challenge in quantum programming is dealing with errors (quantum noise)
during execution. Because quantum resources (e.g., qubits) are scarce, classical error correction techniques applied at the level of the architecture are currently costprohibitive. But while this reality means that quantum programs are almost certain to have errors, there as yet exists no principled means to reason about erroneous behavior. This paper attempts to fill this gap by developing a semantics for erroneous quantum whileprograms, as well as a logic for reasoning about them. This logic permits proving a property we have identified, called robustness, which characterizes possible “distance” between an ideal program and an erroneous one. We have proved the logic sound, and showed its utility on several case studies, notably: (1) analyzing the robustness of noisy versions of the quantum Bernoulli factory (QBF) and quantum walk (QW); (2) demonstrating the (in)effectiveness of different error correction schemes on singlequibit errors; and (3) analyzing the robustness of a faulttolerant version of QBF.
 Quantum algorithms and lower bounds for convex optimization

 Quantum SDP Solvers: Large Speedups, Optimality, and Applications to Quantum Learning

Abstract:
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speedups. We consider SDP instances with constraint matrices, each of dimension , rank at most , and sparsity . The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time , with the error of the solution. This gives an optimal dependence in terms of and a quadratic improvement over previous quantum algorithms (when ). The second algorithms assumes a fully quantum input model in which the input matrices are given as quantum states. We show its run time is , with an upper bound on the tracenorm of all input matrices. In particular the complexity depends only polylogarithmically in and polynomially in .
We apply the second SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given measurements and a supply of copies of an unknown state , we show we can find in time a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as on the measurements, up to error . The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes’ principle from statistical mechanics.
As in previous work, we obtain our algorithm by ‘‘quantizing“ classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for lowrank Hamiltonians, given quantum states encoding these Hamiltonians, with a polylogarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis.
We also develop a ‘‘fast” quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. (SODA 2017). We believe both techniques might be of independent interest.
 Quantum query complexity of entropy estimation

Tongyang Li and Xiaodi Wu.
Accepted by IEEE Transaction on Information Theory.
Abstract:
Estimation of Shannon and R{'e}nyi entropies of unknown discrete distributions is a fundamental problem in statistical property testing and an active research topic in both theoretical computer science and information theory. Tight bounds of the number of samples to estimate these entropies have been established in the classical setting, while little is known about their quantum counterparts. In this paper, we give the first quantum algorithms for estimating R{'e}nyi entropies (Shannon entropy being 1Renyi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for R{'e}nyi entropy estimation for all ,
including a tight bound for the collisionentropy (2R{'e}nyi entropy) and also an analysis for the minentropy case (i.e., ). Moreover, we complement our quantum upper bounds with quantum lower bounds on R{'e}nyi entropy estimation for all .
Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH) on quantum algorithms for distributional property testing, however, with many new technical ingredients. For Shannon entropy estimation, we improve the performance of the BHH framework, especially its error dependence, using Montanaro's approach to estimating the expected output value of a quantum subroutine with bounded variance and giving a finetuned error analysis.
For general R{'e}nyi entropy estimation, we further develop a procedure that recursively approximates R{'e}nyi entropy for a sequence of s, which is in spirit similar to the cooling schedules in simulated annealing.
For special cases like integer and (i.e., the minentropy case), we reduce the entropy estimation problem to the distinctness and the distinctness problems respectively.
Finally, we exploit reductions as well as the polynomial method to obtain our lower bounds.
 RazMcKenzie simulation with the inner product gadget

