Research Overview: Toward End-to-End Quantum Applications

This is an exciting time for quantum computing! With the availability of prototypes of quantum machines, especially the recently established quantum supremacy and two 53-qubit machines from both Google and IBM, it becomes possible for quantum computing researchers to go beyond purely theoretical studies and to investigate the implementation and actual performance of real-world quantum applications. A paramount goal of quantum computing research is to bridge the gap between theoretical studies and the limitation of real-world QC hardware to achieve end-to-end quantum applications.

Research Goals

My research aims to fill the gap between purely theoretical study of quantum applications and experimental implementation of quantum hardware. In particular, I believe that research on end-to-end applications would benefit from end-to-end thinking. My research aims to identify and contribute to the research where the ideas from computer science in general, i.e., computational thinking, can help facilitate the implementation of end-to-end quantum applications. To that end, my research investigates a broad range of perspectives of quantum computing, including its theoretical foundation and applications, applications and system engineering of near-term quantum devices, the software foundation (programming languages and system) of quantum computing, and the computer-aided design for quantum devices. My research has also been published in prestigious venues of all relevant fields such as quantum information (e.g., QIP), theoretical computer science (e.g., STOC, CCC, ICALP), machine learning (e.g., ICML, NeurIPS, AAAI), and programming languages (e.g., POPL, PLDI).

Quantum Applications in Optimization and Machine Learning

Provable Quantum Advantages for Optimization and Machine Learning

My recent research aims to understand the landscape of provable quantum advantages in optimization and machine learning, a major targeted domain of quantum applications. To that end, I have developed quantum algorithms with polynomial speed-ups over classical ones for semidefinite programs (SDPs), general convex optimization, training linear and kernelized classifiers, and estimating volumes of high-dimensional convex bodies. My algorithms also hint at possible exponential quantum speed-ups when using quantum data as inputs/outputs of SDPs and the principal component analysis problem.


  • Sublinear Classical and Quantum Algorithms for General Matrix Games. Tongyang Li, Chunhao Wang, Shouvanik Chakrabarti, Xiaodi Wu. To appear at the 35th AAAI Conference on Artificial Intelligence (AAAI 2021).

  • Quantum algorithm for volume estimation of convex bodies. Shouvanik Chakrabarti, Andrew Childs, Shih-han Hung, Tongyang Li, Chunhao Wang, and Xiaodi Wu. QIP 2020. arXiv:1908.03903.

  • Sublinear quantum algorithms for training linear and kernel-based classifiers. Tongyang Li, Shouvanik Chakrabarti, and Xiaodi Wu. In the proceedings of the 36th International Conference on Machine Learning (ICML 2019). Also available at arXiv:1904.02276.

  • Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning. Fernando G. S. L. Brandao, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. arXiv: 1710.02581v2. QIP 2019. ICALP 2019.

  • Quantum algorithms and lower bounds for convex optimization. Shouvanik Chakrabarti, Andrew M. Childs, Tongyang Li, and Xiaodi Wu. arXiv: 1809.01731. QIP 2019.

Quantum Algorithms for Property Testing

Another well-motivated topic is the property test of quantum and classical distributions. I have closed a long-standing gap between the upper and the lower bound of the sample complexity of testing the whole information of quantum states, so-called quantum tomography, which is a fundamental step to verify the preparation of the experimental setup. I also demonstrated the quantum speed-up in estimating the Shannon and Renyi entropies of classical distributions.


  • Quantum query complexity of entropy estimation. Tongyang Li and Xiaodi Wu. IEEE Transaction on Information Theory, Vol. 65, Issue 5, pages 2899 - 2921, May 2019. arXiv: 1710.06025.

  • Sample-optimal tomography of quantum states. Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu, Nengkun Yu. QIP 2016 and STOC 2016.

Variational Quantum Methods and Quantum Neural-Networks

Quantum Neural-networks (i.e., parameterized quantum circuits) are important and promising candidates for applications of quantum machine learning. My research aims to conduct a comprehensive investigation in this regard, including functionality (e.g., representation learning, generative models), training methods (e.g., landscape characterization, the efficiency of gradient and non-gradient based optimization methods), and separation between quantum and classical neural-networks.


  • Exponentially Many Local Minima in Shallow Quantum Neural Networks. Xuchen You and Xiaodi Wu. Manuscript, 2020.

