Quantum Information

Quantifying quantum states' complexity is a key problem in various subfields of science, from quantum computing to black-hole physics. Motivated by the expected behavior of wormholes in quantum gravity, Brown and Susskind conjectured that the quantum complexity of the state output by a random circuit on n qubits grows linearly as more and more random gates are applied, until saturating after a number of gates exponential in n. We prove this conjecture by studying the dimension of the set of all unitaries that can be accessed with a given arrangement of two-qubit gates. Our core technical contribution is a lower bound on this dimension, using techniques from algebraic geometry and considerations based on Clifford circuits. In the second part of my talk, I'll discuss some thermodynamic and effective information-theoretic aspects of the complexity of quantum states and its growth in quantum many-body systems, establishing a resource theory to capture a notion of quantum complexity and drawing a connection between the concepts of complexity and entropy.

Joint work with: Jonas Haferkamp, Teja Naga Bhavia Kothakonda, Anthony Munson, Jens Eisert, Nicole Yunger Halpern

Zoom Link: https://pitp.zoom.us/j/94288479163?pwd=Nm8wOUdReGhreDErdUpJTzFETlBUUT09

Quantum metrology, which studies parameter estimation in quantum systems, has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on the estimation precision, called the Heisenberg limit, which is achievable in noiseless quantum systems, but is in general not achievable in noisy systems. This talk is a summary of some recent works by the speaker and collaborators on error-corrected quantum metrology. Specifically, we present a necessary and sufficient condition for achieving the Heisenberg limit in noisy quantum systems. When the condition is satisfied, the Heisenberg limit is recovered by a quantum error correction protocol which corrects all noises while maintaining the signal; when it is violated, we show the estimation limit still in general has a constant factor improvement over classical strategies, and is achievable using approximate quantum error correction. Both error correction protocols can be optimized using semidefinite programs. Examples in some typical noisy systems will be provided.

Zoom Link: https://pitp.zoom.us/j/95698542740?pwd=OWJtcTViKzZqNDg1bWk4cDFtaTRxZz09

I will review the history of anyons, particles that are neither bosons nor fermions, starting with their theoretical proposal all the way to their definitive experimental observation over 40 years later.  I will further discuss why the more general idea of non-abelian anyons is of intense interest for quantum computation.  If time permits I will give a status report on some of the current non-abelian anyon experiments.

Zoom Link: https://pitp.zoom.us/j/98698779123?pwd=SWIyak9OZ0dud2ZGcWdoazdkVURHQT09

The vast and growing number of publications in all disciplines of science cannot be comprehended by a single human researcher. As a consequence, researchers have to specialize in narrow subdisciplines, which makes it challenging to uncover scientific connections beyond the own field of research.

In my talk, I will present a possible solution: I demonstrate the development of a semantic network for quantum physics (SemNet), using 750,000 scientific papers and knowledge from books and Wikipedia. I use it in conjunction with an artificial neural network for predicting future research directions. Finally, I show first indications how individual scientists can use SemNet for suggesting and inspiring personalized, out-of-the-box ideas.

I believe that computer-inspired scientific ideas will play a significant role in accelerating scientific progress, and am looking forward hearing your thoughts and ideas about this crucial question.

References
[1] Mario Krenn, Anton Zeilinger, Predicting research trends with semantic and neural networks with an application in quantum physics, PNAS 117(4) 1910-1916 (2020).
[2] IEEE BigData 2021 competition: Science4Cast: https://github.com/iarai/science4cast

Zoom Link: https://pitp.zoom.us/j/92240839439?pwd=LytUTHlMWE9ycjlsUXJkdHRta2c1UT09

Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. However, it is quite challenging to realize deep unitary circuits on near-term quantum devices, and an unbiased reconstruction map is non-trivial to find for arbitrary random unitary ensembles. In this talk, I will discuss our recent progress on combining classical shadow tomography with quantum chaotic dynamics. Particularly, I will introduce two new families of shadow tomography schemes: 1) Hamiltonian-driven shadow tomography and 2) Classical shadow tomography with locally scrambled quantum dynamics. In both works, I’ll derive the unbiased reconstruction map, and analyze the sample complexity. In the Hamiltonian-driven scheme, I will illustrate how to use proper time windows to achieve a more efficient tomography. In the second work, I will demonstrate advantages of shadow tomography in the shallow circuit region. Then I’ll conclude by discussing approximate shadow tomography with local Hamiltonian dynamics, and demonstrate that a single quench-disordered quantum spin chain can be used for approximate shadow tomography.

References:
[1] Hong-Ye Hu, Yi-Zhuang You. “Hamiltonian-Driven Shadow Tomography of Quantum States”. arXiv:2102.10132 (2021)
[2] Hong-Ye Hu, Soonwon Choi, Yi-Zhuang You. “Classical Shadow Tomography with Locally Scrambled Quantum Dynamics”. arXiv: 2107.04817 (2021)

Zoom Link: https://pitp.zoom.us/j/99011187936?pwd=OVU3VkpyZ21YcXRCOW5DOHlnSWlVQT09

Quantum computers are expected to dramatically outperform classical computers for certain computational problems. While there has been extensive previous work for linear dynamics and discrete models, for more complex realistic problems arising in physical and social science, engineering, and medicine, the capability of quantum computing is far from well understood. One fundamental challenge is the substantial difference between the linear dynamics of a system of qubits and real-world systems with continuum, stochastic, and nonlinear behaviors. Utilizing advanced linear algebra techniques and nonlinear analysis, I attempt to build a bridge between classical and quantum mechanics, understand and optimize the power of quantum computation, and discover new quantum speedups over classical algorithms with provable guarantees. In this talk, I would like to cover quantum algorithms for scientific computational problems, including topics such as linear, nonlinear, and stochastic differential equations, with applications in areas such as quantum dynamics, biology and epidemiology, fluid dynamics, and finance.