 Computational Notions of Quantum MinEntropy

Abstract:
Computational notions of entropy have many applications in cryptography and complexity theory. These notions measure how much (min)entropy a source X has from the eyes of a computationally bounded party who may hold certain ?leakage information? B that is correlated with X. In this work, we initiate the study of computational entropy in the quantum setting, where X and/or B may become quantum states and the computationally bounded observer is modelled as a small quantum circuit. Specifically, we investigate to what extent the classical notions of computational entropy generalize to the quantum setting, and whether quantum analogues of classical theorems still hold. For example, we show:
There exist quantum states that are pseudorandom (i.e. they cannot be distinguished from the totally mixed state by polynomialsized quantum circuits) but have actual entropy 0 (i.e. are pure states). In contrast, a classical distribution needs superlogarithmic entropy to be pseudorandom.
We extend the classical Leakage Chain Rule for pseudoentropy to the case where the leakage information B is quantum (while X remains classical). Specifically, if X has pseudoentropy at least k against quantum distinguishers (i.e. it is indistinguishable from some distribution of minentropy at least k by polynomialsized quantum distinguishers) and B consists of at most a logarithmic number l of qubits, then X conditioned on B has pseudoentropy at least kl against quantum distinguishers.
We show that a general form of the classical Dense Model Theorem (interpreted as showing the equivalence between two definitions of pseudorelativeminentropy) does not extend to quantum states.
As an application, we construct the first quantum leakageresilient streamcipher in the quantum bounded storage model, assuming the existence of a quantumsecure PRG.
Along the way, we develop quantum analogues of a number of classical techniques (e.g. the Leakage Simulation Lemma, proven using a Nonuniform MinMax Theorem or Boosting), and also identify classical techniques (namely, Gap Amplification) that do not hold in the quantum setting. Moreover, we introduce a variety of notions that combine quantum information and quantum complexity, which raise a number of directions for future work.
 Simulating large quantum circuits on a small quantum computer

Links:
Abstract:
We explore various strategies for simulating large quantum circuits on a classical computer that has access to a small quantum device. By representing the circuit as a tensor network, we can cut it into smaller parts that can be executed independently by contracting each of the smaller tensor networks. Assuming a partition with not too many edges between different parts can be found, we provide efficient algorithms for simulating such circuits. While in general the simulation cost scales exponentially in the total number of edges between different parts, the size of the quantum memory required scales only linearly in the degree of each part.
 General randomness amplification with nonsignaling security

Abstract:
Highly unpredictable events appear to be abundant in life. However, when modeled rigorously,
their existence in nature is far from evident. In fact, the world can be deterministic
while at the same time the predictions of quantum mechanics are consistent with observations. Assuming that randomness does exist but only in a weak form,
could highly random events be possible? This fundamental question was first raised by Colbeck and Renner (Nature Physics, 8:450–453, 2012). In this work, we answer this question positively, without the various restrictions assumed in the previous works. More precisely, our protocol uses quantum devices, a single weak randomness source quantified by a general notion of nonsignaling minentropy, tolerates a constant amount of device imperfection, and the security is against an allpowerful nonsignaling adversary. Unlike the previous works proving nonsignaling security, our result does not rely on any structural restrictions or independence assumptions.
Thus it implies a stronger interpretation of the dichotomy statement articulated by Gallego et al. (Nature Communications, 4:2654, 2013): ‘‘either our world is fully deterministic or there exist in nature events that are fully random.’’
Note: This is a new work after our QIP 2014 paper, where the security proved is against a quantum, as opposed to nonsignaling, adversary.
 Limitations of semidefinite programs for separable states and entangled games

Aram Harrow, Anand Natarajan, Xiaodi Wu.
Appeared at QIP 2017. Accepted by Communications in Mathematical Physics.
Abstract:
We introduce a method for using reductions to construct integrality
gaps for
semidefinite programs (SDPs). These imply new limitations on the
sumofsquares (SoS) hierarchy in two settings where previously no
round integrality gaps were known:
The set of separable (i.e. unentangled) quantum states, or equivalently, the norm of a matrix.
The set of quantum correlations; i.e. conditional probability
distributions achievable with local measurements on a shared
entangled state.
Integrality gaps for the norm had previously been sought due
to its connection to SmallSet Expansion (SSE) and Unique Games (UG).
In both cases nogo theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of LeeRaghavendraSteurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Aaronson et al.
These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelsontype bounds to restrict quantum correlations, as well as the SDP hierarchies of DohertyParriloSpedalieri, NavascuesPironioAcin and BertaFawziScholz. Indeed a wide range of past work in quantum information can be described as using an SDP on one of the above two problems and our results put broad limits on these lines of argument.
 Invariants of Quantum Programs: Characterizations and Generation

Abstract:
Program invariant is a fundamental notion widely used in program verification and analysis. The aim of this paper is twofold: (i) find an appropriate definition of invariants for quantum programs; and (ii) develop an effective technique of invariant generation for verification and analysis of quantum programs.
Interestingly, the notion of invariant can be defined for quantum programs in two different ways – additive invariants and multiplicative invariants – corresponding to two interpretations of implication in a continuous valued logic: the Lukasiewicz implication and the Godel implication. It is shown that both of them can be used to establish partial correctness of quantum programs.
The problem of generating additive invariants of quantum programs is addressed by reducing it to an SDP (Semidefinite Programming) problem. This approach is applied with an SDP solver to generate invariants of two important quantum algorithms – quantum walk and quantum Metropolis sampling. Our examples show that the generated invariants can be used to verify correctness of these algorithms and are helpful in optimizing quantum Metropolis sampling.
To our knowledge, this paper is the first attempt to define the notion of invariant and to develop a method of invariant generation for quantum programs.
 Tight SoSdegree bounds for approximate Nash equilibria