  • Quantum Wasserstein Generative Adversarial Networks. Shouvanik Chakrabarti, Yiming Huang, Tongyang Li, Soheil Feizi, and Xiaodi Wu. In the Proceedings of the 33rd Annual Conference on Neural Information Processing Systems (NeurIPS 2019).

Differentiable Quantum Programming Languages & Quantum Neuro-Symbolic Applications

Inspired by the emerging paradigm shift from deep learning towards differentiable programming promoted by prominent classical machine learning researchers, I have initiated the formalization of differentiable quantum programming. This project not only provides the auto-differentiation technique for quantum programs, in particular the possibility of using quantum programs to compute the gradients of another quantum program, for the scalability of gradient-based training in quantum machine learning. It also opens the possibility of designing novel quantum “neuro-symbolic” applications that combine program features/synthesis with simple neural-networks. The study of their classical counterpart, although still in its infancy, has already shown great potential and promise over conventional neural-networks. We demonstrated, for the first time, one such example for quantum machine learning.

  • On the Principles of Differentiable Quantum Programming Languages. Shaopeng Zhu, Shih-han Hung, Shouvanik Chakrabarti, and Xiaodi Wu. In the Proceedings of the 41st ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2020).

Software Foundation of Quantum Computing

Quantum Program Analysis & Verification

Quantum programs are error-prone and their verification is challenging due to the limit of standard software assurance techniques in the quantum setting. My research investigates the verification of quantum programs via their static analysis with the help of quantum Hoare logic. A prominent approach for program verification is to generate invariants and inductive assertions, which is already a highly non-trivial task even classically. I made the first proposal of quantum invariants and demonstrated its use in quantum program verification, which can be generated by semidefinite programs (SDPs).

We also investigate algebraic reasoning of quantum programs inspired by the success of classical program analysis based on Kleene algebra. One prominent example of such is the famous Kleene Algebra with Tests (KAT), which has furnished both theoretical insights and practical tools. A few key features of KAT including the idempotent law and the nice properties of classical tests, however, fail to hold in the context of quantum programs due to their unique quantum features, especially in branching. We propose the Non-idempotent Kleena Algebra (NKA) as a natural alternative, and identify complete and sound semantic models for NKA as well as their appropriate quantum interpretations, which hence enables algebraic proofs in NKA of quantum compiler optimization and the normal form of quantum while-programs.


  • Algebraic Reasoning of Quantum Programs via Non-Idempotent Kleene Algebra. Yuxiang Peng, Mingsheng Ying, and Xiaodi Wu. Manuscript, 2020.

  • On the Theory and Practice of Invariant-based Verification of Quantum Programs. Shih-han Hung, Yuxiang Peng, Xin Wang, Shaopeng Zhu, and Xiaodi Wu. Manuscript, 2019.

  • Invariants of Quantum Programs: Characterizations and Generation. Mingsheng Ying, Shenggang Ying, and Xiaodi Wu. POPL 2017.

Formally Verified Software Tool-chain for Quantum Computing

The complexity of quantum computing and the limitations of near-term quantum devices suggest that the development of sophisticated quantum algorithms and clever optimizations is more likely to have mistakes. This calls for verifying every stage of quantum computation, from the software tools used to generate quantum circuits to the architecture and system design.

We are inspired by formal methods applied in safety-critical domains to ensure the correctness of code by construction, especially in the example of CompCert (a C compiler written and proved correct in Coq) and the NSF project of deep specification (a project to develop specifications of software toolchains to prove end-to-end correctness of whole systems). A verified quantum computing stack would ensure that each level of quantum computation is implemented satisfying certain specifications and the correctness of the final system, which would have a wide practical impact. This approach is especially appealing to quantum computing since alternative software assurance techniques are very limited due to the substantial expense involved in the quantum setting. As an important first step, we have built a proved-correct optimizing compiler for quantum algorithms to optimize the gate count, the depth of circuits, etc., while adhering to any architectural constraints. To that end, we developed an infrastructure for reasoning about quantum programs/operations in Coq, which is so expressive and flexible that we recently accomplish an end-to-end implementation of Shor's algorithm, with both classical and quantum parts as well as a formal correctness proof of everything, in Coq.