Reference:
Quantum spectral methods for differential equations, Communications in Mathematical Physics 375, 1427-1457 (2020), https://arxiv.org/abs/1901.00961
High-precision quantum algorithms for partial differential equations, Quantum 5, 574 (2021), https://arxiv.org/abs/2002.07868
Efficient quantum algorithm for dissipative nonlinear differential equations, Proceedings of the National Academy of Sciences 118, 35 (2021), https://arxiv.org/abs/2011.03185
Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance, Quantum 5, 481 (2021), https://arxiv.org/abs/2012.06283

With the rapid development of programmable quantum simulators, the quantum states can be controlled with unprecedented precision. Thus, it opens a new opportunity to explore the strongly correlated phase of matter with new quantum technology platforms. In quantum simulators, one can engineer interactions between the microscopic degree of freedom and create exotic phases of matter that presumably are beyond the reach of natural materials. Moreover, quantum states can be directly measured instead of probing physical properties indirectly via optical and electrical responses of material as done in traditional condensed matter. Therefore, it is pressing to develop new approaches to efficiently prepare and characterize desired quantum states in the novel quantum technology platforms.

In this talk, I will introduce our recent works on the characterization of the topological invariants from a ground state wave function of the topological order phase and the implementation in noisy intermediate quantum devices. First, using topological field theory and tensor network simulations, we demonstrate how to extract the many-body Chern number (MBCN) given a bulk of a fractional quantum Hall wave function [1]. We then propose an ancilla-free experimental scheme for measuring the MBCN without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function [2]. Finally, I will present an unbiased numerical optimization scheme to systematically find the Wilson loop operators given a ground state wave function of a gapped, translationally invariant Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We then use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully characterize topological order from the bulk of a ground state wave function [3]. 

[1] H. Dehghani, Z.P. Cian, M. Hafezi, and M. Barkeshl, Phys. Rev. B 103, 075102
[2] Z.P. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, and M. Hafezi, Phys. Rev. Lett. 126, 050501
[3] Z.P. Cian, M. Hafezi, and M. Barkeshl, Manuscript in preparation.

Logarithmic Sobolev inequalities (LSI) were first introduced by Gross in the 1970s as an equivalent formulation of hypercontractivity. LSI have been well studied in the past few decades and found applications to information theory, optimal transport, and graph theory. Recently matrix-valued LSI have been an active area of research. Matrix-valued LSI of Lindblad operators are closely related to decoherence of open quantum systems.  In this talk, I will present recent results on matrix-valued LSI, in particular a geometric approach to matrix-valued LSI of Lindblad operators. This talk is based on joint work with Li Gao, Marius Junge, and Nicholas LaRacuente.

Sampling from classical probability distributions is an important task with applications in a wide range of fields, including computational science, statistical physics, and machine learning. In this seminar, I will present a general strategy of solving sampling problems on a quantum computer. The entire probability distribution is encoded in a quantum state such that a measurement of the state yields an unbiased sample. I will discuss the complexity of preparing such states in the context of several toy models, where a polynomial quantum speedup is achieved. The speedup can be understood in terms of the properties of classical and quantum phase transitions, which establishes a connection between computational complexity and phases of matter. To conclude, I will comment on the prospects of applying this approach to challenging, real-world tasks.

Trotter-Suzuki formula is a practical and efficient algorithm for Hamiltonian simulation. It has been widely used in quantum chemistry, quantum field theory and condensed matter physics. Usually, its error is quantified by the operator norm distance between the ideal evolution operator and the digital evolution operator. However, recently more and more papers discovered that, even in large Trotter step region, the quantity of interest can still be accurately simulated. These robustness phenomena imply a different approach of analyzing Trotter-Suzuki formulas. In our previous paper, by analyzing the spectral analysis of the effective Hamiltonian, we successfully established refined estimations of digital errors, and thus improved the circuit complexity of quantum phase estimation and digital adiabatic simulation.

Suppressing noise in physical systems is of fundamental importance. As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level. However in the noisy, intermediate-scale quantum (NISQ) era, the complexity and scale required to adopt even the smallest QEC is prohibitive: a single logical qubit needs to be encoded into many thousands of physical qubits. Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression. We take n independently prepared circuit outputs to create a state whose symmetries prevent errors from contributing bias to the expected value. The approach is very well suited for current and near-term quantum devices as it is modular in the main computation and requires only a shallow circuit that bridges the n copies immediately prior to measurement. Using no more than four circuit copies, we confirm error suppression below 10−6 for circuits consisting of several hundred noisy gates (2-qubit gate error 0.5%) in numerical simulations validating our approach. This talk is based on [B. Koczor, Phys. Rev. X 11, 031057] and [B. Koczor, New J. Phys. (accepted), arXiv:2104.00608].

Zoom Link: https://pitp.zoom.us/j/91654758635?pwd=TEtPMmZMNGZya1JOc05KbGt6OUpjdz09

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation. More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion. Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network. The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as non-stabilizer codes over qubits and qudits. For a contractible tensor network, the sequence of contractions also constructs a decoding/encoding circuit. To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct non-trivial codes. These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a 3d stabilizer code, and other stabilizer codes with interesting properties. Surprisingly, we find that the surface code is equivalent to the 2d Bacon-Shor code after "dualizing" its tensor network encoding map.