Links: preprint
Abstract:
Nash equilibria always exist, but are widely conjectured to require time to find that is exponential in the number of strategies, even for twoplayer games. By contrast, a simple quasipolynomial time algorithm, due to Lipton, Markakis and Mehta (LMM), can find approximate Nash equilibria, in which no player can improve their utility by more than by changing their strategy. The LMM algorithm can also be used to find an approximate Nash equilibrium with nearmaximal total welfare. Matching hardness results for this optimization problem were found assuming the hardness of the plantedclique problem (by Hazan and Krauthgamer) and assuming the Exponential Time Hypothesis (by Braverman, Ko and Weinstein).
In this paper we consider the application of the sumsquares (SoS)
algorithm from convex optimization to the problem of optimizing over
Nash equilibria. We show the first unconditional lower bounds on the
number of levels of SoS needed to achieve a constant factor
approximation to this problem. While it may seem that Nash
equilibria do not naturally lend themselves to convex optimization,
we also describe a simple LP (linear programming) hierarchy that can
find an approximate Nash equilibrium in time comparable to that of
the LMM algorithm, although neither algorithm is obviously a
generalization of the other. This LP can be viewed as arising from
the SoS algorithm at levels — matching our lower
bounds. The lower bounds involve a modification of the
BravermanKoWeinstein embedding of CSPs into strategic games and
techniques from sumofsquares proof systems. The upper bound
(i.e. analysis of the LP) uses informationtheory techniques that
have been recently applied to other linear and
semidefiniteprogramming hierarchies.
 Sampleoptimal tomography of quantum states

 An improved semidefinite programming hierarchy for testing entanglement

Aram Harrow, Anand Natarajan, Xiaodi Wu.
Communications in Mathematical Physics, June 2017, Volume 352, Issue 3, pp 881904.
Abstract:
We present a stronger version of the DohertyParriloSpedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing which is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.
 On the Limits of Communication with Nonlocal Resources

Abstract:
We obtain optimal bounds for the problem of conveying classical messages by communication between a sender and receiver who can utilize nonlocal correlations obeying the NonSignaling Principle. These include correlations arising from quantum entanglement, but also include ‘‘superquantum’’ correlations (e.g. those that maximally violate the CHSH inequality). Our result simultaneously simplifies and generalizes the result of Nayak and Salzman (JACM vol.53, issue 1, pp. 184–206).
 Parallel repetition for entangled kplayer games via fast quantum search

Abstract:
We present two parallel repetition theorems for the entangled value of multiplayer, oneround free games (games where the inputs come from a product distribution). Our first theorem shows that for a player free game with entangled value , the fold repetition of has entangled value at most , where is the answer length of any player. In contrast, the best known parallel repetition theorem for the emph{classical} value of twoplayer free games is , due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition.
Our second theorem handles the broader class of free games where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by . We also show that the dependence of the exponent on is necessary: we exhibit a player free game and such that .
Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: in particular, our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic Grover speedup for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.
 MultiSource Randomness Extractors Against Quantum Side Information, and their Applications

Abstract:
We study the problem of constructing multisource extractors in the quantum setting, which extract almost uniform random bits against an adversary who collects quantum side information from several initially independent classical random sources. This is a natural generalization of the two much studied problems of seeded randomness extraction against quantum side information, and classical independent source extractors. With new challenges such as potential entanglement in the side information, it is not a prior clear under what conditions do quantum multisource extractors exist; the only previous work in this setting is [KK12], where the classical innerproduct twosource extractors of [CG88] and [DEOR04] are shown to be quantum secure in the restricted Independent Adversary (IA) Model and entangled Bounded Storage (BS) Model.
In this paper we propose a new model called General Entangled (GE) Adversary Model, which allows arbitrary entanglement in the side information and subsumes both the IA model and the BS model. We proceed to show how to construct GEsecure quantum multisource extractors.
To that end, we propose another model called Onesided Adversary (OA) Model, which is weaker than all the above models. Somewhat surprisingly, we establish an equivalence between strong OAsecurity and strong GEsecurity. As a result, all classical multisource extractors can either directly work, or be modified to work in the GE model at the cost of one extra random source. Thus, our constructions essentially match the best known constructions of classical multisource extractors. This answers several open questions in [KK12,DPVR12].
We also apply our techniques to two important problems in cryptography and distributed computing — privacy amplification and network extractor. Both problems deal with converting secret weak random sources into secret uniform random bits in a communicating environment, with the presence of a passive adversary who has unlimited computational power and can see every message transmitted. We show that as long as the sources have certain amounts of conditional minentropy in our GE model (even with entangled quantum side information), we can design very efficient privacy amplification protocols and network extractors.
 Physical Randomness Extractors