  • A Verified Optimizer for Quantum Circuits. Kesha Hietala, Robert Rand, Shih-Han Hung, Xiaodi Wu, and Michael Hicks. POPL 2021. (GitHub).

  • Verified Optimization in a Quantum Intermediate Representation. Kesha Hietala, Robert Rand, Shih-Han Hung, Xiaodi Wu, and Michael Hicks. Extended Abstract at QPL 2019. Available at arXiv:1904.06319.

Design of Quantum Programming Languages

Quantum circuits, which are simple to understand, have served as the main description language for most quantum applications so far. However, there are also many challenges and limitations when using quantum circuits as the only description language of quantum applications both in theory and in practice. We are inspired to investigate the more appropriate abstractions as well as the language design built on top of these abstractions for various quantum applications, such as variational quantum algorithms and quantum simulation. We are also working on the formulation of useful classical semantic constructs, e.g., recursion and abstract data types, in the context of quantum programming languages.

Quantum Architecture Engineering for NISQ era

Near-term quantum computers are likely to have very restricted hardware resources, where precisely controllable qubits are expensive, error-prone, and scarce. Therefore, application designers for such near-term intermediate-size quantum (NISQ) computers are forced to investigate the best balance of trade-offs among a large number of (potentially heterogeneous) factors specific to the targeted application and quantum hardware.

Noise-Analysis of Quantum Applications

We believe that one way to attack these problems is to set aside a one-size-fits-all approach to fault tolerance and instead consider elevating the question of errors and related architecture-specific resource optimization to the level of the programming language and algorithm design. In particular, inspired by techniques in approximate computing that optimizes computation on unreliable classical hardware, we've built formal semantics and a logic for reasoning about reliability in the presence of noise in quantum computation.


  • Quantitative Robustness Analysis of Quantum Programs. Shih-Han Hung, Kesha Hietala, Shaopeng Zhu, Mingsheng Ying, Michael Hicks, and Xiaodi Wu. POPL 2019.

Meta-Programming Framework for Automating NISQ Application Design

We propose Meta Quantum Circuits with Constraints (MQCC), a meta-programming framework for quantum programs, to assist the balance of trade-offs in NISQ application design. Designers express their application as a succinct collection of normal quantum circuits stitched together by a set of meta-level choice variables, whose values are constrained according to a programmable set of quantitative optimization criteria. MQCC?s compiler automatically generates the appropriate constraints, hands them to a solver (e.g., a Satisfiability Modulo Theories (SMT) solver), and from the solution produces an optimized, runnable program. We demonstrate MQCC's expressiveness through an extensive case study, demonstrating that ideas from previous examples of NISQ application design – such as multi-programming, cost-effective uncomputation, and crosstalk mitigation, as well as their combination – can be readily implemented in MQCC, and produce comparable results.


  • Automating NISQ Application Design with Meta Quantum Circuits with Constraints (MQCC). Haowei Deng, Yuxiang Peng, Michael Hicks, and Xiaodi Wu. Manuscript, 2020.

Leveraging Small Quantum Machines for NISQ Applications

To accelerate NISQ quantum applications on small quantum machines, we designed a systematic framework of decomposing quantum computation into small pieces and then combining simulation results from each piece for the final output. A particular application is, e.g., a practical scheme to implement a 60-qubit quantum computation with only 45-qubit quantum machines.

  • Simulating large quantum circuits on a small quantum computer. Tianyi Peng, Aram Harrow, Maris Ozols, and Xiaodi Wu. Physical Review Letters 125, 150504. Available at arXiv:1904.00102.

Treating NISQ Machines as Analog Machines

Recent research results suggest a promising approach to designing resource-efficient NISQ applications by breaking high-level abstractions, such as quantum circuits, and directly compiling applications to the pulse-level control of quantum machines. This approach essentially treats near-term quantum computers as analog quantum machines. We are inspired to identify qualitative/quantitative advantages of treating near-term quantum computers as analog quantum machines. To that end, we will investigate the robustness and the interpretability of the generated control pulses, as well as the scalability of variational algorithms generating these pulses. Moreover, we aim to study the optimal pulse-control-inspired ansatz design for variational quantum methods, as well as their implementation based on the pulse-level, rather than the gate-level, compilation.