Abstract:
How to generate provably true randomness with minimal assumptions? This question is important not only for the efficiency and the security of information processing, but also for understanding how extremely unpredictable events are possible in Nature. All current solutions require special structures in the initial source of randomness, or a certain independence relation among two or more sources. Both types of assumptions are impossible to test and difficult to guarantee in practice. Here we show how this fundamental limit can be circumvented by extractors that base security on the validity of physical laws and extract randomness from untrusted quantum devices. In conjunction with the recent work of Miller and Shi (arXiv:1402:0489), our physical randomness extractor uses just a single and general weak source, produces an arbitrarily long and nearuniform output, with a closetooptimal error, secure against allpowerful quantum adversaries, and tolerating a constant level of implementation imprecision. The source necessarily needs to be unpredictable to the devices, but otherwise can even be known to the adversary.
Our central technical contribution, the Equivalence Lemma, provides a general principle for proving composition security of untrusteddevice protocols. It implies that unbounded randomness expansion can be achieved simply by crossfeeding any two expansion protocols. In particular, such an unbounded expansion can be made robust, which is known for the first time. Another significant implication is, it enables the secure randomness generation and key distribution using public randomness, such as that broadcast by NIST's Randomness Beacon. Our protocol also provides a method for refuting local hidden variable theories under a weak assumption on the available randomness for choosing the measurement settings.
 Limits of Quantum Oneway Communication by Matrix Hypercontractive Inequality

Yaoyun Shi, Xiaodi Wu, and Wei Yu .
Manuscript, 2013.
 Epsilonnet method for optimizations over separable states

Yaoyun Shi and Xiaodi Wu .
Contributed talk at the 15th Workshop on Quantum Information Processing (QIP 2012). In the Proceedings of the 39th International Colloquium on Automata, Languages and Programming (ICALP 2012). Theoretical Computer Science, 2015 .
Abstract:
We give algorithms for the optimization problem: , where is a Hermitian matrix, and the variable is a bipartite separable quantum state. This problem lies at the heart of several problems in quantum computation and information, such as the complexity of QMA(2). While the problem is NPhard, our algorithms are better than brute force for several instances of interest. In particular, they give PSPACE upper bounds on promise problems admitting a QMA(2) protocol in which the verifier performs only logarithmic number of elementary gate on both proofs, as well as the promise problem of deciding if a bipartite local Hamiltonian has large or small ground energy. For , our algorithm runs in time exponential in . While the existence of such an algorithm was first proved recently by Brandao, Christandl and Yard in Proceedings of the 43rd annual ACM Symposium on Theory of Computation , 343–352, 2011, our algorithm is conceptually simpler.
 Parallel approximation of minmax problems with applications to classical and quantum zerosum games

Gus Gutoski and Xiaodi Wu .
Featured talk at the 15th Workshop on Quantum Information Processing (QIP 2012). In the Proceedings of the 27rd Annual IEEE Conference on Computational Complexity (CCC 2012). Computational Complexity, 22(2):385428, 2013, the special issue of CCC 2012.
Abstract:
This paper presents an efficient parallel algorithm for a new class of minmax problems based on the matrix multiplicative weights update method. Our algorithm can be used to find nearoptimal strategies for competitive twoplayer classical or quantum games in which a referee exchanges any number of messages with one player followed by any number of additional messages with the other. This algorithm considerably extends the class of games which admit parallel solutions, demonstrating for the first time the existence of a parallel algorithm for a game in which one player reacts adaptively to the other. As a consequence, we prove that several competingprovers complexity classes collapse to PSPACE such as QRG, SQG and two new classes called DIP and DQIP. A special case of our result is a parallel approximation scheme for a new class of semidefinite programs whose feasible region consists of tuples of semidefinite matrices that satisfy a ‘‘transcriptlike’’ consistency condition. Applied to this special case, our algorithm yields a direct polynomialspace simulation of multimessage quantum interactive proofs resulting in a firstprinciples proof of QIP=PSPACE.
 Parallelized Solution to Semidefinite Programmings in Quantum Complexity Theory

Xiaodi Wu .
Manuscript, 2010. Note: most of the techniques in this paper are applied and improved in the above paper. However, this paper contains more examples of the applications of such techniques.
Abstract:
In this paper we present an equilibrium value based framework for solving SDPs via the multiplicative weight update method which is different from the one in Kale's thesis [Kale07]. One of the main advantages of the new framework is that we can guarantee the convertibility from approximate to exact feasibility in a much more general class of SDPs than previous result. Another advantage is the design of the oracle which is necessary for applying the multiplicative weight update method is much simplified in the general case. This leads to an alternative and easier solutions to the SDPs used in the previous results QIP(2) PSPACE [JainUW09] and QMAM=PSPACE [JainJUW09]. Furthermore, we provide a generic form of SDPs which can be solved in the similar way. By parallelizing every step in our solution, we are able to solve a class of SDPs in NC. Although our motivation is from quantum computing, our result will also apply to any SDP which satisfies our conditions directly.
In addition to the new framework for solving SDPs, we also provide a novel framework which improves the range of equilibrium value problems that can be solved via the multiplicative weight update method. Before this work we are only able to calculate the equilibrium value where one of the two convex sets needs to be the set of density operators. Our work demonstrates that in the case when one set is the set of density operators with further linear constraints, we are still able to approximate the equilibrium value to high precision via the multiplicative weight update method.
 Equilibrium Value Method for the proof of QIP=PSPACE

Xiaodi Wu .
Manuscript, 2010.
Abstract:
We provide an alternative proof of QIP=PSPACE to the recent breakthrough result [JainJUW09]. Unlike solving some semidefinite program corresponding to the definition of the complexity class, our method starts with one QIPcomplete problem which computes the diamond norm between two admissible quantum channels. The key observation is that we can convert the computation of the diamond norm into the computation of some equilibrium value, which could be interesting for its own sake. The later problem, different from semidefinite programs, is generally more structured and easier to solve. The multiplicative weight update method is also used to solve the equilibrium value problem, however, in a relatively simpler way than the original proof [JainJUW09]. Furthermore, we provide a generalized form of equilibrium value problems which can be solved in the same way as well as comparisons to semidefinite programs.
 NonIdentity Check Remains QMAComplete for Short Circuits

Abstract:
In the NonIdentity Check problem, we are given a quantum circuit as input, and are asked whether it is far away from the identity or not. It is well known that this problem is QMAcomplete [JWB05]. In this note, it is shown that the NonIdentity Check problem remains QMAcomplete for circuits of short depth. Specifically, we prove that for constant depth quantum circuit in which each gate is given to bits of precision, the NonIdentity Check problem is QMAcomplete. It also follows that polylog depth circuit consisting of only gates from any universal gate set is QMAcomplete.
 Exact Quantum Search by Parallel Unitary Discrimination Schemes

(by contribution) Xiaodi Wu and Runyao Duan.
Physical Review A, 78, 012303 (2008). Selected for the July 2008 issue of Virtual Journal of Quantum Information.
 Verifierbased Algorithm for Unsorted Database Search Problem

(by contribution) Xiaodi Wu and Gui Lu Long.
International Journal of Quantum Information, vol.5 no.4, pp 597  604 (2007)
Abstract:
Unsorted database search problem is an important science and engineering problem. We proposed quantum algorithm to solve the problem by dividing the database binarily and then use a probabilistic verifier algorithm to determine if the item is within one part of the divided database. We analyzed the computational complexity of this algorithm, and found that in general, the number of steps is proportional to that in the standard Grover algorithm. However in some cases, it is less than that of the Grover algorithm.
 Data Processing and software realization of wavelengths of spectrum during CCD measurement (in Chinese)

(by contribution) Xiaodi Wu, Yongtao Huang and Xinkun Ma.
Experimental Technology and Management, vol. 24, no. 4 p4851 (2006)
DOI: 10.3969/j.issn.10024956.2007.04.015