Cryptography & Security in the Quantum Regime

Tamper-resilient Cryptography under Physical Assumptions

Devices, classical or quantum, are subject to tampering in cryptographic settings, especially due to the proliferation of side-channel attacks. These attacks exploit the fact that de- vices leak information to the outside world not just through input-output interaction, but through physical characteristics of computation such as power consumption, timing, and electro-magnetic radiation. Research on this topic explores two possible solutions to protect cryptographic systems from (quantum) side-channel attacks.

  • Device-independent Cryptography: In this setting, none of the devices can be a priori trusted: security is based solely on simple tests performed by the honest users on the input-output behavior of their devices. The security analysis guarantees that any set of devices meeting reasonable physical assumptions (e.g., spatial separations of devices that prohibit mutual communication) and conducting an acceptable interaction will lead to a secure outcome, regardless of the actual process inside the devices. My research focuses on generating uniform random bits in this setting under minimal assumptions. Check the recent review in Nature (Certified randomness in quantum physics) about this research direction and my work.

  • Leakage-resilient Cryptography: Modern computing environment, such as cloud computing where computation no longer takes place on private machines under our control, makes a lot of classical cryptographic systems victims to all kinds of side-channel attacks. The leakage can be quantum. Collecting and storing quantum side information is technically much less challenging than building fully-fledged quantum computers, and could become realistic in the near future. My research focuses on studying the role of quantum side information in both information-theoretical and computational settings. In a recent result, I initiated the study of computational notions of entropy in the quantum setting and developed the first quantum leakage-resilient cryptographic protocol.

Both device-independent and leakage-resilient cryptography can be deemed as tamper-resilient cryptography under physical assumptions. My future plan is to bring these cryptographic designs closer to practice, with better efficiency and broader functionality.


  • Computational Notions of Quantum Min-Entropy. Yi-Hsiu Chen, Kai-Min Chung, Ching-Yi Lai, Salil P. Vadhan, and Xiaodi Wu. QCrypt 2017. arXiv: 1704.07309.

  • General randomness amplification with non-signaling security. Kai-Min Chung, Yaoyun Shi, and Xiaodi Wu. QIP 2017.

  • Multi-Source Randomness Extractors Against Quantum Side Information, and their Applications. Kai-Min Chung, Xin Li, and Xiaodi Wu. arXiv:1411.2315.

  • Physical Randomness Extractors. Kai-Min Chung, Yaoyun Shi, and Xiaodi Wu. Plenary talk at QIP 2014.

(Practical) Delegation and Verification of Quantum Computation

In a recent breakthrough, Mahadev constructed a classical verification of quantum computation (CVQC) protocol for a classical client to delegate decision problems in BQP to an untrusted quantum prover under computational assumptions. We explore further the feasibility of CVQC with the more general sampling problems in BQP and with the desirable blindness property, and contribute affirmative solutions to both. We also investigate the lightweight verification of quantum supremacy, where all existing protocols, either based on classical simulation or public-key quantum cryptography, are too expensive to serve the purpose of practical verification. We propose a circuit-obfuscation-based verification scheme for quantum supremacy that is scalable and has some complexity-theoretic support based on the quantum minimum equivalent circuit problem (QMECP). We also implement a prototype of our circuit obfuscator which has the desired empirical performance against the attack from all the off-the-shelf tools.


  • Constant-round Blind Classical Verification of Quantum Sampling. Kai-Min Chung, Yi Lee, Han-Hsuan Lin, and Xiaodi Wu. Manuscript, 2020.

  • Scalable Verification of Quantum Supremacy based on Circuit Obfuscation. Shouvanik Chakrabarti, Chi-Ning Chou, Kai-Min Chung, and Xiaodi Wu. Manuscript, 2020.

Mechanized and Automated Security Analysis of Cryptographic Systems under Quantum Attacks

The emergence of quantum computing technology has promised unprecedented improvement in our computational ability, which, however, also leads to quantum attacks that would put many security techniques for modern communication in peril in the not-too-distant future. The defense against quantum attacks should ideally be deployed in the near future to protect today's secret information from future quantum attacks, especially in security-critical domains like hardware signatures or block-chains, both with very long life cycles.

Inspired by the success of the development of formal methods in the security analysis for large, real-world cryptographic systems, we are excited to develop and apply formal method techniques in quantum cryptography for automated security analysis of cryptographic systems under quantum attacks. Formally generated security analysis will provide not only efficient and high assurance proofs that can replace the tedious and error-prone analysis for experts, but also independently verifiable proofs that can be used by security practitioners without much quantum knowledge. One of our ultimate goals is to formally verify candidates of cryptographic systems from the NIST quantum-secure cryptography standardization.

Computer-aided Design of Quantum Devices

The design complexity/productivity of quantum devices will conceivably significantly increase as the quantum devices become more complicated. A lot of existing quantum devices are relatively small, and their design is mostly done by manual analysis. Achieving this requires leading experts on this topic, and is usually is time-consuming, and error-prone. Scaling up this approach for slightly larger or more complicated quantum devices seems very challenging, and less effective. A natural candidate to replace human manual design is computer-aided design. Inspired by the success of classical hardware description languages (HDLs) as an integral part of electronic design automation (EDA) systems, we are inspired to develop quantum HDL (QHDL) for the design of quantum devices. To that end, we will develop formal abstractions of various experimental platforms for quantum devices (e.g., superconducting qubits, neutral atoms), building on top of which we will develop design automation, automatic control generation, and formal verification. We will also directly link the precise description of quantum devices to the high-level quantum programming languages on the software developer's side, which will enable a hardware-software co-design framework for quantum applications.

Quantum Sensing Networks

The demand for highly accurate position and time information, provided by localization and synchronization methods, respectively, is growing rapidly. However, classical localization and synchronization methods are approaching their limits. Utilizing quantum properties promises to push these boundaries beyond classical limitations and provide unprecedented accuracy. We aim to develop theoretical and practical methodologies for the design and analysis of quantum localization and synchronization networks. These methodologies consist of statistical models and distributed algorithms to harness quantum phenomena for beyond-classical localization and synchronization.

Earlier Theoretical Studies in Quantum Information and Computation

Entanglement, Quantum Correlations, and Sum-of-Squares

Research on this topic studies many aspects of one of the key quantum features, entanglement and non-locality. I attack this topic by exploring a surprising connection between quantum information and the sum-of-squares (SOS) proof in approximation algorithms and the famous Unique Games Conjecture (UGC). This connection allows one to leverage technical advances in one field to apply to the other. Specific problems that I am working on include the characterization of separable (unentangled) states, the complexity of quantum Merlin-Arthur games with unentangled provers (QMA(2)), the possibility of a quantum-inspired approach to attack the UGC.


  • Limitations of semidefinite programs for separable states and entangled games. Aram Harrow, Anand Natarajan, Xiaodi Wu. QIP 2017. Communications in Mathematical Physics, Volume 366, Issue 2, pp 423–468, 2019.

  • Tight SoS-degree bounds for approximate Nash equilibria. Aram Harrow, Anand Natarajan, Xiaodi Wu. CCC 2016.

  • 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 881– 904.

  • Epsilon-net method for optimizations over separable states. Yaoyun Shi and Xiaodi Wu. QIP 2012, ICALP 2012. Theoretical Computer Science, Volume 598, Pages 51–63.

Quantum Computational Complexity

Interactive proof systems have been a central model in complexity theory with applications ranging from the PCP theorem in the hardness of approximation to cryptography. It studies problems with efficiently verifiable proofs via interactions between a polynomial-time verifier and all-powerful provers, where the verifier determines the validity of the proofs. My main contribution on this topic is the development of the Equilibrium Value Method to obtain space-efficient simulations of quantum interactive proof systems, including QIP=PSPACE, QRG(2)=PSPACE. Recently, I have been working on the quantum variant of the PCP theorem in the interactive proof setting. As a concrete first step, I have obtained a parallel repetition result for entangled k-player games.


  • Parallel repetition for entangled k-player games via fast quantum search. Kai-Min Chung, Xiaodi Wu and Henry Yuen. CCC 2015.

  • Epsilon-net method for optimizations over separable states. Yaoyun Shi and Xiaodi Wu. QIP 2012, ICALP 2012. Theoretical Computer Science, Volume 598, Pages 51-63.

  • Parallel approximation of min-max problems with applications to classical and quantum zero-sum games Gus Gutoski and Xiaodi Wu. QIP 2012, CCC 2012. Computational Complexity, 22(2):385-428, 2013, the special issue of CCC 2012.

  • Equilibrium Value Method for the proof of QIP=PSPACE. Xiaodi Wu . arXiv:1004.0264.